Rounding Up

William Zahner, Understanding the Role of Language in Math Classrooms ROUNDING UP: SEASON 3 | EPISODE 17 How can educators understand the relationship between language and the mathematical concepts and skills students engage with in their classrooms? And how might educators think about the mathematical demands and the language demands of tasks when planning their instruction?  In this episode, we discuss these questions with Bill Zahner, director of the Center for Research in Mathematics and Science Education at San Diego State University. BIOGRAPHY Bill Zahner is a professor in the mathematics department at San Diego State University and the director of the Center for Research in Mathematics and Science Education. Zahner's research is focused on improving mathematics learning for all students, especially multilingual students who are classified as English Learners and students from historically marginalized communities that are underrepresented in STEM fields. RESOURCES Teaching Math to Multilingual Learners, Grades K–8 by Kathryn B. Chval, Erin Smith, Lina Trigos-Carrillo, and Rachel J. Pinnow National Council of Teachers of Mathematics Mathematics Teacher: Learning and Teaching PK– 12 English Learners Success Forum SDSU-ELSF Video Cases for Professional Development The Math Learning Center materials Bridges in Mathematics curriculum Bridges in Mathematics Teachers Guides [BES login required] TRANSCRIPT Mike Wallus: How can educators understand the way that language interacts with the mathematical concepts and skills their students are learning? And how can educators focus on the mathematics of a task without losing sight of its language demands as their planning for instruction? We'll examine these topics with our guest, Bill Zahner, director of the Center for Research in Mathematics and Science Education at San Diego State University.  Welcome to the podcast, Bill. Thank you for joining us today. Bill Zahner: Oh, thanks. I'm glad to be here. Mike: So, I'd like to start by asking you to address a few ideas that often surface in conversations around multilingual learners and mathematics. The first is the notion that math is universal, and it's detached from language. What, if anything, is wrong with this idea and what impact might an idea like that have on the ways that we try to support multilingual learners? Bill: Yeah, thanks for that. That's a great question because I think we have a common-sense and strongly held idea that math is math no matter where you are and who you are. And of course, the example that's always given is something like 2 plus 2 equals 4, no matter who you are or where you are. And that is true, I guess [in] the sense that 2 plus 2 is 4, unless you're in base 3 or something. But that is not necessarily what mathematics in its fullness is. And when we think about what mathematics broadly is, mathematics is a way of thinking and a way of reasoning and a way of using various tools to make sense of the world or to engage with those tools [in] their own right. And oftentimes, that is deeply embedded with language.  Probably the most straightforward example is anytime I ask someone to justify or explain what they're thinking in mathematics. I'm immediately bringing in language into that case. And we all know the old funny examples where a kid is asked to show their thinking and they draw a diagram of themselves with a thought bubble on a math problem. And that's a really good case where I think a teacher can say, “OK, clearly that was not what I had in mind when I said, ‘Show your thinking.'”  And instead, the demand or the request was for a student to show their reasoning or their thought process, typically in words or in a combination of words and pictures and equations. And so, there's where I see this idea that math is detached from language is something of a myth; that there's actually a lot of [language in] mathematics. And the interesting part of mathematics is often deeply entwined with language. So, that's my first response and thought about that.  And if you look at our Common Core State Standards for Mathematics, especially those standards for mathematical practice, you see all sorts of connections to communication and to language interspersed throughout those standards. So, “create viable arguments,” that's a language practice. And even “attend to precision,” which most of us tend to think of as, “round appropriately.” But when you actually read the standard itself, it's really about mathematical communication and definitions and using those definitions with precision. So again, that's an example, bringing it right back into the school mathematics domain where language and mathematics are somewhat inseparable from my perspective here. Mike: That's really helpful. So, the second idea that I often hear is, “The best way to support multilingual learners is by focusing on facts or procedures,” and that language comes later, for lack of a better way of saying it. And it seems like this is connected to that first notion, but I wanted to ask the question again: What, if anything, is wrong with this idea that a focus on facts or procedures with language coming after the fact? What impact do you suspect that that would have on the way that we support multilingual learners? Bill: So, that's a great question, too, because there's a grain of truth, right? Both of these questions have simultaneously a grain of truth and simultaneously a fundamental problem in them. So, the grain of truth—and an experience that I've heard from many folks who learned mathematics in a second language—was that they felt more competent in mathematics than they did in say, a literature class, where the only activity was engaging with texts or engaging with words because there was a connection to the numbers and to symbols that were familiar. So, on one level, I think that this idea of focusing on facts or procedures comes out of this observation that sometimes an emergent multilingual student feels most comfortable in that context, in that setting.  But then the second part of the answer goes back to this first idea that really what we're trying to teach students in school mathematics now is not simply, or only, how to apply procedures to really big numbers or to know your times tables fast. I think we have a much more ambitious goal when it comes to teaching and learning mathematics. That includes explaining, justifying, modeling, using mathematics to analyze the world and so on. And so, those practices are deeply tied with language and deeply tied with using communication. And so, if we want to develop those, well, the best way to do that is to develop them, to think about, “What are the scaffolds? What are the supports that we need to integrate into our lessons or into our designs to make that possible?”  And so, that might be the takeaway there, is that if you simply look at mathematics as calculations, then this could be true. But I think our vision of mathematics is much broader than that, and that's where I see this potential. Mike: That's really clarifying. I think the way that you unpack that is if you view mathematics as simply a set of procedures or calculations, maybe? But I would agree with you. What we want for students is actually so much more than that.  One of the things that I heard you say when we were preparing for this interview is that at the elementary level, learning mathematics is a deeply social endeavor. Tell us a little bit about what you mean by that, Bill. Bill: Sure. So, mathematics itself, maybe as a premise, is a social activity. It's created by humans as a way of engaging with the world and a way of reasoning. So, the learning of mathematics is also social in the sense that we're giving students an introduction to this way of engaging in the world. Using numbers and quantities and shapes in order to make sense of our environment.  And when I think about learning mathematics, I think that we are not simply downloading knowledge and sticking it into our heads. And in the modern day where artificial intelligence and computers can do almost every calculation that we can imagine—although your AI may do it incorrectly, just as a fair warning [laughs]—but in the modern day, the actual answer is not what we're so focused on. It's actually the process and the reasoning and the modeling and justification of those choices. And so, when I think about learning mathematics as learning to use these language tools, learning to use these ways of communication, how do we learn to communicate? We learn to communicate by engaging with other people, by engaging with the ideas and the minds and the feelings and so on of the folks around us, whether it's the teacher and the student, the student and the student, the whole class and the teacher. That's where I really see the power. And most of us who have learned, I think can attest to the fact that even when we're engaging with a text, really fundamentally we're engaging with something that was created by somebody else. So, fundamentally, even when you're sitting by yourself doing a math word problem or doing calculations, someone has given that to you and you think that that's important enough to do, right?  So, from that stance, I see all of teaching and learning mathematics is social. And maybe one of our goals in mathematics classrooms, beyond memorizing the times tables, is learning to communicate with other people, learning to be participants in this activity with other folks. Mike: One of the things that strikes me about what you were saying, Bill, is there's this kind of virtuous cycle, right? That by engaging with language and having the social aspect of it, you're actually also deepening the opportunity for students to make sense of the math. You're building the scaffolds that help kids communicate their ideas as opposed to removing or stripping out the language. That's the context in some ways that helps them filter and make sense. You could either be in a vicious cycle, which comes from removing the language, or a virtuous cycle. And it seems a little counterintuitive because I think people perceive language as the thing that is holding kids back as opposed to the thing that might actually help them move forward and make sense. Bill: Yeah. And actually that's one of the really interesting pieces that we've looked at in my research and the broader research is this question of, “What makes mathematics linguistically complex?” is a complicated question. And so sometimes we think of things like looking at the word count as a way to say, “If there are fewer words, it's less complex, and if there are more words, it's more complex.” But that's not totally true. And similarly, “If there's no context, it's easier or more accessible, and if there is a context, then it's less accessible.”  And I don't see these as binary choices. I see these as happening on a somewhat complicated terrain where we want to think about, “How do these words or these contexts add to student understanding or potentially impede [it]?” And that's where I think this social aspect of learning mathematics—as you described, it could be a virtuous cycle so that we can use language in order to engage in the process of learning language. Or, the vicious cycle is, you withhold all language and then get frustrated when students can't apply their mathematics. That's maybe the most stereotypical answer: “My kids can do this, but as soon as they get a word problem, they can't do it.” And it's like, “Well, did you give them opportunities to learn how to do this? [laughs] Or is this the first time?” Because that would explain a lot. Mike: Well, it's an interesting question, too, because I think what sits behind that in some ways is the idea that you're kind of going to reach a point, or students might reach a point, where they're “ready” for word problems.  Bill: Right. Mike: And I think what we're really saying is it's actually through engaging with word problems that you build your proficiency, your skillset that actually allows you to become a stronger mathematician. Bill: Mm-hmm. Right. Exactly. And it's a daily practice, right? It's not something that you just hold off to the end of the unit, and then you have the word problems, but it's part of the process of learning. And thinking about how you integrate and support that. That's the key question that I really wrestle with. Not trivial, but I think that's the key and the most important part of this. Mike: Well, I think that's actually a really good segue because I wanted to shift and talk about some of the concrete or productive ways that educators can support multilingual learners. And in preparing for this conversation, one of the things that I've heard you stress is this notion of a consistent context. So, can you just talk a little bit more about what you mean by that and how educators can use that when they're looking at their lessons or when they're writing lessons or looking at the curriculum that they're using? Bill: Absolutely. So, in our past work, we engaged in some cycles of design research with teachers looking at their mathematics curriculum and opportunities to engage multilingual learners in communication and reasoning in the classroom. And one of the surprising things that we found—just by looking at a couple of standard textbooks—was a surprising number of contexts were introduced that are all related to the same concept. So, the concept would be something like rate of change or ratio, and then the contexts, there would be a half dozen of them in the same section of the book. Now, this was, I should say, at a secondary level, so not quite where most of the Bridges work is happening. But I think it's an interesting lesson for us that we took away from this. Actually, at the elementary level, Kathryn Chval has made the same observation.  What we realized was that contexts are not good or bad by themselves. In fact, they can be highly supportive of student reasoning or they can get in the way. And it's how they are used and introduced. And so, the other way we thought about this was: When you introduce a context, you want to make sure that that context is one that you give sufficient time for the students to understand and to engage with; that is relatable, that everyone has access to it; not something that's just completely unrelated to students' experiences. And then you can really leverage that relatable, understandable context for multiple problems and iterations and opportunities to go deeper and deeper.  To give a concrete example of that, when we were looking at this ratio and rate of change, we went all the way back to one of the fundamental contexts that's been studied for a long time, which is motion and speed and distance and time. And that seemed like a really important topic because we know that that starts all the way back in elementary school and continues through college-level physics and beyond. So, it was a rich context. It was also something that was accessible in the sense that we could do things like act out story problems or reenact a race that's described in a story problem. And so, the students themselves had access to the context in a deep way.  And then, last, that context was one that we could come back to again and again, so we could do variations [of] that context on that story. And I think there's lots of examples of materials out there that start off with a core context and build it out. I'm thinking of some of the Bridges materials, even on the counting and the multiplication. I think there's stories of the insects and their legs and wings and counting and multiplying. And that's a really nice example of—it's accessible, you can go find insects almost anywhere you are. Kids like it. [Laughs] They enjoy thinking about insects and other icky, creepy-crawly things. And then you can take that and run with it in lots of different ways, right? Counting, multiplication, division ratio, and so on. Mike: This last bit of our conversation has me thinking about what it might look like to plan a lesson for a class or a group of multilingual learners. And I know that it's important that I think about mathematical demands as well as the language demands of a given task. Can you unpack why it's important to set math and language development learning goals for a task, or a set of tasks, and what are the opportunities that come along with that, if I'm thinking about both of those things during my planning? Bill: Yeah, that's a great question. And I want to mark the shift, right? We've gone from thinking about the demands to thinking about the goals, and where we're going to go next.  And so, when I think about integrating mathematical goals—mathematical learning goals and language learning goals—I often go back to these ideas that we call the practices, or these standards that are about how you engage in mathematics. And then I think about linking those back to the content itself. And so, there's kind of a two-piece element to that. And so, when we're setting our goals and lesson planning, at least here in the great state of California, sometimes we'll have these templates that have, “What standard are you addressing?,” [Laughs] “What language standard are you addressing?,” “What ELD standard are you addressing?,” “What SEL standard are you addressing?” And I've seen sometimes teachers approach that as a checkbox, right? Tick, tick, tick, tick, tick. But I see that as a missed opportunity—if you just look at this like you're plugging things in—because as we started with talking about how learning mathematics is deeply social and integrated with language, that we can integrate the mathematical goals and the language goals in a lesson. And I think really good materials should be suggesting that to the teacher. You shouldn't be doing this yourself every day from scratch. But I think really high-quality materials will say, “Here's the mathematical goal, and here's an associated language goal,” whether it's productive or receptive functions of language. “And here's how the language goal connects the mathematical goal.”  Now, just to get really concrete, if we're talking about an example of reasoning with ratios—so I was going back to that—then it might be generalized, the relationship between distance and time. And that the ratio of distance and time gives you this quantity called speed, and that different combinations of distance and time can lead to the same speed. And so, explain and justify and show using words, pictures, diagrams. So, that would be a language goal, but it's also very much a mathematical goal.  And I guess I see the mathematical content, the practices, and the language really braided together in these goals. And that I think is the ideal, and at least from our work, has been most powerful and productive for students. Mike: This is off script, but I'm going to ask it, and you can pass if you want to.  Bill: Mm-hmm. Mike: I wonder if you could just share a little bit about what the impact of those [kinds] of practices that you described [have been]—have you seen what that impact looks like? Either for an educator who has made the step and is doing that integration or for students who are in a classroom where an educator is purposely thinking about that level of integration? Bill: Yeah, I can talk a little bit about that. In our research, we have tried to measure the effects of some of these efforts. It is a difficult thing to measure because it's not just a simple true-false test question type of thing that you can give a multiple-choice test for.  But one of the ways that we've looked for the impact [of] these types of intentional designs is by looking at patterns of student participation in classroom discussions and seeing who is accessing the floor of the discussion and how. And then looking at other results, like giving an assessment, but deeper than looking at the outcome, the binary correct versus incorrect. Also looking at the quality of the explanation that's provided. So, how [do] you justify an answer? Does the student provide a deeper or a more mathematically complete explanation?  That is an area where I think more investigation is needed, and it's also very hard to vary systematically. So, from a research perspective—you may not want to put this into the final version [laughs]—but from a research perspective, it's very hard to fix and isolate these things because they are integrated. Mike: Yeah. Yeah. Bill: Because language and mathematics are so deeply integrated that trying to fix everything and do this—“What caused this water to taste like water? Was it the hydrogen or the oxygen?”—well, [laughs] you can't really pull those apart, right? The water molecule is hydrogen and oxygen together. Mike: I think that's a lovely analogy for what we were talking about with mathematical goals and language goals. That, I think, is really a helpful way to think about the extent to which they're intertwined with one another. Bill: Yeah, I need to give full credit to Vygotsky, I think, who said that. Mike: You're— Bill: Something. Might be Vygotsky. I'll need to check my notes. Mike: I think you're in good company if you're quoting Vygotsky.  Before we close, I'd love to just ask you a bit about resources. I say this often on the podcast. We have 20 to 25 minutes to dig deeply into an idea, and I know people who are listening often think about, “Where do I go from here?” Are there any particular resources that you would suggest for someone who wanted to continue learning about what it is to support multilingual learners in a math classroom? Bill: Sure. Happy to share that.  So, I think on the individual and collective level—so, say, a group of teachers—there's a beautiful book by Kathryn Chval and her colleagues [Teaching Math to Multilingual Learners, Grades K–8] about supporting multilingual learners and mathematics. And I really see that as a valuable resource. I've used that in reading groups with teachers and used that in book studies, and it's been very productive and powerful for us. Beyond that, of course, I think the NCTM [National Council of Teachers of Mathematics] provides a number of really useful resources. And there are articles, for example, in the [NCTM journal] Mathematics Teacher: Learning and Teaching PK– 12 that could make for a really wonderful study or opportunity to engage more deeply.  And then I would say on a broader perspective, I've worked with organizations like the English Learners Success Forum and others. We've done some case studies and little classroom studies that are accessible on my website [SDSU-ELSF Video Cases for Professional Development], so you can go to that. But there's also from that organization some really valuable insights, if you're looking at adopting new materials or evaluating things, that gives you a principled set of guidelines to follow. And I think that's really helpful for educators because we don't have to do this all on our own. This is not a “reinvent the wheel at every single site” kind of situation. And so, I always encourage people to look for those resources.  And of course, I will say that the MLC materials, the Bridges in Mathematics [curriculum], I think have been really beautifully designed with a lot of these principles right behind them. So, for example, if you look through the Teachers Guides on the Bridges in Mathematics [BES login required], those integrated math and language and practice goals are a part of the design. Mike: Well, I think that's a great place to stop. Thank you so much for joining us, Bill. This has been insightful, and it's really been a pleasure talking with you. Bill: Oh, well, thank you. I appreciate it. Mike: And that's a wrap for Season 3 of Rounding Up. I want to thank all of our guests and the MLC staff who make these podcasts possible, as well as all of our listeners for tuning in. Have a great summer, and we'll be back in September for Season 4.  This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center | www.mathlearningcenter.org  

Tisha Jones, Assessment as a Shared Journey: Cultivating Partnerships with Families & Caregivers ROUNDING UP: SEASON 3 | EPISODE 16 Families and caregivers play an essential role in students' success in school and in shaping their identities as learners. Therefore, establishing strong partnerships with families and caregivers is crucial for equitable teaching and learning. This episode is designed to help educators explore the importance of collaborating with families and caregivers and learn strategies for shifting to asset-based communication. BIOGRAPHY Tisha Jones is the senior manager of assessment at The Math Learning Center. Previously, Tisha taught math to elementary and middle school students as well as undergraduate and graduate math methods courses at Georgia State University. TRANSCRIPT Mike Wallus: As educators, we know that families and caregivers play an essential role in our students' success at school. With that in mind, what are some of the ways we can establish strong partnerships with caregivers and communicate about students' progress in asset-based ways? We'll explore these questions with MLC's [senior] assessment manager, Tisha Jones, on this episode of Rounding Up.  Welcome back to the podcast, Tisha. I think you are our first guest to appear three times. We're really excited to talk to you about assessment and families and caregivers.  Tisha Jones: I am always happy to talk to you, Mike, and I really love getting to share new ideas with people on your podcast.  Mike: So, we've titled this episode “Assessment as a Shared Journey with Families & Caregivers,” and I feel like that title—especially the words “shared journey”—say a lot about how you hope educators approach this part of their practice.  Tisha: Absolutely.  Mike: So, I want to start by being explicit about how we at The Math Learning Center think about the purpose of assessment because I think a lot of the ideas and the practices and the suggestions that you're about to offer flow out of that way that we think about the purpose.  Tisha: When we think about the purpose of assessment at The Math Learning Center, what sums it up best to me is that all assessment is formative, even if it's summative, which is a belief that you'll find in our Assessment Guide. And what that means is that assessment really is to drive learning. It's for the purpose of learning. So, it's not just to capture, “What did they learn?,” but it's, “What do they need?,” “How can we support kids?,” “How can we build on what they're learning?” over and over and over again. And so, there's no point where we're like, “OK, we've assessed it and now the learning of that is in the past.” We're always trying to build on what they're doing, what they've learned so far.  Mike: You know, I've also heard you talk about the importance of an asset-focused approach to assessment. So, for folks who haven't heard us talk about this in the past, what does that mean, Tisha?  Tisha: So that means starting with finding the things that the kids know how to do and what they understand instead of the alternative, which is looking for what they don't know, looking for the deficits in their thinking. We're looking at, “OK, here's the evidence for all the things that they can do,” and then we're looking to think about, “OK, what are their opportunities for growth?” Mike: That sounds subtle, but it is so profound a shift in thinking about what is happening when we're assessing and what we're seeing from students. How do you think that change in perspective shifts the work of assessing, but also the work of teaching?  Tisha: When I think about approaching assessment from an asset-based perspective—finding the things that kids know how to do, the things that kids understand—one, I am now on a mission to find their brilliance. I am just this brilliance detective. I'm always looking for, “What is that thing that this kid can shine at?” That's one, and a different way of thinking about it just to start with.  And then I think the other thing, too, is, I feel like when you find the things that they're doing, I can think about, “OK, what do I need to know? What can I do for them next to support them in that next step of growth?”  Mike: I think that sounds fairly simple, but there's something very different about thinking about building from something versus, say, looking for what's broken.  Tisha: For sure. And it also helps build relationships, right? If you approach any relationship from a deficit perspective, you're always focusing on the things that are wrong. And so, if we're talking about building stronger relationships with kids, coming from an asset-based perspective helps in that area too.  Mike: That's a great pivot point because if we take this notion that the purpose of assessment is to inform the ways that we support student learning, it really seems like that has a major set of implications for how and what and even why we would communicate with families and caregivers.  So, while I suspect there isn't a script for the type of communication, are there some essential components that you'd want to see in an asset-focused assessment conversation that an educator would have with a family or with their child's caregivers?  Tisha: Well, before thinking about a singular conversation, I want to back it up and think about—over the course of the school year. And I think that when we start the communication, it has to start before that first assessment. It has to start before we've seen a piece of kids' work. We have to start building those relationships with families and caregivers. We need to invite them into this process. We need to give them an opportunity to understand what we think about assessment. How are we approaching it? When we send things home, and they haven't heard of things like “proficiency” or “meeting current expectations”—those are common words that you'll see throughout the Bridges assessment materials—if parents haven't seen that, if families and caregivers haven't heard from you on what that means for you in your classroom at your school, then they have questions. It feels unfamiliar. It feels like, “Wait, what does this mean about how my child is doing in your class?”  And so, we want to start this conversation from the very beginning of the school year and continue it on continuously. And it should be this open invitation for them to participate in this process too, for them to share what they're seeing about their student at home, when they're talking about math or they're hearing how their student is talking about math. We want to know those things because that informs how we approach the instruction in class.  Mike: Let's talk about that because it really strikes me that what you're describing in terms of the meaning of proficiency or the meaning of meeting expectations—that language is likely fairly new to families and caregivers.  And I think the other thing that strikes me is, families and caregivers have their own lived experience with assessment from when they were children, perhaps with other children. And that's generally a mixed bag at best. Folks have this set of ideas about what it means when the teacher contacts them and what assessment means. So, I really hear what you're saying when you're talking about, there's work that educators need to do at the start of the year to set the stage for these conversations.  Let's try to get a little bit specific, though. What are some of the practices that you'd want teachers to consider when they're thinking about their communication?  Tisha: So, I think that starting at the very beginning of the year, most schools do some sort of a curriculum night. I would start by making sure that assessment is a part of that conversation and making sure that you're explaining what assessment means to you. Why are you assessing? What are the different ways that you're assessing? What are some things that [families and caregivers] might see coming home? Are they going to see feedback? Are they going to see scores from assessments? But how were you communicating progress? How do they know how their student is doing? And then also that invitation, right then and there, to be a part of this process, to hear from them, to hear their concerns or their ideas around feedback or the things that they've got questions about.  I would also suggest … really working hard to have that asset-based lens apply to parents and families and caregivers. I know that I have been that parent that was the last one to sign up for the parent teacher conferences, and I'm sending the apologetic email, and I'm begging for a special time slot. So, it didn't mean that I didn't care about my kids. It didn't mean that I didn't care about what they were doing. I was swamped. And so, I think we want to keep finding that asset-based lens for parents and caregivers in the same way that we do for the students.  And then making sure that you're giving them good news, not just bad news. And then making sure when you're sending any communication about how a student is doing, try to be concrete about what you're seeing, right? So, trying to say, “These are the things where I see your child's strengths. These are the strengths that I'm seeing from your student. And these are the areas where we're working on to grow. And this is what we're doing here at school, and this is what you can do to support them at home.” Mike: I was really struck by a piece of what you said, Tisha, when you really made the case for not assuming that the picture that you have in your mind as an educator is clear for families when it comes to assessment. So, really being transparent about how you think about assessment, why you're assessing, and the cadence of when parents or families or caregivers could expect to hear from you and what they could expect as well.  I know for a fact that if my teacher called my family when I was a kid, generally there was a look that came across their face when they answered the phone. And even if it was good news, they didn't think it was good news at the front end of that conversation.  Tisha: I've been there. I had my son's fifth grade teacher call me last year, and I was like, “Oh, what is this?” [laughs] Mike: One of the things that I want to talk about before we finish this conversation is homework. I want to talk a little bit about the purpose of homework. We're having this conversation in the context of Bridges in Mathematics, which is the curriculum that The Math Learning Center publishes. So, while we can't talk about how all folks think about homework, we can talk about the stance that we take when it comes to homework: what its purpose is, how we imagine families and caregivers can engage with their students around it.  Can you talk a little bit about our perspective on homework? How we think about its value, how we think about its purpose? And then we can dig a little bit into what it might look like at home, but let's start with purpose and intent.  Tisha: So, we definitely recognize that there are lots of different ideas about homework, and I think that shows in how we've structured homework through our Bridges units. Most of the time, it's set up so that there's a homework [assignment] that goes with every other session, but it's still optional. So, there's no formal expectation in our curriculum that homework is given on a nightly basis or even on an every-other-night basis. We really have left that up to the schools to determine what is best practice for their population. And I think that is actually what's really the most important thing is, understanding the families and caregivers and the situations that are in your building, and making determinations about homework that makes sense for the students that you're serving. And so, I think we've set homework up in a way that makes it so that it's easy for schools to make those decisions.  Mike: One of the things that I'm thinking about is that—again, I'm going to be autobiographical—when I was a kid, homework went back, it was graded, and it actually counted toward my grade at the end of the semester or the quarter or what have you. And I guess I wonder if a school or a district chose to not go about that, to not have homework necessarily be graded, I wonder if some families and caregivers might wonder, “What's the purpose?” I think we know that there can be a productive and important purpose—even if educators aren't grading homework and adding it to a percentage that is somehow determining students' grades, that it can actually still have purpose. How do you think about the purpose of homework, regardless of whether it's graded or not?  Tisha: So first off, I would just like to advocate not grading homework if I can.  Mike: You certainly can, yeah.  Tisha: [laughs] Mike: Let's talk about that.  Tisha: I think that, one, if we're talking about this idea of putting this score into an average grade or this percentage grade, I think that this is something that has so many different circumstances for kids at home. You have some students who get lots and lots of help. You get some students who do not have help available to them.  Another experience that has been very common when I was teaching was that I would get messages where it was like, “We were doing homework. The kid was in tears, I was in tears. This was just really hard.” And that's just not—I don't ever want that scenario for any student, for any family, for any caregiver, for anybody trying to support a child at home. I used to tell them, “If you are getting to the point where it's that level of frustration, please just stop and send me a message, write it on the homework. Just communicate something that [says,] ‘This was too hard' because that's information now that I can use.”  And so, for me, I think about [how] homework can be an opportunity for students to practice some skills and concepts and things that they've learned at home. It's an opportunity for parents, families, caregivers to see some of the things that the kids are working on at school.  Mike: What do you think is meaningful for homework? And I have kind of two bits to that. What do you think is meaningful for the child? And then, what do you think might be meaningful for the interaction between the child and their family or caregiver? What's the best case for homework? When you imagine a successful or a productive or a meaningful experience with homework at home between child and family and caregiver, what's that look like?  Tisha: Well, one of the things that I've heard families say is, “I don't know how to help my child with blank.” So, then I think it is, “Well, how do we support families and caregivers in knowing what [to] do with homework when we don't know how to tell them what to do?”  So, to me, it's about, how can we restructure the homework experience so that it's not this, “I have to tell you how to do it so you can get the right answer so you can get the grade.” But it's like, “How can I get at more of your thinking? How can I understand then what is happening or what you do know?” So, “We can't get to the answer. OK. So tell me about what you do know, and how can we build from there? How can we build understanding?” And that way it maybe will take some of the pressure off of families and caregivers to help their child get to the right answer.  Mike: What hits me is we've really come full circle with that last statement you made because you could conceivably have a student who really clearly understands a particular problem that might be a piece of homework, [who] might have some ideas that are on the right track, but ultimately perhaps doesn't get to a fully clear answer that is perfect. And you might have a student who at a certain point in time, maybe [for them] the context or the problem itself is profoundly challenging.  And in all of those cases, the question, “Tell me what you do know” or “Tell me what you're thinking” is still an opportunity to draw out the students' ideas and to focus on the assets. Even if the work as you described it is to get them to think about, “What are the questions that are really causing me to feel stuck?” That is a productive move for a family and a caregiver and a student to engage in, to kind of wonder about, “What's going on here that's making me feel stuck?” Because then, as you said, all assessment is formative.  Tisha: Mm-hmm. Mike: That homework that comes back is functioning as a formative assessment, and it allows you to think about your next moves, how you build on what the student knows, or even how you build on the questions that the student is bringing to you.  Tisha: And that's such a great point, too, is there's really more value in them coming back with an incomplete assignment or there's, I don't know, maybe “more value” is not the right way to say it. But there is value in kids coming back with an incomplete assignment or an attempted assignment, but they weren't sure how to get through all the problems—as opposed to a parent who has told their student what to do to get to all of the right answers. And so, now they have all these right answers, but it doesn't really give you a clear picture of what that student actually does understand.  So, I'd much rather have a student attempt the homework and stop because they got too stuck, because now I know that, than having a family [member] or a caregiver—somebody working with that student—feel like if they don't have all of the right answers, then it's a problem.  Mike: I think that's really great guidance, both for teachers as they're trying to set expectations and be transparent with families. But also I think it takes that pressure off of families or caregivers who feel like their work when homework shows up, is to get to a right answer. It just feels like a much more healthy relationship with homework and a much more healthy way to think about the value that it has.  Tisha: Well, in truth, it's a healthier relationship with math overall, right? That math is a process. It's not just—the value is not in just this one right answer or this paper of right answers, but it's really in, “How do we deepen our understanding?,” “How do we help students deepen their understanding and have this more positive relationship with math?” And I think that creating these homework struggles between families and caregivers and the children does not support that end goal of having a more positive relationship with math overall.  Mike: Which is a really important part of what we're looking for in a child's elementary experience.  Tisha: Absolutely.  Mike: I think that's a great place to stop. Tisha Jones, thank you so much for joining us. We would love to have you back at some time. It has been a pleasure talking with you.  Tisha: It's been great talking to you, too, Mike.  Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center | www.mathlearningcenter.org  

Ryan Flessner, What If I Don't Understand Their Thinking? ROUNDING UP: SEASON 3 | EPISODE 15 “What do I do if I don't understand my student's strategy?” This is a question teachers grapple with constantly, particularly when conferring with students during class. How educators respond in moments like these can have a profound impact on students' learning and their mathematical identities. In this episode, we talk with Ryan Flessner from Butler University about what educators can say or do when faced with this situation. BIOGRAPHY Ryan Flessner is a professor of teacher education in the College of Education at Butler University in Indianapolis, Indiana. He holds a PhD in curriculum and instruction with an emphasis in teacher education from the University of Wisconsin–Madison; a master of arts in curriculum and teaching from Teachers College, Columbia University; and a bachelor of science in elementary education from Butler University. Prior to his time at the university level, he taught grades 3–7 in Indianapolis; New York City; and Madison, Wisconsin. RESOURCES Nearpod Pear Deck GeoGebra  Magma Math TRANSCRIPT Mike Wallus: “What do I do if I don't understand my student's strategy?” This is a question teachers grapple with constantly, particularly when conferring with students during class. How we respond in moments like these can have a profound impact on our students' learning and their mathematical identities. Today we'll talk with Ryan Flessner from Butler University about what educators can say or do when faced with this very common situation.  Welcome to the podcast, Ryan. Really excited to talk to you today. Ryan Flessner: Thanks, Mike. I'm flattered to be here. Thank you so much for the invitation. Mike: So, this experience of working with a student and not being able to make sense of their solution feels like something that almost every teacher has had. And I'll speak for myself and say that when it happens to me, I feel a lot of anxiety. And I just want to start by asking, what would you say to educators who are feeling apprehensive or unsure about what to do when they encounter a situation like this? Ryan: Yeah, so I think that everybody has that experience. I think the problem that we have is that teachers often feel the need to have all of the answers and to know everything and to be the expert in the room. But as an educator, I learned really quickly that I didn't have all the answers. And to pretend like I did put a lot of pressure on me and made me feel a lot of stress and would leave me answering children by saying, “Let me get back to you on that.” And then I would scurry and try and find all the answers so I could come back with a knowledgeable idea. And it was just so much more work than to just simply say, “I don't know. Let's investigate that together.” Or to ask kids, “That's something interesting that I'm seeing you do. I've never seen a student do that before. Can you talk to me a little bit about that?” And just having that ability to free myself from having to have all the answers and using that Reggio-inspired practice—for those who know early childhood education—to follow the child, to listen to what he or she or they say to us and try to see. I can usually keep up with a 7- or an 8-year-old as they're explaining math to me. I just may never have seen them notate something the way they did. So, trying to ask that question about, “Show me what you know. Teach me something new.” The idea that a teacher could be a learner at the same time I think is novel to kids, and I think they respond really well to that idea. Mike: So, before we dig in a little bit more deeply about how teachers respond to student strategies if they don't understand, I just want to linger and think about the assumptions that many educators, myself included, might bring to this situation. Assumptions about their role, assumptions about what it would mean for a student if they don't know the answer right away. How do you think about some of the assumptions that are causing some of that anxiety for us? Ryan: Yeah. When the new generation of standards came out, especially in the field of math, teachers were all of a sudden asked to teach in a way that they themselves didn't learn. And so, if you have that idea that you have to have all the answers and you have to know everything, that puts you in a really vulnerable spot because how are we supposed to just magically teach things we've never learned ourselves? And so, trying to figure out ways that we can back up and try and make sense of the work that we're doing with kids, for me that was really helpful in understanding what I wanted from my students. I wanted them to make sense of the learning. So, if I hadn't made sense of it yet, how in the world could I teach them to make sense of it? And so we have to have that humility to say, “I don't know how to do this. I need to continue my learning trajectory and to keep going and trying to do a little bit better than the day that I did before.” I think that teachers are uniquely self-critical and they're always trying to do better, but I don't know if we necessarily are taught how to learn once we become teachers. Like, “We've already learned everything we have to do. Now we just have to learn how to teach it to other people.” But I don't think we have learned everything that we have to learn. There's a lot of stuff in the math world that I don't think we actually learned. We just memorized steps and kind of regurgitated them to get our A+ on a test or whatever we did.  So, I think having the ability to stop and say, “I don't know how to do this, and so I'm going to keep working at it, and when I start to learn it, I'm going to be able to ask myself questions that I should be asking my students.” And just being really thoughtful about, “Why is the child saying the thing that she is?,” “Why is she doing it the way that she's doing it?,” “Why is she writing it the way that she's writing it?” And if I can't figure it out, the expert on that piece of paper is the child [herself], so why wouldn't I go and say, “Talk to me about this.”? I don't have to have all the answers right off the cuff. Mike: In some ways, what you were describing just there is a real nice segue because I've heard you say that our minds and our students' minds often work faster than we can write, or even in some cases faster than we can speak. I'm wondering if you can unpack that. Why do you think this matters, particularly in the situation that we're talking about? Ryan: Yeah, I think a lot of us, especially in math, have been conditioned to get an answer. And nobody's really asked us “Why?” in the past. And so, we've done all of the thinking, we give the answer, and then we think the job is done. But with a lot of the new standards, we have to explain why we think that way. And so, all those ideas that just flurried through our head, we have to now articulate those either in writing on paper or in speech, trying to figure out how we can communicate the mathematics behind the answer.  And so, a lot of times I'll be in a classroom, and I'll ask a student for an answer, and I'll say, “How'd you get that?” And the first inclination that a lot of kids have is, “Oh, I must be wrong if a teacher is asking me why.” So, they think they're wrong. And so I say, “No, no, no. It's not that you're wrong. I'm just curious. You came to that answer, you stopped and you looked up at the ceiling for a while and then you came to me and you said the answer is 68. How did you do that?” A child will say something like, “Well, I just thought about it in my head.” And I say, “Well, what did you think about in your head?” “Well, my brain just told me the answer was 68.”  And we have to actually talk to kids. And we have to teach them how to talk to us—that we're not quizzing them or saying that they're wrong or they didn't do something well enough—that we just want them to communicate with us how they're going about finding these things, what the strategies are. Because if they can communicate with us in writing, if they can communicate on paper, if they can use gestures to explain what they're thinking about, all of those tell us strengths that they bring to the table. And if I can figure out the strengths that you have, then I can leverage those strengths as I address needs that arise in my classroom. And so, I really want to create this bank of information about individual students that will help me be the best teacher that I can be for them. And if I can't ask those questions and they can't answer those questions for me, how am I going to individualize my instruction in meaningful ways for kids? Mike: We've been talking a little bit about the teacher experience in this moment, and we've been talking about some of the things that a person might say.  One of the things that I'm thinking about before we dig in a little bit deeper is, just, what is my role? How do you think about the role of a teacher in the moment when they encounter thinking from a student that they don't quite understand […] yet? Part of what I'm after is, how can a teacher think about what they're trying to accomplish in that moment for themselves as a learner and also for the learner in front of them? How would you answer that question? Ryan: When I think about an interaction with a kid in a moment like that, I try to figure out, as the teacher, my goal is to try and figure out what this child knows so that I can continue their journey in a forward trajectory. Instead of thinking about, “They need to go to page 34 because we're on page 33,” just thinking about, “What does this kid need next from me as the teacher?”  What I want them to get out of the situation is I want them to understand that they are powerful individuals, that they have something to offer the conversation and not just to prove it to the adult in the room. But if I can hear them talk about these ideas, sometimes the kids in the classroom can answer each other's questions. And so, if I can ask these things aloud and other kids are listening in, maybe because we're in close proximity or because we're in a small-group setting, if I can get the kids to verbalize those ideas sometimes one kid talking strikes an idea in another kid. Or another kid will say, “I didn't know how to answer Ryan when he asked me that question before, but now that I hear what it sounds like to answer that type of a question, now I get it, and I know how I would say it if it were my turn.”  So, we have to actually offer kids the opportunity to learn how to engage in those moments and how to share their expertise so others can benefit from their expertise and use that in a way that's helpful in the mathematical process. Mike: One of the most practical—and, I have to say, freeing—things that I've heard you recommend when a teacher encounters student work and they're still trying to make sense of it, is to just go ahead and name it. What are some of the things you imagine that a teacher might say that just straight out name the fact that they're still trying to understand a student's thinking? Tell me a little bit about that. Ryan: Well, I think the first thing is that we just have to normalize the question “Why?” or “Tell me how you know that.” If we normalize those things—a lot of times kids get asked that question when they're wrong, and so it's an [immediate] tip of the hat that “You're wrong, now go back and fix it. There's something wrong with you. You haven't tried hard enough.” Kids get these messages even if we don't intend for them to get them. So, if we can normalize the question “Tell me why you think that” or “Explain that to me”—if we can just get them to see that every time you give me an answer whether it's right or wrong, I'm just going to ask you to talk to me about it, that takes care of half of the problem.  But I think sometimes teachers get stuck because—and myself being one of them—we get stuck because we'll look at what a student is doing and they do something that we don't anticipate. Or we say, “I've shown you three different ways to get at this problem, different strategies you can use, and you're not using any of them.” And so, instead of getting frustrated that they're not listening to us, how do we use that moment to inquire into the things that we said obviously aren't useful, so what is useful to this kid? How is he attacking this on his paper?  So, I often like to say to a kid, “Huh, I noticed that you're doing something that isn't up on our anchor chart. Tell me about this. I haven't seen this before. How can you help me understand what you're doing?” And sometimes it's the exact same thinking as other strategies that kids are using. So, I can pair kids together and say, “Huh, you're both talking about it in the same way, but you're writing it differently on paper.” And so, I think about how I can get kids just to talk to me and tell me what's happening so that I can help give them a notation that might be more acceptable to other mathematicians or to just honor the fact that they have something novel and interesting to share with other kids. Other questions I talk about are, I will say, “I don't understand what's happening here, and that's not your fault, that's my fault. I just need you to keep explaining it to me until you say something that strikes a chord.” Or sometimes I'll bring another kid in, and I'll have the kids listen together, and I'll say, “I think this is interesting, but I don't understand what's going on. Can you say it to her? And then maybe she'll say it in a way that will make more sense to me.” Or I'll say, “Can you show me on your paper—you just said that—can you show me on your paper where that idea is?” Because a lot of times kids will think things in their head, but they don't translate it all onto the paper. And so, on the paper, it's missing a step that isn't obvious to the viewer of the paper. And so, we'll say, “Oh, I see how you do that. Maybe you could label your table so that we know exactly what you're talking about when you do this. Or maybe you could show us how you got to 56 by writing 8 times 7 in the margin or something.”  Just getting them to clarify and try to help us understand all of the amazing things that are in their head. I will often tell them too, “I love what you're saying. I don't see it on your paper, so I just want you to say it again. And I'm going to write it down on a piece of paper that makes sense to me so that I don't forget all of the cool things that you said.” And I'll just write it using more of a standard notation, whether that's a ratio table or a standard US algorithm or something. I'll write it to show the kid that thing that you're doing, there's a way that people write that down. And so, then we can compare our notations and try and figure out “What's the thing that you did?,” “How does that compare to the thing that I did?,” “Do I understand you clearly now?” to make sure that the kid has the right to say the thing she wants to say in the way that she wants to say it, and then I can still make sense of it in my own way. It's not a problem for me to write it differently as long as we're speaking the same language. Mike: I want to mark something really important, and I don't want it to get lost for folks. One of the things that jumped out is the moves that you were describing. You could potentially take up those moves if you really were unsure of how a student were thinking, if you had a general notion but you had some questions, or if you totally already understood what the student was doing. Those are questions that aren't just reserved for the point in time when you don't understand—they're actually good questions regardless of whether you fully understand it or don't understand it at all. Did I get that right? Ryan: Yes. I think that's exactly the point. One thing that I am careful of is, sometimes kids will ask me a question that I know the answer to, and there's this thing that we do as teachers where we're like, “I'm not sure. Why don't you help me figure that out?”—when the kid knows full well that you know the answer.  And so, trying not to patronize kids with those questions, but to really show that I'm asking you these questions, not because I'm patronizing you. I'm asking these questions because I am truly curious about what you're thinking inside and all of the ideas that surround the things that you've written on your paper, or the things that you've said to your partner, to truly honor that the more I know about you, the better teacher I can be for you. Mike: So, in addition to naming the situation, one of the things that jumped out for me—particularly as you were talking about the students—is, what do you think the impact is on a student's thinking? But also their mathematical identity, or even the set of classroom norms, when they experience this type of questioning or these [types] of questions? Ryan: So, I think I talked a little bit about normalizing the [questions] “Why?” or “How do you know that?” And so, just letting that become a classroom norm I think is a sea-changing moment for a lot of classrooms—that the conversation is just different if the kids know they have to justify their thinking whether they're right or wrong. Half the time, if they are incorrect, they'll be able to correct themselves as they're talking it through with you. So, kids can be freed up when they're allowed to use their expertise in ways that allow them to understand that the point of math is to truly make sense of it so that when you go out into the world, you understand the situation, and you have different tools to attack it.  So, what's the way that we can create an environment that allows them to truly see themselves as mathematical thinkers? And to let them know that “Your grades in other classes don't tell me much about you as a mathematician. I want to learn what really works for you, and I want to try and figure out where you struggle. And both of those things are important to me because we can use them in concert with each other. So, if I know the things you do well, I can use those to help me build a plan of instruction that will take you further in your understandings.”  I think that one of the things that is really important is for kids to understand that we don't do math because we want a good grade. I think a lot of people think that the point of math is to get a good grade or to pass a test or to get into the college that you want to get into, or because sixth grade teachers want you to know this. I really want kids to understand that math is a fantastic language to use out in the world, and there are ways that we can interpret things around us if we understand some pretty basic math. And so how do we get them to stop thinking that math is about right answers and next year and to get the job I want? Well, those things may be true, but that's not the real meaning of math. Math is a way that we can live life. And so, if we don't help them understand the connections between the things that they're doing on a worksheet or in a workbook page, if we don't connect those things to the real world, what's the meaning? What's the point for them? And how do we keep them engaged in wanting to know more mathematics?  So, really getting kids to think about who they are as people and how math can help them live the life that they want to live. Creating classroom environments that have routines in place that support kids in thinking in ways that will move them forward in their mathematical understanding. Trying to help them see that there's no such thing as “a math person” or “not a math person.” That everybody has to do math. You do math all the time. You just might not even know that you're doing math. So, I think all of those ideas are really important. And the more curious I can be about students, maybe the more curious they'll be about the math. Mike: You're making me think that this experience of making sense of someone else's reasoning has a lot of value for students. And I'm wondering how you've seen educators have students engage and make sense of their peer strategies. Ryan: Yeah. One of the things that I love to see teachers doing is using students' work as the conversation starter. I often, in my classroom, when I started doing this work, I would bring children up to the overhead projector or the document camera. And they would kind of do a show and tell and just say, “I did this and then I did this, and then I did this thing next.” And I would say, “That's really great, thank you.” And I'd bring up the next student. And it kind of became a show-and-tell-type situation. And I would look at the faces of the other kids in the room, and they would kind of just either be completely checked out or sitting there like raising their hand excitedly—“I want to share mine, I want to share mine.” And what I realized was, that there was really only one person who was engaged in that show-and-tell manner, and that was the person who was sharing their work. And so, I thought, “How can I change that?” So, I saw a lot of really amazing teachers across my career. And the thing that I saw that I appreciated the most is that when a piece of student work is shared, the person who really shouldn't talk is the person who created the work because they already know the work. What we need to do as a group is we need to investigate, “What happened here on this paper?” “Why do you think they made the moves that they made? And how could that help us understand math, our own math, in a different way?” And so, getting kids to look in at other kids' work, and not just saying, “Oh, Mike, how do you understand Ryan's work?” It's “Mike, can you get us started?” And then you say the first thing, and then I say, “OK, let's stop. Let's make sure that we've got this right.” And then we go to the kid whose work it is and say, “Are we on the right track? Are we understanding what you're …?” So, we're always checking with that expert. We're making sure they have the last word, because It's not my strategy. I didn't create it. Just because I'm the teacher doesn't mean you should come and ask me about this because this is Mike's strategy. So go and ask the person who created that.  So, trying to get them to understand that we all need to engage in each other's work. We all need to see the connections. We can learn from each other. And there's an expectation that everyone shares, right? So, it's not just the first kid who raises his hand. It's “All of you are going to get a chance to share.” And I think the really powerful thing is I've done this work even with in-service teachers. And so, when we look at samples of student work, what's fascinating is it just happens naturally because the kid's not in the room. We can't have that kid do a show and tell. We have to interpret their work. And so, trying to look at the kid's work and imagine, “What are the types of things we think this child is doing?,” “What do we think the strengths are on this paper?,” “What questions would you ask?,” “What would you do next?,” is such an interesting thing to do when the child isn't in the room. But when I'm with students, it's just fascinating to watch the kid whose work is on display just shine, even though they're not saying a word, because they just say, “Huh.” They get it. They understand what I did and why I did it.  I think that it's really important for us not just to have kids walk up to the board and do board work and just solve a problem using the steps that they've memorized or just go up and do a show and tell, [but] to really engage everyone in that process so that we're all learning. We're not just kind of checking out or waiting for our turn to talk. Mike: OK, you were talking about the ways that an educator can see how a student was thinking or the ways that an educator could place student work in front of other students and have them try to make sense of it. I wonder if there are any educational technology tools that you've seen that might help an educator who's trying to either understand their students' thinking or put it out for their students to understand one another's thinking. Ryan: Yeah, there's so many different pieces of technology and things out there. It's kind of overwhelming to try and figure out which one is which. So, I mean, I've seen people use things like Nearpod or Pear Deck—some of those kind of common technologies that you'll see when people do an educational technology class or a workshop at a conference or something. I've seen a lot of people lately using GeoGebra to create applets that they can use with their kids. One that I've started using a lot recently is Magma Math. Magma Math is great. I've used this with teachers and professional development situations to look at samples of student work because the thing that Magma has that I haven't seen in a lot of other technologies is there's a playback function. So, I can look at a static piece of finished work, but I can also rewind, and as the child works in this program, it records it. So, I can watch in real time what the child does. And so, if I can't understand the work because things are kind of sporadically all over the page, I can just rewatch the order that the child put something onto the page. And I think that's a really great feature.  There's just all these technologies that offer us opportunities to do things that I couldn't do at the beginning of my career or I didn't know how to do. And the technology facilitates that. And it's not just putting kids on an iPad so they can shoot lasers at the alien that's invading by saying, “8 times 5 is 40,” and the alien magically blows up. How does that teach us anything? But some of these technologies really allow us to dig deeply into a sample of work that students have finished or inquire into, “How did that happen and why did that happen?” And the technologies are just getting smarter and smarter, and they're listening to teachers saying, “It would be really helpful if we could do this or if we could do that.” And so, I think there are a lot of resources out there—sometimes too many, almost an embarrassment of riches. So, trying to figure out which ones are the ones that are actually worth our time, and how do we fund that in a school district or in a school so that teachers aren't paying for these pieces out of their pocket. Mike: You know what? I think that's a great place to stop. Ryan, thank you so much for joining us. It has been an absolute pleasure talking with you. Ryan: It's always great to talk to you, Mike. Thanks for all you do. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center | www.mathlearningcenter.org  

Dr. Cathery Yeh, Supporting Neurodiverse Students in Elementary Mathematics Classrooms ROUNDING UP: SEASON 3 | EPISODE 14 What meaning does the term neurodiverse convey and how might it impact a student's learning experience?  And how can educators think about the work of designing environments and experiences that support neurodiverse students learning mathematics?  In this episode, we discuss these questions with Dr. Cathery Yeh, a professor in STEM education from the University of Texas at Austin.  BIOGRAPHY Dr. Cathery Yeh is an assistant professor in STEM education and a core faculty member in the Center for Asian American Studies from the University of Texas at Austin. Her research examines the intersections of race, language, and disability to provide a nuanced analysis of the constructions of ability in mathematics classrooms and education systems. TRANSCRIPT Mike Wallus: What meaning does the term neurodiverse convey and how might that language impact a student's learning experience? In this episode, we'll explore those questions. And we'll think about ways that educators can design learning environments that support all of their students. Joining us for this conversation is Dr. Cathery Yeh, a professor in STEM education from the University of Texas at Austin.  Welcome to the podcast, Cathery. It's really exciting to have you with us today. Cathery Yeh: Thank you, Mike. Honored to be invited. Mike: So, I wonder if we can start by offering listeners a common understanding of language that we'll use from time to time throughout the episode. How do you think about the meaning of neurodiversity? Cathery: Thank you for this thoughtful question. Language matters a lot. For me, neurodiversity refers to the natural variation in our human brains and our neurocognition, challenging this idea that there's a normal brain. I always think of… In Texas, we just had a snow day two days ago. And I think of, just as, there's no two snowflakes that are the same, there's no two brains that are exactly the same, too. I also think of its meaning from a personal perspective. I am not a special educator. I was a bilingual teacher and taught in inclusive settings. And my first exposure to the meaning of neurodiversity came from my own child, who—she openly blogs about it—as a Chinese-American girl, it was actually really hard for her to be diagnosed. Asian Americans, 1 out of 10 are diagnosed—that's the lowest of any ethnic racial group. And I'll often think about when… She's proud of her disabled identity. It is who she is. But what she noticed that when she tells people about her disabled identity, what do you think is the first thing people say when she says, “I'm neurodivergent. I have ADHD. I have autism.” What do you think folks usually say to her? The most common response? Mike: I'm going to guess that they express some level of surprise, and it might be associated with her ethnic background or racial identity. Cathery: She doesn't get that as much. The first thing people say is, they apologize to her. They say, “I'm sorry.” Mike: Wow. Cathery: And that happens quite a lot. And I say that because–and then I connected back to the term neurodiversity—because I think it's important to know its origins. It came about by Judy Singer. She's a sociologist. And about 30 years ago, she coined the term neurodiversity as an opposition to the medical model of understanding people and human difference as deficits. And her understanding is that difference is beautiful. All of us think and learn and process differently, and that's part of human diversity. So that original definition of neurodiversity was tied to the autism rights movement. But now, when we think about the term, it's expanded to include folks with ADHD, dyslexia, dyscalculia, mental health, conditions like depression, anxiety, and other neuro minorities like Tourette syndrome, and even memory loss. I wanted to name out all these things because sometimes we're looking for a really clean definition, and definitions are messy. There's a personal one. There's a societal one of how we position neurodiversity as something that's deficit, that needs to be fixed. But it's part of who one is. But it's also socially constructed. Because how do you decide when a difference becomes a difference that counts where you qualify as being neurodiverse, right? So, I think there's a lot to consider around that. Mike: You know, the answer that you shared is really a good segue because the question I was going to ask you involves something that I suspect you hear quite often is people asking you, “What are the best ways that I can support my neurodiverse students?” And it occurs to me that part of the challenge of that question is it assumes that there's this narrow range of things that you do for this narrow range of students who are different. The way that you just talked about the meaning of neurodiversity probably means that you have a different kind of answer to that question when people ask it. Cathery: I do get this question quite a lot. People email it to me, or they'll ask me. That's usually the first thing people ask. I think my response kind of matches my pink hair question. When they ask me the question, I often ask a question back. And I go, “How would you best educate Chinese children in math?” And they're like, “Why would you ask that?” The underlining assumption is that all Chinese children are the same, and they learn the same ways, they have the same needs, and also that their needs are different than the research-based equity math practices we know and have done 50–60 years of research that we've highlighted our effective teaching practices for all children. We've been part of NCTM for 20 years. We know that tasks that promote reasoning and problem solving have been effectively shown to be good for all. Using a connecting math representation—across math representations in a lesson—is good for all. Multimodal math discourse, not just verbal, written, but embodied in part who we are and, in building on student thinking, and all those things we know. And those are often the recommendations we should ask. But I think an important question is how often are our questions connecting to that instead? How often are we seeing that we assume that certain students cannot engage in these practices? And I think that's something we should prioritize more. I'm not saying that there are not specific struggles or difficulties that the neurodiversity umbrella includes, which includes ADHD, dyslexia, autism, bipolar disorder, on and on, so many things. I'm not saying that they don't experience difficulties in our school environment, but it's also understanding that if you know one neurodiverse student—you know me or my child—you only know one. That's all you know. And by assuming we're all the same, it ignores the other social identities and lived experiences that students have that impact their learning.  So, I'm going to ask you a question. Mike: Fire away. Cathery: OK. What comes to your mind when you hear the term “neurodiverse student”? What does that student look like, sound like, appear like to you? Mike: I think that's a really great question. There's a version of me not long ago that would have thought of that student as someone who's been categorized as special education, receiving special education services, perhaps a student that has ADHD. I might've used language like “students who have sensory needs or processing.” And I think as I hear myself say some of those things that I would've previously said, what jumps out is two things: One is I'm painting with a really broad brush as opposed to looking at the individual student and the things that they need. And two is the extent to which painting with a broad brush or trying to find a bucket of strategies that's for a particular group of students, that that really limits my thinking around what they can do or all the brilliance that they may have inside them. Cathery: Thank you for sharing that because that's a reflection I often do. I think about when I learned about my child, I learned about myself. How I automatically went to a deficit lens of like, “Oh, no, how are we going to function in the world? How's she going to function in the world?” But I also do this prompt quite a lot with teachers and others, and I ask them to draw it. When you draw someone, what do you see? And I'll be honest, kind of like drawing a scientist, we often draw Albert Einstein. When I ask folks to draw what a neurodiverse student looks like, they're predominantly white boys, to be honest with you. And I want to name that out. It's because students of color, especially black, brown, native students—they're disproportionately over- and under-identified as disabled in our schooling. Like we think about this idea that when most of us associate autism or ADHD mainly as part of the neurodiversity branch and as entirely within as white boys, which often happens with many of the teachers that I talk to and parents. We see them as needing services, but in contrast, when we think about, particularly our students of color and our boys—these young men—there's often a contrast of criminalization in being deprived of services for them. And this is not even what I'm saying. It's been 50 years of documented research from the Department of Ed from annual civil rights that repeatedly shows for 50 years now extreme disproportionality for disabled black and Latinx boys, in particular from suspension, expulsion, and in-school arrests. I think one of the most surprising statistics for me that I had learned recently was African-American youth are five times more likely to be misdiagnosed with conduct disorder before receiving the proper diagnosis of autism spectrum disorder. And I appreciate going back to that term of neurodiversity because I think it's really important for us to realize that neurodiversity is an asset-based perspective that makes us shift from looking at it as the student that needs to be fixed, that neurodiversity is the norm, but for us to look at the environment. And I really believe that we cannot have conversations about disability without fully having conversations about race, language, and the need to question what needs to be fixed, particularly not just our teaching, but our assessment practices. For example, we talk about neurodiversities around what we consider normal or abnormal, which is based on how we make expectations around what society thinks. One of the things that showed up in our own household—when we think about neurodiversity or assessments for autism—is this idea of maintaining eye contact. That's one of the widely considered autistic traits. In the Chinese and in the Asian household, and also in African communities, making eye contact to an adult or somebody with authority? It is considered rude. But we consider that as one of the characteristics when we engage in diagnostic tools. This is where I think there needs to be more deep reflection around how one is diagnosed, how a conversation of disability is not separate from our understanding of students and their language practices, their cultural practices. What do we consider normative? Because normative is highly situated in culture and context. Mike: I would love to stay on this theme because one of the things that stands out in that last portion of our conversation was this notion that rather than thinking about, “We need to change the child.” Part of what we really want to think about is, “What is the work that we might do to change the learning environment?” And I wonder if you could talk a bit about how educators go about that and what, maybe, some of the tools could be in their toolbox if they were trying to think in that way. Cathery: I love that question of, “What can we as teachers do? What's some actionable things?” I really appreciate Universal Design for Learning framework, particularly their revised updated version, or 3.0 version, that just came out, I think it was June or July of this year. Let me give you a little bit of background about universal design. And I'm sure you probably already know. I've been reading a lot around its origins. It came about [in the] 1980s, we know from cast.org. But I want to go further back, and it really builds from universal design and the work of architecture. So universal design was coined by a disabled architect. His name was Ronald Mace. And as I was reading his words, it really helped me better understand what UDL is. We know that UDL— Universal Design for Learning and universal design—is about access. Everybody should have access to curriculum. And that sounds great, but I've also seen classrooms where access to curriculum meant doing a different worksheet while everybody else is engaging in small group, whole group problem-based learning.  Access might mean your desk is in the front of the room where you're self-isolated—where you're really close to the front of the board so you can see it really well—but you can't talk to your peers. Or that access might mean you're in a whole different classroom, doing the same set of worksheets or problems, but you're not with your grade-level peers.  And when Ronald Mace talks about access, he explained that access in architecture had already been a focus in the late 1900s, around 1998, I think. But he said that universal design is really about the longing. And I think that really shifted the framing. And his argument was that we need to design a place, an environment where folks across a range of bodies and minds feel a sense of belonging there. That we don't need to adapt—the space was already designed for you. And that has been such a transformative perspective: That it shouldn't be going a different route or doing something different, because by doing that, you don't feel like you belong. But if the space is one where you can take part equally and access across the ways you may engage, then you feel a sense of belonging. Mike: The piece of what you said that I'm really contemplating right now is this notion of belonging. What occurs to me is that approaching design principles for a learning environment or a learning experience with belonging in mind is a really profound shift. Like asking the question, “What would it mean to feel a sense of belonging in this classroom or during this activity that's happening?” That really changes the kinds of things that an educator might consider going through a planning process. I'm wondering if you think you might be able to share an example or two of how you've seen educators apply universal design principles in their classrooms in ways that remove barriers in the environment and support students' mathematical learning. Cathery: Oh gosh, I feel so blessed. I spend… Tomorrow I'm going to be at a school site all day doing this. UDL is about being responsive to our students and knowing that the best teaching requires us to listen deeply to who they are, honor their mathematical brilliance, and their agency. It's about honoring who they are. I think where UDL ups it to another level, is it asks us to consider who makes the decision. If we are making all the decisions of what is best for that student, that's not fully aligned with UDL. The heart of UDL, it's around multiple ways for me to engage, to represent and express, and then students are given choice. So, one of the things that's an important part of UDL is honoring students' agency, so we do something called “access needs.” At the start of a lesson, we might go, “What do you need to be able to fully participate in math today?” And kids from kindergarten to high school or even my college students will just write out what they need. And usually, it's pretty stereotypical: “I want to talk to someone when I'm learning.” “I would like to see it and not just hear it.” And then you continually go back and you ask, “What are your access needs? What do you need to fully participate?”  So students are reflecting on their own what they need to be fully present and what they believe is helpful to create a successful learning environment. So that's a very strong UDL principle—that instead of us coming up with a set of norms for our students, we co-develop that. But we're co-developing it based on students reflecting on their experience in their environment. In kindergarten, we have children draw pictures. As they get older, they can draw, they can write. But it's this idea that it's an ongoing process for me to name out what I need to be fully present. And oftentimes, they're going to say things that are pretty critical. It's almost always critical, to be honest with you, but that's a… I would say that's a core component of UDL. We're allowing students to reflect on what they need so they can name it for themselves, and then we can then design that space together. And along the way, we have kids that name, “You know what? I need the manipulatives to be closer.” That would not come about at the start of me asking about access needs. But if we did a lesson, and it was not close by, they'll tell me. So it's really around designing an environment where they can fully participate and be their full selves and feel a sense of belonging. So, that's one example.  Another one that we've been doing is teachers and kids who have traditionally not participated the most in our classrooms or have even engaged in pullout intervention. And we'll have them walk around school, telling us about their day. “Will you walk me through your day and tell me how you feel in each of these spaces, and what are your experiences like?” And again, we're allowing the students to name out what they need. And then they're naming out… Oftentimes, with the students that we're at, where I'm working in mostly multilingual spaces, they'll say, “Oh, I love this teacher because she allows us to speak in Spanish in the room. It's OK.” So that's going back to ideas of action, expression, engagement, where students are allowed a trans language. That's one of the language principles.  But we're allowing students and providing spaces and really paying close attention to: “How do we decide how to maximize participation for our students with these set of UDL guidelines? How we are able to listen and make certain decisions on how we can strengthen their participation, their sense of belonging in our classrooms.” Mike: I think what's lovely about both of those examples—asking them to write or draw what they need or the description of, “Let's walk through the day. Let's walk through the different spaces that you learn in or the humans that you learn with”—is one, it really is listening to them and trying to make meaning of that and using that as your starting point. I think the other piece is that it makes me think that it's something that happens over time. It might shift, you might gain more clarity around the things that students need or they might gain more clarity around the things that they need over time. And those might shift a little bit, or it might come into greater focus. Like, “I thought I needed this” or “I think I needed this, but what I really meant was this.” There's this opportunity for kids to refine their needs and for educators to think about that in the designs that they create. Cathery: I really appreciate you naming that because it's all of that. It's an ongoing process where we're building a relationship with our students for us to co-design what effective teaching looks like—that it's not a one size fits all. It's disrupting this idea that what works for one works for all. It's around supporting our students to name out what they need. Now, I'm almost 50. I struggle to name out what I need sometimes, so it's not going to happen in, like, one time. It's an ongoing process. And what we need is linked to context, so it has to be ongoing. But there's also in the moments as well. And it's the heart of good teaching in math, when you allow students to solve problems in the ways that make sense to them, that's UDL by design. That's honoring the ideas of multiplicity in action, expression. When you might give a context-based problem and you take the numbers away and you give a set of number choices that students get to choose from. That is also this idea of UDL because there's multiple ways for them to engage. So there are also little things that we do that… note how they're just effective teaching. But we're honoring this idea that children should have agency. All children can engage in doing mathematics. And part of learning mathematics is also supporting our students to see the brilliance in themselves and to leverage that in their own teaching and learning. Mike: Yeah. Something else that really occurred to me as we've been talking is the difference between the way we've been talking about centering students' needs and asking them to help us understand them and the process that that kind of kicks off. I think what strikes me is that it's actually opening up the possibilities of what might happen or the ways that a student could be successful as opposed to this notion that “You're neurodiverse, you fit in this bucket. There's a set of strategies that I'm going to do just for you,” and those strategies might actually limit or constrict the options you have. For example, in terms of mathematics, what I remember happening very often when I was teaching is, I would create an open space for students to think about ways that they could solve problems. And at the time, often what would happen is kids who were characterized as neurodiverse wouldn't get access to those same strategies. It would be kind of the idea that “This is the way we should show them how to do it.” It just strikes me how different that experience is. I suspect that that was done with the best of intentions, but I think the impact unfortunately probably really didn't match the intent. Cathery: I love how you're being honest. I did the same thing when I was teaching, too, because we were often instructed to engage in whole-group instruction and probably do a small-group pullout. That was how I was taught. And when the same kids are repeatedly pulled out because we're saying that they're not able to engage in the instruction. I think that part of UDL is UDL is a process, realizing that if students are not engaging fully in the ways that we had hoped, instead of trying to fix the child, we look at the environment and think about what changes we need to make in tier one. So whole-group instruction, whole-group participation first to see how we can maximize their participation. And it's not one strategy, because it depends; it really depends. I think of, for example, with a group of teachers in California and Texas now, we've been looking at how we can track participation in whole-group settings. And we look at them across social demographics, and then we started to notice that when we promote multimodal whole-group participation, like kids have access to manipulatives even during whole-group share out. Or they have visuals that they can point to, their participation and who gets to participate drastically increase. So there's many ways in which, by nature, we engage in some narrow practices because, too, oftentimes whole group discussion is almost completely verbal and, at times, written, and usually the teacher's writing. So it's going back to the idea of, “Can we look at what we want our students to do at that moment? So starting on the math concept and practices, but then looking at our students and when they're not participating fully, it's not them. What are the UDL principles and things that I know and strategies that I have with my colleagues that I can make some small shifts?” Mike: You know, one of the things that I enjoy most about the podcast is that we really can take a deep dive into some big ideas, and the limitation is we have 20 minutes to perhaps a half hour. And I suspect there are a lot of people who are trying to make meaning of what we're talking about and thinking about, “How might I follow up? How might I take action on some of the ideas?” So I want to turn just for a little while to resources, and I'm wondering if there are resources that you would suggest for a listener who wants to continue learning about universal design in a mathematics classroom? Cathery: Oh, my goodness, that's such a hard question because there's so many. Some good ones overall: I would definitely encourage folks to dive into the UDL guidelines—the 3.0 updates. They're amazing. They're so joyful and transformative that they even have, one of the principles is centering joy in play, and for us to imagine that, right? Mike: Yes! Cathery: What does that mean to do that in a math classroom? We can name out 50 different ways. So how often do we get to see that? So, I would highly encourage folks to download that, engage in deep discussion because it was a 2.2 version for, I think, quite a few years. I would also lean into a resource that I'm glad to email later on so it's more easily accessible. I talked about access needs, this idea of asking students, asking community members, asking folks to give this opportunity to name out what they need. It's written by a colleague, Dr. Daniel Reinholz and Dr. Samantha Ridgway. It's a lovely reading, and it focuses specifically in STEM but I think it's a great place to read. I would say that Dr. Rachel Lambert's new book on UDL math is an excellent read. It's a great joyful read to think about. I'm going to give one shout out to the book called the Year of the Tiger: An Activist's Life. It's by Alice Wong. I encourage that because how often do we put the word activism next to disability? And Alice Wong is one of the most amazing humans in the world, and it's a graphic novel. So it's just joyful. It's words with poetry and graphic novel mixed together to see the life of what it means to be a disabled activist and how activism and disability goes hand in hand. Because when you are disabled and multi-marginalized, you are often advocating for yourself and others. It's amazing. So I'll stop there. There's endless amounts. Mike: So for listeners, we'll link the resources that Cathery was talking about in our show notes. I could keep going, but I think this is probably a great place to stop. I want to thank you so much for joining us. It's really been a pleasure talking with you. Cathery: Thank you. Thank you. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center | www.mathlearningcenter.org

Assessment in the Early Years Guest: Shelly Scheafer ROUNDING UP: SEASON 3 | EPISODE 13 Mike (00:09.127) Welcome to the podcast Shelley. Thank you so much for joining us today. Shelly (00:12.956) Thank you, Mike, for having me. Mike (00:16.078) So I'd like to start with this question. What makes the work of assessing younger children, particularly students in grades K through two, different from assessing students in upper elementary grades or even beyond? Shelly (00:30.3) There's a lot to that question, Mike. I think there's some obvious things. So effective assessment of our youngest learners is different because obviously our pre-K, first, even our second grade students are developmentally different from fourth and fifth graders. So when we think about assessing these early primary students, we need to use appropriate assessment methods that match their stage of development. For example, when we think of typical paper pencil assessments and how we often ask students to show their thinking with pictures, numbers and words, our youngest learners are just starting to connect symbolic representations to mathematical ideas, let alone, you know, put letters together to make words. So When we think of these assessments, we need to take into consideration that primary students are in the early stages of development with respect to their language, their reading, and their writing skills. And this in itself makes it challenging for them to fully articulate, write, sketch any of their mathematical thinking. So we often find that with young children in reviews, you know, individual interviews can be really helpful. But even then, there's some drawbacks. Some children find it challenging, you know, to be put on the spot, to show in the moment, you know, on demand, you know, what they know. Others, you know, just aren't fully engaged or interested because you've called them over from something that they're busy doing. Or maybe, you know, they're not yet comfortable with the setting or even the person doing the interview. So when we work with young children, we need to recognize all of these little peculiarities that come with working with that age. We also need to understand that their mathematical development is fluid, it's continually evolving. And this is why Shelly (02:47.42) they often or some may respond differently to the same proper question, especially if the setting or the context is changed. We may find that a kindergarten student who counts to 29 on Monday may count to 69 or even 100 later in the week, kind of depending on what's going on in their mind at the time. So this means that assessment with young children needs to be frequent. informative and ongoing. So we're not necessarily waiting for the end of the unit to see, aha, did they get this? You know, what do we do? You know, we're looking at their work all of the time. And fortunately, some of the best assessments on young children are the observations in their natural setting, like times when maybe they're playing a math game or working with a center activity or even during just your classroom routines. And it's these authentic situations that we can look at as assessments to help us capture a more accurate picture of their abilities because we not only get to hear what they say or see what they write on paper, we get to watch them in action. We get to see what they do when they're engaged in small group activities or playing games with friends. Mike (04:11.832) So I wanna go back to something you said and even in particular the way that you said it. You were talking about watching or noticing what students can do and you really emphasize the words do. Talk a little bit about what you were trying to convey with that, Shelley. Shelly (04:27.548) So young children are doers. When they work on a math task, they show their thinking and their actions with finger formations and objects. And we can see if a student has one-to-one correspondence when they're counting, if they group their objects, how they line them up, do they tag them, do they move them as they count them. They may not always have the verbal skills to articulate their thinking, but we can also attend to things like head nodding, finger counting, and even how they cluster or match objects. So I'm going to give you an example. So let's say that I'm watching some early first graders, and they're solving the expression 6 plus 7. And the first student picks up a number rack or a rec and rec. And if you're not familiar with a number rack, it's a tool with two rows of beads. And on the first row, there are five red beads and five white beads. And on the second row, there's five red beads and five white beads. And the student solving six plus seven begins by pushing over five red beads in one push and then one more bead on the top row. And then they do the same thing for the seven. They push over five red beads and two white beads. And they haven't said a word to me. I'm just watching their actions. And I'm already able to tell, hmm, that student could subitize a group of five, because I saw him push over all five beads in one push. And that they know that six is composed of five and one, and seven is composed of five and two. And they haven't said a word. I'm just watching what they're doing. And then I might watch the student, and they see it. I see him pause, know, nothing's being said, but I start to notice this slight little head nodding. Shelly (06:26.748) And then they say 13 and they give me the answer and they're really pleased. I didn't get a lot of language from them, but boy, did I get a lot from watching how they solve that problem. And I want to contrast that observation with a student who might be solving the same expression six plus seven and they might go six and then they start popping up one finger at a time while counting seven, eight. 9, 10, 11, 12, 13. And when they get seven fingers held up, they say 13 again. They've approached that problem quite differently. But again, I get that information that they understood the equation. They were able to count on starting with six. And they kept track of their count with their fingers. And they knew to stop when seven fingers were raised. And I might even have a different student that solves the problem by thinking, hmm, and they talk to themselves or they know I'm watching and they might start talking to me. And they say, well, 6 plus 6 is 12 and 7 is 1 more than 6. So the answer is 16 or 13. And if this were being done on a paper pencil as an assessment item or they were answering on some kind of a device, all I would know about my students is that they were able to get the correct answer. I wouldn't really know a lot about how they got the answer. What skills do they have? What was their thinking? And there's not a lot that I can work with to plan my instruction. Does that kind of make sense? Mike (08:20.84) Absolutely. I think the, the way that you described this really attending to behaviors, to gestures, to the way that kids are interacting with manipulatives, the self-talk that's happening. It makes a ton of sense. And I think for me, when I think back to my own practice, I wish I could wind the clock back because I think I was attending a lot to what kids were saying. and sometimes they're written communication, and there was a lot that I could have also taken in if I was attending to those things in a little bit more depth. It also strikes me that this might feel a little bit overwhelming for an educator. How do you think about what an educator, let me back that up. How can an educator know what they're looking for? Shelly (09:17.5) to start, Mike, by honoring your feelings, because I do think it can feel overwhelming at first. But as teachers begin to make informal observations, really listening to you and watching students' actions as part of just their daily practice, something that they're doing, you know, just on a normal basis, they start to develop these kind of intuitive understandings of how children learn, what to expect them to do, what they might say next if they see a certain actions. And after several years, let's say teaching kindergarten, if you've been a kindergarten teacher for four, five, six, 20, you know, plus years, you start to notice these patterns of behavior, things that five and six year olds seem to say and think and do on a fairly consistent basis. And that kind of helps you know, you know, what you're looking at. But before you say anything, I know that isn't especially helpful for teachers new to the profession or new to a grade level. And fortunately, we have several researchers that have been, let's say, kid watching for 40, I don't know, 50 years, and they have identified stages through which most children pass as they develop their counting skills or maybe strategies for solving addition and subtraction problems. And these stages are laid out as progressions of thinking or actions that students exhibit as they develop understanding over periods of time. listeners might, you know, know these as learning progressions or learning trajectories. And these are ways to convey an idea of concept in little bits of understanding. So. When I was sharing the thinking and actions of three students solving six plus seven, listeners familiar with cognitively guided instruction, CGI, they might have recognized the sequence of strategies that children go through when they're solving addition and subtraction problems. So in my first student, they didn't say anything but gave me an answer. Shelly (11:40.068) was using direct modeling. We saw them push over five and one beads for six and then five and two beads for seven and then kind of pause at their model. And I could tell, you know, with their head nodding that they were counting quietly in their head, counting all the beads to get the answer. And, you know, that's kind of one of those first stages that we see and recognize with direct modeling. And that gives me information on what I might do with a student. coming next time, I might work on the second strategy that I conveyed with my second student where they were able to count on. They started with that six and then they counted seven more using their fingers to keep track of their count and got the answer. And then that third kind of level in that progression as we're moving of understanding. was shown with my third student when they were able to use a derived fact strategy. The student said, well, I know that 6 plus 6 is 12. I knew my double fact. And then I used that relationship of knowing that 7 is 1 more than 6. And so that's kind of how we move kids through. And so when I'm watching them, I can kind of pinpoint where they are and where they might go next. And I can also think about what I might do. And so it's this knowledge of development and progressions and how children learn number concepts that can help teachers recognize the skills as they emerge, as they begin to see them with their students. And they can use those, you know, to guide their instruction for that student or, you know, look at the class overall and plan their instruction or think about more open-ended kinds of questions that they can ask that recognize these different levels that students are working with. Mike (13:39.17) You know, as a K-1 teacher, I remember that I spent a lot of my time tracking students with things like checklists. You know, so I'd note if students quote unquote had or didn't have a skill. And I think as I hear you talk, that feels fairly oversimplified when we think about this idea of developmental progressions. How do you suggest that teachers approach capturing evidence of student learning, Shelly (14:09.604) well, I think it's important to know that if, you know, it takes us belief. We have to really think about assessment and children's learning is something that is ongoing and evolving. And if we do, it just kind of becomes part of what we can do every day. We can look for opportunities to observe students skills in authentic settings. Many in the moment. types of assessment opportunities happen when we pose a question to the class and then we kind of scan looking for a response. Maybe it's something that we're having them write down on their whiteboard or maybe it's something where they're showing the answer with finger formations or we're giving a thumbs up or a thumbs down, know, kind of to check in on their understanding. We might not be checking on every student, but we're capturing the one, you know, a few. And we can take note because we're doing this on a daily basis of who we want to check in with. What do we want to see? We can also do a little more formal planning when we draw from what we're going to do already in our lesson. Let's say, for example, that our lesson today includes a dot talk or a number talk, something that we're going to write down. We're going to record student thinking. And so during the lesson, the teacher is going to be busy facilitating the discussion, recording the students thinking, you know, and making all of those notes. But if we write the child's name, kind of honor their thinking and give it that caption on that public record, at the end of the lesson, you know, we can capture a picture, just, you use our phone, use an iPad, quickly take a picture of that student's thinking, and then we can record that. you know, where we're keeping track of our students. So we have, OK, another moment in time. And it's this collection of evidence that we keep kind of growing. We can also, you by capturing these public records, note whose voice and thinking were elevating in the classroom. So it kind of gives us how are they thinking and who are we listening to and making sure that we're kind of spreading that out and hearing everyone. Shelly (16:31.728) I think, Mikey, you checklists that you used. Yeah, and even checklists can play a role in observation and assessments when they have a focus and a way to capture students' thinking. So one of the things we did in Bridges 3rd edition is we designed additional tools for gathering and recording information during workplaces. Mike (16:35.501) I did. Shelly (16:56.208) That's a routine where students are playing games and or engaged with partners doing some sort of a math activity. And we designed these based on what we might see students do at these different games and activities. And we didn't necessarily think about this is something you're going to do with every student. You know, or even, you know, in one day because these are spanned out over a period of four to six weeks where they can go to these games. And we might even see the students go to these activities multiple times. And so let's say that kindergarten students are playing something like the game Beat You to 10, where they're spinning a spinner, they're counting cubes, and they're trying to race their partner to collect 10 cubes. And with an activity like that, I might just want to focus on students who I still want to see, do they have one-to-one correspondence? Are they developing cardinality? Are they able to count out a set? And so those might, you know, of objects, you know, based on the number, they spin a four, can they count out four? And those might be kinds of skills that you might have had typically on a checklist, right, Mike, for kindergarten? But I could use this activity to kind of gap. gather that note and make any comments. So just for those kids I'm looking at or maybe first graders are playing a game like sort the sum where they're drawing two different dominoes and they're supposed to find how many they have in all. And so with a game like that, I might focus on what are their strategies? Are they counting all the dots? Are they counting on from the dot? And if one set of the dots on one side and then counting on the other. Are they starting with the greater number or the most dots? Are they starting with the one always on the left? Or I might even see they might instantly recognize some of those. So I might know the skills that I want to look for with those games and be making notes, which kind of feels checklist-like. But I can target that time to do it on students I want that information by thinking ahead of time. Shelly (19:18.684) What can I get by watching, observing these students at these games? trust, I mean, as you know, young children love it. Older children love it. When the teacher goes over and wants to watch them play, or even better, wants to engage in the game play with them, but I can still use that as an assessment. Mike (19:39.32) think that's really helpful, Shelly, for a couple reasons. First, I think it helps me rethink, like you said, one, getting really a lot clearer on like, love the, I'm gonna back that up. I think one of the things that you said was really powerful is thinking about not just the assessment tools that might be within your curriculum, but looking at the task itself that you're gonna have students engage with, be it a game or a, Shelly (19:39.356) and Mike (20:07.96) project or some kind of activity and really thinking like, what can I get from this as a person who's trying to make sense of students thinking? And I think my checklist suddenly feels really different when I've got a clear vision of like, what can I get from this task or this game that students are playing and looking for evidence of that versus feeling like I was pulling kids over one-on-one, which I think I would still do because there's some depth that I might want to capture. But it it changes the way that I think about what I might do and also what I might get out of a task So that that really resonates for me Shelly (20:47.066) Yeah, and I think absolutely, you know, I didn't want to make individual interviews or anything sound bad because we can't do them. just, you there's the downfall of, you know, kids comfort level with that and ask them to do something on demand. But we do want more depth and it's that depth that, you know, we know who we want more depth on because of these informal types of observations that we're gathering on a daily basis in our class. You know, might, says something and we take note I want to touch bases with that thinking or I think I'm going to go observe that child during that workplace or maybe we're seeing some things happening during a game and instead of you know like stopping the game and really doing some in-depth interview with the student at that moment because you need more information I can might I might want to call them over and do that more privately at a different time so you're absolutely on there's a place there's a place for you know both Mike (21:42.466) The other thing that you made me think about is the extent to which, like one of the things that I remember thinking is like, I need to make sure if a student has got it or not got it. And I think what you're making me think can really come out of this experience of observing students in the wild, so to speak, when they're working on a task or with a partner is that I can gather a lot more evidence about the application of that idea. I can see the extent to which students are. doing something like counting on in the context of a game or a task. And maybe that adds to the evidence that I gather in a one-on-one interview with them. But it gives me a chance to kind of see, is this way of thinking something that students are applying in different contexts, or did it just happen at that one particular moment in time when I was with them? So that really helps me think about, I think, how those two... maybe different ways of assessing students, be it one-on-one or observing them and seeing what's happening, kind of support one another. Shelly (22:46.268) think you also made me think, you know, really hit on this idea that students, like I said, you their learning is evolving over time. And it might change with the context so that they, you know, they show us that they know something in one context with these numbers or this, you know, scenario. But they don't necessarily always see that it applies across the board. I mean, they don't, you know, make these. generalizations. That's something that we really have to work with students to develop. they're also, they're young children. Think about how quickly a three-year-old and a four-year-old change, you know, the same five to six, six to seven. I mean, they're evolving all the time. And so we want to get this information for them on a regular basis. You know, a unit of instruction may be a month or more long. And a lot can happen in that time. So we want to make sure that we continue to check in with them and help them to develop if needed or that we advance them. know, we nudge them along. We challenge them with maybe a question. Will that apply to every number? So a student discovers, when we add one to every number, it's like saying the next number. So six and one more is seven and eight and one more is nine. And you can challenge them, ooh, does that always work? What if the number was 22? What if it was 132? Would it always work? you know, when you're checking in with kids, you have those opportunities to keep them thinking, to help them grow. Mike (24:23.426) I want to pick up on something that we haven't necessarily said aloud, but I'd like to explore it. You know, looking at young students work from an asset-based perspective, particularly with younger students, I think I often had points in time where there felt like so much that I needed to teach them. And sometimes I felt myself focusing on what they couldn't do. Looking back, I wish I had thought about my work as noticing the assets, the strategies, the ways of thinking. that they were accumulating. Are there practices you think support an asset-based approach to assessment with young learners? Shelly (25:06.278) think probably the biggest thing we can do is broaden our thinking about assessment. The National Council of Teachers of Mathematics wrote in Catalyzing Change in Early Childhood and Elementary Mathematics that the primary purpose of assessment is to gather evidence of children's thinking, understanding, and reasoning to inform both instructional decisions and student and teaching learning. If we consider assessments and observations as tools to inform our instruction, we need to pay attention to the details of the child's thinking. And when we're paying attention to the details, what the child is bringing to the table, what they can do, that's where our focus goes. So the question becomes, what is the student understanding? What assets do they bring to the task? It's no longer, can they do it or can they not do it? And when we know, when we're focusing on just what that student can do, and we have some understanding of learning progressions, how students learn, then we can place what they're doing kind of on that trajectory, in that progression, and that becomes knowledge. And with that knowledge, then we can help students move along the progression to develop more developed understanding. For example, again, if I go back to my six plus seven and we notice that a student is direct modeling, they're counting out each of the sets and counting all, we can start to nudge them toward counting on. We might cover, you know, they were using that number rec, we might cover the first row and say, you just really showed me a good physical representation of six plus seven. And I kind of noticed that you were counting the beads to see how many were there. I'm wondering if I cover this first row. How many beads am I covering? Hmm. I wonder, could you start your counting at six? You know, we can kind of work with what they know. And I can do that because of Shelly (27:31.928) I haven't, I've focused on where they are in that progression and where that development is going. And I kind of have a goal of where I want students to go, you know, to further their thinking. Not that being in one place is right or wrong, or yes they can do it, no they can't. It's my understanding of what assets they bring that I can build on. Is that kind of what you're after? Mike (27:58.51) It is, and I think you also addressed something that again has gone unsaid, but I think you, you, you unpacked it there, which is assessment is really designed to inform my instruction. And I think the example you offered us a really lovely one where, we have a student who's direct modeling and they're making sense of number in a certain way and their strategy reflects that. And that helps us think about the kinds of nudges we can offer. that might shift that thinking or press them to make sense of numbers in a different way. That really the assessment is, it is a moment in time, but it also informs the way that you think about what you're gonna do next to keep nudging that student's thinking. Shelly (28:44.348) Exactly, and we have to know that if we have 20 students, they all might be, you know, have 20 little plans that they're on, 20 little pathways of their learning. And so we need to think about everybody, you know. So we're going to ask questions that help them do them, and we're going to honor their thinking. And then we can, you know, like so again, I'm going go back to like doing that dot talk with those students. And so I'm honoring all these different ways that students are finding the total number of dots. And then I'm asking them to look for what's the same within their thinking so that other students also can serve to nudge kids, to have them let them try and explore a different idea or, ooh, can we try that Mike's way and see if we can do that? hmm, what do you notice about? how Mike solved the problem and how Shelly solved the problem. Where is their thinking the same? Where is it different? And so we're honoring everybody's place of where they're at, but they're still learning from each other. Mike (29:51.224) You know, you have made multiple mentions to this idea of progressions or trajectories, and I'm wondering if there are resources that have informed your thinking about assessment at the early ages. Is there anything you would invite listeners to engage with if they wanted to continue learning, Shelley? Shelly (30:13.008) I think Mike, had that question earlier, so just pause this for a second. Okay. I know you will. I just know it's right here. Mike (30:16.558) That's okay, no worries. We'll cut every single bit of this out and it will sound supernatural. Yeah, yeah. Mike (30:39.854) Brent's over here multitasking. Shelly (30:41.85) OK. OK, I'm just making sure that I'm not going to blow it. I think you're spot on. I think I thought we skipped something. No, it's up here. Mike (30:53.132) Okay, just pick up whenever you're ready. Shelly (30:55.108) Yeah, I just have too many notes here. Shelly (31:10.084) OK, I've got it. Do you want to ask the question again? Mike (31:12.258) Go for it. Absolutely. Yep. Are there resources you'd invite listeners to engage with if they wanted to keep learning, Shelly? Shelly (31:28.368) You phrased that a little bit different. What I answered was, what are some of the resources that helped you build an understanding of children's developmental progressions? Do you have that question? Or I can jump on from what you asked, too. Mike (31:35.5) Okay. Yeah, let let. No, no, no, let me let me ask the question that way. Shelly (31:42.202) Okay. Mike (31:45.774) Okay, how did we have it in the thing? Can you say it one more time and I'll say it back in the question? Shelly (31:50.768) What are some of the resources that help to build an understanding of children's developmental progressions? Mike (31:56.504) Perfect. What are some of the resources that helped you build an understanding of children's developmental progression, Shelley? Shelly (32:05.34) Honestly, I can say that I learned a lot from the students I taught in my classroom. My roots run deep in early childhood. And I can also proudly say that I have a career-long relationship with the Math Learning Center and Bridges Curriculum, which has always been developmentally appropriate curriculum for young learners. And with that said, I think I stand on the back of giants. practitioner researchers for early childhood who have spent decades observing children and recording their thinking. I briefly mentioned Cognitively Guided Instruction, which features the research of Thomas Carpenter and his team. And their book, Children's Mathematics, is a great guide for K-5 teachers. I love it because I mean, the recent edition has QR codes where you can watch teachers and students in action. You can see some interviews. You can see some classroom lessons. And they also wrote young children's mathematics on cognitive-guided instruction in early childhood education. So I mean, they're just a great resource. Another teacher researcher. is Kathy Richardson, and some listeners may know her from her books, the developing number concept series or number talks in the primary classroom. And she also wrote a book called How Children Learn Number Concepts, Guide to the Critical Learning Phases, which targets pre-kindergarten through grade four. And I love that Kathy writes. in her acknowledgments that this work is the culmination of more than 40 years working with children and teachers observing, wondering, discussing, reading, and thinking. Shelly (34:11.692) It is. So spot on to the observations and the things that I noticed in my own teaching, but it's also still one of the most referenced resources that I use. And if podcasts had a video, I would be able to hold up and show you my dog eared book with sticky notes coming out the all the sides because it is just something that. just resonates with me again. And then I think also maybe less familiar. Mike (36:38.958) I think you mentioned giants and those are some gigantic folks in the world of mathematics education. The other piece that I think really resonates for me is I had a really similar experience with both CGI and Kathy Richardson in that a lot of what they're describing are the things that I was seeing in classrooms. What it really helped me do is understand how to place that behavior and what the meaning of it was in terms of students understanding of mathematics. And it also helped me think about that as an asset that then I could build on. Shelley, I think this is probably a great place to stop, but I wanna thank you so much for joining us. It has really been a pleasure talking with you. Mike (37:28.95) Say thank you again, but definitively. Mike (37:35.48) Brent, how do you feel about that? Mike (37:41.966) do you want to jump in? Yeah, feel free. Mike (38:04.194) is Is there a question I could ask that would set you up? Mike (38:14.574) can work that in to a new ending. Mike (38:42.872) Do you, there something that you want to add though, Shelley? Cause we can, we can edit it, edit content in, and we can sequence content in too. So if there's something that mattered to you, we can absolutely add it. Mike (39:14.542) Let's do a question like that then. Mike (39:35.02) What if we see... Mike (39:40.578) Why don't you, why don't, Yeah, why don't you say it? Go ahead and say it the way that you you it was going to flow out and then we'll we can edit this in definitely. Mike (39:59.191) Okay. Okay, go for it. Yeah, yeah. Mike (40:06.676) can I tell you this is one of the smoothest podcast recordings we have had? There's nothing to be sorry about. Mike (40:18.562) There, okay, I was, can you ask the question again, Mike? So that it's clean. Mike (40:29.752) Are there resources you would invite our listeners to engage with if they want to continue learning? Mike (42:26.328) I think that's a great place to stop. Shelly Schaefer, thank you so much for joining us. Mike (42:38.602) That was perfect. Yeah, fantastic. I'm gonna cut roll. No, there's nothing to be sorry about © 2025 The Math Learning Center | www.mathlearningcenter.org

Dr. Victoria Jacobs, Examining the Meaning and Purpose of our Questions ROUNDING UP: SEASON 3 | EPISODE 12 Mike (00:03): The questions educators ask their students matter. They can have a profound impact on students' thinking and the shape of their mathematical identities. Today we're examining different types of questions, their purpose and the meaning students make of them. Joining us for this conversation is Dr. Vicki Jacobs from the University of North Carolina Greensboro. Welcome to the podcast, Vicki. I'm really excited to talk with you today. Vicki (00:33): Thanks so much for having me. I'm excited to be here. Mike (00:36): So you've been examining the ways that educators use questioning to explore the details of students' thinking. And I wonder if we could start by having you share what drew you to the topic. Vicki (00:47): For me, it all starts with children's thinking because it's absolutely fascinating, but it's also mathematically rich. And so a core part of good math instruction is when teachers elicit children's ideas and then build instruction based on that. And so questioning obviously plays a big role in that, but it's hard. It's hard to do that well in the moment. So I found questioning to explore children's thinking to be a worthwhile thing to spend time thinking about and working on. Mike (01:17): Well, let's dig into the ideas that have emerged from that work. How can teachers think about the types of questions that they might ask their students? Vicki (01:24): Happy to share. But before I talk about what I've learned about questioning, I really need to acknowledge some of the many people that have helped me learn about questioning over the years. And I want to give a particular shout out to the teachers and researchers in the wonderful cognitively guided instruction or CGI community as well as my long-term research collaborators at San Diego State University. And more recently, Susan Sen. This work isn't done alone, but what have we learned about teacher questioning across a variety of projects? I'll share two big ideas and the first relates to the goals of questioning and the second addresses more directly the types of questions teachers might ask. So let's start with the goals of questioning because there are lots of reasons teachers might ask questions in math classrooms. And one common way to think about the goal of questioning is that we need to direct children to particular strategies during problem solving. (02:23): So if children are stuck or they're headed down a wrong path, we can use questions to redirect them so that they can get to correct answers with particular strategies. Sometimes that may be okay, but when we only do that, we're missing a big opportunity to tap into children's sense-making. Another way to think about the goal of questioning is that we're trying to explore children's thinking during problem solving. So think about a math task where multiple strategies are encouraged and children can approach problem solving in any way that makes sense to. So we can then ask questions that are designed to reveal how children are thinking about the problem solving, not just how well they're executing our strategies. And we can ask these questions when children are stuck, but also when they solve problems correctly. So this shift in the purpose of questioning is huge. And I want to share a quote from a teacher that I think captures the enormity of this shift. (03:26): She's a fifth grade teacher, and what she said was the biggest thing I learned from the professional development was not asking questions to get them to the answers so that I could move them up a strategy, but to understand their thinking. That literally changed my world. It changed everything. So I love this quote because it shows how transformative this shift can be because when teachers become curious about how children are thinking about problem solving, they give children more space to problem solve in multiple ways, and then they can question to understand and support children's ideas. And these types of questions are great because they increase learning opportunities for both children and teachers. So children get more opportunities to learn how to talk math in a way that's meaningful to them because they're talking about their own ideas and they also get to clarify what they did think more about important math that's embedded in their strategies and sometimes to even self-correct. And then as teachers, these types of questions give us a window into children's understandings, and that helps us determine our next steps. Questioning can have a different and powerful purpose when we shift from directing children toward particular strategies to exploring their mathematical thinking. Mike (04:54): I keep going back to the quote that you shared, and I think the details of the why and kind of the difference in the experience for students really jump out. But I'm really compelled by what that teacher said to you about how it changes everything. And I wonder if we could just linger there for a moment and you could talk about some of the things that you've seen happen for educators who have that kind of aha moment in the same way that that teacher did, how that impacts the work that they're doing with children or how they see themselves as an educator. Vicki (05:28): That's a great question. I think it's freeing in some way because it changes how educators think about what their next steps are. Every teacher has lots of pressures from standards and sometimes pacing guides and grade level teams that are working on the same page, all sorts of things that are a big part of teaching. But it puts the focus back on children and children's thinking and that my next steps should then come from there. And so in some ways, I think it gives a clearer direction for how to navigate all those various pressures that teachers have. Mike (06:14): I love that. Let's talk about part two. Vicki (06:17): Sure. So if we have the goal of questioning to explore children's thinking, how do we decide what questions to ask? So first of all, there's never a best question. There are many questioning frameworks out there that can provide lots of ideas, but what we've found is that the most productive questions always start with what children say and do. So that means I can't plan all my questions in advance, and instead I have to pay close attention to what children are saying and doing during problem solving. And to help us with that, we found a distinction between inside questions and outside questions. And that distinction has been really useful to us and also usable even during instruction. So inside questions are questions that explore details that are part of inside children's current strategies. And outside questions are questions that focus on strategies or representations that are not what children have done and may even be linked to how we as teachers are thinking about problem solving. (07:26): So I promised an example, and this is from our recent research project on teaching and learning about fractions. And we asked teachers to think about a child's written strategy for a fraction story problem. And the problem was that there are six children equally sharing four pancakes, and they need to figure out how much pancake each child can get. So we're going to talk about Joy's strategy for solving this problem. She is a fourth grader who solved the problem successfully, but in a complex and rather unconventional way. So I'm going to describe her strategy as a reminder. We have six children sharing four pancakes. So she drew the four pancakes. She split the first three pancakes into fourths and distributed the pieces to the six children, and that works out to two fourths for each child. But now she has a problem because she has one pancake left and fourths aren't going to work anymore because that's not enough pieces for her six children. (08:23): So she split the pancake first into eighths and then into 20 fourths and distributed those pieces. So each child ends up receiving two fourths, one eighth and one 24th. And when you put all those amounts together, they equal the correct amount of two thirds pancake per child. But Joy left her answer in pieces as two fourths, one eighth and one 24th, and she wrote those fractions in words rather than using symbols. Okay, so there's a lot going on in this strategy. And the specific strategy doesn't matter so much for our conversation, but the situation does. Here we have a child who has successfully solved the problem, but how she solved it and how she represented her answer are different than what we as adults typically do. So we ask teachers to think about what kind of follow-up conversation would you want to have with joy? (09:23): What types of questions would you want to ask her? And there were these two main questioning approaches, what we call inside questioning and outside questioning. So let's start with outside questioning. These teachers focused on improving Joy's strategy. So they ask follow-up questions like, is there another way you can share the four pancakes with six children? Or is your strategy the most efficient way you could share the pancakes? Or is there a way to cut bigger servings that would be more efficient? So given the complexity of Joy's strategy, we can appreciate these teachers' goals of helping joy move to a more efficient strategy. But all of these questions are pushing her to use a different strategy. So we considered them outside questions because they were outside of her current strategy. And outside questions can sometimes be productive, but they tend to get overused. And when we use them a lot, they can communicate to kids that what they're actually doing was wrong and that it needs fixing. (10:29): So let's think about the other approach of inside questioning. These teachers started by exploring what Joy had done in all of its complexity. And they ask a variety of questions. Usually it started with a general question, can you tell me what you did? But then they zoomed in on some of the many details. So for examples, they've asked how she split the pancakes. They offered questions like, why did you split the first three pancakes into four pieces? Or Tell me about the last pancake. That was the one that she split into eights and 20 fourths. Or they might ask about how she knew how to name each of the fractional amounts, especially the one 24th, because that's something that many children might've struggled with. And then there were questions about a variety of other details. Some of them are hard to explain without showing you a picture of the strategy, but the point is that the teachers took seriously what Joy had done and elevated it to the focus of the conversation. So Joy had a chance to share her reasoning and reflect on it, and the teachers could better understand Joy's approach to problem solving. So we found this distinction between inside and outside questioning to be useful to teachers and even in the midst of instruction because teachers can quickly check in with themselves. Am I asking an inside question or an outside question? Mike (11:49): Well, I have so many questions about inside and outside questions, but I want to linger on inside questions. What I found myself thinking is that for the learner, there are benefits for building number sense or conceptual understanding. The other thing that strikes me is that inside questions are also an opportunity to support students' math identity. And I wonder if that's something that you've seen in your work with teachers and with students. Vicki (12:14): Absolutely. I love your question. One of my favorite things about inside questions is that children see that their ideas are being taken seriously. And that's so empowering. It helps children believe that they can do math and that they are in charge of their mathematical thinking. I'll share a short story that was memorable for me, and this was from a while ago when I was in graduate school. So I was working on a research project and we were conducting problem solving interviews with young children. And our job was to document their strategies. So if we could see exactly what they did, we were told to write down the strategy and move on. But if we needed to clarify something, we could ask follow up questions. I was working with a first grader who had just spent a really long time solving a story problem. He had solved it successfully, and he had done that by joining many, many, many unifix cubes into a very long train. (13:10): And then he had counted them by ones multiple times. So he had been successful. I could tell exactly what he had done. So I started to move on to the next problem. So this young child looked at me a little incredulous and simply asked, don't you want to know how I did it? And he had come from a class where his math thinking was valued, and talking about children's thinking was a regular part of what they did. So he couldn't quite understand why this adult was not interested in how he had thought about the problem. Well, I was a little embarrassed and of course backtracked and listened to his full explanation. But the interaction stuck with me because it showed me how empowering it was for children to truly be listened to as math thinkers. And I think that's something we want for all children. Mike (14:00): The other thing that's hitting me in that story and in the story of joy is mea culpa. I am a person who has lived in the cult of efficiency where I looked at a student's work and my initial thought was, how do I nip the edges of this to get to more efficiency? But I really am struck by it how different the idea of asking the student to explain their thinking or the why behind it. I find myself thinking about joy, and it appears that she was intent on making sure that there were equal shares for each person. So there's ways that she could build to a different level of efficiency. But I think recognizing that there's something here that is really important to note about how and why she chose that, that would feel really meaningful as a learner. Vicki (14:44): I agree. I think what I like about inside questions is that they encourage us to, that children's thinking makes sense, even if it's different than how we think about it. It's our job to figure out how it makes sense. And then to build from there. Mike (15:03): Can you just say more about that? That feels like kind of a revelation. Vicki (15:08): Well, if we start with how kids are thinking and we take that seriously and we make that the center of the conversation, then we're acknowledging to the student and to ourselves that the child has something meaningful to bring to this conversation. And so we need to figure out how the child is thinking all the kind of kernels of mathematical strength in that thinking. And then yes, we can build from there, but we start with where they are as opposed to how we might solve the problem. Mike (15:49): If you were to offer educators a universal inside question or a few sentence frames for inside questions, is it possible to construct something like that that's generic or do you have other advice for us? Vicki (16:02): So that's a nice trick question. I wish it were that easy. I don't really think there are any universal inside questions. Perhaps the only universal one I can think of is something like, how did you solve this problem? It's a great general open-ended question. That's a good starter question in most situations. But the really powerful questions generally come from noticing mathematically important details in children's strategies. So a sentence stem that has been helpful in our work is, I noticed blank, so I wonder blank. Obviously questions don't have to be phrased exactly like this, but the idea is that we pick something that the child has done in their strategy and ask a question about the child's thinking behind that strategy detail. And that keeps us honest because the question absolutely has to begin with something in the child's strategy rather than inadvertently kind of slipping into our strategy. Mike (17:04): Vicki, what do you think about the purpose of outside questions? Are there circumstances where we would want to ask our students an outside question? Vicki (17:12): Absolutely. Sometimes we need to push children's thinking or share particular ideas, and that's okay. It's not that all outside questions are bad, it's just that we tend to overuse them and we could use them at more productive times. And by that I mean that we generally want to understand children's thinking before nudging their thinking forward with outside questions. So let's go back to the earlier example of Joy. Who was solving that problem about six children sharing four pancakes. And we had the two groups of teachers that had the different approaches to follow up questioning. There was the outside questioning that immediately zeroed in on improving Joy's strategy and the inside questioning that spent time exploring Joy's reasoning behind her strategy. So I'm thinking of two specific teachers right now. One generally took the outside questioning approach and the other inside questioning approach. And what was interesting about this pair was that they both asked the same outside question, could Joy partition the pancakes in a different way? (18:19): But they asked this question at different times and the timing really matters. So the teacher who took an outside questioning approach wanted to begin her conversation that way. She wanted to ask Joy, could she partition in a different way? But in contrast, the teacher who took an inside questioning approach wanted to ask Joy lots of questions about the details of her existing strategy, and then posed this very same question at the end to see if Joy had some new ideas for partitioning after their conversation about her existing strategy. And that feels really different to children. So the exact same question can send children different messages when outside questions are posed. First they communicate to children that what they did was wrong and needs fixing. But when outside questions are posed after a conversation about their thinking, it communicates a puzzle or a problem to be solved. (19:17): And children often are better equipped to consider this new problem having thoroughly discussed their own strategy. So I guess when I think about outside questions, I think of timing and amount. We generally want to start with inside questions, and we want most of our questions to be inside questions, but some outside questions can be productive. It's just that we overuse them. I want to mention one other thing about outside questions, and I think we often need fewer outside questions than we think we do, as long as we have space for children to learn from other children's thinking. So think about a typical lesson structure like launch, explore, discuss where children solve problems independently. And then the lesson concludes with a whole class discussion where children share their strategies and reflect on their problem solving. Will these sharing sessions serve as natural outside questions? Because children get to think about strategies that are outside of their own, but in a way that doesn't point to their own strategy as lacking in some way. So outside questions definitely have a place we just need to think about when we ask them and how many of them are really necessary. Mike (20:34): That is really helpful. I find myself thinking about my own process when I'm working on a problem, be it mathematical or organizational or what have you. When someone asks me to talk about how I've thought about it, engaging in that process in some ways primes me, right? Because I've gotten clearer on my own thinking. I suspect that the person who's asking me the question is also clearer on that, which allows them to ask a different kind of outside question if and when they get to the point. So there's the benefit for the learner in that their clarifying their own thinking. There's the benefit in the educator who's engaging with the learner and getting just a much clearer sense of how that thinking was happening. And I suspect that leads to an outside question that's much more productive. Vicki (21:16): It's a win-win situation. Mike (21:18): Absolutely. This conversation has been wonderful. The challenge of having a podcast, of course, is that we've got about 20 to 25 minutes to talk about a really big idea that has profound implications for teachers. If someone wanted to pick up on the things we've been talking about today, where would you start, Vicki? Vicki (21:38): I would encourage them to go talk to children. Children's thinking is so mathematically rich and it's so fascinating. So be curious about their thinking. Ask questions, ask those inside questions. Don't worry about asking the best question. It doesn't exist, but ask questions and then children will be your guides. They'll help you know where to go next. The other thing I would suggest is these journeys are always best done with your colleagues. And so get a colleague together and think about questioning together what we were talking about earlier with joy strategy teachers. We're looking at students' written work. That's a great place to practice. You can look at children's written work and talk together to figure out what types of conversations do you want to have with this child afterwards. Mike (22:28): I think that's a great place to stop. I want to thank you so much for joining us today, Vicki, it has really been a pleasure talking with you. Vicki (22:34): That was fun. Thanks for having me. Mike (22:39): This podcast is brought to you by the Math Learning Center and the Meyer Math Foundation dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center | www.mathlearningcenter.org  

Dr. Karisma Morton, Understanding and Supporting Math Identity   ROUNDING UP: SEASON 3 | EPISODE 11 In this episode, we will explore the connection between identity and mathematics learning. We'll examine the factors that may have shaped our own identities and those of our students. We'll also discuss ways to practice affirming students' identities in mathematics instruction. BIOGRAPHIES Dr. Karisma Morton is an assistant professor of mathematics education at the University of North Texas. Her research explores elementary preservice teachers' ability to teach mathematics in equitable ways, particularly through the development of their critical racial consciousness. Findings from her research have been published in the Journal for Research in Mathematics Education and Educational Researcher. ​ RESOURCES The Impact of Identity in K–8 Mathematics: Rethinking Equity-Based Practices by Julia Aguirre, Karen Mayfield-Ingram, and Danny Martin Rough Draft Math: Revising to Learn by Amanda Jansen Olga Torres' “Rights of the Learner” framework Cultivating Mathematical Hearts: Culturally Responsive Mathematics Teaching in Elementary Classrooms by Maria del Rosario Zavala and Julia Maria Aguirre TRANSCRIPT Mike Wallus: If someone asked you if you were good at math, what would you say, and what justification would you provide for your answer? Regardless of whether you said yes or no, there are some big assumptions baked into this question. In this episode, we're talking with Dr. Karisma Morton about the ways the mathematics identities we formed in childhood impact our instructional practices as adults and how we can support students' mathematical identity formation in the here and now.  Welcome to the podcast, Karisma. I am really excited to be talking with you about affirming our students' mathematics identities. Karisma: Oh, I am really, really excited to be here, Mike. Thank you so much for the invitation to come speak to your audience about this. Mike: As we were preparing for this podcast, one of the things that you mentioned was the need to move away from this idea that there are math people and nonmath people. While it may seem obvious to some folks, I'm wondering if you can talk about why is this such an important thing and what type of stance educators might adopt in its place? Karisma: So, the thing is, there is no such thing as a math person, right? We are all math people. And so, if we want to move away from this idea, it means moving away from the belief that people are inherently good or bad at math. The truth is, we all engage in mathematical activity every single day, whether we realize it or not. We are all mathematicians. And so, the key is, as math teachers, we want to remove that barrier in our classrooms that says that only some students are math capable.  In the math classroom, we can begin doing that by leveraging what students know mathematically, how they experience mathematics in their daily life. And then we as educators can then incorporate some of those types of activities into the everyday learning of math in our classrooms. So, the idea is to get students to realize they are capable math doers, that they are math people. And you're showing them the evidence that they are by bringing in what they're already doing. And not just that they are math doers, but that those peers that are also engaged in the classroom with them are capable math doers. And so, breaking down those barriers that say that some students are and some students aren't is really key. So, we are all math people. Mike: I love that sentiment. You know, I've seen you facilitate an activity with educators that I'm hoping that we could replicate on the podcast. You asked educators to sort themselves into one of four groups that best describe their experience when they were a learner of mathematics. And I'm wondering if you could read the categories aloud and then I'm going to ask our listeners to think about the description that best describes their own experiences. Karisma: OK, great. So, there are four groups. And so, if you believe that your experience is one where you dreaded math and you had an overall bad experience with it, then you would choose group 1. If you believe that math was difficult but you could solve problems with tutoring or help, then you would select group 2. If you found that math was easy because you were able to memorize and follow procedures but you had to practice a lot, then you'd be in group 3. And finally, if you had very few difficulties with math or you were kind of considered a math whiz, then you would select group 4. Mike: I had such a strong reaction when I participated in this activity for the first time. So, I have had my own reckoning with this experience, but I wonder what impact you've seen this have on educators. Why do it? What's the impact that you hope it has for someone who's participating? Karisma: Yeah. So, I would say that a key part of promoting that message that we started off talking about is for teachers to go back, to reflect. We have to have that experience of thinking about what it was like for us as math learners. Because oftentimes we go into the classroom and we're like, “All right, I got to do this thing.” But we don't take a minute to reflect: “What was it like for me as a math learner?” And I wanted to first also say that I did not develop this activity. This is not a Karisma original. I did see this presented at a math teacher-educator conference about five years ago by Jennifer Ward. I think she's at Kennesaw State [University] right now. But the premise is the same: We want to give teachers an opportunity to reflect over their own experiences as math learners as a good starting place for helping them to identify with each other and also with the students that they're teaching. And so, whenever I have this activity done, I have each of the participants reflect. And then they have conversations around why they chose what they chose. And this is the opportunity for them to have what we call “windows,” “mirrors,” and “sliding glass doors,” right? So, you either can see yourself in another person's experience and feel like, “Oh, I'm not alone here,” especially if it were a negative experience. Or you may get to see or take a glimpse into what someone else has experienced that was very different from your own and really get a chance to understand what it was like for them. They may have been the math whiz, and you're looking at them like they're an alien that fell from the sky because you're like, “How did that happen,” right? But you can begin to have those kinds of conversations: “Why was it like this for you?” and “It wasn't like that for me.” Or “It was the same for me, but what did it look like in your instance versus my instance?”  I honestly feel like sometimes people don't realize that their experience is not necessarily unique, especially if it's coming from a math trauma perspective. Some people don't want to talk about their experience because they feel like it was just theirs. But they sometimes can begin to realize that, “Hey, you had that experience too, and let's kind of break down what that means.” Do you want to be that type of teacher? Do you want to create the type of environment where you felt like you weren't a capable math doer? So powerful, powerful exercise. I encourage your listeners to try it with a group of friends or colleagues at work and really have that conversation. Mike: Gosh, I'm just processing this. One of the things that I keep going back to is you challenging us to discard the idea that some people are inherently good at math and other people are not. And I'm making a connection that if I'm a person who identified with group 1, where I dreaded math and it was really a rough experience, what does it mean for me to discard the idea that some people are inherently good or inherently not good at math versus if I identified as a person who was treated as the math whiz and it came easy for me, again, what's required for me?  It feels like there's things that we can agree with on the surface. We can agree that people are not good inherently at mathematics. But I find myself really thinking about how my own experience actually colors my beliefs and my actions, how agreeing to that on the surface and then really digging into how your own experience plays out in your practice or the ways that you interact with kids. There's some work to be done there, it seems like. Karisma: Absolutely. You hit the nail on the head there. It's important to do that work. It's really important for us to take that moment to reflect and think about how our own experience may be impacting how we're teaching mathematics to children. Mike: I think that's a great place to make a shift and talk about areas where teachers could take action to cultivate a positive mathematics identity for kids. I wonder if we can begin by talking about expectations and norms when it comes to problem solving. Karisma: Yes. So, Julia Aguirre, Karen Mayfield-Ingram, and Danny Martin wrote this amazing book, called The Impact of Identity in K–8 Mathematics: Rethinking Equity-Based Practices. And one of those equity-based practices is affirming math learners' identities. And so, one of the ways we can do this in the math classroom is when having students engaged in problem solving. And so, one of the things that we want to be thinking about when we are having students engaged in math problem solving is we want to be promoting students' persistence and reasoning during problem solving. And you might wonder, “Well, what does that actually look like?”  Well, it might be helpful to see what it doesn't look like, right? So, in the typical math classroom, we often see an emphasis on speed: who got it done quickly, who got it done first, who even got it done within the time allotted. And then also this idea of competition. So, that is really hard for kids because we all need time to process and think through our problem-solving strategies. And if we're putting value on speed, and we're putting value on competition, are we in fact putting value on a problem-solving strategy or the process of problem-solving? So, one way to affirm math learners' identities is to move away from this idea of speed and competition and foster the type of environment where we're valuing students' persistence with the problem. We're valuing students' processes in solving a problem, how they're reasoning, how they're justifying their steps or their solutions' strategies, as opposed to who's getting done quickly.  Another thing to be thinking about is reframing making mistakes. There's so many great resources about this. What comes to mind immediately is Rough Draft Math by Amanda Jansen, which is really helping us to reframe the idea that we can make some mistakes, and we can revise our thinking. We can revise our reasoning, and that's perfectly OK.  Olga Torres' “Rights of the Learner” framework talks a lot about the right to make a mistake is one of the four rights of the learner in the mathematics classroom. And so, when having kids engaged in problem-solving and mathematics, mistakes should be seen more like what Olga Torres calls “celebrations,” because there are opportunities for learning to occur. We can focus on this mistake and think about and problem-solve through the mistake. “Well, how did we get here?” Use it as a moment that all students can benefit from. And so, kids then become less afraid to make mistakes because they're not ridiculed or made to feel less than because they've done so. Instead, it empowers them to know that “Hey, I made this mistake, but in actuality, this is going to help me learn. And it's also going to help my classmates.” Mike: I suspect a lot of those moments, people really appreciate when there's the “aha!” or the “oh!” What was happening before that might've been some struggle or some misconceptions or a mistake. You're making me think that we kind of have to leave space for those mistakes or those misconceptions to emerge if we really want to have those “aha!”s or those “oh!”s in our classroom. Karisma: That's exactly right. And imagine if you are the one who's like, “Oh!”—what that does for your self-confidence. And even having your peers recognize that you've come to this answer or this understanding. It almost becomes like a collective win if you have fostered a type of environment where it's less about me against you and more about all of us learning together. Mike: The other thing that came to me is that I'm thinking back to the four groups. I would've identified as a person who would fit into group 2, meaning that there were definitely points where math was difficult for me, but I could figure it out with tutoring or with help from a teacher. I start to wonder now how much of my perception was about the fact that it just took me a little bit longer to process and think about it. So, it wasn't that math was difficult. It was that I was measuring my sense of myself in mathematics around whether I was the first person, or I was fast, or I got it right away, or I got it right the first time, as opposed to really thinking about, “Do I understand this?” And to me, that really feels connected to what you're saying, which is the way that we as teachers value students' actions, their rough-draft attempts, their mistakes, and position those as part of the process—that can have a really concrete impact on how I think about myself and also how I think about what it is to do math.  Well, let's shift again and talk about another area where educators could support positive identity. I'm thinking about the ways that they can engage with students' background knowledge and their life experiences. Karisma: Hmm, yeah. This is a huge one. And this really, again, comes back to recognizing that our students are whole human beings. They have experiences that we should want to leverage in the math classroom, that they don't need to keep certain parts of themselves at the door when they come in. And so, how do we take advantage of what our students are bringing to the table? And so, we want to be thinking a lot about, “Well, who is the student?” “What do they know?” “What other identities do they hold?” “What's important to them?” “What kinds of experiences do they have in their everyday life that I can bring into the math classroom?” “What are their strengths?” “What do they enjoy doing?” The truth of the matter is really great teachers do this all the time, you know? You know who your students are for the most part, right? And students come to us with a whole host of experiences that we want to leverage and come with all sorts of experiences that we could use in the math classroom. I think oftentimes we don't think about making connections between those things and how to connect them to the mathematics that's happening in the classroom. So, oftentimes we don't necessarily see a reason to connect what we know about our students to mathematics. And so, it's really just a simple extra step because really amazing teachers—which I know they're amazing teachers that are listening right now—you know who your students are. So how do we take what we know about them and bring that into the mathematics learning? Again, as with problem solving, what is it that we want to stay away from? We want to be staying away from connecting math identity only with correct answers and how fast a kid is at solving a problem. Their math identity shouldn't be dependent on how many items they got correct on an assessment. It should be more about, “Well, what is it that they know? And how are we able to use this in the math classroom?” Mike: You're making me think about how oftentimes there's this distinction that happens in people's minds between school math and math that happens everywhere in the real world. Part of what I hear you suggesting is that when you help kids connect to their real world, you're actually doing them another service and that you're helping them see, like, “Oh, these lived experiences that I might not have called mathematics, they are,” right? “I do mathematics. I'm a doer.” And part of our work in bringing that in is helping them see what's already there. Karisma: I love that. Helping them see what's already there. That's exactly right. Mike: Well, before we go, I'm wondering if you could talk about some of the resources that have informed your thinking about this and that you think might also help a person who's listening who wants to keep learning. Karisma: Yeah. There's a lot of great resources out there. The one that I rely on heavily is The Impact of Identity in K–8 Mathematics: Rethinking Equity-Based Practices. I really like this book because it's very accessible. It does a really great job of setting the stage for why we need to be thinking about equity-based practices. And I really enjoy how practical things are. So, the book goes through describing what a representative lesson would look like. And so, it's a really nice blueprint for teachers as they're thinking about students' identities and how to promote positive math identity amongst their students. And then I think we also mentioned Rough Draft Math by Amanda Jansen, which is a good read. And then there's also a new book that came out recently, Cultivating Mathematical Hearts: Culturally Responsive [Mathematics] Teaching in Elementary Classrooms. And this book goes even deeper by having vignettes and having specific classroom examples of what teaching in this kind of way can look like. So those are three resources off the top of my head that you could dig into and have book clubs at your schools and engage with your fellow educators and grow together. Mike: I think that's a great place to stop. Thank you so much for joining us today. This has really been a pleasure. Karisma: Oh, it's been a pleasure talking to you too. Thank you so much for this opportunity. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center | www.mathlearningcenter.org  

Sue Kim and Myuriel Von Aspen, Building Productive Partnerships   ROUNDING UP: SEASON 3 | EPISODE 10 In this episode, we examine the practice of building productive student partnerships. We'll talk about ways  educators can cultivate joyful and productive partnerships and the role the educator plays once students are engaged with their partner.  BIOGRAPHIES Sue Kim is an advocate for children's thinking and providing them a voice in learning mathematics. She received her teaching credential and master of education from Biola University in Southern California. She has been an educator for 15 years and has taught and coached across TK–5th grade classrooms including Los Angeles Unified School District and El Segundo Unified School District as well as several other Orange County, California, school districts.  Myuriel von Aspen believes in fostering collaborative partnerships with teachers with the goal of advancing equitable, high-quality learning opportunities for all children. Myuriel earned a master of arts in teaching and a master of business administration from the University of California, Irvine and a bachelor of science in computer science from Florida International University. She currently serves as a math coordinator of the Teaching, Learning, and Instructional Leadership Collaborative. ​ RESOURCES Catalyzing Change in Early Childhood and Elementary Mathematics by National Council of Teachers of Mathematics Purposeful Play by Kristine Mraz, Alison Porcelli, and Cheryl Tyler  Hands Down, Speak Out: Listening and Talking Across Literacy and Math K–5 by Kassia Omohundro Wedekind and Christy Hermann Thompson TRANSCRIPT Mike Wallus: What are the keys to establishing productive student partnerships in an elementary classroom? And how can educators leverage the learning that happens in partnerships for the benefit of the entire class? We'll explore these and other questions with Sue Kim and Myuriel von Aspen from the Orange County Office of Education on this episode of Rounding Up.  Well, hi, Sue and Myuriel. Welcome to the podcast. Myuriel von Aspen: Hi, Mike.  Sue Kim: Thanks for having us. Mike: Thrilled to have you both.  So, I first heard you two talk about the power of student partnerships in a context that involved counting collections. And during that presentation, you all said a few things that I have been thinking about ever since. The first thing that you said was that neuroscience shows that you can't really separate emotions from the way that we learn. And I wonder what do you mean when you say that and why do you think it's important when we're thinking about student partnerships? Myuriel: Yes, absolutely. So, this idea comes directly from neuroscience research, the idea that we cannot build memories without emotions. I'm going to read to you a short quote from the NCTM [National Council of Teachers of Mathematics] publication Catalyzing Change in Early Childhood and Elementary Mathematics that says, “Emerging evidence from neuroscience strongly shows that one cannot separate the learning of mathematics content from children's views and feelings toward mathematics.”  So, to me, what that says is that how children feel has a huge influence on their ability to learn math and also on how they feel about themselves as learners of math. So, depending on how they feel, they might be willing to engage in the content or not. And so, as they're engaging in counting collections and they're enjoying counting and they feel joyful and they're doing this with friends, they will learn better because they enjoy it, and they care about what they're doing and what they're learning. Mike: You know, this is a nice segue to the other thing that has been on my mind since I heard you all talk about this because I remember you said that students don't think about a task like counting collections as work, that they see it as play. And I wonder what you think the ramifications of that are for how we approach student partnership? Sue: Yeah, you know, I've been in so many classrooms across TK through fifth [grade], and when I watch kids count collections, we see joy, we see engagement in these ways. But I've also been thinking about this idea of how play is even defined, in a way, since you asked that question that they think of it as play.  Kristine Mraz, teacher, author, and a consultant, has [coauthored] a book called Purposeful Play. And I remember this was the first time I hear about this reference about Vivian Paley, an American early childhood educator and researcher, stress through her career, the importance of play for children when she discovered in her work that play's actually a very complex activity and that it is indeed hard work. It's the work of kids. It's the work of what children do. That's their life, in a sense. And so, something I've been thinking about is how kids perceive play is different than how adults perceive play. And so, they take it with seriousness. There is a complex, very intentionality behind things that they do and say. And so, when we are in our session, and we reference Megan Franke, she says that when young people are engaging with each other's ideas, what they're able to do is mathematically important. But it's also important because they're learning to learn together. They're learning to hear each other. They're developing social and emotional skills as they try and navigate and negotiate each other's ideas. And I think for kids that this could be considered play, and I think that's so fascinating because it's so meaningful to them. And even in a task like counting, they're doing all these complex things. But as adults we see them, and we're like, “Oh, they're playing.” But they are really thinking deeply about some of these ideas while they're developing these very critical skills that we need to give opportunities for them to develop. Myuriel: I like that idea of leaning into the play that you consider maybe not as serious, but they are. Whether they're playing seriously or not, that you might take that opportunity to make it into a mathematical question or a mathematical reflection. Sue: I totally agree with you. And taking it back to that question that you asked, Mike, about, “How do we approach student partnerships then?” And I think that we need to approach it with this lens of curiosity while we let kids engage in these ways and opportunities of learning to hear each other and develop these social-emotional skills, like we said. And so, when you see kids that we think are “playing” or they're building a tower: How might we enter that space with a lens of curiosity? Because to them, I think it's serious work. We can't just think, “Oh, they're not really in the task” or “They're not doing what they were supposed to do.” But how do we lean into that space with a lens of curiosity as Megan reminded us to do, to see what mathematical things we can tap into? And I think that kids always rise to the occasion. Mike: I love that. So, let's talk about how educators can cultivate joyful and productive student partnerships. I'm going to guess that as is often the case, this starts by examining existing beliefs that I might have and some of my expectations. Sue: Yeah, I think it really begins with your outlook and your identity as a teacher. What's your outlook on what's actually possible for kids in your class? Do you believe that kids as young as 4-year-olds can take on this responsibility of engaging with each other in these intelligent ways? Unless we begin there and we really think and reflect and examine what our beliefs are about that, I think it's hard to go and move beyond that, if that makes sense.  And like what we just talked about, it's being open to the curiosity of what could be the capacity of how kids learn. I've seen enough 4-year-olds in TK classrooms doing these big things. They always blow my mind, blow my expectations, when opportunities are given to them and consistently given to them. And it's a process, right? They're not going to start on day one doing some of these more complex things. But they can learn from one another, and they also learn from you as a teacher because they are really paying attention. They are attending to some of these complex ideas that we put in front of them. Mike: Well, you hit on the question that I was thinking about. Because I remember you saying that part of nurturing partnerships starts with a teacher and perhaps a pair of children at a table. Can you all paint a picture of what that might look like for educators who are listening? Sue: Yeah, so actually in one of the most recent classrooms, I went in, and this teacher allowed me to partner with her in this work. She wanted to be able to observe and do it in a structured way so that she could pick up on some details of noticing the things that kids were doing. And so, she would have a collection out, or they got to choose. She was really good about offering choice to kids, another way to really engage them. And so, they would choose. They would come together. And then she started just taking some anecdotal notes on what she heard kids saying, what she saw them doing, what they had to actually navigate through some of the things, the stuck moments that came up.  From that, we were able to develop, “OK, what are some goals? We noticed Students A and B doing this and speaking in these ways. What might be the next step that we might want to put into a mini lesson or model out or have them actually share with the class what they were working on mathematically?” Whether it was organization, or how they decided they wanted to represent their count, how they counted and things like that.  And so, it was just this really natural process that took place that we were able to really lean into and leverage that kids really responded to because it wasn't someone else's work or a page from a textbook. It was their work, their collection that was meaningful to them and they had a true voice and a stake in that work. Mike: I feel like there have been points in time where my understanding of building groups was almost like an engineering problem, where you needed to model what you wanted kids to do and have them rehearse it so specifically. But I think what sits at the bottom of that approach is more about compliance. And what I loved about what you described, Sue, is a process where you're building on the mathematical assets that kids are showing you during their time together—but also on the social assets that they're showing you. So, in that time when you might be observing a pair or a partnership playing together, working together with something like counting collections, you have a chance to observe the mathematics that's happening. You also have a chance to observe the social assets that you see happening. And you can use that as a way to build for that group, but also to build for the larger group of children. And that just feels really profoundly different than, I think, how I used to think about what it was to build partnerships that were “effective.” Myuriel: You know, Mike, I think it's not only compliance. It's also that control. And what it makes me think about is, when we want to model ourselves what we want students to do, instead of—exactly what you said, looking at what they're doing and bringing that knowledge, those skills, that wisdom that's in the room from the students to show to others so that they feel like their knowledge counts. The teacher is not only the only authority or the only source of knowledge in the room—we bring so much, and we can learn from each other. So, I think it's so much more productive and so effective in developing the identity of students when you are showing something that they're doing to their peers versus you as an adult telling them what to do. Mike: Yeah. Are there any particular resources that you all have found helpful for crafting mini lessons as students are learning about how to become a partnership or to be productive in a partnership? Myuriel: Yes. One book that I love, it's not specific to counting collections, but it does provide opportunities for teachers to create micro-lessons when students are listening and talking to each other. It's Hands Down, Speak Out: Listening and Talking Across Literacy and Math K–5 by Kassia [Omohundro] Wedekind and Christy [Hermann] Thompson. And the reason why I love this book is because it provides, again, these micro-lessons depending on what the teacher is noticing, whether it is that the teacher is noticing that students need support listening to each other or maybe making their ideas clear. Or maybe students need to learn how to ask questions more effectively or even reflect on setting and reflecting on the goals that they have as partners. It does provide ideas for teachers to create those micro-lessons based on what the teacher is noticing. Sue: Yeah, I guess I want to add to that, Mike, as well, the resources that Myuriel said. But also, I think this is something I really learned along the process of walking alongside this teacher, was looking at partnerships through a mathematical lens and then a social lens. And so, the mini lesson could be birthed out of watching kids in one day. It might be a social lens thinking about, “They were kind of stuck because they wanted to choose different collections. What might we do about that?” And that kind of is tied to this problem-solving type of skill and goal that we would want kids to work on. That's definitely something that's going to come up as kids are working in partnerships. These partnerships are not perfect and pristine all the time. I think that's the nature of the job. And just as humans, they're learning how to get along, they're learning how to communicate and navigate and negotiate these things. And I think those are beautiful opportunities for kids and for teachers, then, to really lean into as goals, as mini lessons that can be out of this. And these mini lessons don't have to be long and drawn out. They can be a quick 5-, 10-minute thing. Or you can pause in the middle of counting and kind of spotlight the fact that “Mike and Brent had this problem, but we want to learn from them because they figured out how to solve it. And this is how. Let's listen to what happened.” So, these natural, not only places in a lesson that these opportunities for teaching can pop up, but that these mini lessons come straight from kids and how they are interacting and how they are taking up partnerships, whether it be mathematical or social. Mike: I think you're helping me address something that if I'm transparent about was challenging for me when I was a classroom teacher. I got a little bit nervous about what was happening and sometimes I would shut things down if I perceived partnerships to be, I don't know, overwhelming or maybe even messy. But you're making me think now that part of this work is actually noticing what are the assets that kids have in their social interactions in the way that they're playing together, collaborating together, the mathematics? And I think that's a big shift in my mind from the way that I was thinking about this work before. And I wonder, first of all, is this something that you all notice that teachers sometimes are challenged by? And two, how you talk to someone who's struggling with that question of like, “Oh my gosh, what's happening in my classroom?” Myuriel: Yes, I can totally understand how teachers might get overwhelmed. We hear this from, not only from teachers trying to do the work of counting collections, but even just using tools for students to problem-solve because it does get messy. I like the way Sue keeps emphasizing how it will be messy. When you have rich mathematical learning happening, and you're using tools and collections and you have 30 students having conversations, it definitely will get messy. But I would say that something that teachers can do to mitigate some of that messiness is to think about the logistics ahead of time and be intentional about what you are planning to do. So, some of the things that they may want to think about is: How are students going to access the counting collections? Where are you going to [put] the tools that they're going to be using? Where physically in the classrooms will students get together to have collections so that they have enough room to spread out and record and talk to each other? And just like Sue was mentioning: How do I partner students so that they do have a good experience, and they support each other? So, all of these things that might cost a bit of chaos if you don't think about them, you can actually think about each one of those ahead of time so that you do have a plan for each one of those.  Another thing that teachers may want to consider thinking about is, what do they want to pay attention to when they are facilitating or walking around? There's a lot that they need to pay attention to. Just like Sue mentioned, it is important for them to pay attention to something because you want to bring what's in the room to connect it and have these mini lessons of what students actually need. And also, thinking about after the counting collections: What worked and what didn't? And what changes do I want to make next time when I do this again? Just so that there is a process of improvement every time. Because as Sue had mentioned, it's not going to happen on day one. You are learning as a teacher, and the students are learning. So, everybody in that room is learning to make this a productive and joyful experience. Sue: Yeah, and another thing that I would definitely remind teachers about is that there's actually research out there about how important it is for kids to engage with one another's mathematical ideas. I'm so thankful that people are researching out there doing this work for us. And this goes along with what Myuriel was saying, but the expectations that we put on ourselves as teachers sometimes are too far. We're our biggest critique-ers of the work that we do. And of course we want things to go well, but to make it more low-risk for yourself. I think that when we lower those stakes, we're more prone to let kids take ownership of working together in these ways, to use language and communication that makes sense while doing math and using these cognitive abilities that are still in the process of developing. And I think they need to remember that it takes time to develop, and it's going to get there. And kids are going to learn. Kids are going to do some really big things with their understanding. But giving [yourself] space, the time to learn along with your students, I think is very critical so that you feel like it's manageable. You feel like you can do it again the next day. Mike: Tell me a little bit about how you have seen educators use things like authentic images or even video to help their students make sense of what it means to work in a partnership. What have you seen teachers do? Sue: Yeah. Not to mention how that is one sure way to get kids engaged. I don't know if you've been in a room full of first graders or kindergartners, but if you put a video image up that's them counting and showing how they are thinking about things, they are one-hundred-percent there with you. They love being acknowledged and recognized as being the doers and the sensemakers of mathematics. And it goes into this idea of how we position kids competently, and this is another way that we can do that. But capturing student thinking in photos or a short clip has really been a powerful tool to get kids to engage in each other's ideas in a deeper way. I think it allows teachers and students to pause and slow down and really focus in on the skill of noticing. I think people forget that noticing is a skill you have to teach. And you have to give opportunities for kids to actually do these things so they can see mathematically what's happening within the freeze-frame of this image, of this collection, and how we might ask questions to help facilitate and guide their thinking to think deeply about these ideas. And so, I've seen teachers use them with partners, and they may say, “Hey, here's one way that they were counting. How do you think they counted within the frame of this picture or this photo that we took?” And then kids will have these conversations. They'll engage mathematically what they think, and then they might show the video clip of the students actually counting. And they get to make predictions. They get to navigate the language around what they think. And it's just, again, been a really nice tool that has then branched out into whole-group discussions. So, you can use it with partnerships and engage certain kids in specific ways, but then being able to utilize that and leverage that in whole-group settings has really been powerful to see. Myuriel: I also recently observed a teacher with pictures, showing students different tools that different partners were using and having those discussions about, “Why did this tool work and why didn't this one?” or “What will you have to do if your collection gets bigger?” So, it is a great opportunity to really show from what they're using and having those discussions about what works and what doesn't, and “Why would I use this versus this?” from their own work. Mike: Myuriel, what you made me wonder is if you could apply this same idea of using video or images to help support some of those social goals that we were talking about for students as well. Myuriel: I think that you could. I can just imagine that if you see two students working together and supporting each other or asking some good questions and being curious, you could record them and then show that to the others to ask them what they're noticing. “How are these two students supporting each other in their learning?” Even “How are they being kind to each other when they make a mistake?” So, there is so much power in using video for not just the mathematical skills, but also for the social skills. Sue: Myuriel, when you're talking, you're reminding me about two particular students that we have watched, and we have recorded video around, actually, when they came to a disagreement.  There was this one instance when a couple of students came to a disagreement about what to call the next number of the sequence. And that was a really cool moment because we actually discovered, “Wow, these two peers had enough trust in each other to pause, to listen to both sides.” And then when it came time to actually call the number and the sequence, the other student actually trusted enough and listened to the reasoning of the other student to say, “OK, I'm going to go along with you, and I think that should be what the sequence is.” And it was just a really neat opportunity and—that this teacher actually showed in front of kids just to see what kids would say in response to that particular moment. Myuriel: It was actually one very cute, but very interesting moment when you see that second student who's listening to the other one. And actually at first she kind of argued with him a little bit about, “No, it's not this number.” But the second time around, when she counted, she paused right at that same spot where she had trouble before, and she set the number that he had suggested the earlier time so that you see that she's listening, she's considering someone else's ideas, and she's learning the correct sequence. Yes, that was really amazing to see. Sue: So, it's the sequence of numbers that they're working on, but think about all the social aspects of what is happening and developing, and I think that they're addressing it and that they're having to engage with [it]. It's [a] very complex situation that they're learning a lot of skills around in that very moment. Mike: You know, I wonder how an educator might think about their role once students are actually engaged with a partner. How do you all think about goals, or the role of the teacher, once students are working with a partner? Sue: I think that one of the things we're really thinking about and being more intentional about is: When do we actually interject, or when do we as teachers actually say something? When and how do we make those decisions? And for several years now, I've really taken on this notion that we are facilitators. Yes, we're teachers. But more than anything, we are facilitators of the students in our class, and we want to really give them the opportunity to work through some of these ideas. And we will have set up partnerships based on what we've seen and notes that we took as kids have been working. But it's an ever-innovated process, I think. And I think something that's always going to be on the forefront is that idea: How are we facilitating? How are we deciding when we want to say something or interject, and why? And what is it that we are trying to get kids to think about? Because I think we need to help students realize that they are always in the driver's seat of what they're doing, especially if they're in a partnership. And there are targeted things that we can have them maybe think about when we drop a question based on what we're noticing. Or maybe when they're stuck, and they're in the middle of negotiating something. But I really think that it starts there with us kind of thinking about: What is our role? Is it OK that we step back and we just watch even if they have to problem-solve through something that feels like, “Oh, I don't know if they're going to get through that moment.” But we've got to let them. We've got to give them opportunities to do that without having to rescue them every single time. Myuriel: And you're right, Sue, we've seen it so many times when if you just bite your tongue, 10 seconds later, it's happening, right? They're helping each other, and they get to the idea that you thought you had to bring up to them. But they were able to resolve it. So, if we only allow that time for them to process the idea or to revise their thinking or to allow the other partner to support their partner, it will happen. Sue: Yeah, and I think that doesn't mean that we can't set kids up. I've seen teachers launch the lesson with something a partner did before yesterday, and they will have referred to a protocol or something they're working on. And then as facilitators, we can then go out, and we might already be thinking about, “Oh, I want to be watching these two partnerships today”—having in mind, “OK, this is my target idea for them, my target goal for them.” So, there are definite ways that we can frame and decide who we want to watch and observe, but while in the balance of letting kids do what they're going to do and what the expectation of being surprised. Because kids always surprise us with their brilliance. Mike: Yeah, there's multiple things that came to mind as I was listening to you all talk about this. The first one is how it's possible to inadvertently condition kids to see the teacher coming and look and stop and potentially look for the teacher to say something. We actually do want to avoid that. We want to see their thinking.  The other piece is the difference between, as you said, potentially dropping a question and interjecting, as you said, Myuriel, biting your tongue and letting them persist through—whether it's an idea they're grappling with or a struggle for what to do next—that there's so much information in those moments that we can learn or that might help us think about what's next. It's a challenge, I think, because math culture in the United States is such that we're kind of trained to see something that looks like a mistake. “Let's get in there.” And I hear you giving people permission to say, “Actually, it's OK to step back and watch their thinking and watch them try to make sense of things because there's a big payoff there.” Sue: Absolutely. Yeah.  Myuriel: Yes. And, Mike, I think we as teachers—you feel the need of having to address every single “mistake” per either individual student or per partnership. And sometimes you feel like, “I have 30 students, how can I possibly do that?” And I think that's where the power of doing a share out from what you've observed, bringing everyone together, learning from what was in the room, right? Because just like Sue was saying, it's not that you don't ever set up kids with knowledge of what you've observed, but you bring the power. It's what you're bringing, what's in the room, what you've noticed. But you share it out, or you have students share it out, with everyone so that everyone is moving forward. Mike: I have a follow-up question for you all about goals for partnerships. I'm wondering how you think about the potential for partnerships as a way to help develop language, be it academic or social, for students. Are there particular practices that you imagine educators could take up if language development was one of their goals? Myuriel: I'm so glad you're asking that question because I don't think we can learn math without language. I don't think we can learn anything without language. And I think that working in partnerships provides such an authentic, meaningful way of developing language because students are in conversations with each other. And we know that conversation is one way that ideas develop conversations or even sharing your thinking. Sometimes we notice that as students are sharing their thinking, and they're listening to themselves, they catch themselves making a mistake, and they are able to revise their thinking based on what they are saying. So again, I think it is the perfect opportunity for students to mathematically learn counting sequence or socially learn how to negotiate and make sense of what they're going to represent, when they're counting, or to explain their thinking. And we know, of course, that one of the mathematical practices is justifying, explaining your thinking. So, it's important to provide those opportunities for students to do that in this kind of structural way. I also think that working in partnerships provides this opportunity for teachers to listen and notice if there's any language that students are starting to use that can be shared with others. So again, this idea that you hear it from someone in the room and that's going to help everybody else grow. Or that if students are doing something and you can name it, provide those terms to students. So, for example, just like I mentioned, somebody's explaining their thinking and through that they change their mind. They revised their thinking. Actually sharing that with the whole class and naming it: “Oh, they were revising their thinking” or sharing how they were explaining something with academic language so that others can also use that language as they're explaining their own thinking. So, I think that those are powerful ways to provide opportunities for everyone's academic language or social skills through language to be developed. Sue: Yeah, I think that another big idea that comes out of that language piece is just how kids are learning to make sense of how to be partners, especially our younger students, our younger mathematicians. They're really needing to figure out like, “Oh, what does it mean to take turns to speak about this and how I use my words in this way versus another?” And I think that's another big opportunity for kids to build those skills because we can't just assume that kids come into our classrooms knowing how to talk in these ways, how to address each other, how to engage respectfully, that they can disagree respectfully, even in partnerships. And we want them to have the time and space to be able to develop those skills through language as well. Mike: You know, I think the mental movie that I have for the point in time after children have engaged in any kind of partnership task, be it counting collections or something else, has really shifted. Because I think beforehand the way the movie ended was potentially sharing a student's representation if they had represented something on a piece of paper that showed what they had physically done with their things. And I still think that's valid and important, particularly if that's one of your goals.  But you're making me think a lot more about the potential of images of students at work as they're going through the process or video and how closing, or potentially opening the next time, with that really just kind of expands this idea of what's happening. Being able to look at a set of hands that are on a set of materials or in the process of moving materials or listening to language that's emerging from students in the form of a short video. There's a lot of richness that you could capture, and it's also a little bit more of a diverse way of showing what's going on. And it feels like another way to really position what you're doing—not just the output in the form of the paper representation—but what you're actually doing is valuable, and it's a contribution. And I think that just feels like there's a lot of potential in what you all are describing. Sue: I think you hit the nail on the head. We're trying, and it's hard work. But to be open to these ideas, to these possibilities. And like you said, it's positioning kids so drastically different than how we've been doing it for so many years. And how you're actually inviting kids to be contributors of this work that they are now. They have the knowledge. They are the ones that hold the knowledge in the room. And how we frame kids and what they're doing is I think very critical because kids learn from that, and kids have so many things to offer that we need to really be able to think about how we want to create those opportunities for kids. Myuriel: And, Mike, something that you said also made me think of just like we want to provide those opportunities for students to be creative and to show what they know. What you were talking about, having this new perspective, makes me think about also teachers being creative with how they use counting collections, right? There isn't just the one way. It doesn't mean that at the end of every counting collection, I have to have a share out right at the end and decide at that moment. I could start the day that way. I could start the next session that way. I could use a video. I could use a picture. I could have students share it. So, you can get creative. And I think that's the beauty also, because I think as a teacher, it's not only the students that are learning; you are learning along with them. Mike: That's a great place to stop. This has been an absolutely fabulous conversation. Thank you both so much for joining us. Myuriel: Thank you. Thank you so much for this opportunity. Sue: Thank you. Thanks for having us. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center | www.mathlearningcenter.org  

Dr. Kasi Allen, Breaking the Cycle of Math Trauma   ROUNDING UP: SEASON 3 | EPISODE 9 If you are an educator, you've likely heard people say things like “I'm a math person.” While this may make you cringe, if you dig a bit deeper, many people can identify specific experiences that convinced them that this was true. In fact, some of you might secretly wonder if you are a math person as well. Today we're talking with Dr. Kasi Allen about math trauma: what it is and how educators can take steps to address it. BIOGRAPHY Kasi Allen serves as the vice president of learning and impact at The Ford Family Foundation. She holds a PhD degree in educational policy and a bachelor's degree in mathematics and its history, both from Stanford University. RESOURCES “Jo Boaler Wants Everyone to Love Math” — Stanford Magazine R-RIGHTS Learning to Love Math by Judy Willis TRANSCRIPT Mike Wallus: If you're an educator, I'm almost certain you've heard people say things like, “I am not a math person.” While this may make you cringe, if you dig a bit deeper, many of those folks can identify specific experiences that convinced them that this was true. In fact, some of you might secretly wonder if you're actually a math person. Today we're talking with Dr. Kasi Allen about math trauma: what it is and how educators can take steps to address it.  Well, hello, Kasi. Welcome to the podcast. Kasi Allen: Hi, Mike. Thanks for having me. Great to be here. Mike: I wonder if we could start by talking about what drew you to the topic of math trauma in the first place? Kasi: Really good question. You know, I've been curious about this topic for almost as long as I can remember, especially about how people's different relationships with math seem to affect their lives and how that starts at a very early age. I think it was around fourth grade for me probably, that I became aware of how much I liked math and how much my best friend and my sister had an absolutely opposite relationship with it—even though we were attending the same school, same teachers, and so on. And I really wanted to understand why that was happening. And honestly, I think that's what made me want to become a high school math teacher. I was convinced I could do it in a way that maybe wouldn't hurt people as much. Or it might even make them like it and feel like they could do anything that they wanted to do.  But it wasn't until many years later, as a professor of education, when I was teaching teachers how to teach math, that this topic really resurfaced for me [in] a whole new way among my family, among my friends. And if you're somebody who's taught math, you're the math emergency person. And so, I had collected over the years stories of people's not-so-awesome experiences with math. But it was when I was asked to teach an algebra for elementary teachers course, that was actually the students' idea. And the idea of this course was that we'd help preservice elementary teachers get a better window into how the math they were teaching was planting the seeds for how people might access algebra later.  On the very first day, the first year I taught this class, there were three sections. I passed out the syllabus; in all three sections, the same thing happened. Somebody either started crying in a way that needed consoling by another peer, or they got up and left, or both. And I was just pretty dismayed. I hadn't spoken a word. The syllabi were just sitting on the table. And it really made me want to go after this in a new way. I mean, something—it just made me feel like something different was happening here. This was not the math anxiety that everybody talked about when I was younger. This was definitely different, and it became my passion project: trying to figure how we disrupt that cycle. Mike: Well, I think that's a good segue because I've heard you say that the term “math anxiety” centers this as a problem that's within the person. And that in fact, this isn't about the person. Instead, it's about the experience, something that's happened to people that's causing this type of reaction. Do I have that right, Kasi? Kasi: One hundred percent. And I think this is really important. When I grew up and when I became a teacher, I think that was an era when there was a lot of focus on math anxiety, the prevalence of math anxiety. Sheila Tobias wrote the famous book Overcoming Math Anxiety. This was especially a problem among women. There were dozens of books. And there were a number of problems with that work at the time, and that most of the research people were citing was taking place outside of math education. The work was all really before the field of neuroscience was actually a thing. Lots of deficit thinking that something is wrong with the person who is suffering this anxiety. And most of these books were very self-helpy. And so, not only is there something wrong with you, but you need to fix it yourself. So, it really centers all these negative emotions around math on the person that's experiencing the pain, that something's wrong with them.  Whereas math trauma really shifts the focus to say, “No, no, no. This reaction, this emotional reaction, nobody's born that way.” Right? This came from a place, from an experience. And so, math trauma is saying, “No, there's been some series of events, maybe a set of circumstances, that this individual began to see as harmful or threatening, and that it's having long-lasting adverse effects. And that those long-lasting effects, this kind of triggering that starts to happen, is really beginning to affect that person's functioning, their sense of well-being when they're in the presence, in this case, of mathematics.” And I think the thing about trauma is just that. And I have to say in the early days of my doing this research, I was honestly a little bit hesitant to use that word because I didn't want to devalue some of the horrific experiences that people have experienced in times of war, witnessing the murder of a parent or something. But it's about the brain. It's how the brain is responding to the situation. And what I think we know now, even more than when I started this work, is that there is simply trauma [in] everyday life. There are things that we experience that cause our brains to be triggered. And math is unfortunately this subject in school that we require nearly every year of a young person's life. And there are things about the way it's been taught over time that can be humiliating, ridiculing; that can cause people to have just some really negative experiences that then they carry with them into the next year. And so that's really the shift. The shift is instead of labeling somebody as math anxious—“Oh, you poor thing, you better fix yourself”—it's like, “No, we have some prevalence of math trauma, and we've got to figure out how people's experiences with math are causing this kind of a reaction in their bodies and brains.” Mike: I want to take this a little bit further before we start to talk about causes and solutions. This idea that you mentioned of feeling under threat, it made me think that when we're talking about trauma, we are talking about a physiological response. Something is happening within the brain that's being manifested in the body. And I wonder if you could talk just a little bit about what happens to people experiencing trauma? What does that feel like in their body? Kasi: So, this is really important and our brains have evolved over time. We have this incredible processing capacity, and it's coupled with a very powerful filter called the amygdala. And the amygdala [has been] there from eons ago to protect us. It's the filter that says, “Hey, do not provide access to that powerful processor unless I'm safe, unless my needs are met. Otherwise, I gotta focus on being well over here.” So, we're not going to give access to that higher-order thinking unless we're safe. And this is really important because modern imaging has given us really new insights into how we learn and how our body is reacting when our brain gets fired in this way.  And so, when somebody is experiencing math trauma, you know it. They sweat. Their face turns red. They cry. Their body and brain are telling them, “Get out. Get away from this thing. It will hurt you.” And I just feel like that is so important for us to remember because the amygdala also becomes increasingly sensitive to repeat negativity. So, it's one thing that you have a bad day in math, or you maybe have a teacher that makes you feel not great about yourself. But day after day, week after week, year after year, that messaging can start to make the amygdala hypersensitive to these sorts of situations. Is that what you were getting at with your question? Mike: It is. And I think you really hit on something. There's this idea of repeat negativity causing increased sensitivity, I think has real ramifications for classroom culture or the importance of the way that I show up as an educator. It's making me think a lot about culture and norms related to math in schools. I'm starting to wonder about the type of traumatizing traditions that we've had in math education that might contribute to this type of experience. What does that make you think? Kasi: Oh, for sure. Unfortunately, I think the list is a little long of the things that we may have been doing completely inadvertently. Everybody wants their students to have a great experience, and I actually think our practices have evolved. But culturally, I think there are some things about math that contribute to these “traumatizing traditions,” is what I've called them.  Before we go there, I do want to say just one other thing about this trauma piece, and that is that we've learned about some things about trauma in childhood. And a lot of the trauma in childhood is about not a single life-altering event. But childhood trauma is often about these things that happened repeatedly where a child was being ridiculed, being treated cruelly. And it's about that repetition that is really seeding that trauma so deeply and that sense that they can't stop it, that they don't have control to stop the thing that is causing them pain or suffering. So, I just wanted to make sure that I tagged that because I think there is something about what we've learned about the different forms of childhood trauma that's especially salient in this situation.  And so, I'll tie it to your question, which is, think about some of the things we've done in math historically. We don't do them in every place, but the ability grouping that has happened over time, it seems to go in and out of fashion. When a kid is told they're in the lower class, “Oh, this is something you're not good [at]—the slower math.” We often use speed to measure understanding, and so smarter is not faster. And there's some great quotes, Einstein among them. So that's a thing. When you gotta do it right now, it has to be one-hundred-percent right. It has to be superfast. We've often prioritized individual work over collaboration. So, you're all alone in this. In fact, if you're working with others, somehow that's cheating as opposed to collaborating.  We teach kids tricks rather than teaching them how to think. And I think we deprive kids of the opportunity to have an idea. It's really hard to get excited about something where all you're doing is reproducing—reproducing something that somebody else thought of as quickly as possible and [it] needs to be one-hundred-percent [accurate]. You don't get to bring your own spin to it. And so, we focus on answers rather than people's reasoning behind the answers. That can be something that happens as well. And I think one of the things that's always gotten me is that there's only one way. Not only is there only one right answer, but there's only one way to get there, which also contributes to this idea of having to absorb somebody else's thinking rather than actualizing your own.  And I absolutely know that most teachers are working to not do as much of these things in their math classrooms. And I want to be sure in having this conversation that—you know, I'm a lover of education and teachers, I taught teachers for many years. This is not about the teachers so much as the sort of culture of math and math education that we were all brought up in. And we've got to figure out how to make math something more so that kids can see themselves in it. And that it's not something that happens in a vacuum and is this performance course rather than a class where you get to solve cool problems that no one knows the exact answer to, or there's the exact right way, or that you get to get your own questions answered. Things you wonder about. That it's a chance to explore.  So, I mean, ultimately, I think we just know that there's a lot of negativity that happens around math, and we accept it. And that is perhaps the most traumatizing tradition of all because that kind of repeat negativity we know affects the amygdala. It affects people's ability to access math in the long run. So, we gotta have neutral or better. Mike: So, in the field of psychology, there's this notion of generational trauma, and it's passed from generation to generation. And you're making me wonder if we're facing something similar when it comes to the field of math education. I'm wondering what you think educators might be able to do to reclaim math for themselves, especially if they're a person who potentially does have a traumatic mathematics experience and maybe some of the ways that they might create a different type of experience for their students. Kasi: Yeah, let's talk about each of those. I'm going to talk about one, the multigenerational piece, and then let's talk about how we can help ourselves and our students. One is, I think it's really very possible that that's what we're looking at in terms of math trauma. Culturally, I think we've known for a while that this is happening, with respect to math, that—you know, I've had parents come to back-to-school night and tell me that they're just not a math family. And even jokingly say, “Oh, we're all bad at math, don't be too hard on us,” and all the other things. And so, kids inherit that. And it's very common for kids to have the same attitude towards math that their parents do and also that their teachers do.  And that's where I think in my mind, I really want to help every elementary teacher fall in love with math because if we look at the data, I think of any undergraduate major, it's those who major in education who report the highest rates of math anxiety and math trauma. And so, when you think about folks who feel that way about math, then being in charge of teaching it to kids in the early years, that's a lot to carry. And so, we want to give those teachers and anyone who has had this experience with math an opportunity to reclaim, regroup. And in my experience, what I've found is actually simply shifting the location of the problem is a really strong first step. When people understand that they actually aren't broken, that the feelings that they have about math don't reflect some sort of flaw in them as a human, but that it's a result of something they've experienced, a lot is unlocked. And most folks that I have worked with over my time working on this issue, they know. They know exactly the moment. They know the set of experiences that led to the reactions that they feel in their body. They can name it, and with actually fairly startling detail. So, in my teaching—and I think this is something anybody can do—is they would write a “mathography.” What is the story of your life through a math lens? What has been the story of your relationship with math over the course of your life and what windows does that give you into the places where you might need to heal? We've never had more tools to go back and sort of relearn areas of math that we thought we couldn't learn. And so often the trauma points are as math becomes more abstract. So many people have something that happened around fractions or multidigit multiplication and division. When we started—we get letters involved in math. I had somebody say, “Math was great as long as it was numbers. Then we got letters involved, and it was terrible.” And so, if people can locate, “This is where I had the problem. It's not me. I can go back and relearn some things.” I feel like that's a lot of the healing, and that, in fact, if I'm a teacher or if I'm a parent, I love my kids, whether they're my children or my students, and I'm going to work on me so that they have a better experience than I had. And I've found so many teachers embrace that idea and go to work. So, some of the things that can happen in classrooms that I think fall from this is that, first of all, the recognition that emotional safety, you can't have cognition and problem solving without it. If you have kids in your classroom who have had these negative experiences in math, you're going to need to help them unpack those and level set in order to move on. And “mathography” is also a good tool for that. Some people use breathing.  Making sure that when you encounter kids that are exhibiting math anxiety, that you help them localize the problem outside of them. No one is born with math anxiety. It's the math of school that creates it. And if we ignore it, it's just going to get worse. So, some people feel like they can kind of smooth it over. I think we need to give kids the tools to unpack it and move beyond it. But it's so widespread, and I've encountered teachers who were afraid to go there. It's like the Pandora's box. My advice to them is that if you'll open the box and heal what's inside, the teaching becomes much easier. Whereas if you don't, you're fighting that uphill battle all the time.  You know, students will feel more safe in classrooms where mistakes are opportunities to learn; where they're not a bad thing and where they see each other as resources, where they are not alone, and where they can collaborate and really take responsibility for each other's learning. So, some of the most powerful classrooms I've seen where there were a lot of kids who had very negative experiences with math, a teacher had succeeded in creating this learning environment, this community of learners where all the kids seem to recognize that somebody would have a good day, someone else would have a not good day, but it would be their turn for a good day a few days from now. [chuckles] So, we're all just going to take care of each other as we go.  I think some things that teachers can keep a particular eye on is being sure that kids are given authentic work to do in math. It's really easy to start giving kids what we've called busywork, but work that really isn't engaging their brain. And it turns out that that boredom cycle triggers the negativity cycle, which can actually get your amygdala operating in a way that is not as far from trauma as we might all like to think. And so, while it isn't the same kind of math trauma that we're talking about here, it does affect the amygdala. And so that's something we should be aware of. And so, this is something—I think kids should learn about their brains in school. I don't know if it's the math teacher's job. But if they haven't learned about their brains yet, when you get them, I would recommend teaching kids about their brains, teaching them strategies for when they feel that kind of shutdown, that headache, like “I can't think.” Because most of the time, they actually can't. And they need to have some kind of reset.  Another tip, just in terms of disrupting that trauma cycle in the classroom, is that by the time kids get to be third, fourth grade and up, they know who is good at math, or they've labeled each other. You know, “Who's good at math? Who's struggled?” Even if they are not tracked and sorted, they've assessed each other. Sometimes they've put those labels on themselves. And so, if a teacher has the skills to assign competence to those students that may be being labeled as low status mathematically in their classroom—and it takes a teacher that knows their students well. But if you happen to see that a student that maybe has low status with computation, but wow, they are really good at developing the visuals for a math problem, or they're really great at illustrating a story or drawing others out in a collaborative group, but finding an area of competence that's authentic.  Sorry to go on and on. I could sit here and talk to you about this all day, but those are some of the things I would recommend. Mike: Well, I think there's a few things that jump out, and I wanted to take them in little bits. I'm going to try to summarize, and then I want to come back and pick these up a little bit. So, one of the pieces that you named really struck a chord with me, which is recognizing as an educator that I have a story about mathematics that is playing out maybe just under the level of consciousness that bubbles up here and there. When you mentioned the traumatic experiences, my head went back to third grade with multiplication tables, and I can see myself sitting in the seat. And when you mentioned fractions, again, I could see myself facing the board in third grade looking down at a workbook where we were supposed to be adding fractions with denominators that were not common. And I had this moment of just dread in my stomach because I remember just thinking, “I don't know what is happening at all.” And I'll say biographically, I think I spent the first seven or eight years of my teaching career carrying those things with me in the way that I approach students. I knew that they weren't good for me, but I didn't really have a compelling sense of what could be different until I actually took some mathematics education courses and really started to understand mathematics and how children's ideas develop. And it did allow me to decenter the problem for myself and say, “Actually, I can make a lot of meaning out of mathematics.” What I experienced was not mathematics. It was memorizing a bunch of stuff and practicing a bunch of procedures. This idea that decentering where the problem is from the educator or in classrooms from the student, really, really feels powerful. I think it's a huge gift that we can give to our students and also to ourselves. The other piece that I'm really thinking about is this idea of positioning students and finding competency. That really stands out as something that I could attend to as a classroom teacher. I suspect that people who are listening can think about their own class of students. You as an educator probably know who the other kids think of as good at math, and I suspect you also know who they think isn't good at math. Knowing that kids know those stories as well, I could do something about that. I could look at the students who have low status and think about ways that I could raise them up. That feels really tangible. I could take and start thinking about that when I ask students to share their ideas, and I could do that tomorrow. It doesn't take a master's-level course in mathematics to do that. Does that make sense, Kasi? Kasi: I love all of that so much. One hundred percent. You know, when I was observing teachers—and this tended to happen more with elementary teachers just because of their own histories with math as you were saying here—but the difference between saying, “OK, everybody, we get to do math now. Clear your desks!” and “OK, everybody, I know it's hard, but it's time for math. We're strong. We're going to do it.” But there is this underlying kind of, “I don't really like this either, but we gotta do it.” as opposed to “We're going to discover something new today!” And so really just kind of listening to some of those implicit messages in the words that we choose, that's something we can change in a moment as well. Mike: Well, I think you and I could probably go on and on and continue this conversation for a long time. If I'm someone who's listening, are there resources you would recommend for someone who wants to continue learning about these ideas? Kasi: Yeah, absolutely. For me, the OG of this line of thinking is Jo Boaler, who most math teachers will know. She's the first person I ever heard use the word “math-traumatized.” And before I embarked and dove deeper into my math trauma research, I went down to Stanford and met with her, and she was wonderful and encouraging of, like, “Oh, no, no, no. Go, go, go, go. This is great.”  There's a woman named Ebony McGee, who's the founder of R-RIGHTS. [She was] a professor at Vanderbilt. She's doing some work with math identity that I think touches on this subject in a valuable way. I mean, I think this whole area of developing positive math identity is tightly connected to the math trauma work. And honestly, anyone who is doing work around child trauma and neuroscience and how we are seeing the development of the brain is going to provide some interesting resources.  I have to say, my all-time favorite is a book that I believe [...] is out of print, so it might be a thrift books purchase. But Dr. Judy Willis wrote a book called Learning to Love Math. Looks like you might be familiar with it. And I really think she did a lovely job in that book in a way that is absolutely targeting teachers to help us see how these very small actions that we take in the classroom could make a really big difference in terms of how our students see and experience the subject that we care about so much. Mike: I think that's a great place to stop. Thank you so much for joining us, Kasi. It's really been a pleasure talking with you. Kasi: Oh, my goodness, Mike, thank you so much. It's really been an honor to be here. Thanks for having me. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center | www.mathlearningcenter.org

ROUNDING UP: SEASON 3 | EPISODE 8 As a field, mathematics education has come a long way over the past few years in describing the ways students come to understand number, quantity, place value, and even fractions. But when it comes to geometry, particularly concepts involving shape, it's often less clear how student thinking develops. Today, we're talking with Dr. Rebecca Ambrose about ways we can help our students build a meaningful understanding of geometry. BIOGRAPHIES Rebecca Ambrose researches how children solve mathematics problems and works with teachers to apply what she has learned about the informal strategies children employ to differentiate and improve instruction in math. She is currently a professor at the University of California, Davis in the School of Education. RESOURCES Geometry Resources Curated by Dr. Ambrose Seeing What Others Cannot See Opening the Mind's Eye  TRANSCRIPT Mike Wallus: As a field, mathematics education has come a long way over the past few years in describing the ways that students come to understand number, place value, and even fractions. But when it comes to geometry, especially concepts involving shape, it's often less clear how student thinking develops. Today, we're talking with Dr. Rebecca Ambrose about ways we can help our students build a meaningful understanding of geometry.  Well, welcome to the podcast, Rebecca. Thank you so much for joining us today. Rebecca Ambrose: It's nice to be here. I appreciate the invitation. Mike: So, I'd like to start by asking: What led you to focus your work on the ways that students build a meaningful understanding of geometry, particularly shape? Rebecca: So, I taught middle school math for 10 years. And the first seven years were in coed classrooms. And I was always struck by especially the girls who were actually very successful in math, but they would tell me, “I like you, Ms. Ambrose, but I don't like math. I'm not going to continue to pursue it.” And I found that troubling, and I also found it troubling that they were not as involved in class discussion. And I went for three years and taught at an all-girls school so I could see what difference it made. And we did have more student voice in those classrooms, but I still had some very successful students who told me the same thing. So, I was really concerned that we were doing something wrong and that led me to graduate school with a focus on gender issues in math education. And I had the blessing of studying with Elizabeth Fennema, who was really the pioneer in studying gender issues in math education. And as I started studying with her, I learned that the one area that females tended to underperform males on aptitude tests—not achievement tests, but aptitude tests—was in the area of spatial reasoning. And you'll remember those are the tests, or items that you may have had where you have one view of a shape and then you have a choice of four other views, and you have to choose the one that is the same shape from a different view. And those particular tasks we see consistent gender differences on. I became convinced it was because we didn't give kids enough opportunity to engage in that kind of activity at school. You either had some strengths there or not, and because of the play activity of boys, that may be why some of them are more successful at that than others.  And then the other thing that informed that was when I was teaching middle school, and I did do a few spatial activities, kids would emerge with talents that I was unaware of. So, I remember in particular this [student,] Stacy, who was an eighth-grader who was kind of a good worker and was able to learn along with the rest of the class, but she didn't stand out as particularly interested or gifted in mathematics. And yet, when we started doing these spatial tasks, and I pulled out my spatial puzzles, she was all over it. And she was doing things much more quickly than I could. And I said, “Stacy, wow.” She said, “Oh, I love this stuff, and I do it at home.” And she wasn't the kind of kid to ever draw attention to herself, but when I saw, “Oh, this is a side of Stacy that I didn't know about, and it is very pertinent to mathematics. And she needs to know what doorways could be open to her that would employ these skills that she has and also to help her shine in front of her classmates.” So, that made me really curious about what we could do to provide kids with more opportunities like that little piece that I gave her and her classmates back in the day. So, that's what led me to look at geometry thinking. And the more that I have had my opportunities to dabble with teachers and kids, people have a real appetite for it. There are always a couple of people who go, “Ooh.” But many more who are just so eager to do something in addition to number that we can call mathematics. Mike: You know, I'm thinking about our conversation before we set up and started to record the formal podcast today. And during that conversation you asked me a question that involved kites, and I'm wondering if you might ask that question again for our listeners. Rebecca: I'm going to invite you to do a mental challenge. And the way you think about it might be quite revealing to how you engage in both geometric and spatial reasoning. So, I invite you to picture in your mind's eye a kite and then to describe to me what you're seeing. Mike: So, I see two equilateral triangles that are joined at their bases—although as I say the word “bases,” I realize that could also lead to some follow-up questions. And then I see one wooden line that bisects those two triangles from top to bottom and another wooden line that bisects them along what I would call their bases. Rebecca: OK, I'm trying to imagine with you. So, you have two equilateral triangles that—a different way of saying it might be they share a side? Mike: They do share a side. Yes. Rebecca: OK. And then tell me again about these wooden parts. Mike: So, when I think about the kite, I imagine that there is a point at the top of the kite and a point at the bottom of the kite. And there's a wooden piece that runs from the point at the top down to the point at the bottom. And it cuts right through the middle. So, essentially, if you were thinking about the two triangles forming something that looked like a diamond, there would be a line that cut right from the top to the bottom point. Rebecca: OK. Mike: And then, likewise, there would be another wooden piece running from the point on one side to the point on the other side. So essentially, the triangles would be cut in half, but then there would also be a piece of wood that would essentially separate each triangle from the other along the two sides that they shared. Rebecca: OK. One thing that I noticed was you used a lot of mathematical ideas, and we don't always see that in children. And I hope that the listeners engaged in that activity themselves and maybe even stopped for a moment to sort of picture it before they started trying to process what you said so that they would just kind of play with this challenge of taking what you're seeing in your mind's eye and trying to articulate in words what that looks like. And that's a whole mathematical task in and of itself. And the way that you engaged in it was from a fairly high level of mathematics.  And so, one of the things that I hope that task sort of illustrates is how a.) geometry involves these images that we have. And that we are often having to develop that concept image, this way of imagining it in our visual domain, in our brain. And almost everybody has it. And some people call it “the mind's eye.” Three percent of the population apparently don't have it—but the fact that 97 percent do suggests for teachers that they can depend on almost every child being able to at least close their eyes and picture that kite. I was strategic in choosing the kite rather than asking you to picture a rectangle or a hexagon or something like that because the kite is a mathematical idea that some mathematicians talk about, but it's also this real-world thing that we have some experiences with.  And so, one of the things that that particular exercise does is highlight how we have these prototypes, these single images that we associate with particular words. And that's our starting point for instruction with children, for helping them to build up their mathematical ideas about these shapes. Having a mental image and then describing the mental image is where we put language to these math ideas. And the prototypes can be very helpful, but sometimes, especially for young children, when they believe that a triangle is an equilateral triangle that's sitting on, you know, the horizontal—one side is basically its base, the word that you used—they've got that mental picture. But that is not associated with any other triangles. So, if something looks more or less like that prototype, they'll say, “Yeah, that's a triangle.” But when we start showing them some things that are very different from that, but that mathematicians would call triangles, they're not always successful at recognizing those as triangles. And then if we also show them something that has curved sides or a jagged side but has that nice 60-degree angle on the top, they'll say, “Oh yeah, that's close enough to my prototype that we'll call that a triangle.”  So, part of what we are doing when we are engaging kids in these conversations is helping them to attend to the precision that mathematicians always use. And that's one of our standards. And as I've done more work with talking to kids about these geometric shapes, I realize it's about helping them to be very clear about when they are referring to something, what it is they're referring to. So, I listen very carefully to, “Are they saying ‘this' and ‘that' and pointing to something?” That communicates their idea, but it would be more precise as like, I have to ask you to repeat what you were telling me so that I knew exactly what you were talking about. And in this domain, where we don't have access to a picture to point to, we have to be more precise. And that's part of this geometric learning that we're trying to advance. Mike: So, this is bringing a lot of questions for me. The first one that I want to unpack is, you talked about the idea that when we're accessing the mind's eye, there's potentially a prototype of a shape that we see in our mind's eye. Tell me more about what you mean when you say “a prototype.” Rebecca: The way that that word is used more generally, as often when people are designing something, they build a prototype. So, it's sort of the iconic image that goes with a particular idea. Mike: You're making me think about when I was teaching kindergarten and first grade, we had colored pattern blocks that we use quite often. And often when we talked about triangles, what the students would describe or what I believed was the prototype in their mind's eye really matched up with that. So, they saw the green equilateral triangle. And when we said trapezoid, it looked like the red trapezoid, right? And so, what you're making me think about is the extent to which having a prototype is useful, but if you only have one prototype, it might also be limiting. Rebecca: Exactly. And when we're talking to a 3- or a 4-year-old, and we're pointing to something and saying, “That's a triangle,” they don't know what aspect of it makes it a triangle. So, does it have to be green? Does it have to be that particular size? So, we'll both understand each other when we're talking about that pattern block. But when we're looking at something that's much different, they may not know what aspect of it is making me call it a triangle” And they may experience a lot of dissonance if I'm telling them that—I'm trying to think of a non-equilateral triangle that we might all, “Oh, well, let's”—and I'm thinking of 3-D shapes, like an ice cream cone. Well, that's got a triangular-ish shape, but it's not a triangle. But if we can imagine that sort of is isosceles triangle with two long sides and a shorter side, if I start calling that a triangle or if I show a child that kind of isosceles triangle and I say, “Oh, what's that?” And they say, “I don't know.” So, we have to help them come to terms with that dissonance that's going to come from me calling something a triangle that they're not familiar with calling a triangle. And sadly, that moment of dissonance from which Piaget tells us learning occurs, doesn't happen enough in the elementary school classroom. Kids are often given equilateral triangles or maybe a right triangle. But they're not often seeing that unusual triangle that I described. So, they're not bumping into that dissonance that'll help them to work through, “Well, what makes something a triangle? What counts and what doesn't count?” And that's where the geometry part comes in that goes beyond just spatial visualization and using your mind's eye, but actually applying these properties and figuring out when do they apply and when do they not apply. Mike: I think this is probably a good place to shift and ask you: What do we know as a field about how students' ideas about shape initially emerge and how they mature over time? Rebecca: Well, that's an interesting question because we have our theory about how they would develop under the excellent teaching conditions, and we haven't had very many opportunities to confirm that theory because geometry is so overlooked in the elementary school classroom. So, I'm going to theorize about how they develop based on my own experience and my reading of the literature on very specific examples of trying to teach kids about squares and rectangles. Or, in my case, trying to see how they describe three-dimensional shapes that they may have built from polydrons. So, their thinking tends to start at a very visual level. And like in the kite example, they might say, “It looks like a diamond”—and you actually said that at one point—but not go farther from there.  So, you decomposed your kite, and you decomposed it a lot. You said it has two equilateral triangles and then it has those—mathematicians would call [them] diagonals. So, you were skipping several levels in doing that. So, I'll give you the intermediate levels using that kite example. So, one thing a child might say is that “I'm seeing two short sides and two long sides.” So, in that case, they're starting to decompose the kite into component parts. And as we help them to learn about those component parts, they might say, “Oh, it's got a couple of different angles.” And again, that's a different thing to pay attention to. That's a component part that would be the beginning of them doing what Battista called spatial structuring. Michael Battista built on the van Hiele levels to try to capture this theory about how kids' thinking might develop. So, attention to component parts is the first place that we see them making some advances.  And then the next is if they're able to talk about relationships between those component parts. So, in the case of the kite, they might say, “Oh, the two short sides are equal to each other”—so, there's a relationship there—“and they're connected to each other at the top.” And I think you said something about that. “And then the long sides are also connected to each other.” And that's looking at how the sides are related to the other sides is where the component parts start getting to become a new part. So, it's like decomposing and recomposing, which is part of all of mathematics.  And then the last stage is when they're able to put the shapes themselves into the hierarchy that we have. So, for example, in the kite case, they might say, “It's got four sides, so it's a quadrilateral. But it's not a parallelogram because none of the four sides are parallel to each other.” So now I'm not just looking at component parts and their relations, but I'm using those relations to think about the definition of that shape. So, I would never expect a kid to be able to tell me, “Oh yeah, a kite is a quadrilateral that is not a parallelogram,” and then tell me about the angles and tell me about the sides without a lot of experience describing shapes. Mike: There are a few things that are popping out for me when I'm listening to you talk about this. One of them is the real importance of language and attempting to use language to build a meaningful description or to make sense of shape. The other piece that it really makes me think about is the prototypes, as you described them, are a useful starting place. They're something to build on.  But there's real importance in showing a wide variety of shapes or even “almost-shapes.” I can imagine a triangle that is a triangle in every respect except for the fact that it's not a closed shape. Maybe there's an opening or a triangle that has wavy sides that are connected at three points. Or an obtuse triangle. Being able to see multiple examples and nonexamples feels like a really important part of helping kids actually find the language but also get to the essence of, “What is a triangle?” Tell me if I'm on point or off base when I'm thinking about that, Rebecca. Rebecca: You are right on target. And in fact, Clements and Sarama wrote a piece in the NCTM Teaching Children Mathematics in about 2000 where they describe their study that found exactly what you said. And they make a recommendation that kids do have opportunities to see all kinds of examples. And one way that that can happen is if they're using dynamic geometry software. So, for example, Polypad, I was just playing with it, and you can create a three-sided figure and then drag around one of the points and see all these different triangles. And the class could have a discussion about, “Are all of these triangles? Well, that looks like a weird triangle. I've never seen that before.” And today I was just playing around with the idea of having kids create a favorite triangle in Polypad and then make copies of it and compose new shapes out of their favorite triangle. What I like about that task, and I think can be a design principle for a teacher who wants to play around with these ideas and get creative with them, is to give kids opportunities to use their creativity in making new kinds of shapes and having a sense of ownership over those creations. And then using those creations as a topic of conversation for other kids. So, they have to treat their classmates as contributors to their mathematics learning, and they're all getting an opportunity to have kind of an aesthetic experience. I think that's the beauty of geometry. It's using a different part of our brain. Thomas West talks about Seeing What Others Cannot See, and he describes people like Einstein and others who really solved problems visually. They didn't use numbers. They used pictures. And Ian Robertson talks about Opening the Mind's Eye. So, his work is more focused on how we all could benefit from being able to visualize things. And actually, our fallback might be to engage our mind's eye instead of always wanting to talk [chuckles] about things.  That brings us back to this language idea. And I think language is very important. But maybe we need to stretch it to communication. I want to engage kids in sharing with me what they notice and what they see, but it may be embodied as much as it is verbal. So, we might use our arms and our elbow to discuss angle. And well, we'll put words to it. We're also then experiencing it in our body and showing it to each other in a different way than [...] just the words and the pictures on the paper. So, people are just beginning to explore this idea of gesture. But I have seen, I worked with a teacher who was working with first graders and they were—you say, “Show us a right angle,” and they would show it to us on their body. Mike: Wow. I mean, this is so far from the way that I initially understood my job when I was teaching geometry, which was: I was going to teach the definition, and kids were going to remember that definition and look at the prototypical shape and say, “That's a triangle” or “That's a square.” Even this last bit that you were talking about really flips that whole idea on its head, right? It makes me think that teaching the definitions before kids engage with shapes is actually having it backwards. How would you think about the way that kids come to make meaning about what defines any given shape? If you were to imagine a process for a teacher helping to build a sense of triangle-ness, talk about that if you wouldn't mind. Rebecca: Well, so I'm going to draw on a 3-D example for this, and it's actually something that I worked with a teacher in a third grade classroom, and we had a lot of English language learners in this classroom. And we had been building polyhedra, which are just three-dimensional shapes using a tool called the polydrons. And our first activities, the kids had just made their own polyhedra and described them. So, we didn't tell them what a prism was. We didn't tell them what a pyramid was or a cube. Another shape they tend to build with those tools is something called an anti-prism, but we didn't introduce any of those terms to them. They were familiar with the terms triangle and square, and those are within the collection of tools they have to work with. But it was interesting to me that their experience with those words was so limited that they often confused those two. And I attributed it to all they'd had was maybe a few lessons every year where they were asked to identify, “Which of these are triangles?” They had never even spoken that word themselves. So, that's to have this classroom where you are hearing from the kids and getting them to communicate with each other and the teacher as much as possible. I think that's part of our mantra for everything. But we took what they built. So, they had all built something, and it was a polyhedra. That was the thing we described. We said it has to be closed. So, we did provide them with that definition. You have to build a closed figure with these shapes, and it needs to be three-dimensional. It can't be flat. So, then we had this collection of shapes, and in this case, I was the arbiter. And I started with, “Oh wow, this is really cool. It's a pyramid.” And I just picked an example of a pyramid, and it was the triangular pyramid, made out of four equilateral triangles. And then I pulled another shape that they had built that was obviously not any—I think it was a cube. And I said, “Well, what do you think? Is this a pyramid?” And they'd said, “No, that's not a pyramid.” “OK, why isn't it?” And by the way, they did know something about pyramids. They'd heard the word before. And every time I do this with a class where I say, “OK, tell me, ‘What's a pyramid?'” They'll tell me that it's from Egypt. It's really big. So, they're drawing on the Egyptian pyramids that they're familiar with. Some of them might say a little something mathematical, but usually it's more about the pyramids they've seen maybe in movies or in school.  So, they're drawing on that concept image, right? But they don't have any kind of mathematical definition. They don't know the component parts of a pyramid. So, after we say that the cube is not a pyramid, and I say, “Well, why isn't it?,” they'll say, “because it doesn't have a pointy top.” So, we can see there that they're still drawing on the concept image that they have, which is valid and helpful in this case, but it's not real defined. So, we have attention to a component part. That's the first step we hope that they'll make. And we're still going to talk about which of these shapes are pyramids. So, we continued to bring in shapes, and they ended up with, it needed to have triangular sides. Because we had some things that had pointy tops, but it wasn't where triangles met. It would be an edge where there were two sloped sides that were meeting there. Let's see. If you can imagine, while I engage your mind's eye again, a prism, basically a triangular prism with two equilateral triangles on each end, and then rectangles that attach those two triangles. Mike: I can see that. Rebecca: OK. So, usually you see that sitting on a triangle, and we call the triangles the base. But if you tilt it so it's sitting on a rectangle, now you've got something that looks like a tent. And the kids will say that. “That looks like a tent.” “OK, yeah, that looks like a tent.” And so, that's giving us that Level 1 thinking: “What does it look like?” “What's the word that comes to mind?” And—but we've got those sloped sides, and so when they see that, some of them will call that the pointy top because we haven't defined pointy top. Mike: Yes. Rebecca: But when I give them the feedback, “Oh, you know what, that's not a pyramid.” Then the class started talking about, “Hmm, OK. What's different about that top versus this other top?” And so, then they came to, “Well, it has to be where triangles meet.” I could have introduced the word vertex at that time. I could have said, “Well, we call any place where sides meet a vertex.” That might be [a] helpful word for us today. But that's where the word comes from what they're doing, rather than me just arbitrarily saying, “Today I'm going to teach you about vertices. You need to know about vertices.” But we need a word for this place where the sides meet. So, I can introduce that word, and we can be more precise now in what we're talking about. So, the tent thing didn't have a vertex on top. It had an edge on top. So now we could be precise about that. Mike: I want to go back, and I'm going to restate the thing that you said for people who are listening, because to me, it was huge. This whole idea of “the word comes from the things that they are doing or that they are saying.” Did I get that right? Rebecca: Yeah, that the precise terminology grows out of the conversation you're having and helps people to be clear about what they're referring to. Because even if they're just pointing at it, that's helpful. And especially for students whose first language might not be English, then they at least have a reference. That's why it's so hard for me to be doing geometry with you just verbally. I don't even have a picture or a thing to refer to. But then when I say “vertex” and we're pointing to this thing, I have to try as much as I can to help them distinguish between, “This one is a vertex. This one is not a vertex.” Mike: You brought up earlier supporting multilingual learners, particularly given the way that you just modeled what was a really rich back-and-forth conversation where children were making comparisons. They were using language that was very informal, and then the things that they were saying and doing led to introducing some of those more precise pieces of language. How does that look when you have a group of students who might have a diverse set of languages that they're speaking in the same classroom? Rebecca: Well, when we do this in that environment, which is most of the time when I'm doing this, we do a lot of pair-share. And I like to let kids talk to the people that they communicate best with so that if you have two Spanish speakers, for example, they could speak in Spanish to each other. And ideally the classroom norms have been established so that that's OK. But that opportunity to hear it again from a peer helps them to process. And it slows things down. Like, often we're just going so fast that people get lost. And it may be a language thing; it may be a concept thing. So, whatever we can do to slow things down and let kids hear it repeatedly—because we know that that repeated input is very helpful—and from various different people. So, what I'll often do, if I want everybody to have an opportunity to hear about the vertex, I'm going to invite the kids to retell what they understood from what I said. And then that gives me an opportunity to assess those individuals who are doing the retell and also gives the other students a chance to hear it again. It's OK for them to see or hear the kind of textbook explanation for vertex in their preferred language. But again, only when the class has been kind of grappling with the idea, it's not the starting point. It emerges as needed in that heat of instruction. And you don't expect them to necessarily get it the first time around. That's why these building tasks or construction tasks can be done at different levels. So, we were talking about the different levels the learner might be at. Everybody can imagine a kite, and everybody could draw a kite. So, I'm sort of differentiating my instruction by giving this very open-ended task, and then I'm trying to tune into what am I seeing and hearing from the different individuals that can give me some insight into their geometrical reasoning at this point in time. But we're going to keep drawing things, and we're going to keep building things, and everybody's going to have their opportunity to advance. But it's not in unison. Mike: A few things jumped out. One, as you were describing the experiences that you can give to students, particularly students who might have a diversity of languages in the same classroom, it strikes me that this is where nonverbal communication like gesturing or using a visual or using a physical model really comes in handy.  I think the other piece that I was reminded of as I was listening to you is, we have made some progress in suggesting that it's really important to listen to kids' mathematical thinking. And I often think that that's taken root, particularly as kids are doing things like adding or subtracting. And I think what you're reminding [me] is, that holds true when it comes to thinking about geometry or shape; that it's in listening to what kids are saying, that they're helping us understand, “What's next?” “Where do we introduce language?” “How can we have kids speaking to one another in a way that builds a set of ideas?”  I think the big takeaway for me is that sometimes geometry has kind of been treated like this separate entity in the world of elementary mathematics. And yet some of the principles that we find really important in things like number or operation, they still hold true. Rebecca: Definitely, definitely. And again, as I said, when you are interested in getting to know your children, seeing who's got some gifts in this domain will allow you to uplift kids who might otherwise not have those opportunities to shine. Mike: I think that's a great place to stop. Rebecca, thank you so much for joining us. It's been a pleasure talking to you. Rebecca: This has really been fun. And I do want to mention one thing: that I have developed a list of various articles and resources. Most of them come from NCTM, and I can make that available to you so that people who are interested in learning more can get some more resources. Mike: That's fantastic. We'll link those to our show notes. Thank you again very much for helping us make sense of this really important set of concepts. Rebecca: You're welcome. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 3 | Episode 7 – Number Sense Guest: Dr. James Brickwedde Mike Wallus: Carry the 1, add a 0, cross multiply. All of these are phrases that educators heard when they were growing up. This language is so ingrained we often use it without even thinking. But what's the long-term impact of language like this on our students' number sense? Today we're talking with Dr. James Brickwedde about the impact of language and the ways educators can use it to cultivate their students' number sense.  Welcome to the podcast, James. I'm excited to be talking with you today. James Brickwedde: Glad to be here. Mike: Well, I want to start with something that you said as we were preparing for this podcast. You described how an educator's language can play a critical role in helping students think in value rather than digits. And I'm wondering if you can start by explaining what you mean when you say that. James: Well, thinking first of primary students, so kindergarten, second grade, that age bracket; kindergartners, in particular, come to school thinking that numbers are just piles of 1s. They're trying to figure out the standard order. They're trying to figure out cardinality. There are a lot of those initial counting principles that lead to strong number sense that they are trying to integrate neurologically. And so, one of the goals of kindergarten, first grade and above is to build the solid quantity sense—number sense—of how one number is relative to the next number in terms of its size, magnitude, et cetera. And then as you get beyond 10 and you start dealing with the place value components that are inherent behind our multidigit numbers, it's important for teachers to really think carefully of the language that they're using so that, neurologically, students are connecting the value that goes with the quantities that they're after. So, helping the brain to understand that 23 can be thought of not only as that pile of 1s, but I can decompose it into a pile of 20 1s and three 1s and eventually that 20 can be organized into two groups of 10. And so, using manipulatives, tracking your language so that when somebody asks, “How do I write 23?” it's not a 2 and a 3 that you put together, which is what a lot of young children think is happening. But rather, they realize that there's the 20 and the 3. Mike: So, you're making me think about the words in the number sequence that we use to describe quantities. And I wonder about the types of tasks or the language that can help children build a meaningful understanding of whole numbers, like say, 11 or 23. James: The English language is not as kind to our learners ( laughs ) as other languages around the world are when it comes to multidigit numbers. We have in English 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And when we get beyond 10, we have this unique word called “eleven” and another unique word called “twelve.” And so, they really are words capturing collections of 1s really then capturing any sort of 10s in 1s relationship. There's been a lot of wonderful documentation around the Chinese-based languages. So, that would be Chinese, Japanese, Korean, Vietnamese, Hmong follows the similar language patterns where when they get after 10, it literally translates as ten 1, ten 2. When they get to 20, it's two ten, two ten 1, two ten 2. And so, the place-value language is inherent in the words that they are saying to describe the quantities. The teen numbers, when you get to 13, a lot of young children try to write 13 as three 1 because they're trying to follow the language patterns of other numbers where you start left to right. And so, they're bringing meaning to something, which of course is not the social convention. So, the teens are all screwed up in terms of English. Spanish does begin to do some regularizing when they get to 16 because of the name diez y seis, so ten 6. But prior to that you have, again, sort of more unique names that either don't follow the order of how you write the number or they're unique like 11 and 12 is. Somali is another interesting language in that—and I apologize to anybody who is fluent in that language because I'm hoping I'm going to articulate it correctly—I believe that there, when they get into the teens, it's one and 10, two and 10, is the literal translation. So, while it may not be the ten 1 sort of order, it still is giving that the fact that there's ten-ness there as you go. So, for the classrooms that I have been in and out of both as my own classroom years ago as well as the ones I still go in and out of now, I try to encourage teachers to tap the language assets that are among their students so that they can use them to think about the English numbers, the English language, that can help them wire that brain so that the various representations, the manipulatives, expanded notation cards or dice, the numbers that I write, how I break the numbers apart, say that 23 is equal to 20 plus 3. All of those models that you're using, and the language that you use to back it up with, is consistent so that, neurologically, those pathways are deeply organized.  Piaget, in his learning theory, talks about young children—this is sort of the 10 years and younger—can only really think about one attribute at a time. So that if you start operating on multidigit numbers, and I'm using digitized language, I'm asking that, kindergartner first, second-grader, to think of two things at the same time. I'm say, moving a 1 while I also mean 10. What you find, therefore, is when I start scratching the surface of kids who were really procedural-bound, that they really are not reflecting on the values of how they've decomposed the numbers or are reconfiguring the numbers. They're just doing digit manipulation. They may be getting a correct answer, they may be very fast with it, but they've lost track of what values they're tracking. There's been a lot of research on kids' development of multidigit operations, and it's inherent in that research about students following … the students who are more fluid with it talk in values rather than in digits. And that's the piece that has always caught my attention as a teacher and helped transform how I talked with kids with it. And now as a professional development supporter of teachers, I'm trying to encourage them to incorporate in their practice. Mike: So, I want to hang on to this theme that we're starting to talk about. I'm thinking a lot about the very digit-based language that as a child I learned for adding and subtracting multidigit numbers. So, phrases like carry the 1 or borrow something from the 6. Those were really commonplace. And in many ways, they were tied to this standard algorithm, where a number was stacked on top of another number. And they really obscured the meaning of addition and subtraction. I wonder if we can walk through what it might sound like or what other models might draw out … some of the value-based language that we want to model for kids and also that we want kids to eventually adopt when they're operating on numbers. James: A task that I give adults, whether they are parents that I'm out doing a family math night with or my teacher candidates that I have worked with, I have them just build 54 and 38, say, with base 10 blocks. And then I say, “How would you quickly add them?” And invariably everybody grabs the tens before they move to the ones. Now your upbringing, my upbringing is the same and still in many classrooms, students are directed only to start with the ones place. And if you get a new 10, you have to borrow and you have to do all of this exchange kinds of things.  But the research shows when school gets out of the way ( chuckles ) and students and adults are operating on more of their natural number sense, people start with the larger and then move to the smaller. And this has been found around the world. This is not just unique to us classrooms that have been working this way. If in the standard algorithms—which really grew out of accounting procedures that needed to save space in ledger books out of the 18th, 19th centuries—they are efficient, space-saving means to be able to accurately compute. But in today's world, technology takes over a lot of that bookkeeping type of thing. An analogy I like to make is, in today's world, Bob Cratchit out of the Christmas Carol, Charles Dickens' character, doesn't have a job because technology has taken over everything that he was in charge of. So, in order for Bob Cratchit to have a job ( laughs ), he does need to know how to compute. But he really needs to think in values.  So, what I try to encourage educators to loosen up their practice is to say, “If I'm adding 54 plus 38, so if you keep those two numbers in your mind ( chuckles ), if I start with the ones and I add 4 and 8, I can get 12. There's no reason if I'm working in a vertical format to not put 12 fully under the line down below, particularly when kids are first learning how to add. But then language-wise, when they go to the tens place, they're adding 50 and 30 to get 80, and the 80 goes under the 12.” Now, many teachers will know that's partial sums. That's not the standard algorithm. That is the standard algorithm. The difference between the shortcut of carrying digits is only a space-saving version of partial sums. Once you go to partial sums in a formatting piece, and you're having kids watch their language, and that's a phrase I use constantly in my classrooms. It's not a 5 and 3 that you are working with, it's a 50 and a 30. So when you move to the language of value, you allow kids to initially, at least, get well-grounded in the partial sums formatting of their work, the algebra of the connectivity property pops out, the number sense of how I am building the quantities, how I'm adding another 10 to the 80, and then the 2, all of that begins to more fully fall into place. There are some of the longitudinal studies that have come out that students who were using more of the partial sums approach for addition, their place value knowledge fell into place sooner than the students who only did the standard algorithm and used the digitized language. So, I don't mind if a student starts in the one's place, but I want them to watch their language. So, if they're going to put down a 2, they're not carrying a 1—because I'll challenge them on that—is “What did you do to the 12 to just isolate the 2? What's left? Oh, you have a 10 up there and the 10 plus the 50 plus the 30 gives me 90.” So, the internal script that they are verbalizing is different than the internal digitized script that you and I and many students still learn today in classrooms around the country. So, that's where the language and the values and the number sense all begin to gel together. And when you get to subtraction, there's a whole other set of language things. So, when I taught first grade and a student would say, “Well, you can't take 8 from 4,” if I still use that 54 and 38 numbers as a reference here. My challenge to them is who said?  Now, my students are in Minnesota. So, Minnesota is at a cultural advantage of knowing what happens in wintertime when temperatures drop below zero ( laughs ). And so, I usually have as a representation model in my room, a number line that swept around the edges of the room that started from negative 35 and went to 185. And so, there are kids who've been puzzling about those other numbers on the other side of zero. And so, somebody pops up and says, “Well, you'll get a negative number.” “What do you mean?” And then they whip around and start pointing at that number line and being able to say, “Well, if you're at 4 and you count back 8, you'll be at negative 4.” So, I am not expecting first-graders to be able to master the idea of negative integers, but I want them to know the door is open. And there are some students in late first grade and certainly in second grade who start using partial differences where they begin to consciously use with the idea of negative integers.  However, there [are] other students, given that same scenario, who think going into the negative numbers is too much of the twilight zone ( laughs ). They'll say, “Well, I have 4 and I need 8. I don't have enough to take 8 from 4.” And another phrase I ask them is, “Well, what are you short?” And that actually brings us back to the accounting reference point of sort of debit-credit language of, “I'm short 4.” “Well, if you're short 4, well just write minus-4.” But if they already have subtracted 30 from 50 and have 20, then the question becomes, “Where are you going to get that 4 from?” “Well, you have 20 cookies sitting on that plate there. I'm going to get that 4 out of the 20.” So again, the language around some of these strategies in subtractions shifts kids to think with alternative strategies and algorithms compared to the American standard algorithm that predominates U.S. education. Mike: I think what's interesting about what you just said, too, is you're making me think about an article. I believe it was Rules That Expire. And what strikes me is that this whole notion that you can't take 8 away from 4 is actually a rule that expires once kids do begin to work in integers. And what you're suggesting about subtraction is, “Let's not do that. Let's use language to help them make meaning of, “Well, what if?” As a former Minnesotan, I can definitely validate that when it's 4 degrees outside and the temperature drops 8 degrees, kids can look at a thermometer and that context helps them understand. I suppose if you're a person listening to this in Southern California or Arizona, that might feel a little bit odd. But I would say that I have seen first-graders do the same thing. James: And if you are more international travelers, as soon as say, people in southern California or southern Arizona step across into Mexico, everything is in Celsius. If those of us in the Northern Plains go into Canada, everything is in Celsius. And so, you see negative numbers sooner ( laughs ) than we do in Fahrenheit, but that's another story. Mike: This is a place where I want to talk a little bit about multiplication, particularly this idea of multiplying by 10. Because I personally learned a fairly procedural understanding of what it is to multiply by 10 or a hundred or a thousand. And the language of “add a zero” was the language that was my internal script. And for a long time when I was teaching, that was the language that I passed along. You're making me wonder how we could actually help kids build a more meaningful understanding of multiplying by 10 or multiplying by powers of 10. James: I have spent a lot of time with my own research as well as working with teachers about what is practical in the classroom, in terms of their approach to this. First of all, and I've alluded to this earlier, when you start talking in values, et cetera, and allow multiple strategies to emerge with students, the underlying algebraic properties, the properties of operations, begin to come to the surface. So, one of the properties is the zero property. What happens when you add a number to zero or a zero to a number? I'm now going to shift more towards a third-grade scenario here. When a student needs to multiply four groups of 30. “I want 34 times,” if you're using the time language. And they'd say, “Well, I know 3 times 4 is 12 and then I just add a zero.” And that's where I as a teacher reply, “Well, I thought 12 plus zero is still 12. How could you make it 120?” And they'd say, “Well, because I put it there.” So, I begin to try to create some cognitive dissonance ( laughs ) over what they're trying to describe, and I do stop and say this to kids: “I see that you recognize a pattern that's happening there, but I want us to explore, and I want you to describe why does that pattern work mathematically?”  So, with addition and subtraction, kids learn that they need to decompose the numbers to work on them more readily and efficiently. Same thing when it comes to multiplication. I have to decompose the numbers somehow. So if, for the moment, you come back to, “If you can visualize the numbers, four groups of 36.” Kids would say, “Well, yeah, I have to decompose the 36 into 30 plus 6.” But by them now exploring how to multiply four groups of 30 without being additive and just adding above, which is an early stage to it. But as they become more abstract and thinking more in multiples, I want them to explore the fact that they are decomposing the 30 into factors  Now, factors isn't necessarily a third-grade standard, right? But I want students to understand that that's how they are breaking that number apart. So, I'm left with 4 times 3 times 10. And if they've explored, in this case the associate of property of multiplication, “Oh, I did that. So, I want to do 4 times 3 because that's easy. I know that. But now I have 12 times 10.” And how can you justify what 12 times 10 is? And that's where students who are starting to move in this place quickly say, “Well, I know 10 tens are 100 and two tens are 20, so it's 120.” They can explain it. The explanation sometimes comes longer than the fact that they are able to calculate it in their heads, but the pathway to understanding why it should be in the hundreds is because I have a 10 times a 10 there. So that when the numbers now begin to increase to a double digit times a double digit. So, now let's make it 42 groups of 36. And I now am faced with, first of all, estimating how large might my number be? If I've gotten students grounded in being able to pull out the factors of 10, I know that I have a double digit times a double digit, I have a factor of 10, a factor of 10. My answer's going to be in the hundreds. How high in the hundreds? In this case with the 42 and 36, 1,200. Because if I grab the largest partial product, then I know my answer is at least above 1,200 or one thousand two hundred. Again, this is a language issue. It's breaking things into factors of 10 so that the powers of 10 are operated on.  So that when I get deeper into fourth grade, and it's a two digit times a three digit, I know that I'm going to have a 10 times a hundred. So, my answer's at least going to be up in the thousands. I can grab that information and use it both from an estimation point of view, but also strategically to multiply the first partial product or however you are decomposing the number. Because you don't have to always break everything down into their place value components. That's another story and requires a visual ( laughs ) work to explain that. But going back to your question, the “add the zero,” or as I have heard, some teachers say, “Just append the zero,” they think that that's going to solve the mathematical issue. No, that doesn't. That's still masking why the pattern works. So, bringing students back to the factors of 10 anchors them into why a number should be in the hundreds or in the thousands. Mike: What occurs to me is what started as a conversation where we were talking about the importance of speaking in value really revealed the extent to which speaking in value creates an opportunity for kids to really engage with some of the properties and the big ideas that are going to be critical for them when they get to middle school and high school. And they're really thinking algebraically as opposed to just about arithmetic. James: Yes. And one of the ways I try to empower elementary teachers is to begin to look at elementary arithmetic through the lens of algebra rather than the strict accounting procedures that sort of emerge. Yes, the accounting procedures are useful. They can be efficient. I can come to use them. But if I've got the algebraic foundation underneath it, when I get to middle school, it is my foundation allows for generative growth rather than a house of cards that collapses, and I become frustrated. And where we see the national data in middle school, there tends to be a real separation between who are able to go on and who gets stuck. Because as you mentioned before the article that the Rules That Expire, too many of them expire when you have to start thinking in rates, ratios, proportionality, et cetera. Mike: So, for those of you who are listening who want to follow along, we do have a visual aid that's attached to the show notes that has the mathematics that James is talking about. I think that's a great place to stop.  Thank you so much for joining us, James, it has really been a pleasure talking with you. James: Well, thanks a lot, Mike. It was great talking to you as well.  Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org 

Rounding Up Season 3 | Episode 6 – Argumentation, Justification & Conjecture Guests: Jody Guarino and Chepina Rumsey Mike Wallus: Argumentation, justification, conjecture. All of these are practices we hope to cultivate, but they may not be practices we associate with kindergarten, first-, and second-graders. What would it look like to encourage these practices with our youngest learners? Today we'll talk about this question with Jody Guarino and Chepina Rumsey, authors of the book Nurturing Math Curiosity with Learners in Grades K–2. Welcome to the podcast, Chepina and Jody. Thank you so much for joining us today. Jody Guarino: Thank you for having us. Chepina Rumsey: Yeah, thank you. Mike: So, I'm wondering if we can start by talking about the genesis of your work, particularly for students in grades K–2. Jody: Sure. Chepina had written a paper about argumentation, and her paper was situated in a fourth-grade class. At the time, I read the article and was so inspired, and I wanted it to use it in an upcoming professional learning that I was going to be doing. And I got some pushback with people saying, “Well, how is this relevant to K–2 teachers?” And it really hit me that there was this belief that K–2 students couldn't engage in argumentation. Like, “OK, this paper's great for older kids, but we're not really sure about the young students.” And at the time, there wasn't a lot written on argumentation in primary grades. So, we thought, “Well, let's try some things and really think about, ‘What does it look like in primary grades?' And let's find some people to learn with.” So, I approached some of my recent graduates from my teacher ed program who were working in primary classrooms and a principal that employed quite a few of them with this idea of, “Could we learn some things together? Could we come and work with your teachers and work with you and just kind of get a sense of what could students do in kindergarten to second grade?” So, we worked with three amazing teachers, Bethany, Rachael, and Christina—in their first years of teaching—and we worked with them monthly for two years. We wanted to learn, “What does it look like in K–2 classrooms?” And each time we met with them, we would learn more and get more and more excited. Little kids are brilliant, but also their teachers were brilliant, taking risks and trying things. I met with one of the teachers last week, and the original students that were part of the book that we've written now are actually in high school. So, it was just such a great learning opportunity for us. Mike: Well, I'll say this, there are many things that I appreciated about the book, about Nurturing Math Curiosity with Learners in Grades K–2, and I think one of the first things was the word “with” that was found in the title. So why “with” learners? What were y'all trying to communicate? Chepina: I'm so glad you asked that, Mike, because that was something really important to us when we were coming up with the title and the theme of the book, the message. So, we think it's really important to nurture curiosity with our students, meaning we can't expect to grow it in them if we're not also growing it in ourselves. So, we see that children are naturally curious and bring these ideas to the classroom. So, the word “with” was important because we want everyone in the classroom to grow more curious together. So, teachers nurturing their own math curiosity along with their students is important to us. One unique opportunity we tried to include in the book is for teachers who are reading it to have opportunities to think about the math and have spaces in the book where they can write their own responses and think deeply along with the vignettes to show them that this is something they can carry to their classroom. Mike: I love that. I wonder if we could talk a little bit about the meaning and the importance of argumentation? In the book, you describe four layers: noticing and wondering, conjecture, justification, and extending ideas. Could you share a brief explanation of those layers? Jody: Absolutely. So, as we started working with teachers, we'd noticed these themes or trends across, or within, all of the classrooms. So, we think about noticing and wondering as a space for students to make observations and ask curious questions. So, as teachers would do whatever activity or do games, they would always ask kids, “What are you noticing?” So, it really gave kids opportunities to just pause and observe things, which then led to questions as well. And when we think about students conjecturing, we think about when they make general statements about observations. So, an example of this could be a child who notices that 3 plus 7 is 10 and 7 plus 3 is 10. So, the child might think, “Oh wait, the order of the addends doesn't matter when adding. And maybe that would even work with other numbers.” So, forming a conjecture like this is, “What I believe to be true.” The next phase is justification, where a student can explain either verbally or with writing or with tools to prove the conjecture. So, in the case of the example that I brought up, 3 plus 7 and 7 plus 3, maybe a student even uses their fingers, where they're saying, “Oh, I have these 3 fingers and these 7 fingers and whichever fingers I look at first, or whichever number I start with, it doesn't matter. The sum is going to be the same.” So, they would justify in ways like that. I've seen students use counters, just explaining it. Oftentimes, they use language and hand motions and all kinds of things to try to prove what they're saying works. Or sometimes they'll find, just really look for, “Can I find an example where that doesn't work?” So, just testing their conjecture would be justifying. And then the final stage, extending ideas, could be extending that idea to all numbers. So, in the idea of addition in the commutative property, and they come to discover that they might realize, “Wait a minute, it also works for 1 plus 9 and 9 plus 1.” They could also think, “Does it work for other operations? So, not just with addition, but maybe I can subtract like that, too. Does that make a difference if I'm subtracting 5, takeaway 2 versus 2 takeaway 5. So, just this idea of, “Now I've made sense of something, what else does it work with or how can I extend that thinking?” Mike: So, the question that I was wondering about as you were talking is, “How do you think about the relationship between a conjecture and students' justification?” Jody: I've seen a lot of kids … so, sometimes they make conjectures that they don't even realize are conjectures, and they're like, “Oh, wait a minute, this pattern's happening, and I think I see something.” And so often they're like, “OK, I think that every time you add two numbers together, the sum is greater than the two numbers.” And so, then this whole idea of justifying … we often ask them, “How could you convince someone that that's true?” Or, “Is that always true?” And now they actually having to take and study it and think about, “Is it true? Does it always work?” Which, Mike, in your question, often leads back to another conjecture or refining their conjecture. It's kind of this cyclical process. Mike: That totally makes sense. I was going to use the words virtuous cycle, but that absolutely helps me understand that. I wonder if we can go back to the language of conjecture, because that feels really important to get clear on and to both understand and start to build a picture of. So, I wonder if you could offer a definition of conjecture for someone who's unfamiliar with the term or talk about how students understand conjecture. Chepina: Yeah. So, a conjecture is based on our exploration with the patterns and observations. So, through that exploration, we might have an idea that we believe to be true. We are starting to notice things and some language that students start to use. Things like, “Oh, that's always going to work” or “Sometimes we can do that.” So, there starts to be this shift toward an idea that they believe is going to be true. It's often a work in progress, so it needs to be explored more in order to have evidence to justify why that's going to be true. And through that process, we can modify our conjecture. Or we might have an idea, like this working idea of a conjecture, that then when we go to justify it, we realize, “Oh, it's not always true the way we thought. So, we have to make a change.” So, the conjecture is something that we believe to be true, and then we try to convince other people. So, once we introduce that with young mathematicians, they tend to latch on to that idea that it's this really neat thing to come up with a conjecture. And so, then they often start to come up with them even when we're not asking and get excited about, “Wait, I have a conjecture about the numbers and story problems,” where that wasn't actually where the lesson was going, but then they get excited about it. And that idea that we can take our patterns and observations, create a conjecture, and have this cyclical thing that happens. We had a second-grade student make what she called a “conjecture cycle.” So, she drew a circle with arrows and showed, “We can have an idea, we can test it, we can revise it, and we can keep going to create new information.” So, those are some examples of where we've seen conjectures and kids using them and getting excited and what they mean. And yeah, it's been really exciting. Mike: What is hitting me is that this idea of introducing conjectures and making them, it really has the potential to change the way that children understand mathematics. It has the potential to change from, “I'm seeking a particular answer” or “I'm memorizing a procedure” or “I'm doing a thing at a discreet point in time to get a discreet answer.” It feels culturally very different. It changes what we're talking about or what we're thinking about. Does that make sense to the two of you? Chepina: Yeah, it does. And I think it changes how they view themselves. They're mathematicians who are creating knowledge and seeking knowledge rather than memorizing facts. Part of it is we do want them to know their facts—but understand them in this deep way with the structure behind it. And so, they're creating knowledge, not just taking it in from someone else. Mike: I love that. Jody: Yeah, I think that they feel really empowered. Mike: That's a great pivot point. I wonder if the two of you would be willing to share a story from a K–2 classroom that could bring some of the ideas we've been talking about to life for people who are listening. Jody: Sure, I would love to. I got to spend a lot of time in these teachers' classrooms, and one of the days I spent in a first grade, the teacher was Rachael Gildea, and she had led a choral count with her first-graders. And they were counting by 10 but starting with 8. So, like, “Eight, 18, 28, 38, 48 … .” And as the kids were counting, Rachael was charting. And she was charting it vertically. So, below 8 was written 18, and then 28. And she wrote it as they counted. And one of her students paused and said, “Oh, they're all going to end with 8.” And Rachael took that student's conjecture. So, a lot of other conjectures or a lot of other ideas were shared. Students were sharing things they noticed. “Oh, looking at the tens place, it's counting 1, 2, 3,” and all sorts of things. But this one, particular student, who said they're all going to end in 8, Rachel took that student's—the actual wording—the language that the student had used, and she turned it into the task that the whole class then engaged in. Like, “Oh, this student thought or thinks it's always going to end in 8. That's her conjecture, how can we prove it?” And I happened to be in her classroom the day that they tested it. And it was just a wild scene. So, students were everywhere: at tables, laying down on the carpet, standing in front of the chart, they were examining it or something kind of standing with clipboards. And there was all kinds of buzz in the classroom. And Rachael was down on the carpet with the students listening to them. And there was this group of girls, I think three of them, that sort of screamed out, “We got it!” And Rachel walked over to the girls, and I followed her, and they were using base 10 blocks. And they showed her, they had 8 ones, little units, and then they had the 10 sticks. And so, one girl would say, they'd say, “Eight, 18, 28,” and one of the girls was adding the 10 sticks and almost had this excitement, like she discovered, I don't know, a new universe. It was so exciting. And she was like, “Well, look, you don't ever change them. You don't change the ones, you just keep adding tens.” And it was so magical because Rachael went over there and then right after that she paused the class and she's like, “Come here everyone, let's listen to these girls share what they discovered.” And all of the kids were sort of huddled around, and it was just magical. And they had used manipulatives, the base 10 blocks, to make sense of the conjecture that came from the coral count. And I thought it was beautiful. And so, I did coral counts in my classroom and never really thought about, “OK, what's that next step beyond, like, ‘Oh, this is exciting. Great things happen with numbers.'” Mike: What's hitting me is that there's probably a lot of value in being able to use students' conjectures as reference points for potential future lessons. I wonder if you have some ideas or if you've seen educators create something like a public space for conjectures in their classroom. Chepina: We've seen amazing work around conjectures with young mathematicians. In that story that Jody was telling us about Rachael, she used that conjecture in the next lesson to bring it together. It fits so perfectly with the storyline for that unit, and the lesson, and where it was going to go next. But sometimes ideas can be really great, but they don't quite fit where the storyline is going. So, we've encouraged teachers and seen this happen in the classrooms we've worked in, where they have a conjecture wall in their classroom, where ideas can be added with Post-it notes have a station where there are Post-it notes and pencil right there. And students can go and write their idea, put their name on it, stick it to the wall. And so, conjectures that are used in the lesson can be put up there, but ones that aren't used yet could be put up there. And so, if there was a lesson where a great idea emerges in the middle, and it doesn't quite fit in, the teacher could say, “That's a great idea. I want to make sure we come back to it. Could you add it to the conjecture wall?” And it gives that validation that their idea is important, and we're going to come back to it instead of just shutting it down and not acknowledging it at all. So, we have them put their names on to share. It's their expertise. They have value in our classroom. They add something to our community. Everyone has something important to share. So, that public space, I think, is really important to nurture that community where everyone has something to share. And we're all learning together. We're all exploring, conjecturing. Jody: And I've been to in those classrooms, that Chepina is referring to with conjecture walls, and kids actually will come in, they'll be doing math, and they'll go to recess or lunch and come back in and ask for a Post-it to add a conjecture like this … I don't know, one of my colleagues uses the word “mathematical residue.” They continue thinking about this, and their thoughts are acknowledged. And there's a space for them. Mike: So, as a former kindergarten, first-grade teacher, I'm seeing a picture in my head. And I'm wondering if you could talk about setting the stage for this type of experience, particularly the types of questions that can draw out conjectures and encourage justification? Jody: Yeah. So, as we worked with teachers, we found so many rich opportunities. And now looking back, those opportunities are probably in all classrooms all the time. But I hadn't realized in my experience that I'm one step away from this ( chuckles ). So, as teachers engaged in instructional routines, like the example of coral counting I shared from Rachael's classroom, they often ask questions like: “What do you notice? Why do you think that's happening? Will that always happen? How do you know? How can you prove it will always work? How can you convince a friend?” And those questions nudge children naturally to go to that next step when we're pushing, asking an advancing question in response to something that a student said. Mike: You know, one of the things that occurs to me is that those questions are a little bit different even than the kinds of questions we would ask if we were trying to elicit a student's strategy or their conceptual understanding, right? In that case, it seems like we want to understand the ideas that were kind of animating a student's strategy or the ideas that they were using or even how they saw a mental model unfolding in their head. But the questions that you just described, they really do go back to this idea of generalizing, right? Is there a pattern that we can recognize that is consistently the same or that doesn't change. And it's pressing them to think about that in a way that's different even than conceptual-based questions. Does that make sense? Jody: It does, and it makes me think about … I believe it's Vicki Jacobs and Joan Case, who do a lot of work with questioning. They ask this question, too: “As a teacher, what did that child say that gave you permission to ask that question?” Where often, I want to take my question somewhere else, but really all of these questions are nudging kids in their own thinking. So, when they're sharing something, it's like, “Well, do you think that will always work?” It's still grounded in what their ideas were but sort of taking them to that next place. Mike: So, one of the things that I'm also wondering about is a scaffold called “language frames.” How do students or a teacher use language frames to support argumentation? Chepina: Yeah, I think that communication is such a big part of argumentation. And we found language frames can help support students to share their ideas by having this common language that might be different than the way they talk about other things with their friends or in other subjects. So, using the language frames as a scaffold that supports students in communicating by offering them a model for that discussion. When I've been teaching lessons, I will have the language written out in a space where everyone can see, and I'll use it to model my discussion. And then students will use it as they're sharing their ideas. And that's been really helpful to get language at all grade levels. Mike: Can you share one or two examples of a language frame? That's something you would use in say, a K, 1, or a 2 classroom, Chepina? Chepina: Yeah. We've had something like, we'll put, “I notice” and then a blank line. (“I notice ______.) And so, we'll have them say, “I notice,” and then they'll fill it in. Or “I wonder” or “I have a different idea.” So, helping to model, “How do you talk in a community of learners when you're sharing ideas? Or if you have a different idea and you're disagreeing.” So, we'll have that actually written out, and we can use it ourselves or help students to restate what they've said using that model so that then they can pick up that language. Mike: One of the things that stands out for me is that these experiences with argumentation and conjecture, they obviously have benefits for individual student's conceptual understanding and for their communication. But I suspect that they also have a real benefit for the class as a collective. Can you talk about the impact that you've seen in K–2 classrooms that are thinking about argumentation and putting some of these practices into place? Jody: Sure. I've been really fortunate to get to spend so much time in classrooms really learning from the teachers that we worked with. And one of the things I noticed about the classrooms is the ongoing curiosity and wonder. Students were always making sense of things and investigating ideas. And the other thing that I really picked up on was how they listened to each other, which, coming from a primary background, is challenging for kids to listen to each other. But they were really attentive and attuned, and they saw themselves as problem-solvers, and they thought their role was to things out. That's just what they do at school. But they thought about other kids in those ways, too. “Well, let me see what other people think” or “Let me hear Chepina's idea because maybe there's something that's useful for me.” So, they really engaged in learning, not as an isolated, sort of, “Myself as a learner,” but as part of a community. The classrooms were also buzzing all the time. There was noise and movement. And the kids, the word I would say is “intellectually engaged.” So, not just engaged, like busy doing things, but really deeply thinking. Chepina: The other thing we've seen that has been also really exciting is the impact on the teachers as they become more curious along with the students. So, in our first group, we had the teachers, the K–2 teachers, and we saw that they started to say things when we would meet because we would meet monthly. And they would start to say things like, “I noticed this, and I wonder if this is what my student was thinking?” So, when they were talking about their own students and their own lessons and the mathematics behind the problems, we saw teachers start to use that language and become more curious, too. So, it's been really exciting to see that aspect as we work with teachers. Mike: So, I suspect that we have many listeners who are making sense of the ideas that you're sharing and are going to want to continue learning about argumentation and conjecture. Are there particular resources that you would recommend that might help an educator continue down this path? Chepina: Yeah. We are both so excited that our first book just came out in May, and we took all the things that we had learned in this project, exploring alongside teachers, and we have more examples. There are strategies, there's examples of the routines that we think it's often we stop too soon. Like, “Here are some ideas of how to keep going with these instructional routines,” and we have templates to support teachers as they take those common routines further. So, we also have some links of our recent articles, and we have some social media pages. We can share those. Mike: That's fabulous. We will post all of those links and also a link to the book that you all have written. I think this is probably a great place to stop. Chepina and Jody, I want to thank you both so much for joining us. It's really been a pleasure talking with you. Jody: Thank you for the opportunity. It's been great to share some of the work that we've learned from classrooms, from students and teachers. Chepina: Yeah. Thank you, Mike. It's been so fun to talk to you. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org 

Rounding Up Season 3 | Episode 05 - Building Asset-Focused Professional Learning Communities Guests: Summer Pettigrew and Megan Williams Mike Wallus: Professional learning communities have been around for a long time and in many different iterations. But what does it look like to schedule and structure professional learning communities that actually help educators understand and respond to their students' thinking in meaningful ways? Today we're talking with Summer Pettigrew and Megan Williams from the Charleston Public Schools about building asset-focused professional learning communities.  Hello, Summer and Megan. Welcome to the podcast. I am excited to be talking with you all today about PLCs. Megan Williams: Hi! Summer Pettigrew: Thanks for having us. We're excited to be here. Mike: I'd like to start this conversation in a very practical place, scheduling. So, Megan, I wonder if you could talk just a bit about when and how you schedule PLCs at your building. Megan: Sure. I think it's a great place to start, too, because I think without the structure of PLCs in place, you can't really have fabulous PLC meetings. And so, we used to do our PLC meetings once a week during teacher planning periods, and the teachers were having to give up their planning period during the day to come to the PLC meeting. And so, we created a master schedule that gives an hour for PLC each morning. So, we meet with one grade level a day, and then the teachers still have their regular planning period throughout the day. So, we were able to do that by building a time for clubs in the schedule. So, first thing in the morning, depending on your day, so if it's Monday and that's third grade, then the related arts teachers—and that for us is art, music, P.E., guidance, our special areas—they go to the third-grade teachers' classrooms. The teachers are released to go to PLC, and then the students choose a club. And so, those range from basketball to gardening to fashion to STEMs. We've had Spanish club before. So, they participate with the related arts teacher in their chosen club, and then the teachers go to their PLC meeting. And then once that hour is up, then the teachers come back to class. The related arts teachers are released to go get ready for their day. So, everybody still has their planning period, per se, throughout the day. Mike: I think that feels really important, and I just want to linger a little bit longer on it. One of the things that stands out is that you're preserving the planning time on a regular basis. They have that, and they have PLC time in addition to it. Summer: Uh-hm. Megan: Correct. And that I think is key because planning time in the middle of the day is critical for making copies, calling parents, calling your doctor to schedule an appointment, using the restroom … those kind of things that people have to do throughout the day. And so, when you have PLC during their planning time, one or the other is not occurring. Either a teacher is not taking care of those things that need to be taken care of on the planning period. Or they're not engaged in the PLC because they're worried about something else that they've got to do. So, building that time in, it's just like a game-changer. Mike: Summer, as a person who's playing the role of an instructional coach, what impact do you think this way of scheduling has had on educators who are participating in the PLCs that you're facilitating? Summer: Well, it's huge. I have experienced going to A PLC on our planning and just not being a hundred percent engaged. And so, I think having the opportunity to provide the time and the space for that during the school day allows the teachers to be more present. And I think that the rate at which we're growing as a staff is expedited because we're able to drill into what we need to drill into without worrying about all the other things that need to happen. So, I think that the scheduling piece has been one of the biggest reasons we've been so successful with our PLCs. Mike: Yeah, I can totally relate to that experience of feeling like I want to be here, present in this moment, and I have 15 things that I need to do to get ready for the next chunk of my day. So, taking away that “if, then,” and instead having an “and” when it comes to PLCs, really just feels like a game-changer. Megan: And we were worried at first about the instructional time that was going to be lost from the classroom doing the PLC like this. We really were, because we needed to make sure instructional time was maximized and we weren't losing any time. And so, this really was about an hour a week where the teachers aren't directly instructing the kids. But it has not been anything negative at all. Our scores have gone up, our teachers have grown. They love the kids, love going to their clubs. I mean, even the attendance on the grade-level club day is so much better because they love coming in. And they start the day really getting that SEL instruction. I mean, that's really a lot of what they're getting in clubs. They're hanging out with each other. They're doing something they love. Mike: Maybe this is a good place to shift and talk a little bit about the structure of the PLCs that are happening. So, I've heard you say that PLCs, as they're designed and functioning right now, they're not for planning. They're instead for teacher collaboration. So, what does that mean? Megan: Well, there's a significant amount of planning that does happen in PLC, but it's not a teacher writing his or her lesson plans for the upcoming week. So, there's planning, but not necessarily specific lesson planning: like on Monday I'm doing this, on Tuesday I'm doing this. It's more looking at the standards, looking at the important skills that are being taught, discussing with each other ways that you do this. “How can I help kids that are struggling? How can I push kids that are higher?” So, teachers are collaborating and planning, but they're not really producing written lesson plans. Mike: Yeah. One of the pieces that you all talked about when we were getting ready for this interview, was this idea that you always start your PLCs with a recognition of the celebrations that are happening in classrooms. I'm wondering if you can talk about what that looks like and the impact it has on the PLCs and the educators who are a part of them. Summer: Yeah. I think our teachers are doing some great things in their classrooms, and I think having the time to share those great things with their colleagues is really important. Just starting the meeting on that positive note tends to lead us in a more productive direction. Mike: You two have also talked to me about the impact of having an opportunity for educators to engage in the math that their students will be doing or looking at common examples of student work and how it shows up in the classroom. I wonder if you could talk about what you see in classrooms and how you think that loops back into the experiences that are happening in PLCs. Summer: Yeah. One of the things that we start off with in our PLCs is looking at student work. And so, teachers are bringing common work examples to the table, and we're looking to see, “What are our students coming with? What's a good starting point for us to build skills, to develop these skills a little bit further to help them be more successful?” And I think a huge part of that is actually doing the work that our students are doing. And so, prior to giving a task to a student, we all saw that together in a couple of different ways. And that's going to give us that opportunity to think about what misconceptions might show up, what questions we might want to ask if we want to push students further, reign them back in a little bit. Just that pre-planning piece with the student math, I think has been very important for us. And so, when we go into classrooms, I'll smile because they kind of look like little miniature PLCs going on. The teacher's facilitating, the students are looking at strategies of their classmates and having conversations about what's similar, what's different. I think the teachers are modeling with their students that productive practice of looking at the evidence and the student work and talking about how we go about thinking through these problems. Mike: I think the more that I hear you talk about that, I flashback to what Megan, what you said earlier about, there is planning that's happening, and there's collaboration. They're planning the questions that they might ask. They're anticipating the things that might come from students. So, while it's not, “I'm writing my lesson for Tuesday,” there is a lot of planning that's coming. It's just perhaps not as specific as, “This is what we'll do on this particular day.” Am I getting that right? Megan: Yes. You're getting that a hundred percent right. Summer has teachers sometimes taken the assessment at the beginning of a unit. We'll go ahead and take the end-of-unit assessment and the information that you gain from that. Just with having the teachers take it and knowing how the kids are going to be assessed, then just in turn makes them better planners for the unit. And there's a lot of good conversation that comes from that. Mike: I mean, in some ways, your PLC design, the word that pops into my head is almost like a “rehearsal” of sorts. Does that analogy seem right? Summer: It seems right. And just to add on to that, I think, too, again, providing that time within the school day for them to look at the math, to do the math, to think about what they want to ask, is like a mini-rehearsal. Because typically, when teachers are planning outside of school hours, it's by themselves in a silo. But this just gives that opportunity to talk about all the possibilities together, run through the math together, ask questions if they have them. So, I think that's a decent analogy, yeah. Mike: Yeah. Well, you know what it makes me think about is competitive sports like basketball. As a person who played quite a lot, there are points in time when you start to learn the game that everything feels so fast. And then there are points in time when you've had some experience when you know how to anticipate, where things seem to slow down a little bit. And the analogy is that if you can kind of anticipate what might happen or the meaning of the math that kids are showing you, it gives you a little bit more space in the moment to really think about what you want to do versus just feeling like you have to react. Summer: And I think, too, it keeps you focused on the math at hand. You're constantly thinking about your next teacher move. And so, if you've got that math in your mind and you do get thrown off, you've had an opportunity, like you said, to have a little informal rehearsal with it, and maybe you're not thrown off as badly. (  laughs ) Mike: Well, one of the things that you've both mentioned when we've talked about PLCs is the impact of a program called OGAP. I'm wondering if you can talk about what OGAP is, what it brought to your educators, and how it impacted what's been happening in PLCs. Megan: I'll start in terms … OGAP stands for ongoing assessment project. Summer can talk about the specifics, but we rolled it out as a whole school. And I think there was power in that. Everybody in your school taking the same professional development at the same time, speaking the same language, hearing the same things. And for us, it was just a game-changer. Summer: Yeah, I taught elementary math for 12 years before I knew anything about OGAP, and I had no idea what I was doing until OGAP came into my life. All of the light bulbs that went off with this very complex elementary math that I had no idea was a thing, it was just incredible. And so, I think the way that OGAP plays a role in PLCs is that we're constantly using the evidence in our student work to make decisions about what we do next. We're not just plowing through a curriculum, we're looking at the visual models and strategies that Bridges expects of us in that unit. We're coupling it with the content knowledge that we get from OGAP and how students should and could move along this progression. And we're planning really carefully around that; thinking about, “If we give this task and some of our students are still at a less sophisticated strategy and some of our students are at a more sophisticated strategy, how can we use those two examples to bridge that gap for more kids?” And we're really learning from each other's work. It's not the teacher up there saying, “This is how you'd solve this problem.” But it's a really deep dive into the content. And I think the level of confidence that OGAP has brought our teachers as they've learned to teach Bridges has been like a powerhouse for us. Mike: Talk a little bit about the confidence that you see from your teachers who have had an OGAP experience and who are now using a curriculum and implementing it. Can you say more about that? Summer: Yeah. I mean, I think about our PLCs, the collaborative part of it, we're having truly professional conversations. It's centered around the math, truly, and how students think about the math. And so again, not to diminish the need to strategically lesson plan and come up with activities and things, but we're talking really complex stuff in PLCs. And so, when we look at student work and we that work on the OGAP progression, depending on what skill we're teaching that week, we're able to really look at, “Gosh, the kid is, he's doing this, but I'm not sure why.” And then we can talk a little bit about, “Well, maybe he's thinking about this strategy, and he got confused with that part of it.” So, it really, again, is just centered around the student thinking. The evidence is in front of us, and we use that to plan accordingly. And I think it just one-ups a typical PLC because our teachers know what they're talking about. There's no question in, “Why am I teaching how to add on an open number line?” We know the reasoning behind it. We know what comes before that. We know what comes after that, and we know the importance of why we're doing it right now. Mike: Megan, I wanted to ask you one more question. You are the instructional leader for the building, the position you hold is principal. I know that Summer is a person who does facilitation of the PLCs. What role do you play or what role do you try to play in PLCs as well? Megan: I try to be present at every single PLC meeting and an active participant. I do all the assessments. I get excited when Summer says we're taking a test. I mean, I do everything that the teachers do. I offer suggestions if I think that I have something valuable to bring to the table. I look at student work. I just do everything with everybody because I like being part of that team. Mike: What impact do you think that that has on the educators who are in the PLC? Megan: I mean, I think it makes teachers feel that their time is valuable. We're valuing their time. It's helpful for me, too, when I go into classrooms. I know what I'm looking for. I know which kids I want to work with. Sometimes I'm like, “Ooh, I want to come in and see you do that. That's exciting.” It helps me plan my day, and it helps me know what's going on in the school. And I think it also is just a non-judgmental, non-confrontational time for people to ask me questions. I mean, it's part of me trying to be accessible as well. Mike: Summer, as the person who's the facilitator, how do you think about preparing for the kind of PLCs that you've described? What are some of the things that are important to know as a facilitator or to do in preparation? Summer: So, I typically sort of rehearse myself, if you will, before the PLC kicks off. I will take assessments, I will take screeners. I'll look at screener implementation guides and think about the pieces of that that would be useful for our teachers if they needed to pull some small groups and re-engage those kids prior to a unit. What I really think is important though, is that vertical alignment. So, looking at the standards that are coming up in a module, thinking about what came before it: “What does that standard look like in second grade?” If I'm doing a third grade PLC: “What does that standard look like in fourth grade?” Because teachers don't have time to do that on their own. And I think it's really important for that collective efficacy, like, “We're all doing this together. What you did last year matters. What you're doing next year matters, and this is how they tie together.” I kind of started that actually this year, wanting to know more myself about how these standards align to each other and how we can think about Bridges as a ladder among grade levels. Because we were going into classrooms, and teachers were seeing older grade levels doing something that they developed, and that was super exciting for them. And so, having an understanding of how our state standards align in that way just helps them to understand the importance of what they're doing and bring about that efficacy that we all really just need our teachers to own. It's so huge. And just making sure that our students are going to the next grade prepared. Mike: One of the things that I was thinking about as I was listening to you two describe the different facets of this system that you've put together is, how to get started. Everything from scheduling to structure to professional learning. There's a lot that goes into making what you all have built successful. I think my question to you all would be, “If someone were listening to this, and they were thinking to themselves, ‘Wow, that's fascinating!' What are some of the things that you might encourage them to do if they wanted to start to take up some of the ideas that you shared?” Megan: It's very easy to crash and burn by trying to take on too much. And so, I think if you have a long-range plan and an end goal, you need to try to break it into chunks. Just making small changes and doing those small changes consistently. And once they become routine practices, then taking on something new. Mike: Summer, how about you? Summer: Yeah, I think as an instructional coach, one of the things that I learned through OGAP is that our student work is personal. And if we're looking at student work without the mindset of, “We're learning together,” sometimes we can feel a little bit attacked. And so, one of the first things that we did when we were rolling this out and learning how to analyze student work is, we looked at student work that wasn't necessarily from our class. We asked teachers to save student work samples. I have folders in my office of different student work samples that we can practice sorting and have conversations about. And that's sort of where we started with it. Looking at work that wasn't necessarily our students gave us an opportunity to be a little bit more open about what we wanted to say about it, how we wanted to talk about it. And it really does take some practice to dig into student thinking and figure out, “Where do I need to go from here?” And I think that allowed us to play with it in a way that wasn't threatening necessarily. Mike: I think that's a great place to stop, Megan and Summer. I want to thank you so much for joining us. It's really been a pleasure talking to both of you. Megan: Well, thank you for having us.  Summer: Yeah, thanks a lot for having us. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 3 | Episode 4 – Making Sense of Unitizing: The Theme That Runs Through Elementary Mathematics Guest: Beth Hulbert Mike Wallus: During their elementary years, students grapple with many topics that involve relationships between different units. This concept, called “unitizing,” serves as a foundation for much of the mathematics that students encounter during their elementary years. Today, we're talking with Beth Hulbert from the Ongoing Assessment Project (OGAP) about the ways educators can encourage unitizing in their classrooms.  Welcome to the podcast, Beth. We are really excited to talk with you today. Beth Hulbert: Thanks. I'm really excited to be here. Mike: I'm wondering if we can start with a fairly basic question: Can you explain OGAP and the mission of the organization? Beth: Sure. So, OGAP stands for the Ongoing Assessment Project, and it started with a grant from the National Science Foundation to develop tools and resources for teachers to use in their classroom during math that were formative in nature. And we began with fractions. And the primary goal was to read, distill, and make the research accessible to classroom teachers, and at the same time develop tools and strategies that we could share with teachers that they could use to enhance whatever math program materials they were using. Essentially, we started by developing materials, but it turned into professional development because we realized teachers didn't have a lot of opportunity to think deeply about the content at the level they teach. The more we dug into that content, the more it became clear to us that content was complicated. It was complicated to understand, it was complicated to teach, and it was complicated to learn. So, we started with fractions, and we expanded to do work in multiplicative reasoning and then additive reasoning and proportional reasoning. And those cover the vast majority of the critical content in K–8. And our professional development is really focused on helping teachers understand how to use formative assessment effectively in their classroom. But also, our other goals are to give teachers a deep understanding of the content and an understanding of the math ed research, and then some support and strategies for using whatever program materials they want to use. And we say all the time that we're a program blind—we don't have any skin in the game about what program people are using. We are more interested in making people really effective users of their math program.  Mike: I want to ask a quick follow-up to that. When you think about the lived experience that educators have when they go through OGAP's training, what are the features that you think have an impact on teachers when they go back into their classrooms?  Beth: Well, we have learning progressions in each of those four content strands. And learning progressions are maps of how students acquire the concepts related to, say, multiplicative reasoning or additive reasoning. And we use those to sort, analyze, and decide how we're going to respond to evidence in student work. They're really maps for equity and access, and they help teachers understand that there are multiple right ways to do some mathematics, but they're not all equal in efficiency and sophistication. Another piece they take away of significant value is we have an item bank full of hundreds of short tasks that are meant to add value to, say, a lesson you taught in your math program. So, you teach a lesson, and you decide what is the primary goal of this lesson. And we all know no matter what the program is you're using that every lesson has multiple goals, and they're all in varying degrees of importance. So partly, picking an item in our item bank is about helping yourself think about what was the most critical piece of that lesson that I want to know about that's critical for my students to understand for success tomorrow.  Mike: So, one big idea that runs through your work with teachers is this concept called “unitizing.” And it struck me that whether we're talking about addition, subtraction, multiplication, fractions, that this idea just keeps coming back and keeps coming up. I'm wondering if you could offer a brief definition of unitizing for folks who may not have heard that term before.  Beth: Sure. It became really clear as we read the research and thought about where the struggles kids have, that unitizing is at the core of a lot of struggles that students have. So, unitizing is the ability to call something 1, say, but know it's worth maybe 1 or 100 or a 1,000, or even one-tenth. So, think about your numbers in a place value system. In our base 10 system, 1 of 1 is in the tenths place. It's not worth 1 anymore, it's worth 1 of 10. And so that idea that the 1 isn't the value of its face value, but it's the value of its place in that system. So, base 10 is one of the first big ways that kids have to understand unitizing. Another kind of unitizing would be money. Money's a really nice example of unitizing. So, I can see one thing, it's called a nickel, but it's worth 5. And I can see one thing that's smaller, and it's called a dime, and it's worth 10. And so, the idea that 1 would be worth 5 and 1 would be worth 10, that's unitizing. And it's an abstract idea, but it provides the foundation for pretty much everything kids are going to learn from first grade on. And when you hear that kids are struggling, say, in third and fourth grade, I promise you that one of their fundamental struggles is a unitizing struggle.  Mike: Well, let's start where you all started when you began this work in OGAP. Let's start with multiplication. Can you talk a little bit about how this notion of unitizing plays out in the context of multiplication?  Beth: Sure. In multiplication, one of the first ways you think about unitizing is, say, in the example of 3 times 4. One of those numbers is a unit or a composite unit, and the other number is how many times you copy or iterate that unit. So, your composite unit in that case could be 3, and you're going to repeat or iterate it four times. Or your composite unit could be 4, and you're going to repeat or iterate it three times. When I was in school, the teacher wrote 3 times 4 up on the board and she said, “Three tells you how many groups you have, and 4 tells you how many you put in each group.” But if you think about the process you go through when you draw that in that definition, you draw 1, 2, 3 circles, then you go 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4. And in creating that model, you never once thought about a unit, you thought about single items in a group.  So, you counted 1, 2, 3, 4, three times, and there was never really any thought about the unit. In a composite unit way of thinking about it, you would say, “I have a composite unit of 3, and I'm going to replicate it four times.” And in that case, every time, say, you stamped that—you had this stamp that was 3—every time you stamped it, that one action would mean 3, right? One to 3, 1 to 3, 1 to 3, 1 to 3. So, in really early number work, kids think 1 to 1. When little kids are counting a small quantity, they'll count 1, 2, 3, 4. But what we want them to think about in multiplication is a many-to-1 action. When each of those quantities happens, it's not one thing, even though you make one action, it's four things or three things, depending upon what your unit is. If you needed 3 times 8, you could take your 3 times 4 and add 4 more, 3 times 4s to that.  So, you have your four 3s and now you need four more 3s. And that allows you to use a fact to get a fact you don't know because you've got that unit and that understanding that it's not by 1, but by a unit. When gets to larger multiplication, we don't really want to be working by drawing by 1s, and we don't even want to be stamping 27 19 times. But it's a first step into multiplication. This idea that you have a composite unit, and in the case of 3 times 4 and 3 times 7, seeing that 3 is common. So, there's your common composite unit. You needed four of them for 3 times 4, and you need seven of them for 3 times 7. So, it allows you to see those relationships, which if you look at the standards, the relationships are the glue. So, it's not enough to memorize your multiplication facts. If you don't have a strong relationship understanding there, it does fall short of a depth of understanding.  Mike: I think it was interesting to hear you talk about that, Beth, because one of the things that struck me is some of the language that you used, and I was comparing it in my head to some of the language that I've used in the past. So, I know I've talked about 3 times 4, but I thought it was really interesting how you used iterations of or duplicated …  Beth: Copies.  Mike: … or copies, right? What you make me think is that those language choices are a little bit clearer. I can visualize them in a way that 3 times 4 is a little bit more abstract or obscure. I may be thinking of that wrong, but I'm curious how you think the language that you use when you're trying to get kids to think about composite numbers matters.  Beth: Well, I'll say this, that when you draw your 3 circles and count 4 dots in each circle, the result is the same model than if you thought of it as a unit of 3 stamped four times. In the end, the model looks the same, but the physical and mental process you went through is significantly different. So, you thought when you drew every dot, you were thinking about 1, 1, 1, 1, 1, 1, 1. When you thought about your composite unit copied or iterated, you thought about this unit being repeated over and over. And that changes the way you're even thinking about what those numbers mean. And one of those big, significant things that makes addition different than multiplication when you look at equations is, in addition, those numbers mean the same thing. You have 3 things, and you have 4 things, and you're going to put them together. If you had 3 plus 4, and you changed that 4 to a 5, you're going to change one of your quantities by 1, impacting your answer by 1. In multiplication, if you had 3 times 4, and you change that 4 to a 5, your factor increases by 1, but your product increases by the value of your composite unit. So, it's a change of the other factor. And that is significant change in how you think about multiplication, and it allows you to pave the way, essentially, to proportional reasoning, which is that replicating your unit.  Mike: One of the things I'd appreciated about what you said was it's a change in how you're thinking. Because when I think back to Mike Wallus, classroom teacher, I don't know that I understood that as my work. What I thought of my work at that point in time was I need to teach kids how to use an algorithm or how to get an answer. But I think where you're really leading is we really need to be attending to, “What's the thinking that underlies whatever is happening?”  Beth: Yes. And that's what our work is all about, is how do you give teachers a sort of lens into or a look into how kids are thinking and how that impacts whether they can employ more efficient and sophisticated relationships and strategies in their thinking. And it's not enough to know your multiplication facts. And the research is pretty clear on the fact that memorizing is difficult. If you're memorizing a hundred single facts just by memory, the likelihood you're not going to remember some is high. But if you understand the relationship between those numbers, then you can use your 3 times 4 to get your 3 times 5 or your 3 times 8. So, the language that you use is important, and the way you leave kids thinking about something is important. And this idea of the composite unit, it's thematic, right? It goes through fractions and additive and proportional, but it's not the only definition of multiplication. So, you've got to also think of multiplication as scaling that comes later, but you also have to think of multiplication as area and as dimensions. But that first experience with multiplication has to be that composite-unit experience.  Mike: You've got me thinking already about how these ideas around unitizing that students can start to make sense of when they're multiplying whole numbers, that that would have a significant impact when they started to think about fractions or rational numbers. Can you talk a little bit about unitizing in the context of fractions, Beth?  Beth: Sure. The fraction standards have been most difficult for teachers to get their heads around because the way that the standards promote thinking about fractions is significantly different than the way most of us were taught fractions. So, in the standards and in the research, you come across the term “unit fraction,” and you can probably recognize the unitizing piece in the unit fraction. So, a unit fraction is a fraction where 1 is in the numerator, it's one unit of a fraction. So, in the case of three-fourths, you have three of the one-fourths. Now, this is a bit of a shift in how we were taught. Most of us were taught, “Oh, we have three-fourths. It means you have four things, but you only keep three of them,” right? We learned about the name “numerator” and the name “denominator.” And, of course, we know in fractions, in particular, kids really struggle.  Adults really struggle. Fractions are difficult because they seem to be a set of numbers that don't have anything in common with any other numbers. But once you start to think about unitizing and that composite unit, there's a standard in third grade that talks about “decompose any fraction into the sum of unit fractions.” So, in the case of five-sixths, you would identify the unit fraction as one-sixth, and you would have 5 of those one-sixths. So, your unit fraction is one-sixth, and you're going to iterate it or copy it or repeat it five times.  Mike: I can hear the parallels between the way you described this work with whole numbers. I have one-fourth, and I've duplicated or copied that five times, and that's what five-fourths is. It feels really helpful to see the through line between how we think about helping kids think about composite numbers and multiplying with whole numbers, to what you just described with unit fractions.  Beth: Yeah, and even the language that language infractions is similar, too. So, you talk about that 5 one-fourths. You decompose the five-fourths into 5 of the one-fourths, or you recompose those 5 one-fourths. This is a fourth-grade standard. You recompose those 5 one-fourths into 3 one-fourths or three-fourths and 2 one-fourths or two-fourths. So, even reading a fraction like seven-eighths says 7 one-eighths, helps to really understand what that seven-eighths means, and it keeps you from reading it as seven out of eight. Because when you read a fraction as seven out of eight, it sounds like you're talking about a whole number over another whole number. And so again, that connection to the composite unit in multiplication extends to that composite unit or that unit fraction or unitizing in multiplication. And really, even when we talk about multiplying fractions, we talk about multiplying, say, a whole number times a fraction “5 times one-fourth.” That would be the same as saying, “I'm going to repeat one-fourth five times,” as opposed to, we were told, “Put a 1 under the 5 and multiply across the numerator and multiply across the denominator.” But that didn't help kids really understand what was happening.  Mike: So, this progression of ideas that we've talked about from multiplication to fractions, what you've got me thinking about is, what does it mean to think about unitizing with younger kids, particularly perhaps, kids in kindergarten, first or second grade? I'm wondering how or what you think educators could do to draw out the big ideas about units and unitizing with students in those grade levels?  Beth: Well, really we don't expect kindergartners to strictly unitize because it's a relatively abstract idea. The big focus in kindergarten is for a student to understand four means 4, four 1s, and 7 means seven 1s. But where we do unitize is in the use of our models in early grades. In kindergarten, the use of a five-frame or a ten-frame. So, let's use the ten-frame to count by tens: 10, 20, 30. And then, how many ten-frames did it take us to count to 30? It took 3. There's the beginning of your unitizing idea. The idea that we would say, “It took 3 of the ten-frames to make 30” is really starting to plant that idea of unitizing 3 can mean 30. And in first grade, when we start to expose kids to coin values, time, telling time, one of the examples we use is, “Whenever was 1 minus 1, 59?” And that was, “When you read for one hour and your friend read for 1 minute less than you, how long did they read?”  So, all time is really a unitizing idea. So, all measures, measure conversion, time, money, and the big one in first grade is base 10. And first grade and second grade [have] the opportunity to solidify strong base 10 so that when kids enter third grade, they've already developed a concept of unitizing within the base 10 system. In first grade, the idea that in a number like 78, the 7 is actually worth more than the 8, even though at face value, the 7 seems less than the 8. The idea that 7 is greater than the 8 in a number like 78 is unitizing. In second grade, when we have a number like 378, we can unitize that into 307 tens and 8 ones, or 37 tens and 8 ones, and there's your re-unitizing. And that's actually a standard in second grade. Or 378 ones. So, in first and second grade, really what teachers have to commit to is developing really strong, flexible base 10 understanding. Because that's the first place kids have to struggle with this idea of the face value of a number isn't the same as the place value of a number.  Mike: Yeah, yeah. So, my question is, would you describe that as the seeds of unitizing? Like conserving? That's the thing that popped into my head, is maybe that's what I'm actually starting to do when I'm trying to get kids to go from counting each individual 1 and naming the total when they say the last 1.  Beth: So, there are some early number concepts that need to be solidified for kids to be able to unitize, right? So, conservation is certainly one of them. And we work on conservation all throughout elementary school. As numbers get larger, as they have different features to them, they're more complex. Conservation doesn't get fixed in kindergarten. It's just pre-K and K are the places where we start to build that really early understanding with small quantities. There's cardinality, hierarchical inclusion, those are all concepts that we focus on and develop in the earliest grades that feed into a child's ability to unitize. So, the thing about unitizing that happens in the earliest grades is it's pretty informal. In pre-K and K, you might make piles of 10, you might count quantities. Counting collections is something we talk a lot about, and we talk a lot about the importance of counting in early math instruction actually all the way up through, but particularly in early math. And let's say you had a group of kids, and they were counting out piles of, say, 45 things, and they put them in piles of 10 and then a pile of 5, and they were able to go back and say, “Ten, 20, 30, 40 and 5.”  So, there's a lot that's happening there. So, one is, they're able to make those piles of 10, so they could count to 10. But the other one is, they have conservation. And the other one is, they have a rope-count sequence that got developed outside of this use of that rope-count sequence, and now they're applying that. So, there's so many balls in the air when a student can do something like that. The unitizing question would be, “You counted 45 things. How many piles of 10 did you have?” There's your unitizing question. In kindergarten, there are students—even though we say it's not something we work on in kindergarten—there are certainly students who could look at that and say, “Forty-give is 4 piles of 10 and 5 extra.” So, when I say we don't really do it in kindergarten, we have exposure, but it's very relaxed. It becomes a lot more significant in first and second grade.  Mike: You said earlier that teachers in first and second grade really have to commit to building a flexible understanding of base 10. What I wanted to ask you is, how would you describe that? And the reason I ask is, I also think it's possible to build an inflexible understanding of base 10. So, I wonder how you would differentiate between the kind of practices that might lead to a relatively inflexible understanding of base 10 versus the kind of practices that lead to a more flexible understanding.  Beth: So, I think counting collections. I already said we talk a lot about counting collections and the primary training. Having kids count things and make groups of 10, focus on your 10 and your 5. We tell kindergarten teachers that the first month or two of school, the most important number you learn is 5. It's not 10, because our brain likes 5, and we can manage 5 easily. Our hand is very helpful. So, building that unit of 5 toward putting two 5s together to make a 10. I mean, I have a 3-year-old granddaughter, and she knows 5, and she knows that she can hold up both her hands and show me 10. But if she had to show me 7, she would actually start back at 1 and count up to 7. So, taking advantage of those units that are baked in already and focusing on them helps in the earliest grades.  And then really, I like materials to go into kids' hands where they're doing the building. I feel like second grade is a great time to hand kids base 10 blocks, but first grade is not. And first-grade kids should be snapping cubes together and building their own units, because the more they build their own units of 5 or 10, the more it's meaningful and useful for them. The other thing I'm going to say, and Bridges has this as a tool, which I really like, is they have dark lines at their 5s and 10s on their base 10 blocks. And that helps, even though people are going to say, “Kids can tell you it's a hundred,” they didn't build it. And so, there's a leap of faith there that is an abstraction that we take for granted. So, what we want is kids using those manipulatives in ways that they constructed those groupings, and that helps a lot. Also, no operations for addition and subtraction. You shouldn't be adding and subtracting without using base 10. So, adding and subtracting on a number line helps you practice not just addition and subtraction, but also base 10. So, because base 10's so important, it could be taught all year long in second grade with everything you do. We call second grade the sweet spot of math because all the most important math can be taught together in second grade.  Mike: One of the things that you made me think about is something that a colleague said, which is this idea that 10 is simultaneously 10 ones and one unit of 10. And I really connect that with what you said about the need for kids to actually, physically build the units in first grade.  Beth: What you just said, that's unitizing. I can call this 10 ones, and I can call this 1, worth 10. And it's more in face in the earliest grades because we often are very comfortable having kids make piles of 10 things or seeing the marks on a base 10 block, say. Or snapping 10 Unifix cubes together, 5 red and 5 yellow Unifix cubes or something to see those two 5s inside that unit of 10. And then also there's your math hand, your fives and your tens and your ten-frames are your fives and your tens. So, we take full advantage of that. But as kids get older, the math that's going to happen is going to rely on kids already coming wired with that concept. And if we don't push it in those early grades by putting your hands on things and building them and sketching what you've just built and transferring it to the pictorial and the abstract in very strategic ways, then you could go a long way and look like you know what you're doing—but don't really. Base 10 is one of those ways we think, because kids can tell you the 7 is in the tens place, they really understand. But the reality is that's a low bar, and it probably isn't an indication a student really understands. There's a lot more to ask.  Mike: Well, I think that's a good place for my next question, which is to ask you what resources OGAP has available, either for someone who might participate in the training, other kinds of resources. Could you just unpack the resources, the training, the other things that OGAP has available, and perhaps how people could learn more about it or be in touch if they were interested in training?  Beth: Sure. Well, if they want to be in touch, they can go to ogapmathllc.com, and that's our website. And there's a link there to send us a message, and we are really good at getting back to people. We've written books on each of our four content strands. The titles of all those books are “A Focus on … .” So, we have “A Focus on Addition and Subtraction,” “A Focus on Multiplication and Division,” “A Focus on Fractions,” “A Focus on Ratios and Proportions,” and you can buy them on Amazon. Our progressions are readily available on our website. You can look around on our website, and all our progressions are there so people can have access to those. We do training all over. We don't do any open training. In other words, we only do training with districts who want to do the work with more than just one person. So, we contract with districts and work with them directly. We help districts use their math program. Some of the follow-up work we've done is help them see the possibilities within their program, help them look at their program and see how they might need to add more. And once people come to training, they have access to all our resources, the item bank, the progressions, the training, the book, all that stuff.  Mike: So, listeners, know that we're going to add links to the resources that Beth is referencing to the show notes for this particular episode. And, Beth, I want to just say thank you so much for this really interesting conversation. I'm so glad we had a chance to talk with you today.  Beth: Well, I'm really happy to talk to you, so it was a good time.  Mike: Fantastic.  Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 3 | Episode 3 – Choice as a Foundation for Student Engagement Guest: Drs. Zandra de Araujo and Amber Candela Mike Wallus: As an educator, I know that offering my students choice has a big impact on their engagement, their identity, and their sense of autonomy. That said, I've not always been sure how to design choice into the activities in my classroom, especially when I'm using curriculum. Today we're talking with Drs. Zandra de Araujo and Amber Candela about some of the ways educators can design choice into their students' learning experiences.  Welcome back to the podcast, Zandra and Amber. It is really exciting to have you all with us today.  Zandra de Araujo: Glad to be back.  Amber Candela: Very excited to be here. Mike: So, I've heard you both talk at length about the importance of choice in students' learning experiences, and I wonder if we can start there. Before we talk about the ways you think teachers can design choice in a learning experience, can we just talk about the “why”? How would you describe the impact that choice has on students' learning experiences? Zandra: So, if you think about your own life, how fun would it be to never have a choice in what you get to do during a day? So, you don't get to choose what chores to do, where to go, what order to do things, who to work with, who to talk to. Schools are a very low-choice environment, and those tend to be punitive when you have a low-choice environment. And so, we don't want schools to be that way. We want them to be very free and open and empowering places. Amber: And a lot of times, especially in mathematics, students don't always enjoy being in that space. So, you can get more enjoyment, engagement, and if you have choice with how to engage with the content, you'll have more opportunity to be more curious and joyful and have hopefully better experiences in math. Zandra: And if you think about being able to choose things in your day makes you better able to make choices. And so, I think we want students to be smart consumers and users and creators of mathematics. And if you're never given choice or opportunity to kind of own it, I think that you're at a deficit. Amber: Also, if we want problem-solving people engaged in mathematics, it needs to be something that you view as something you were able to do. And so often we teach math like it's this pre-packaged thing, and it's just your role to memorize this thing that I give you. You don't feel like it's yours to play with. Choice offers more of those opportunities for kids. Zandra: Yeah, it feels like you're a consumer of something that's already made rather than somebody who's empowered to create and use and drive the mathematics that you're using, which would make it a lot more fun.  Mike: Yeah. You all are hitting on something that really clicked for me as I was listening to you talk. This idea that school, as it's designed oftentimes, is low choice. But math, in particular, where historically it has really been, “Let me show you what to do. Let me have you practice the way I showed you how to do it,” rinse and repeat. It's particularly important in math, it feels like, to break out and build a sense of choice for kids. Zandra: Absolutely. Mike: Well, one of the things that I appreciate about the work that both of you do is the way that you advocate for practices that are both really, really impactful and also eminently practical. And I'm wondering if we can dive right in and have you all share some of the ways that you think about designing choice into learning experiences.  Amber: I feel like I want “eminently practical” on a sticker for my laptop. Because I find that is a very satisfying and positive way to describe the work that I do because I do want it to be practical and doable within the constraints of schooling as it currently is, not as we wish it to be. Which, we do want it to be better and more empowering for students and teachers. But also, there are a lot of constraints that we have to work within. So, I appreciate that.  Zandra: I think that choice is meant to be a way of empowering students, but the goal for the instruction should come first. So, I'm going to talk about what I would want from my students in my classroom and then how we can build choice in. Because choice is kind of like the secondary component. So, first you have your learning goals, your aims as a teacher. And then, “How do we empower students with choice in service of that goal?” So, I'll start with number sense because that's a hot topic. I'm sure you all hear a lot about it at the MLC. Mike: We absolutely do. Zandra: So, one of the things I think about when teachers say, “Hey, can you help me think about number sense?” It's like, “Yes, I absolutely can.” So, our goal is number sense. So, let's think about what that means for students and how do we build some choice and autonomy into that. So, one of my favorite things is something like, “Give me an estimate, and we can Goldilocks it,” for example. So, it could be a word problem or just a symbolic problem and say, “OK, give me something that you know is either wildly high, wildly low, kind of close, kind of almost close but not right. So, give me an estimate, and it could be a wrong estimate or a close estimate, but you have to explain why.” So, it takes a lot of number sense to be able to do that. You have infinitely many options for an answer, but you have to avoid the one correct answer. So, you have to actually think about the one correct answer to give an estimate. Or if you're trying to give a close estimate, you're kind of using a lot of number sense to estimate the relationships between the numbers ahead of time. The choice comes in because you get to choose what kind of estimate you want. It's totally up to you. You just have to rationalize your idea. Mike: That's awesome. Amber, your turn. Amber: Yep. So related to that is a lot of math goes forward. We give kids the problem, and we want them to come up with the answer. A lot of the work that we've been doing is, “OK, if I give you the answer, can you undo the problem?” I'll go multiplication. So, we do a lot with, “What's seven times eight?” And there's one answer, and then kids are done. And you look for that answer as the teacher, and once that answer has been given, you're kind of like, “OK, here. I'm done with what I'm doing.” But instead, you could say, “Find me numbers whose product is 24.” Now you've opened up what it comes to. There's more access for students. They can come up with more than one solution, but it also gets kids to realize that math doesn't just go one way. It's not, “Here's the problem, find the answer.” It's “Here's the answer, find the problem.” And that also goes to the number sense. Because if students are able to go both ways, they have a better sense in their head around what they're doing and undoing. And you can do it with a lot of different problems. Zandra: And I'll just add in that that's not specific to us. Barb Dougherty had really nice article in, I think, Teaching Children Mathematics, about reversals at some point. And other people have shown this idea as well. So, we're really taking ideas that are really high uptake, we think, and sharing them again with teachers to make sure that they've seen ways that they can do it in their own classroom. Mike: What strikes me about both of these is, the structure is really interesting. But I also think about what the output looks like when you offer these kinds of choices. You're going to have a lot of kids doing things like justifying or using language to help make sense of the “why.” “Why is this one totally wrong, and why is this one kind of right?” And “Why is this close, but maybe not exact?” And to go to the piece where you're like, “Give me some numbers that I can multiply together to get to 24.” There's more of a conversation that comes out of that. There's a back and forth that starts to develop, and you can imagine that back and forth bouncing around with different kids rather than just kind of kid says, teacher validates, and then you're done. Zandra: Yeah, I think one of the cool things about choice is giving kids choice means that there's more variety and diversity of ideas coming in. And that's way more interesting to talk about and rationalize and justify and make sense of than a single correct answer or everybody's doing the same thing. So, I think, not only does it give kids more ownership, it has more access. But also, it just gives you way more interesting math to think and talk about. Mike: Let's keep going. Zandra: Awesome. So, I think another one, a lot of my work is with multilingual students. I really want them to talk. I want everybody to talk about math. So, this goes right to what you were just saying. So, one of the ways that we can easily say, “OK, we want more talk.” So how do we build that in through choice is to say, “Let's open up what you choose to share with the class.” So, there have been lots of studies done on the types of questions that teachers ask: tend to be closed, answer-focused, like single-calculation kind of questions. So, “What is the answer? Who got this?” You know, that kind of thing? Instead, you can give students choices, and I think a lot of teachers have done something akin to this with sentence starters or things. But you can also just say, instead of a sentence starter to say what your answer is, “I agree with X because of Y.” You can also say, “You can share an incorrect answer that you know is wrong because you did it, and it did not work out. You can also share where you got stuck because that's valuable information for the class to have.” You could also say, “I don't want to really share my thinking, my solution because it's not done, but I'll show you my diagram.” And so, “Let me show you a visual.” And just plop it up on the screen. So, there are a lot of different things you could share a question that you have because you're not sure, and it's just a related question. Instead of always sharing answers, let kids open up what they may choose to share, and you'll get more kids sharing. Because answers are kind of scary because you're expecting a correct answer often. And so, when you share and open up, then it's not as scary. And everybody has something to offer because they have a choice that speaks to them. Amber: And kids don't want to be wrong. People don't want to be wrong. “I don't want to give you a wrong answer.” And we went to the University of Georgia together, but Les Steffe always would say, “No child is ever wrong. They're giving you an answer with a purpose behind what that answer is. They don't actually believe that's a wrong answer that they're giving you.” And so, if you open up the space … And teachers say, “We want spaces to be safe, we want kids to want to come in and share.” But are we actually structuring spaces in that way? And so, some of the ideas that we're trying to come up with, we're saying that “We actually do value what you're saying when you choose to give us this. It's your choice of offering it up and you can say whatever it is you want to say around that,” but it's not as evaluative or as high stakes as trying to get the right answer and just like, “Am I right? Did I get it right?” And then what the teacher might say after that. Zandra: I would add on that kids do like to give wrong answers if that's what you're asking for. They don't like to give wrong answers if you're asking for a right one and they're accidentally wrong. So, I think back to my first suggestion: If you ask for a wrong answer and they know it's wrong, they're likely to chime right in because the right answer is the wrong answer, and there are multiple, infinite numbers of them. Mike: You know, it makes me think there's this set of ideas that we need to normalize mistakes as being productive things. And I absolutely agree with that. I also think that when you're asking for the right answer, it's really hard to kind of be like, “Oh, my mistake was so productive.” On the other hand, if you ask for an error or a place where someone's stuck, that just feels different. It feels like an invitation to say, “I've actually been thinking about this. I'm not there. I may be partly there. I'm still engaged. This is where I'm struggling.” That just feels different than providing an answer where you're just like, “Ugh.” I'm really struck by that. Zandra: Yeah, and I think it's a culture thing. So, a lot of teachers say to me that “it's hard to have kids work in groups because they kind of just tell each other the answers.” But they're modeling what they experience as learners in the classroom. “I often get told the answers,” that's the discourse that we have in the classroom. So, if you open up the discourse to include these things like, “Oh, I'm stuck here. I'm not sure where to go here.” They get practice saying, “Oh, I don't know what this is. I don't know how to go from here.” Instead of just going to the answer. And I think it'll spread to the group work as well. Mike: It feels like there's value for every other student in articulating, “I'm certain that this one is wrong, and here's why I know that.” There's information in there that is important for other kids. And even the idea of “I'm stuck here,” right? That's really a great formative assessment opportunity for the teacher. And it also might validate some of the other places where kids are like, “Yeah. Me, too.” Zandra: Uh-hm. Amber: Right, absolutely. Mike: What's next, my friend? Amber: I remember very clearly listening to Zandra present about choice, her idea of choice of feedback. And this was very powerful to me. I had never thought about asking my students how they wanted to receive the feedback I'd be giving them on the problems that they solved. And this idea of students being able to turn something in and then say, “This is how I'd like to receive feedback” or “This is the feedback I'd like to receive,” becomes very powerful because now they're the ones in charge of their own learning. And so much of what we do, kids should get to say, “This is how I think that I will grow better, is if you provide this to me.” And so, having that opportunity for students to say, “This is how I'll be a better learner if you give it to me in this way. And I think if you helped me with this part that would help the whole rest of it.” Or “I don't actually want you to tell me the answer. I am stuck here. I just need a little something to get me through. But please don't tell me what the answer is because I still want to figure it out for myself.” And so, allowing kids to advocate for themselves and teaching them how to advocate for themselves to be better learners; how to advocate for themselves to learn and think about “What I need to learn this material and be a student or be a learner in society” will just ultimately help students. Zandra: Yeah, I think as a student, I don't like to be told the answers. I like to figure things out, and I will puzzle through something for a long time. But sometimes I just want a model or a hint that'll get me on the right path, and that's all I need. But I don't want you to do the problem for me or take over my thinking. If somebody asked me, “What do you want?” I might say like, “Oh, a model problem or something like that.” But I don't think we ask kids a lot. We just do whatever we think as an adult. Which is different, because we're not learning it for the first time. We already know what it is. Mike: You're making me think about the range of possibilities in a situation like that. One is I could notice a student who is working through something and just jump in and take over and do the problem for them essentially and say, “Here, this is how you do it.” Or I guess just let them go, let them continue to work through it. But potentially there could be some struggle, and there might be some frustration. I am really kind of struck by the fact that I wonder how many of us as teachers have really thought about the kinds of options that exist between those two far ends of the continuum. What are the things that we could offer to students rather than just “Let me take over” or productive struggle, but perhaps it's starting to feel unproductive? Does that make sense? Zandra: Yeah, I think it does. I mean, there are so many different ways. I would ask teachers to re-center themselves as the learner that's getting feedback. So, if you have a principal or a coach coming into your room, they've watched a lesson, sometimes you're like, “Oh, that didn't go well. I don't need feedback on that. I know it didn't go well, and I could do better.” But I wonder if you have other things that you notice just being able to take away a part that you know didn't go well. And you're like, “Yep, I know that didn't go well. I have ideas for improving it. I don't really want to focus on that. I want to focus on this other thing.” Or “I've been working really hard on discourse. I really want feedback on the student discourse when you come in.” That's really valuable to be able to steer it—not taking away the other things that you might notice, but really focusing in on something that you've been working on is pretty valuable. And I think kids often have these things that maybe they haven't really thought about a lot, but when you ask them, they might think about it. And they might grow this repertoire of things that they're kind of working on personally. Amber: Yeah, and I just think it's getting at, again, we want students to come out of situations where they can say, “This is how I learn” or “This is how I can grow,” or “This is how I can appreciate math better.” And by allowing them to say, “It'd be really helpful if you just gave me some feedback right here” or “I'm trying to make this argument, and I'm not sure it's coming across clear enough,” or “I'm trying to make this generalization, does it generalize?” We're also maybe talking about some upper-level kids, but I still think we can teach elementary students to advocate for themselves also. Like, “Hey, I try this method all the time. I really want to try this other method. How am I doing with this? I tried it. It didn't really seem to work, but where did I make a mistake? Could you help me out with that? Because I think I want to try this method instead.” And so, I think there are different ways that students can allow for that. And they can say: “I know this answer is wrong. I'm not sure how this answer is wrong. Could you please help me understand my thinking or how could I go back and think about my thinking?” Zandra: Yeah. And I think when you said upper level, you meant upper grades. Amber: Yes. Zandra: I assume.  Amber: Yes.  Zandra: OK, yeah. So, for the lower-grade-level students, too, you can still use this. They still have ideas about how they learn and what you might want to follow up on with them. “Was there an easier way to do this? I did all these hand calculations and stuff. Was there an easier way?” That's a good question to ask. Maybe they've thought about that, and they were like, “That was a lot of work. Maybe there was an easier way that I just didn't see?” That'd be pretty cool if a kid asked you that. Mike: Or even just hearing a kid say something like, “I feel really OK. I feel like I had a strategy. And then I got to this point, and I was like, ‘Something's not working.'” Just being able to say, “This particular place, can you help me think about this?” That's the kind of problem-solving behavior that we ultimately are trying to build in kids, whether it's math or just life. Amber: Right, exactly. And I need, if I want kids to be able … because people say, “I sometimes just want a kid to ask a question.” Well, we do need to give them choice of the question they ask. And that's where a lot of this comes from is, what is your goal as a teacher? What do you want kids to have choice in? If I want you to have choice of feedback, I'm going to give you ideas for what that feedback could be, so then you have something to choose from. Mike: OK, so we've unpacked quite a few ideas in the last bit. I wonder if there are any caveats or any guidance that you would offer to someone who's listening who is maybe thinking about taking up some of these practices in their classroom? Zandra: Oh, yeah. I have a lot. Kids are not necessarily used to having a lot of choice and autonomy. So, you might have to be gentle building it in because it's overwhelming. And they actually might just say, “Just tell me what to do,” because they're not used to it. It's like when you're get a new teacher and they're really into explaining your thinking, and you've never had to do that. Well, you've had 10 years of schooling or however many years of schooling that didn't involve explaining your thinking, and now, all of a sudden, “I'm supposed to explain my thinking. I don't even know what that means. What does that look like? We never had to do that before.”  So maybe start small and think about some things like, “Oh, you can choose a tool or two that helps you with this problem. So, you can use a multiplication table, or you can use a calculator or something to use. You can choose. There are all these things out. You can choose a couple of tools that might help you.” But start small. And you can give too many choices. There's like choice overload. It's like when I go on Amazon, and there are way too many reviews that I have to read for a product, and I never end up buying anything because I've read so many reviews. It's kind of like that. It could get overwhelming. So purposeful, manageable numbers of choices to start out with is a good suggestion. Amber: And also, just going back to what Zandra said in the beginning, is making sure you have a purpose for the choice. And so, if you just are like, “Oh, I'm having choice for choice's sake.” Well, what is that doing? Is that supporting the learning, the mathematics, the number sense, the conceptual understanding, and all of that? And so, have that purpose going in and making sure that the choices backtrack to that purpose. Zandra: Yeah. And you could do a little choice inventory. You could be like, “Huh, if I was a student of my own class today, what would I have gotten to choose? If anything? Did I get to choose where I sat, what utensil I used? What type of paper did I use? Which problems that I did?” Because that's a good one. All these things. And if there's no choice in there, maybe start with one. Mike: I really love that idea of a “choice inventory.” Because I think there's something about really kind of walking through a particular day or a particular lesson that you're planning or that you've enacted, and really thinking about it from that perspective. That's intriguing. Zandra: Yeah, because really, I think once you're aware of how little choice kids get in a day … As an adult learner, who has presumably a longer attention span and more tolerance and really likes math, I've spent my whole life studying it. If I got so little choice and options in what I did, I would not be a well-behaved, engaged student. And I think we need to remember that when we're talking about little children. Mike: So, last question, is there research in the field or researchers who have done work that has informed the kind of thinking that you have about choice? Zandra: Yeah, I think we're always inspired by people who come before us, so it's probably an amalgamation of different things. I listen to a lot of podcasts, and I read a lot of books on behavioral economics and all kinds of different things. So, I think a lot of those ideas bleed into the work in math education. In terms of math education, in particular, there have been a lot of people who have really influenced me, like Marian Small's work with parallel tasks and things like that. I think that's a beautiful example of choice. You give multiple options for choice of challenge and see which ones the students feel like is appropriate instead of assigning them competence ahead of time. So, that kind of work has really influenced me. Amber: And then just, our team really coming together; Sam Otten and Zandra and their ideas and collaborating together. And like you mentioned earlier, that Barb Dougherty article on the different types of questions has really been impactful. More about opening up questions, but it does help you think about choice a little bit better. Mike: I think this is a great place to stop. Zandra, Amber, thank you so much for a really eye-opening conversation. Zandra: Thank you for having us.  Amber: Thanks for having us. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 3 | Episode 2 – Responsive Curriculum Guest: Dr. Corey Drake Mike Wallus: When it comes to curriculum, educators are often told to implement with “fidelity.” But what does fidelity mean? And where does that leave educators who want to be responsive to students in their classrooms? Today we're talking with Dr. Corey Drake about principles for responsive curriculum use that invite educators to respond to the students in their classrooms while still implementing curriculum with integrity. Mike: One of the age-old questions that educators grapple with is how to implement a curriculum in ways that are responsive to the students in their classroom. It's a question I thought a lot about during my years as a classroom teacher, and it's one that I continue to discuss with my colleague at MLC, Dr. Corey Drake. As a former classroom teacher and a former teacher educator who only recently began working for an organization that publishes curriculum, Corey and I have been trying to carve out a set of recommendations that we hope will help teachers navigate this question. Today on the podcast, we'll talk about this question of responsive curriculum use and offer some recommendations to support teachers in the field. Mike: Welcome back to the podcast, Corey. I'm excited to have you with us again. Corey Drake: It's great to be with you again. Mike: So, I've been excited about this conversation for a while because this question of, “What does it mean to be responsive to students and use a curriculum?” is something that teachers have been grappling with for so long, and you and I often hear phrases like “implementation with fidelity” used when folks are trying to describe their expectations when a curriculum's adopted. Corey: Yeah, I mean, I think this is a question teachers grapple with. It's a question I've been grappling with for my whole career, from different points of view from when I was a classroom teacher and a teacher educator and now working at The Math Learning Center. But I think this is the fundamental tension: “How do you use a set of published curriculum materials while also being responsive to your students?” And I think ideas like implementation with fidelity didn't really account for the responsive-to-your-students piece. Fidelity has often been taken up as meaning following curriculum materials, page by page, word for word, task for task. We know that's not actually possible. You have to make decisions, you have to make adaptations as you move from a written page to an enacted curriculum. But still the idea of fidelity was to be as close as possible to the written page. Whereas ideas like implementation with integrity or responsive curriculum use are starting with what's written on the page, staying consistent with the key ideas of what's on the page, but doing it in a way that's responsive to the students who are sitting in front of you. And that's really kind of the art and science of curriculum use. Mike: Yeah, I think one of the things that I used to think was that it was really a binary choice between something like fidelity, where you were following things in what I would've described as a lockstep fashion. Or the alternative, which would be, “I'm going to make everything up.” And you've helped me think, first of all, about what might be some baseline expectations from a large-scale curriculum. What are we actually expecting from curriculum around design, around the audience that it's written for? I wonder if you could share with the audience some of the things that we've talked about when it comes to the assets and also the limitations of a large-scale curriculum. Corey: Yeah, absolutely. And I will say, when you and I were first teachers probably, and definitely when we were students, the conversation was very different. We had different curriculum materials available. There was a very common idea that good teachers were teachers who made up their own curriculum materials, who developed all of their own materials. But there weren't the kinds of materials out there that we have now. And now we have materials that do provide a lot of assets, can be rich tools for teachers, particularly if we release this expectation of fidelity and instead think about integrity. So, some of the assets that a high-quality curriculum can bring are the progression of ideas, the sequence of ideas and tasks that underlies almost any set of curriculum materials; that really looks at, “How does student thinking develop across the course of a school year?” And what kinds of tasks, in what order, can support that development of that thinking. Corey: That's a really important thing that individual teachers or even teams of teachers working on their own, that would be very hard for them to put together in that kind of coherent, sequential way. So, that's really important. A lot of curriculum materials also bring in many ideas that we've learned over the last decades about how children learn mathematics: the kinds of strategies children use, the different ways of thinking that children bring. And so, there's a lot that both teachers and students can learn from using curriculum materials. At the same time, any published set of large-scale curriculum materials are, by definition, designed for a generic group of students, a generic teacher in a generic classroom, in a generic community. That's what it means to be large scale. That's what it means to be published ahead of time. So, those materials are not written for any specific student or teacher or classroom or community. Corey: And so, that's the real limitation. It doesn't mean that the materials are bad. The materials are very good. But they can't be written for those specific children in that specific classroom and community. That's where this idea that responsive curriculum use and equitable instruction always have to happen in the interactions between materials, teachers, and students. Materials by themselves cannot be responsive. Teachers by themselves cannot responsibly develop the kinds of ideas in the ways that curriculum can, the ways they can when using curriculum as a tool. And, of course, students are a key part of that interaction. And so, it's really thinking about those interactions among teachers, students, and materials and thinking about, “What are the strengths the materials bring? What are the strengths the teacher brings?” The teacher brings their knowledge of the students. The teacher brings their knowledge of the context. And the students bring, of course, their engagement and their interaction with those materials. And so, it's thinking about the strengths they each bring to that interaction, and it's in those interactions that equitable and responsive curriculum use happens. Mike: One of the things that jumps out from what you said is this notion that we're not actually attempting to fix “bad curriculum.” We're taking the position that curriculum has a set of assets, but it also has a set of limitations, and that's true regardless of the curriculum materials that you're using. Corey: Absolutely. This is not at all about curriculum being bad or not doing what it's supposed to do. This assumes that you're using a high-quality curriculum that does the things we just talked about that has that progression of learning, those sequences of tasks that brings ideas about how children learn and how we learn and teach mathematics. And then, to use that well and responsibly, the teacher then needs to work in ways, make decisions to enact that responsibly. It's not about fixing the curriculum. It's about using the curriculum in the most productive and responsive ways possible. Mike: I think that's good context, and I also think it's a good segue to talk about the three recommendations that we want educators to consider when they're thinking about, “What does it mean to be responsive when you're using curriculum?” So, just to begin with, why don't we just lay them out? Could you unpack them, Corey? Corey: Yeah, absolutely. But I will say that this is work you and I have developed together and looking at the work of others in the field. And we've really come up with, I think, three key criteria for thinking about responsive curriculum use. One is that it maintains the goals of the curriculum. So again, recognizing that one of the strengths of curriculum is that it's built on this progression of ideas and that it moves in a sequential way from the beginning of the year to the end of the year. We want teachers to be aware of, to understand what the goals are of any particular session or unit or year, and to stay true to those goals, to stay aligned with those goals. But at the same time, doing that in ways that open up opportunities for voice and choice and sensemaking for the specific students who are in front of them in that classroom. And then the last is, we're really concerned with and interested in supporting equitable practice. And so, we think about responsive curriculum use as curriculum use that reflects the equity-based practices that were developed by Julia Aguirre and her colleagues. Mike: I think for me, one of the things that hit home was thinking about this idea that there's a mathematical goal and that goal is actually part of a larger trajectory that the curriculum's designed around. And when I've thought about differentiation in the past, what I was really thinking about was replacement that fundamentally altered the instructional goal. And I think the challenge in this work is to say, “Am I clear on the instructional goal? And do the things that I'm considering actually maintain that for kids or are they really replacing them or changing them in a way that will alter or impact the trajectory?” Corey: I think that's such a critical point. And it's not easy work. It's not always clear even in materials that have a stated learning goal or learning target for a session. There's still work to do for the teacher to say, “What is the mathematical goal? Not the activity, not the task, but what is the goal? What is the understanding I'm trying to support for my students as they engage in this activity?” And so, you're right. I think the first thing is, teachers have to be super clear about that because all the rest of the decisions flow from understanding, “What is the goal of this activity, what are the understandings that I am trying to develop and support with this session? And then I can make decisions that are enhancing and providing access to that goal, but not replacing it. I'm not changing the goal for any of my students. I'm not changing the goal for my whole group of students. Instead, I'm recognizing that students will need different ways into that mathematics. Students will need different kinds of supports along the way. But all of them are reaching toward or moving toward that mathematical goal.” Mike: Yeah. When I think about some of the options, like potentially, number choice; if I'm going to try to provide different options in terms of number choice, is that actually maintaining a connection to the mathematical goal, or have I done something that altered it? Another thing that occurs to me is the resources that we share with kids for representation, be it manipulatives or paper, pencil, even having them talk about it—any of those kinds of choices. To what extent do they support the mathematical goal, or do they veer away from it? Corey: Yeah, absolutely. And there are times when different numbers or different tools or different models will alter the mathematical goal because part of the mathematical goal is to learn about a particular tool or a particular representation. And there are other times when having a different set of numbers or a different set of tools or models will only enhance students' access to that mathematical goal because maybe the goal is understanding something like two-digit addition and developing strategies for two-digit addition. Well then, students could reach that goal in a lot of different ways. And some students will be working just with decade numbers, and some students will be working with decades and ones, and some students will need number pieces, and others will do it mentally. But if the goal is developing strategies, developing your understanding of two-digit addition, then all of those choices make sense, all of those choices stay aligned with the goal. Corey: But if the goal is to understand how base ten pieces work, then providing a different model or telling students they don't need to use that model would, of course, fundamentally alter the goal. So, this is why it's so critically important that we support teachers in understanding, making sense of the goal, figuring out how do they figure that out. How do you open a set of curriculum materials, look at a particular lesson, and understand what the mathematical goal of that lesson is? And it's not as simple as just looking for the statement of the learning goal and the learning target. But it's really about, “What are the understandings that I think will develop or are intended to develop through this session?” Mike: I feel like we should talk a little bit about context, because context is such a powerful tool, right? If you alter the context, it might help kids surface some prior knowledge that they have. What I'm thinking about is this task that exists in Bridges where we're having kids look at a pet store where there are arrays of different sorts and kinds of dog foods or dog toys or cat toys. And I remember an educator saying to me, “I wonder if I could shift the context.” And the question that I asked her is, “If you look at this image that we're using to launch the task, what are the particular parts of that image that are critical to maintain if you're going to replace it with something that's more connected to your students?” Corey: Connecting to your students, using context to help students access the mathematics, is so important and such an empowering thing for teachers and students. But you're asking exactly the right question. And of course, that all relates to, “What's the mathematical goal?” Again. Because if I know that, then I can look for the features of the context that's in the textbook and see the ways in which that context was designed to support students in reaching that mathematical goal. But I can also look at a different context that might be more relevant to my students, that might provide them better access to the mathematics. And I can look at that context through the lens of that mathematical goal and see, “Does this context also present the kinds of features that will help my students understand and make sense of the mathematical goal?” And if the answer is yes, and if that context is also then more relevant to my students or more connected to their lives, then great. That's a wonderful adaptation. That's a great example of responsive curriculum use. If now I'm in a context that's distracting or leading me away from the mathematical goal, that's where we run into adaptations that are less responsive and less productive. Mike: Well, and to finish the example, the conversation that this led to with this educator was she was talking about looking for bodegas in her neighborhood that her children were familiar with, and we end up talking quite a bit about the extent to which she could find images from the local bodega that had different kinds of arrays. She was really excited. She actually did end up finding an image, and she came back, and she shared that this really had an impact on her kids. They felt connected to it, and the mathematical goal was still preserved. Corey: I love that. I think that's a great example. And I think the other thing that comes up sometimes when we present these ideas, is maybe you want to find a different context that is more relevant to your students that they know more about. Sometimes you might look at a context that's presented in the textbook and say, “I really love the mathematical features here. I really see how knowing something about this context could help my students reach the mathematical goal, but I'm going to have to do some work ahead of time to help my students understand the context, to provide them some access to that, to provide them some entry points.” So, in your example, maybe we're going to go visit a pet store. Maybe we're going to look at images from different kinds of stores and notice how things are arranged on shelves, and in arrays, and in different combinations. So, I think there are always a couple of choices. One is to change the context. One is to do some work upfront to help your students access the context so that they can then use that context to access the mathematics. But I think in both cases, it's about understanding the goal of the lesson and then understanding how the features of the context relate to that goal. Mike: Let's shift and just talk about the second notion, this idea of opening up space for students' voice or for sensemaking when you're using curriculum. For me at least, I often try to project ideas for practice into a mental movie of myself in a classroom. And I wonder if we could work to help people imagine what this idea of opening space for voice or sensemaking might look like. Corey: I think a lot of times those opportunities for opening up voice and choice and sensemaking are not in the direct, action steps or the direct instructions to teachers within the lesson, but they're kind of in the in-between. So, “I know I need to introduce this idea to my students, but how am I going to do that? What is that going to look like? What is that going to sound like? What are students going to be experiencing?” And so, asking yourself that question as the lesson plays out is, I think, where you find those opportunities to open up that space for student voice and choice. It's often about looking at that and saying, “Am I going to tell students this idea? Or am I going to ask them? Are students going to develop their strategy and share it with me or turn it in on a piece of paper? Or are they going to turn and talk to a partner? Are they going to share those ideas with a small group, with a whole group? What are they going to listen for in each other's strategies? How am I going to ask them to make connections across those strategies? What kinds of tools am I going to make available to them? What kinds of choices are they going to have throughout that process?” Corey: And so, I think it's having that mental movie play through as you read through the lesson and thinking about those questions all the way through. “Where are my students going to have voice? How are they going to have choice? How are students going to be sensemaking?” And often thinking about, “Where can I step back, as the teacher, to open up that space for student voice or student choice?” Mike: You're making me think about a couple things. The first one that really jumped out was this idea that part of voice is not necessarily always having the conversation flow from teacher to student, but having a turn and talk, or having kids listen to and engage with the ideas that their partners are sharing is a part of that idea that we're creating space for kids to share their ideas, to share their voice, to build their own confidence around the mathematics. Corey: Absolutely. I think that, to me, is the biggest difference I see when I go into different classrooms. “Whose voice am I hearing most often? And who's thinking do I know about when I've spent 20 minutes in a classroom?” And there are some classrooms where I know a lot about what the teacher's thinking. I don't know a lot about what the students are thinking. And there are other classrooms where I can tell you something about the thinking of every one of the students in that room after 10 minutes in that classroom because they're constantly turning and talking and sharing their ideas. Student voice isn't always out loud either, right? Students might be sharing their ideas in writing, they might be sharing their ideas through gestures or through manipulating models, but the ideas are communicating their mathematical thinking. Really, student communication might be an even better way to talk about that because there are so many different ways in which students can express their ideas. Mike: Part of what jumped out is this notion of, “What do you notice? What do you wonder?” Every student can notice, every student can wonder. So, if you share a context before you dive right into telling kids what's going to happen, give them some space to actually notice and wonder about what's going on, generate questions, that really feels like something that's actionable for folks. Corey: I think you could start every activity you did with a, “What do you notice? What do you wonder?” Students always have ideas. Students are always bringing resources and experiences and ideas to any context, to any task, to any situation. And so, we can always begin by accessing those ideas and then figuring out as teachers how we might build on those ideas, where we might go from there. I think even more fundamentally is just this idea that all students are sensemakers. All students bring brilliance to the classroom. And so ,what we need to do is just give them the opportunities to use those ideas to share those ideas, and then we as teachers can build on those ideas. Mike: Before we close this conversation, I want to spend time talking about responsive curriculum use being a vehicle for opening up space for equity-based practice. Personally, this is something that you've helped me find words for. There were some ideas that I had an intuitive understanding of. But I think helping people name what we mean when we're talking about opening space for equity-based practice is something that we might be able to share with folks right now. Can you share how a teacher might take up this idea of creating space for equity-based practice as they're looking at lessons or even a series of lessons? Corey: Yeah, absolutely. And I think student voice and choice are maybe outcomes of equity-based practices. And so, in a similar way, I think teachers can begin by looking at a lesson or a series of lessons and thinking about those spaces and those decisions in between the action steps. And again, asking a series of questions. The equity-based practices aren't a series of steps or rules, but really like a lens or a series of questions that as a teacher, you might ask yourself as you prepare for a lesson. So, “Who is being positioned as mathematically capable? Who's being positioned as having mathematically important ideas? Are all of my students being positioned in that way? Are some of my students being marginalized? And if some of my students are being marginalized, then what can I do about that? How could I physically move students around so that they're not marginalized? How can I call attention to or highlight a certain student's ideas without saying that those ideas are the best or only ideas? But saying, ‘Look, this student, who we might not have recognized before as mathematically capable and brilliant, has a really cool idea right now.'” Corey: You and I have both seen video from classrooms where that's done brilliantly by these small moves that teachers can make to position students as mathematically brilliant, as having important or cool or worthwhile ideas, valuable ideas to contribute. So, I think it's those kinds of decisions that make such a difference. Those decisions to affirm learners' identities. Those aren't big changes in how you teach. Those are how you approach each of those interactions minute by minute in the classroom. How do you help students recognize that they are mathematicians, that they each bring valuable ideas to the classroom? And so, it's more about those in-between moments and those moments of interaction with students where these equity-based practices come to life. Mike: You said a couple things that I'm glad that you brought out, Corey. One of them is this notion of positioning. And the other one that I think is deeply connected is this idea of challenging places where kids might be marginalized. And I think one of the things that I've been grappling with lately is that there's a set of stories or ideas and labels that often follow kids. There are labels that we affix to kids within the school system. There are stories that exist around the communities that kids come from, their families. And then there are also the stories that kids make up about one another, the ideas that carry about, “Who's good at math? Who's not? Who has ideas to share? Who might I listen to, and who might I not?” And positioning, to me, has so much opportunity as a practice to help press back against those stories that might be marginalizing kids. Corey: I think that's such an important point. And I think, along with that is the recognition that this doesn't mean that you, as the individual teacher, created those stories or believed stories or did anything to perpetuate those stories—except if you didn't act to disrupt them. Because those stories come from all around us. We hear Pam Seda and Julia Aguirre and people like that saying, “They're the air we breathe. They're the smog we live in. Those stories are everywhere. They're in our society, they're in our schools, they're in the stories students tell and make up about each other.” And so, the key to challenging marginality is not to say, “Well, I didn't tell that story, I don't believe that story. But those stories exist, and they affect the children in my classroom, so what am I going to do to disrupt them? What am I going to do? Because I know the stories that are told about certain students, even if I'm not the one telling them, I know what those stories are. So how am I going to disrupt them to show that the student who the story or the labels about that student are, that they are not as capable, or they are behind or struggling or ‘low students.' What am I going to do to disrupt that and help everyone in our classroom community see the brilliance of that child, understand that that child has as much to contribute as anybody else in the math classroom?” And that's what it means to enact equity-based practices. Mike: You're making me think about an interview we did earlier this year with Peter Liljedahl, and he talked about this idea. He was talking about it in the context of grouping, but essentially what he was saying is that kids recognize the stories that are being told in a classroom about who's competent and who's not. And so, positioning, in my mind, is really thinking about—and I've heard Julia Aguirre say it this way—“Who needs to shine? Whose ideas can we bring to the center?” Because what I've come to really have a better understanding of, is that the way I feel about myself as a mathematician and the opportunities that exist within a classroom for me to make sense of math, those are really deeply intertwined. Corey: Yes, yes, absolutely. We are not focusing on marginality or identity just because it makes people feel good, or even just because it's the right thing to do. But actually in the math classroom, your identity and the expectations and the way you're positioned in that classroom fundamentally affect what you have opportunities to learn and the kinds of math you have access to. And so, we will do this because it's the right thing to do and because it supports math learning for all students. And understanding the role of identity and marginality and positioning in student learning is critically important. Mike: You're making me think about a classroom that we visited earlier this year, and it was a really dynamic math discussion. There was a young man, I'll call him David, and he was in a multilingual classroom. And I'm thinking back on what you said. At one point you said, “I can go into a classroom, and I can have a really clear idea of what the teacher understands, and perhaps less so with the kids.” In this case, I remember leaving thinking, “I really clearly understand that David has a deep conceptual understanding of the mathematics.” And the reason for that was, he generally volunteered to answer every single question. And it was interesting. It's not because the educator in the classroom was directing all of the questions to him, but I really got the sense that the kids, when the question was answered, were to almost turn their bodies because they knew he was going to say something. And it makes me think David is a kid who, over time, not necessarily through intention, but through the way that status works in classrooms, he was positioned as someone who really had some ideas to share, and the kids were listening. The challenge was, not many of them were talking. And so, the question is, “How do we change that? Not because anyone has any ill intent toward those other children, but because we want them to see themselves as mathematicians as well.” Corey: Yeah, absolutely. And that is part of what's tricky about this is that that's so important is that I think for many years we've talked about opening up the classroom for student talk and student discourse. And we do turn and talks, and we do think pair shares. And we've seen a lot of progress, I think, in seeing those kinds of things in math classrooms. And I think the next step to that is to do those with the kind of intentionality and awareness that you were just demonstrating there; which is to say, “Well, who's talking and how often are they talking? And what sense are people making of the fact that David is talking so much? What sense are they making? What stories are they telling about who David is as a mathematician? But also who they are as mathematicians. And what does it mean to them that even though there are lots of opportunity for students talk in that classroom, it's dominated by one or maybe two students. And so, we have opened it up for student discourse, but we have more work to do. We have more work to say, “Who's talking, and what sense are they making, and what does that look like over time? And how is mathematical authority distributed? How is participation distributed across the class? And, in particular, with intentionality toward disrupting some of those narratives that have become entrenched in classrooms and schools.” Mike: I think that's a great place for us to stop. I want to thank you again for joining us, Corey. It was lovely to have you back on the podcast. Corey: Thanks. It was great to be with you. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org 

Rounding Up Season 3 | Episode 1 – Grouping Practices That Promote Efficacy and Knowledge Transfer Guest: Dr. Peter Liljedahl Mike Wallus: We know from research that student collaboration can have a powerful impact on learning. That said, how we group students for collaboration matters—a lot. Today we're talking with Dr. Peter Liljedahl, author of “Building Thinking Classrooms in Mathematics,” about how educators can form productive, collaborative groups in their classrooms. Mike: Hello, Peter. Welcome to the podcast. Peter Liljedahl: Thanks for having me. Mike: So, to offer our listeners some background, you've written a book, called “Building Thinking Classrooms in Mathematics,” and I think it's fair to say that it's had a pretty profound impact on many educators. In the book, you address 14 different practices. And I'm wondering if you could weigh in on how you weigh the importance of the different practices that you addressed? Peter: Well, OK, so, first of all, 14 is a big number that publishers don't necessarily like. When we first started talking with Corwin about this, they were very open. But I know if you think about books, if there's going to be a number in the title, the number is usually three, five or seven. It's sometimes eight—but 14 is a ridiculous number. They can't all be that valuable. What's important about the fact that it's 14, is that 14 is the number of core practices that every teacher does. That's not to say that there aren't more or less for some teachers, but these are core routines that we all do. We all use tasks. We all create groups for collaboration. We all have the students work somewhere. We all answer questions. We do homework, we assign notes, we do formative, summative assessment. We do all of these things. We consolidate lessons. We launch lessons. Peter: These are sort of the building blocks of what makes our teaching. And through a lot of time in classrooms, I deduced this list of 14. Robert Kaplinsky, in one of his blog posts, actually said that he thinks that that list of 14 probably accounts for 95 percent of what happens in classrooms. And my research was specifically about, “How do we enact each of those 14 so that we can maximize student thinking? So, what kind of tasks get students to think, how can we create groups so that more thinking happens? How can we consolidate a lesson so we get more thinking? How can we do formative and summative assessments so the students are thinking more?” So, the book is about responding to those 14 core routines and the research around how to enact each of those to maximize thinking. Your question around which one is, “How do we put weight on each of these?” Peter: They're all important. But, of course, they're not all equally impactful. Building thinking classrooms is most often recognized visually as the thing where students are standing at whiteboards working. And, of course, that had a huge impact on student engagement and thinking in the classroom, getting them from sitting and working at desks to getting them working at whiteboards. But in my opinion, it's not the most impactful. It is hugely impactful, but the one that actually makes all of thinking classroom function is how we form collaborative groups, which is chapter two. And it seems like that is such an inconsequential thing. “We've been doing groups for forever, and we got this figured out. We know how to do this. But … do we really? Do we really have it figured out?” Because my research really showed that if we want to get students thinking, then the ways we've been doing it aren't working. Mike: I think that's a great segue. And I want to take a step back, Peter. Before we talk about grouping, I want to ask what might be an obvious question. But I wonder if we can talk about the “why” behind collaboration. How would you describe the value or the potential impact of collaboration on students' learning experiences? Peter: That's a great question. We've been doing collaborative work for decades. And by and large, we see that it is effective. We have data that shows that it's effective. And when I say “we,” I don't mean me or the people I work with. I mean “we, in education,” know that collaboration is important. But why? What is it about collaboration that makes it effective? There are a lot of different things. It could be as simple as it breaks the monotony of having to sit and listen. But let's get into some really powerful things that collaboration does. Number one, about 25 years ago, we all were talking about metacognition. We know that metacognition is so powerful and so effective, and if we get students thinking about their thinking, then their thinking actually improves. And metacognition has been shown time and time again to be impactful in learning. Some of the listeners might be old enough to remember the days where we were actually trying to teach students to be metacognitive, and the frustration that that created because it is virtually impossible. Peter: Being reflective about your thinking while you're thinking is incredibly hard to do because it requires you to be both present and reflective at the same time. We're pretty good at being present, and we're pretty good about reflecting on our experiences. But to do both simultaneously is incredibly hard to do. And to teach someone to do it is difficult. But I think we've also all had that experience where a student puts up their hand, and you start walking over to them, and just as you get there, they go, “Never mind.” Or they pick up their book, and they walk over to you, and just as they get to you, they just turn around and walk back. I used to tell my students that they're smarter when they're closer to me. But what's really going on there is, as they've got their hand up, or as they're walking across the room toward you as a teacher, they're starting to formulate their thoughts to ask a question. Peter: They're preparing to externalize their thinking. And that is an incredibly metacognitive process. One of the easiest forms of metacognition, and one of the easiest ways to access metacognition, is just to have students collaborate. Collaborating requires students to talk. It requires them to organize their thoughts. It requires them to prepare their thinking and to think about their thinking for the purposes of externalization. It is an incredibly accessible way of creating metacognition in your classroom, which we already know is effective. So, that's one reason I think collaboration is really, really vital. Peter: Another one comes from the work on register. So, register is the level of sophistication with which we speak about something. So, if I'm in a classroom, and I'm talking to kindergarten students, I set a register that is accessible to them. When I talk to my undergraduates, I use a different register. My master's students, my Ph.D. students, my colleagues, I'm using different registers. I can be talking about the same thing, but the level of sophistication with which I'm going to talk about those things varies depending on the audience. And as much as possible, we try to vary our register to suit the audience we have. But I think we've also all had that instructor who's completely incapable of varying their register, the one who just talks at you as if you're a third-year undergraduate when you're really a Great Eight student. And the ability to vary our register to a huge degree is going to define what makes us successful as a teacher. Can we meet our learners where they're at? Can we talk to them from the perspective that they're at? Now we can work at it, and very adept teachers are good at it. But even the best teachers are not as good at getting their register to be the same as students. Peter: So, this is another reason collaboration is so effective. It allows students to talk and be talked to at their register, which is the most accessible form of communication for them. And I think the third reason that collaboration is so important is the difference between what I talk in my book about the difference between absolute and tentative knowledge. So, I'm going to make two statements. You tell me which one is more inviting to add a comment to. So, statement number one is, “This is how to do it, or this is what I did.” That's statement number one. Statement number two is, “I think that one of the ways that we may want to try, I'm wondering if this might work.” Which one is more inviting for you to contribute to? Mike: Yes, statement number two, for many, many reasons, as I'm sitting here thinking about the impact of those two different language structures. Peter: So, as teachers, we tend to talk in absolutes. The absolute communication doesn't give us anything to hold onto. It's not engaging. It's not inviting. It doesn't bring us into the conversation. It's got no rough patches—it's just smooth. But when that other statement is full of hedging, it's tentative. It's got so many rough patches, so many things to contribute to, things I want to add to, maybe push back at or push further onto. And that's how students talk to each other. When you put them in collaborative groups, they talk in tentative discourse, whereas teachers, we tend to talk in absolutes. So, students are always talking to each other like that. When we put them in collaborative groups, they're like, “Well, maybe we should try this. I'm wondering if this'll work. Hey, have we thought about this? I wonder if?” And it's so inviting to contribute to. Mike: That's fascinating. I'm going to move a little bit and start to focus on grouping. So, in the book, you looked really closely at the way that we group students for collaborative problem-solving and how that impacts the way students engage in a collaborative effort. And I'm wondering if you could talk a little bit about the type of things that you were examining. Peter: OK. So, you don't have to spend a lot of time in classrooms before you see the two dominant paradigms for grouping. So, the first one we tend to see a lot at elementary school. So, that one is called “strategic grouping.” Strategic grouping is where the teacher has a goal, and then they're going to group their students to satisfy that goal. So, maybe my goal is to differentiate, so I'm going to make ability groups. Or maybe my goal is to increase productivity, so I'm going to make mixed-ability groups. Or maybe my goal is to just have peace and quiet, so I'm going to keep those certain students apart. Whatever my goal is, I'm going to create the groups to try to achieve that goal, recognizing that how students behave in the classroom has a lot to do with who they're partnered with. So that's strategic grouping. It is the dominant grouping paradigm we see in elementary school. Peter: By the time we get to high school, we tend to see more of teachers going, “Work with who you want.” This is called “self-selected groupings.” And this is when students are given the option to group themselves any way they want. And alert: They don't group themselves for academic reasons, they group themselves for social reasons. And I think every listener can relate to both of those forms of grouping. It turns out that both of those are highly ineffective at getting students to think. And ironically, for the exact same reason. We surveyed hundreds of students who were in these types of grouping settings: strategic grouping or self-selected groupings. We asked one question, “If you knew you were going to work in groups today, what is the likelihood you would offer an idea?” That was it. And 80 percent of students said that they were unlikely or highly unlikely to offer an idea, and that was the exact same, whether they were in strategic groupings or self-selected groupings. The data cut the same. Mike: That's amazing, Peter. Peter: Yeah, and it's for the same reason it turns out; that whether students were being grouped strategically or self-selected, they already knew what their role was that day. They knew what was expected of them. And for 80 percent of the students, their role is not to think. It's not to lead. Their role is to follow, right? And that's true whether they're grouping themselves socially, where they already know the social hierarchy of this group, or they're being grouped strategically. We interviewed hundreds of students. And after grade 3, every single student could tell us why they were in the group this teacher placed them in. They know. They know what you think of them. You're communicating very clearly what you think their abilities are through the way you group them, and then they live down to that expectation. So, that's what we were seeing in classrooms was that strategic grouping may be great at keeping the peace. And self-selected grouping may be fabulous for getting students to stop whining about collaboration. But neither of them was effective for getting students to think. In fact, they were quite the opposite. They were highly ineffective for getting students to think. Mike: So, I want to keep going with this. And I think one of the things that stood out for me as I was reading is, this notion that regardless of the rationale that a teacher might have for grouping, there's almost always a mismatch between what the teacher's goals are and what the student's goals are. I wonder if you could just unpack this and maybe explain this a bit more. Peter: So, when you do strategic grouping, do you really think the students are with the students that they want to be with? One of the things that we saw happening in elementary school was that strategic grouping is difficult. It takes a lot of effort to try to get the balance right. So, what we saw was teachers largely doing strategic grouping once a month. They would put students into a strategic group, and they would keep them in that group for the entire month. And the kids care a lot about who they're with, when you're going to be in a group for a month. And do you think they were happy with everybody that was in that group? If I'm going to be with a group of students for a month, I'd rather pick those students myself. So, they're not happy. You've created strategic groupings. And, by definition, a huge part of strategic grouping is keeping kids who want to be together away from each other. Peter: They're not happy with that. Self-selected groupings, the students are not grouping themselves for academic reasons. They're just grouping themselves for social reasons so that they can socialize, so they talk, so they can be off topic, and all of these things. And yes, they're not complaining about group work, but they're also not being productive. So, the students are happy. But do you think the teacher's happy? Do you think the teacher looks out across that room and goes, “Yeah, there were some good choices made there.” No, nobody's happy, right? If I'm grouping them strategically, that's not matching their goals. That's not matching their social goals. When they're grouping themselves in self-selected ways, that's matching their social goals but not matching my academic goals for them. So, there's always going to be this mismatch. The teacher, more often than not, has academic goals. The students, more often than not, have social goals. There are some overlaps, right? There are students who are like, “I'm not happy with this group. I know I'm not going to do well in this group. I'm not going to be productive.” And there are some teachers who are going, “I really need this student to come out of the shell, so I need to get them to socialize more.” But other than that, by and large, our goals as teachers are academic in nature. The goals as students are social in nature. Mike: I think one of the biggest takeaways from your work on grouping, for me at least, was the importance of using random groups. And I have to admit, when I read that there was a part of me thinking back to my days as a first-grade teacher that felt a little hesitant. As I read, I came to think about that differently. But I'm wondering if you can talk about why random groups matter, the kind of impact that they have on the collaborative experience and the learning experience for kids. Peter: Alright, so going back to the previous question. So, we have this mismatch. And we have also that 80 percent of students are not thinking; 80 percent of students are entering into that group, not prepared to offer an idea. So those are the two problems that we're trying to address here. So, random groups … random wasn't good enough. It had to be visibly random. The students had to see the randomness because when we first tried it, we said, “Here's your random groups.” They didn't believe we were being random. They just thought we were being strategic. So, it has to be visibly random, and it turns out it has to be frequent as well. About once every 45 to 75 minutes. See, when students are put into random groups, they don't know what their role is. So, we're solving this problem. They don't know what their role is. When we started doing visibly random groups frequently, within three weeks we were running that same survey. Peter: “If you know you're going to work in groups today, what is the likelihood you would offer an idea?” Remember the baseline data was that 80 percent of students said that they were unlikely or highly unlikely, and, all of a sudden, we have a hundred percent of students saying that they're likely or highly likely. That was one thing that it solved. It shifted this idea that students were now entering groups willing to offer an idea, and that's despite 50 percent of them saying, “It probably won't lead to a solution, but I'm going to offer an idea.” Now why is that? Because they don't know what their role is. So, right on the surface, what random groups does, is it shatters this idea of preconceived roles and then preconceived behaviors. So, now they enter the groups willing to offer an idea, willing to be a contributor, not thinking that their role is just to follow. But there's a time limit to this because within 45 to 75 minutes, they're going to start to fall into roles. Peter: In that first 45 minutes, the roles are constantly negotiated. They're dynamic. So, one student is being the leader, and the others are being the follower. And now, someone else is a leader, the others are following. Now everyone is following. They need some help from some external source. Now everyone is leading. We've got to resolve that. But there is all of this dynamicism and negotiation going on around the roles. But after 45 to 75 minutes, this sort of stabilizes and now you have sort of a leader and followers, and that's when we need to randomize again so that the roles are dynamic and that the students aren't falling into sort of predefined patterns of non-thinking behavior. Mike: I think this is fascinating because we've been doing some work internally at MLC around this idea of status or the way that … the stories that kids tell about one another or the labels that kids carry either from school systems or from the community that they come from, and how those things are subtle. They're unspoken, but they often play a role in classroom dynamics in who gets called on. What value kids place on a peer's idea if it is shared. What you're making me think is there's a direct line between this thing that we've been thinking about and what happens in small groups as well. Peter: Yeah, for sure. So, you mentioned status. I want to add to that identity and self-efficacy and so on and so forth. One of the interesting pieces of data that came out of the research into random groups was, we were interviewing students several weeks into this. And we were asking them questions around this, and the students were saying things like, “Oh, the teacher thinks we're all the same, otherwise they wouldn't do random groups. The teacher thinks we're all capable, otherwise they wouldn't do random groups.” So, what we're actually talking about here is that we're starting—just simply through random groups—to have a positive impact on student self-efficacy. One of the things that came out of this work, that I wrote about in a separate paper, was that we've known for a long time that student self-efficacy has a huge impact on student performance. But how do we increase, how do we improve student self-efficacy? Peter: There are a whole bunch of different ways. The work of Bandura on this is absolutely instrumental. But it comes down to a couple of things. From a classroom teacher perspective, the first thing, in order for a student to start on this journey from low self-efficacy to high self-efficacy, they have to encounter a teacher who believes in them. Except students don't listen to what we say. They listen to what we do. So, simply telling our students that we have confidence in them doesn't actually have much impact. It's how we show them that we have confidence in them. And it turns out that random groups actually have a huge impact on that. By doing the random groups, we're actually showing the kids that we believe in them and then they start to internalize this. So that's one thing. The work of Bandura about how we can start to shift student self-efficacy through mastery experiences, where they start to, for example, be successful at something. And that starts to have an impact that is amplified when students start to be successful in front of others, when they are the ones who are contributing in a small group. And that group is now successful. And that success is linked in some small or great part to your contributions; that self-efficacy is amplified because not only am I being successful, I'm being successful in a safe environment, but in front of others. Peter: Now, self-efficacy contributes to identity, and identity has an interesting relationship with status. And you mentioned status. So, self-efficacy is what I think of myself. Status is what others think of me. I can't control my status. I can't shift my status. Status is something that is bestowed on me by others. And, of course, it's affected by their interactions with me in collaborative spaces. So, how they get to see me operate is going to create a status for me, on me, by others. But the status gets to be really nicely evenly distributed in thinking classrooms when we're doing these random groups because everybody gets to be seen as capable. They all get to be someone who can be mathematical and someone who can contribute mathematically. Mike: I want to shift back for a moment to this idea of visibly random groups. This idea that for kids, they need to believe that it's not just a strategic grouping that I've called random for the sake of the moment. What are some of the ways that you've seen teachers visibly randomize their groups so that kids really could see the proof was right out there in front of them? Peter: So, we first started with just cards. So, we got 27 kids. We're going to use playing cards, we're going to have three aces, three 2S, three 3s, three 4s, and so on. We would just shuffle the deck, and the kids would come and take a card. And if you're a 4, you would go to the board that has a 4 on it. Or maybe that fourth 4 is there, so to speak. We learned a whole bunch of things. It has to be visible. And however way we do it, the randomization doesn't just tell them what group they're in, it tells them where to go. That's an efficiency thing. You don't want kids walking around the classroom looking for their partners and then spending 5 minutes deciding where they want to work. Take a card, you got a 7, you go to the 7 board. You got an ace, you go to the ace board. Peter: And that worked incredibly well. Some teachers already had Popsicle sticks in their classroom, so they started using those: Popsicle sticks with students' names. So, they would pull three Popsicle sticks and they would say, “OK, these students are together. These students are together.” At first, we didn't see any problems with that. That seemed to be pretty isomorphic … to using a playing card. Some teachers got frustrated with the cards because with a card, sometimes what happens is that they get ripped or torn or they don't come back. Or they come back, and they're sweaty or they're hot. And it's like, “OK, where were you keeping this card? I don't want to know. It's hot, it's dirty.” They got ink on it. The cards don't come back. The kids are swapping cards. And teachers were frustrated by this. So, they started using digital randomizers, things like Flippity and ClassDojo and Picker Wheel and Team Shake and Team Maker. Peter: There were tons of these digital randomizers, and they all work pretty much the same. But there was a bit of a concern that the students may not perceive the randomness as much in these methods. And you can amplify that by, for example, bringing in a fuzzy [die], a big one, and somebody gets to roll it. And if a 5 comes up, they get to come up and hit the randomized button five times. And now there's a greater perception of randomness that's happening. With Flippity, that turns out actually it'd be true. Turns out that the first randomization is not purely random, and the kids spot that pattern. And we thought, “OK, perfect. That's fine. As long as the students perceive it's random, that it is truly random, that the teacher isn't somehow hacking this so that they are able to impose their own bias into this space.” So, it's seemingly random, but not purely random. And everything was running fine until about six to eight months ago. I was spending a lot of time in classrooms. I think in the last 14 months I've been in 144 different classrooms, co-teaching or teaching. So, I was spending a lot of time in classrooms, and for efficiency's sake, a lot of these teachers were using digital randomizers. And then I noticed something. It had always been there, but I hadn't noticed it. This is the nature of research. It's also the nature of just being a fly on the wall, or someone who's observing a classroom or a teacher. There's so much to notice we can't notice it all. So, we notice the things that are obvious. The more time we spend in spaces, the more nuanced things we're able to notice. And about six to eight months ago, I noticed something that, like I said, has always been there, but I had never really noticed it. Peter: Teacher hits a randomized button, and all the students are standing there watching, waiting for the randomized groups to appear on the screen. And then somebody goes, “Ugh.” It's so small. Or somebody laughs. Or somebody's like, “Nooo.” And it's gone. It's in a moment, it's gone. Sometimes others snicker about it, but it's gone. It's a flash. And it's always been there, and you think it's not a big deal. Turns out it's a huge deal because this is a form of micro-bullying. This is what I call it, “micro-bullying.” Because when somebody goes, “Ugh,” everybody in the room knows who said it. And looking at the screen, they know who they said it about. And this student, themself, knows who said it, and they know that they're saying it about them. And what makes this so much worse than other overt forms of bullying is that they also are keenly aware that everybody in the room just witnessed and saw this happen, including the teacher. Peter: And it cuts deeply. And the only thing that makes bullying worse is when bullying happens in front of someone who's supposed to protect you, and they don't; not because we're evil, but because it's so short, it's so small, it's over in a flash. We don't really see the magnitude of this. But this has deep psychological effects and emotional effects on these students. Not just that they know that this person doesn't like them. But they know that everybody knows that they don't like them. And then what happens on the second day? The second day, whoever's got that student, that victimized student in their group, when the randomization happens, they also go, “Ugh,” because this has become acceptable now. This is normative. Within a week, this student might be completely ostracized. And it's just absolutely normal to sort of hate on this one student. Peter: It's just not worth it. It cuts too deeply. Now you can try to stop it. You can try to control it, but good luck, right? I've seen teachers try to say, “OK, that's it. You're not allowed to say anything when the randomization happens. You're not allowed to cheer, you're not allowed to grunt, you're not allowed to groan, you're not allowed to laugh. All you can do is go to your boards.” Then they hit the random, and immediately you hear someone go, “Ugh.” And they'll look at them, and the student will go, “What? That's how I breathe.” Or “I stubbed my toe where I thought of something funny.” It's virtually impossible to shut it down because it's such a minor thing. But seemingly minor. In about 50 percent of elementary classrooms that I'm in, where a teacher uses that digital randomizer, you don't hear it. But 50 percent you do. Almost 100 percent of high school classrooms I'm in you hear some sort of grunt or groan or complaint. Peter: It's not worth it. Just buy more cards. Go to the casino, get free cards. Go to the dollar store, get them cheap. It's just not worth it. Now, let's get back to the Popsicle stick one. It actually has the same effect. “I'm going to pull three names. I'm going to read out which three names there are, and I'm going to drop them there.” And somebody goes, “Ugh.” But why does this not happen with cards? It doesn't happen with cards because when you take that card, you don't know what group you're in. You don't know who else is in your group. All you know is where to go. You take that card, you don't know who else is in your group. There's no grunting, groaning, laughing, snickering. And then when you do get to the group, there might be someone there that you don't like working with. So, the student might go, “Ugh.” But now there's no audience to amplify this effect. And because there's no audience, more often than not, they don't bother going, “Ugh.” Go back to the cards, people. The digital randomizers are fast and efficient, but they're emotionally really traumatizing. Mike: I think that's a really subtle but important piece for people who are thinking about doing this for the first time. And I appreciate the way that you described the psychological impact on students and the way that using the cards engineers less of the audience than the randomizer [do]. Peter: Yeah, for sure. Mike: Well, let's shift a little bit and just talk about your recommendations for group size, particularly students in kindergarten through second grade as opposed to students in third grade through fifth grade. Can you talk about your recommendations and what are the things that led you to them? Peter: First of all, what led to it? It was just so clear, so obvious. The result was that groups of three were optimal. And that turned out to be true every setting, every grade. There are some caveats to that, and I'll talk about that in a minute. But groups of three were obvious. We saw this in the data almost immediately. Every time we had groups of three, we heard three voices. Every time we heard groups of four, we heard three voices. When we had groups of five, we heard two voices on task, two voices off task, and one voice was silent. Groups of three were just that sort of perfect, perfect group size. It took a long time to understand why. And the reason why comes from something called “complexity theory.” Complexity theory tells us that in order for a group to be productive, it has to have a balance between diversity and redundancy. Peter: So, redundancy is the things that are the same. We need redundancy. We need things like common language, common notation, common vocabulary, common knowledge. We need to have things in common in order for the collaboration to even start. But if all we have is redundancy, then the group is no better than the individual. We also have to have diversity. Diversity is what every individual brings to the group that's different. And the thing that happens is, when the group sizes get larger, the diversity goes up, but redundancy goes down. And that's bad. And when the group sizes get smaller, the redundancy goes up, but the diversity goes down. And that's bad. Groups of three seem to have this perfect balance of redundancy and diversity. It was just the perfect group size. And if you reflect on groups that you've done in your settings, whatever that setting was, you'll probably start to recognize that groups of three were always more effective than groups of four. Peter: But we learned some other things. We learned that in K–2, for example, groups of three were still optimal, but we had to start with groups of two. Why? Because very young children don't know how to collaborate yet. They come to school in kindergarten, they're still working in what we call “parallel,” which means that they'll happily stand side by side at a whiteboard with their own marker and work on their own things side by side. They're working in parallel. Eventually, we move them to a state that we call “polite turn-taking.” Polite turn-taking is we can have two students working at a whiteboard sharing one marker, but they're still working independently. So, “It's now your turn and you're working on your thing, and now it's my turn, I'm working on my thing.” Eventually, we get them to a state of collaboration. And collaboration is defined as “when what one student says or does affects what the other student says or does.” Peter: And now we have collaboration happening. Very young kids don't come to school naturally able to collaborate. I've been in kindergarten classrooms in October where half the groups are polite turn-taking, and half the groups are collaborating. It is possible to accelerate them toward that state. But I've also been in grade 2 classrooms in March where the students are still working in parallel or turn-taking. We need to work actively at improving the collaboration that's actually happening. Once collaboration starts to happen in those settings, we nurtured for a while and then we move to groups of three. So, I can have kindergartens by the end of the year working in groups of three, but I can't assume that grade 2s can do it at the beginning of the year. It has a lot to do with the explicit efforts that have been made to foster collaboration in the classroom. And having students sit side by side and pair desks does not foster collaboration. It fosters parallel play. Peter: So, we always say that “K–2, start with groups of two, see where their level of collaboration is, nurture that work on it, move toward groups of three.” The other setting that we had to start in groups of two were alternate ed settings. Not because the kids can't collaborate, but because they don't trust yet. They don't trust in the process in the educational setting. We have to nurture that. Once they start to trust in working in groups of two, we can move to groups of three. But the data was clear on this. So, if you have a classroom, and let's say you're teaching grade 6, and you don't have a perfect multiple of three, what do you do? You make some groups of two. So, rather than groups of four, make some groups of two. Keep those groups of two close to each other so that they may start to collaborate together. Peter: And that was one of the ironies of the research: If I make a group of four, it's a Dumpster fire. If I make two groups of two and put them close to each other, and they start to talk to each other, it works great. You start with groups of two. So, having some extra groups of two is handy if you're teaching in high school or any grade, to be honest. But let's say you have 27 students on your roster, but only 24 are there. There's going to be this temptation to make eight groups of three. Don't do it. Make nine groups, have a couple of groups of two. Because the minute you get up and running, someone's going to walk in late. And then when they walk in late, it's so much easier to plug them into a group of two than to have them waiting for another person to come along so that they can pair them or to make a group of four. Mike: Yeah, that makes sense. Before we close, Peter, I want to talk about two big ideas that I really wish I would've understood more clearly when I was still in the classroom. What I'm thinking about are the notion of crossing social boundaries and then also the concept of knowledge mobility. And I'm wondering if you could talk about each of them in turn and talk about how they relate to one another. Peter: Certainly. So, when we make our groups, when we make groups, groups are very discreet. I think this comes from that sort of strategic grouping, or even self-selected groupings where the groups are really separate from each other. There are very well-defined boundaries around this group, and everything that happens, happens inside that group, and nothing happens between groups. In fact, as teachers, we often encourage that, and we're like, “No, do your own work in your group. Don't be talking to the other groups.” Because the whole purpose of doing strategic groups is to keep certain kids away from each other, and that creates a very non-permeable boundary between the groups. But what if we can make these boundaries more porous, and so that knowledge actually starts to flow between the groups. This is what's called “knowledge mobility,” the idea that we don't actually want the knowledge to be fixed only inside of a group. Peter: The smartest person in the room is the room. We got to get that knowledge moving around the room. It's not groups, it's groups among groups. So, how can we get what one group is achieving and learning to move to another group that's maybe struggling? And this is called “knowledge mobility.” The easiest way to increase this is we have the students working at vertical whiteboards. Working at vertical whiteboards creates a space where passive knowledge mobility is really easy to do. It's really easy to look over your shoulder and see what another group is doing and go, “Oh, let's try that. They made a table of values. Let's make a table of values. Or they've done a graph, or they drew a picture” or whatever. “We'll steal an idea.” And that idea helps us move forward. And that passive can also lead to more active, where it's like, “I wonder what they're doing over there?” Peter: And then you go and talk to them, and the teacher can encourage this. And both of these things really help with mobilizing knowledge, and that's what we want. We don't want the only source of knowledge to be the teacher. Knowledge is everywhere. Let's get that moving around the room within groups, between groups, between students. And that's not to say that the students are copying. We're not encouraging copying. And if you set the environment up right, they don't copy. They're not going to copy. They'll steal an idea, “Oh, let's organize our stuff into a table of values,” and then it's back to their own board and working on that. And the other way that we help make these boundaries more porous is by breaking down the social barriers that exist within a classroom. All classrooms have social barriers. They could be gender, race. They could be status-based. Peter: There are so many things that make up the boundaries that exist within classrooms. There are these social structures that exist in schools. And one of the things that random groups does is it breaks down these social barriers because we're putting students together that wouldn't normally be together. And our data really reveals just how much that happens; that after three weeks, the students are coming in, they're socializing with different students, students that hadn't been part of their social structure before. They're sitting together outside of class. I see this at the university where students are coming in, they almost don't know each other at all. Or they're coming in small groups that are in the same class. They know each other from other courses, and within three, four weeks, I'm walking through the hallways at the university and I'm seeing them sitting together, working together, even having lunch together in structures that didn't exist on day one. There are so many social structures, social barriers in classrooms. And if we can just erode those barriers, those group structures are going to become more and more porous, and we're creating more community, and we're reducing the risk that exists within those classrooms. Mike: I think the other piece that jumps out for me is when I go back to this notion of one random grouping, a random grouping that shifts every 45 to 75 minutes. This idea of breaking those social boundaries—but also, really this idea that knowledge mobility is accelerated jumps out of those two practices. I can really see that in the structure and how that would encourage that kind of change. Peter: Yeah. And it encourages both passively and actively. Passive in the sense that students can look over the shoulder, active that they can talk to another group. But also passively from the teacher perspective, that random groups does a lot of that heavy lifting. But I can also encourage it actively when a group asks a question. Rather than answering their question, looking around the room going, “You should go talk to the sevens over there.” Or “We're done. What do we do next?” “Go talk to the fours. They know what's next.” That, sort of, “I as a teacher can be passive and let the random groups do a lot of the heavy lifting. But I can also be active and push knowledge around the room. By the way, I respond to students' questions.” Mike: Well, and I think what also strikes me is you're really distributing the authority mathematically to the kids as well. Peter: Yeah, so we're displacing status, we're increasing identity. We're doing all sorts of different things that are de-powering the classroom, decentralizing the classroom. Mike: Well, before we go, Peter, I'm wondering if there are any steps that you'd recommend to an educator who's listening. They want to start to dabble, or they want to take up some of the ideas that we've talked about. Where would you invite people to make a start? Peter: So, first of all, one of the things we found in our research was small change is no change. When you make small changes, the classroom as a system will resist that. So, go big. In building thinking classrooms, random groups is not a practice that gets enacted on its own. It's enacted with two other practices: thinking tasks, which is chapter one of my book, random groups, which is chapter two. And then, getting the students working at vertical whiteboards. These are transformational changes to the classroom. What we're doing in doing that is we're changing the environment in which we're asking students to behave differently. Asking students to behave differently in exactly the same environment that they behaved a certain way for five years already is almost impossible to do. If you want them to behave differently, if you want them to start to think, you're going to have to create an environment that is more conducive to thinking. Peter: So, that's part of it. The other thing is, don't do things by half measures. Don't start doing, “Well, we're going to do random groups on Mondays, but we're going to do strategic groups the rest of the days,” or something like this. Because what that communicates to students is that the randomness is something that you don't really value. Go big. We're doing random groups. We're always doing random groups. Have the courage. Yes, there's going to be some combinations that you're going to go, “Uh-oh.” And some of those are going to be really uh-oh combinations. But you're also going to have way more situations where you go and then it turns out to be amazing. So, have that courage. Go with the random groups and do it persistently and consistently. Because there is going to be resistance. The students are going to resist this thing because at least when you're being strategic, you're being thoughtful about it. Peter: But this feels like too much chance. And they start to attribute, they start to map their emotions around being placed in strategic groups, which were often for a month, into this setting. And what we need to do is, we need to show that this is not that by being consistent, doing it randomly, doing it frequently, so they start to realize that this is different. This is not the kind of grouping structures that have happened in the past. And do it. Do it consistently, persistently. Do it for at least 10 days before you start to really see and really reap those benefits. Mike: I think that's a really great place to stop. Thank you so much for joining us on the podcast, Peter. It really has been a pleasure chatting with you. Peter: Thanks so much. It's been a great conversation. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 18 – Counting Collections Guest: Danielle Robinson and Melissa Hedges Mike Wallus: Earlier this season, we released an episode focused on the complex and interconnected set of concepts that students engage with as they learn to count. In this follow-up episode, we're going to examine a powerful routine called “counting collections.” We'll be talking with Danielle Robinson and Dr. Melissa Hedges from the Milwaukee Public Schools about counting collections and the impact that this routine can have on student thinking.  Mike: Well, welcome to the podcast, Danielle and Melissa. I can't tell you how excited I am to talk with y'all about the practice of counting collections.  Danielle Robinson and Melissa Hedges: Thanks for having us. Yes, we're so excited to be here. Mike: I want to start this conversation by acknowledging that the two of you are actually part of a larger team of educators who really took this work on counting collections. You introduced it in the Milwaukee Public Schools. And, Melissa, I think I'll start with you. Can you take a moment to recognize the collaborators who have been a part of this work? Melissa: Absolutely. In addition to Danielle and myself, we are fortunate to work with three other colleagues: Lakesha King, Krista Beal, and Claire Madden. All three are early childhood coaches that actively support this work as well. Mike: So, Danielle, I wonder for some folks if we can help them see this practice more clearly. Can you spend time unpacking, what does counting collections look like in a classroom? If I walked in, what are some of the things that I might see? Danielle: Yeah, I think what's really amazing about counting collections is there might be some different ways that you might see counting collections happening in the classroom. When you walk into a classroom, you might see some students all over. Maybe they're sitting at tables, maybe they're on the carpet. And what they're doing is they're actually counting a baggie of objects. And really their job is to answer this question, this very simple but complicated question of, “How many?” And they get to decide how they want to count. Not only do they get to pick what they want to count, but they also get to pick their strategy of how they actually want to count that collection. They can use different tools. They might be using bowls or plates. They might be using 10-frames. They might be using number paths. You might see kiddos who are counting by ones. Danielle: You might see kids who are making different groupings. At times, you might also see kiddos [who] are in stations, and you might see a small group where a teacher is doing counting collections with a few kiddos. You might see them working with partners. And I think the beautiful piece of this and the unique part of counting collections within Milwaukee Public Schools is that we've been able to actually pair the counting trajectory from Doug Clements and Julie Sarama with counting collections where teachers are able to do an interview with their students, really see where they're at in their counting so that the kids are counting a just right collection for them—something that's not too easy, something that's not too hard, but something that is available for them to really push them in their understanding of counting. So, you're going to see kids counting different sizes. And we always tell the teachers it's a really beautiful moment when you're looking across the classroom and as a teacher, you can actually step back and know that every one of your kids are getting what they need in that moment. Because I think oftentimes, we really don't ever get to feel like that, where we feel like, “Wow, all my kids are getting what they need right now, and I know that I am providing the scaffolds that they need.” Mike: So, I want to ask you a few follow-ups, if I might, Danielle. Danielle: Yeah, of course.  Mike: There's a bit of language that you used initially where I'm paraphrasing. And tell me where I get this wrong. You use the language “simple yet complicated,” I think. Am I hearing that right? Danielle: I did. I did, yeah. Mike: Tell me about that. Danielle: I think it's so interesting because a lot of times when we introduce this idea of counting collections with our teachers, they're like, “Wait a minute, so I'm supposed to give this baggie of a bunch of things to my students, and they just get to go decide how they want to count it?” And we're like, “Yeah, that is absolutely what we're asking you to do.” And they feel nervous because this idea of the kids, they're answering how many, but then there's all these beautiful pieces a part of it. Maybe kids are counting by ones, maybe they're deciding that they want to make groups, maybe they're working with a partner, maybe they're using tools. It's kind of opened up this really big, amazing idea of the simple question of how many. But there's just so many things that can happen with it. Mike: There's two words that kept just flashing in front of my eyes as I was listening to you talk. And the words were access and differentiation. And I think you didn't explicitly say those things, but they really jump out for me in the structure of the task and the way that a teacher could take it up. Can you talk about the way that you think this both creates access and also the places where you see there's possibility for differentiation? Danielle: For sure. I'm thinking about a couple classrooms that I was in this week and thinking about once we've done the counting trajectory interview with our kiddos, you might have little ones who are still really working with counting to 10. So, they have collections that they can choose that are just at that amount of about 10. We might have some kiddos who are really working kind of in that range of 20 to 40. And so, we have collections that children can choose from there. And we have collections all the way up to about 180 in some cases. So, we kind of have this really nice, natural scaffold within there where children are told, “Hey, you can go get this just right color for you.” We have red collections, blue collections, green and yellow. Within that also, the children get to decide how they want to count. Danielle: So, if they are still really working on that verbal count sequence, then we allow them to choose to count by ones. We have tools for them, like number paths to help do that. Maybe we've got our kiddos who are starting to really think about this idea of unitizing and making groups of 10s. So, then what they might do is they might take a 10-frame and they might fill their 10-frame and then actually pour that 10-frame into a bowl, so they know that that bowl now is a collection of 10. And so, it's this really nice idea of helping them really start to unitize and to make different groupings. And I think the other beautiful piece, too, is that you can also partner. Students can work together and actually talk about counting together. And we found that that really supports them, too, of just that collaboration piece, too. Mike: So, you kind of started poking around the question that I was going to ask Melissa.   Danielle and Melissa: ( laugh ) Mike: You said the word “unitizing,” which is the other thing that was really jumping out because I taught kindergarten and first grade for about eight years. And in my head, immediately all of the different trajectories that kids are on when it comes to counting, unitizing, combining … those things start to pop out. But, Melissa, I think what you would say is there is a lot of mathematics that we can build for kids beyond say K–2, and I'm wondering if you could talk a little bit about that. Melissa: Absolutely. So before I jump to our older kids, I'm just going to step back for a moment with our kindergarten, first- and second-graders. And even our younger ones. So, the mathematics that we know that they need to be able to count collections, that idea of cardinality, one-to-one correspondence, organization—Danielle did a beautiful job explaining how the kids are going to grab a bag, figure out how to count, it's up to them—as well as this idea of producing a set, thinking about how many, being able to name how many. The reason why I wanted to go back and touch on those is that we know that as children get older and they move into third, fourth, and fifth grade, those are understandings that they must carry with them. And sometimes those ideas aren't addressed well in our instructional materials. So, the idea of asking a first- and second-grader to learn how to construct a unit of 10 and know that 10 ones is one 10 is key, because when we look at where place value tends to fall apart in our upper grades. My experience has been it's fifth grade, where all of a sudden we're dealing with big numbers, we're moving into decimals, we're thinking about different size units, we've got fractions. There's all kinds of things happening.  Melissa: So, the idea of counting collections in the early elementary grades helps build kids' number sense, provides them with that confidence of magnitude of number. And then as they move into those either larger collections or different ways to count, we can make beautiful connections to larger place values. So, hundreds, thousands, ten thousands. Sometimes those collections will get big. All those early number relationships also build. So, those early number relationships, part-whole reasoning that numbers are composed and decomposed of parts. And then we've just seen lots really, really fun work about additive and multiplicative thinking. So, in a third-, fourth-, fifth-grade classroom, what I used to do is dump a cup full of lima beans in the middle of the table and say, “How many are there?” And there's a bunch there. So, they can count by ones. It's going to take a long time. And then once they start to figure out, “Oh wait, I can group these.” “Well, how many groups of five do you have?” And how we can extend to that from that additive thinking of five plus five plus five plus five to then thinking about and extending it to multiplicative thinking. So, I think the extensions are numerous.  Mike: There's a lot there that you said, and I think I wanted to ask a couple follow-ups. First thing that comes to mind is, we've been interviewing a guest for a different podcast … and this idea that unitizing is kind of a central theme that runs really all the way through elementary mathematics and certainly beyond that. But I really am struck by the way that this idea of unitizing and not only being able to unitize, but I think you can physically touch the units, and you can physically re-unitize when you pour those things into the cup. And it's giving kids a bit more space with the physical materials themselves before you step into something that might be more abstract. I'm wondering if that's something that you see as valuable for kids and maybe how you see that play out? Melissa: Yes, it's a great question. I will always say when we take a look at our standard base 10 blocks, “The person that really understands the construction of those base 10 blocks is likely the person [who] invented them.” They know that one little cube means one, and that all of a sudden these 10 cubes are fused together and we hold it up and we say, “Everybody, this is 10 ones. Repeat, one 10. What we find is that until kids have multiple experiences and opportunities over time to construct units beyond one, they really won't do it with deep understanding. And again, that's where we see it fall apart when they're in the fourth and fifth grade. And they're struggling just to kind of understand quantity and magnitude. So, the idea and the intentionality behind counting collections and the idea of unitizing is to give kids those opportunities that to be quite honest—and no disrespect to the hardworking curriculum writers out there—it is a tricky, tricky, tricky idea to develop in children through paper and pencil and workbook pages. Melissa: I think we have found over time that it's the importance of going, grabbing, counting, figuring it out. So, if my collection is bears, does that collection of 10 bears look the same as 10 little sharks look the same as 10 spiders? So, what is this idea of 10? And that they do it over and over and over and over again. And once they crack the code—that's the way I look at it—once our first- and second-graders crack the code of counting collections, they're like, “Oh, this is not hard at all.” And then they start to play with larger units. So, then they'll go, “Oh, wait, I can combine two groups of 10. I just found out that's 20. Can I make more 20s?” So, then we're thinking about counting not just by ones, not just by 10s, but by larger units. And I think that we've seen that pay off in so many tremendous ways. And certainly on the affective side, when kids understand what's happening, there's just this sense of joy and excitement and interest in the work that they do, and I actually think they see themselves learning. Mike: Danielle, do you want to jump in here?  Danielle: I think to echo that, I just recently was speaking with some teachers. And the principal was finally able to come and actually see counting collections happening. And what was so amazing is these were K–5 kiddos, 5-year-olds who were teaching the principal about what they were doing. This was that example where we want people to come in, and the idea is what are you learning? How do you know you've learned it, thinking about that work of Hattie? And these 5-year-olds were telling him exactly what they were learning and how they were learning it and talking about their strategies. And I just felt so proud of the K–5 teacher who shared that with me because her principal was blown away and was seeing just the beauty that comes from this routine. Mike: We did an episode earlier this year on place value, and the speaker did a really nice job of unpacking the ideas around it. I think what strikes me, and at this point I might be sounding a bit like a broken record, is the extent to which this practice makes place value feel real. These abstract ideas around unitizing. And I think, Melissa, I'm going back to something you said earlier where you're like, “The ability to do this in an abstract space where you potentially are relying on paper and pencil or even drawing, that's challenging.” Whereas this puts it in kids' hands, and you physically re-unitize something, which is such a massive deal. This idea that one 10 and 10 ones have the same value even though we're looking at them differently, simultaneously. That's such a big deal for kids, and it just really stands out for me as I hear you all talk. Melissa: I had the pleasure of working with a group of first-grade teachers the other day, and we were looking at student work for a simple task that the kids were asked to do. I think it was 24 plus seven, and so it was just a very quick PLC. Look at this work. Let's think about what they're doing. And many of the children had drawn what the teachers referred to as sticks and circles or sticks and dots. And I said, “Well, what do those sticks and dots mean?” Right? “Well, of course the stick is the 10 and the dot is the one.” And I said, “There's lots of this happening,” I said, “Let's pause for a minute and think, ‘To what degree do you think your children understand that that line means 10 and that dot means one? And that there's some kind of a connection, meaningful connection for them just in that drawing.'” It got kind of quiet, and they're like, “Well, yep, you're right. You're right. They probably don't understand what that is.” And then one of the teachers very beautifully said, “This is where I see counting collections helping.” It was fantastic. Mike: Danielle, I want to shift and ask you a little bit about representation. Just talk a bit about the role of representing the collection once the counting process and that work has happened. What do you all ask kids to do in terms of representation and can you talk a little bit about the value of that? Danielle: Right, absolutely. I think one thing that as we continue to go through in thinking about this routine and the importance of really helping our students make sense and count meaningfully, I think we will always go back to our math teaching framework that's been laid out for us through “Taking Action,” “Principles to Action,” “Catalyzing Change.” And really thinking about the power of using multiple representations. And how, just like you said, we want our students to be able to be physically unitizing, so we have that aspect of working with our actual collections. And then how do we help our students understand that “You have counted your collection. Now what I want you to do is, I want you to actually visually represent this. I want you to draw how you counted.” And so, what we talk about with the kids is, “Hey, how you have counted. If you have counted by ones, I should be able to see that on your paper. I should be able to look at your paper, not see your collection and know exactly how you counted. If you counted by tens, I should be able to see, ‘Oh my gosh, look, that's their bowl. I see their bowls, I see their plates, I see their tens inside of there.'”  Danielle: And to really help them make those connections moving back and forth between those representations. And I think that's also that piece, too, for them that then they can really hang their hat on. “This is how I counted. I can draw a picture of this. I can talk about my strategy. I can share with my friends in my classroom.” And then that's how we like to close with our counting collections routine is really going through and picking a piece of student work and really highlighting a student's particular strategy. Or even just highlighting several and being like, “Look at all this work they did today. Look at all of this mathematical thinking.” So, I think it's a really important and powerful piece, especially with our first- and second-graders, too. We really bring in this idea of equations, too. So, this idea of, “If I've counted 73, and I've got my seven groups of 10, I should have 10 plus 10 plus 10, right? All the way to 70. And then adding my three.” So, I think it's just a continuous idea of having our kids really developing that strong understanding of meaningful counting, diving into place value. Mike: I'm really struck by the way that you described the protocol where you said you're asking kids to really clearly make sure that what they're doing aligns with their drawing. The other piece about that is it feels like one, that sets kids up to be able to share their thinking in a way where they've got a scaffold that they've created for themself. The other thing that it really makes me think about is how if I'm a teacher and I'm looking at student work, I can really use that to position that student's idea as valuable. Or position that student's thinking as something that's important for other people to notice or attend to. So, you could use this to really raise a student's ideas status or raise the student status as well. Does that actually play out in a reality? Danielle: It does actually. So, a couple of times what I will do is I will go into a classroom. And oftentimes it can be kind of a parent for which students may just not have the strongest mathematical identity or may not feel that they have a lot of math agency in the space. And so, one thing that I will really intentionally do and work with the teacher to do is, “You know what? We are going to share that little one's work today. We're going to share that work because this is an opportunity to really position that child as a mathematician and to position that child as someone who has something to offer. And the fact that they were able to do this really hard work.” So, that is something that is very near and dear to us to really help our teachers think of these different ways to ensure that this is a routine that is for all of our children, for each and every child that is in that space. So, that is absolutely something that we find power in and seek to help our teachers find as well. Mike: Well, I would love for each of you to just weigh in on this next question. What has really come to mind is how different this experience of mathematics is from what a lot of adults and unfortunately what a lot of kids might experience in elementary school. I'm wondering if both of you would talk a bit about what does this look like in classrooms? How does this impact the lived experience of kids and their math identities? Can you just talk a little bit about that? Melissa: I can start. This is Melissa. So, we have four beliefs on our little math team that we anchor our work around every single day. And we believe that mathematics should be humanizing, healing, liberating and joyful. And so, we talk a lot about when you walk into a classroom, how do you know that mathematics instruction is humanizing, which means our children are placed at the center of this work? It's liberating. They see themselves in it. They're able to do it. It's healing. Healing for the teacher as well as for the student. And healing in that the student sees themselves as capable and able to do this, and then joyful that it's just fun and interesting and engaging. I think, over time, what we've seen is it helps us see those four beliefs come to life in every single classroom that's doing it. When that activity is underway and children are engaged and interested, there's a beautiful hum that settles over the room. And sometimes you have to remind the teacher step back, take a look at what is happening. Melissa: Those guys are all engaged. They're all interested. They're all doing work that matters to them because it's their work, it's their creation. It's not a workbook page, it's not a fill in the blank. It's not a do what I do. It's, you know what? “We have faith in you. We believe that you can do this,” and they show us time and time again that they can.  Danielle: I'll continue to echo that. Where for Milwaukee Public Schools and in the work that we are seeking to do is really creating these really transformative math spaces for, in particular, our Black and brown children. And really just making sure that they are seeing themselves as mathematicians, that they see themselves within this work, and that they are able to share their thinking and have their brilliance on display. And also, to work through the mathematical processes, too, right? This routine allows you to make mistakes and try a new strategy.  Danielle: I had this one little guy a couple months ago, he was working in a pretty large collection, and I walked by him and he was making groups of two, and I was like, “Oh, what are you working on?” And he's like, “I'm making groups of two.” And I thought to myself, I was like, “Oh boy, that's going to take him a long time” cause they had a really big collection. And I kind of came back around and he had changed it and was making groups of 10. So, it really creates a space where they start to calibrate and they are able to engage in that agency for themselves. I think the last piece I'd like to add is to really come to it from the teacher side as well … is that what Melissa spoke about was those four beliefs. And I think what we've also found is that county collections has been really healing for our teachers, too. We've had teachers who have actually told us that this helped me stay in teaching. I found a passion for mathematics again that I thought I'd lost. And I think that's another piece that really keeps us going is seeing not only is this transformative for our kids, cause they deserve the best, but it's also been really transformative for our teachers as well to see that they can teach math in a different way.  Mike: Absolutely, and I think you really got to this next transition point that I had in mind when I was thinking about this podcast, which is, listening to the two of you, it's clear that this is an experience that can be transformative mathematically and in terms of what a child or even a teacher's lived experience with mathematics is. Can you talk a little bit about what might be some very first steps that educators might take to get started with this? Danielle: Absolutely. I think one thing, as Melissa and I were kind of thinking about this, is someone who is like, “Oh my gosh, I really want to try this.” I think the first piece is to really take stock of your kiddos. If you're interested in diving into the research of Clements and Sarama and working with the county trajectory, we would love for you to Google that and go to learningtrajectories.org. But I think the other piece is to even just do a short little interview with your kids. Ask each of your little ones, “Count as high as you can for me and jot down what you're noticing.” Give them a collection of 10 of something. It could be counters, it could be pennies. See how they count that group of 10. Are they able to have that one-to-one? Do they have that verbal count sequence? Do they have that cardinality? Can they tell you that there is 10 if you ask them again, “How many?”? Danielle: If they can do that, then go ahead and give them 31. Give them 31 of something. Have them count and kind of just see the range of kiddos that you have and really see where is that little challenge I might want to give them. I think another really nice piece is once you dive into this work, you are never going to look at the dollar section different. You are always just start gathering things like pattern blocks. I started with noodles. That is how I started counting collections in my classroom. I used a bunch of erasers that I left over from my prize box. I use noodles, I use beads, bobby pins, rocks, twigs. I mean, start kind of just collecting. It doesn't have to be something that you spend your money on. This can be something that you already use, things that you have. I think that's one way that you can kind of get started. Then also, procedures, procedures, procedures, like go slow to go fast. Once you've got your collections, really teach your kids how to respect those collections. Anchor charts are huge. We always say, when I start this with 4-year-olds, our first lesson is, “This is how we open the bag today. This is how we take our collections out.” So, we always recommend go slow to go fast, really help the kids understand how to take care of the collections, and then they'll fly from there.  Mike: So, Melissa, I think this is part two of that question, which is, when you think about the kinds of things that helped you start this work and sustain this work in the Milwaukee Public Schools, do you have any recommendations that you think might help other folks? Melissa: Yeah. My first entry point into learning about counting collections other than through an incredibly valued colleague [who] learned about it at a conference, was to venture into the TED. I think it's TED, the teacher resource site, and that was where I found some initial resources around how do we do this? We were actually getting ready to teach a course that at the time Danielle was going to be a student in, and we knew that we wanted to do this thing called counting collection. So, it's like, “Well, let's get our act together on this.” So, we spent a lot of time looking at that. There's some lovely resources in there. And since the explosion of the importance of early mathematics has happened in American mathematical culture, which I think is fantastic, wonderful sites have come up. One of our favorites that we were talking about is Dreme. D-R-E-M-E, the Dreme website. Fantastic resources. Melissa: The other one Danielle mentioned earlier, it's just learningtrajectories.org. That's the Clements and Sarama research, which, 15 years ago, we were charged as math educators to figure out how to get that into the hands of teachers, and so that's one of the ways that they've done that. A couple of books that come to mind is the [“Young Children's Mathematics: Cognitively Guided Instruction in Early Childhood Education”]. Fantastic. If you don't have it and you're a preschool teacher and you're interested in math, get it. And then of course, the “Choral Counting & Counting Collections” book by Franke, Kazemi, Turrou. Yeah, so I think those are some of the big ones. If you want just kind of snippets of where to go, go to the Dreme, D-R-E-M-E, and you'll get some lovely, lovely hits. There's some very nice videos. Yeah, just watch a kid count ( laughs ). Mike: I think that's a great place to stop. I can't thank you two enough for joining us. It has really been a pleasure talking with both of you. Danielle: Thank you so much.  Melissa: Thanks for your interest in our work. We really appreciate it. Mike: With the close of this episode, we are at the end of season two for Rounding Up, and I want to just thank everyone who's been listening for your support, for the ways that you're taking these ideas up in your own classrooms and schools. We'll be taking the summer off to connect with new speakers, and we'll be back with season three this fall. In the meantime, if you have topics or ideas that you'd like for us to talk about, let us know. You can reach out to us at mikew@mathlearningcenter.org. What are some things you'd like us to talk about in the coming year? Have a great summer. We'll see you all in the fall.  Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 17 – Spatial Reasoning Guest: Dr. Robyn Pinilla Mike Wallus: Spatial reasoning can be a nebulous concept, and it's often hard for many educators to define. In this episode, we're talking about spatial reasoning with Dr. Robyn Pinilla from the University of Texas, El Paso. We'll examine the connections between spatial reasoning and other mathematical concepts and explore different ways that educators can cultivate this type of reasoning with their students. Mike: Welcome to the podcast, Robyn. I'm really excited to be talking with you about spatial reasoning. Robyn Pinilla: And I am excited to be here. Mike: Well, let me start with a basic question. So, when we're talking about spatial reasoning, is that just another way of saying that we're going to be talking about ideas that are associated with geometry? Or are we talking about something bigger? Robyn: It's funny that you say it in that way, Mike, because geometry is definitely the closest mathematical content that we see in curricula, but it is something much bigger. So, I started with the misconception and then I used my own experiences to support that idea that this was just geometry because it was my favorite math course in high school because I could see the concepts modeled and I could make things more tangible. Drawing helped me to visualize some of those concepts that I was learning instead of just using a formula that I didn't necessarily understand. So, at that time, direct instruction really ruled, and I'm unsure what the conceptual understandings of my teachers even were because what I recall is doing numbers 3 through 47 odds in the back of the book and just plugging through these formulas. But spatial reasoning allows us to develop our concepts in a way that lead to deeper conceptual understanding. I liked geometry, and it gave me this vehicle for mathematizing the world. But geometry is really only one strand of spatial reasoning. Mike: So, you're already kind of poking around the question that I was going to ask next, which is the elevator description of, “What do we mean when we talk about spatial reasoning and why does it matter? Why is it a big deal for students?” Robyn: So, spatial reasoning is a notoriously hard to define construct that deals with how things move in space. It's individually how they move in space, in relation to one another. A lot of my ideas come from a network analysis that [Cathy] Bruce and colleagues did back in 2017 that looked at the historical framing of what spatial reasoning is and how we talk about it in different fields. Because psychologists look at spatial reasoning. Mathematics educators look at spatial reasoning. There [are] also connections into philosophy, the arts. But when we start moving toward mathematics more specifically, it does deal with how things move in space individually and in relation to one another. So, with geometry, whether the objects are sliding and transforming or we're composing and decomposing to create new shapes, those are the skills in two-dimensional geometry that we do often see in curricula. But the underlying skills are also critical to everyday life, and they can be taught as well. Robyn: And when we're thinking about the everyday constructs that are being built through our interactions with the world, I like to think about the GPS on our car. So, spatial reasoning has a lot of spatial temporal processes that are going on. It's not just thinking about the ways that things move in relation to one another or the connections to mathematics, but also the way that we move through this world, the way that we navigate through it. So, I'll give a little example. Spatial temporal processes have to do with us running errands, perhaps. How long does it take you to get from work to the store to home? And how many things can you purchase in the store knowing how full your fridge currently is? What pots and pans are you going to use to cook the food that you purchase, and what volume of that food are you and your family going to consume? So, all those daily tasks involve conceptions of how much space things take. And we could call it capacity, which situates nicely within the measurement domain of mathematics education. But it's also spatial reasoning, and it extends further than that. Mike: That is helpful. I think you opened up my understanding of what we're actually talking about, and I think the piece that was really interesting is how in that example of “I'm going to the grocery store, how long will it take? How full is my fridge? What are the different tools that I'll use to prepare? What capacity do they have?” I think that really helped me broaden out my own thinking about what spatial reasoning actually is. I wonder if we could shift a bit and you could help unpack for educators who are listening, a few examples of tasks that kids might encounter that could support the development of spatial reasoning. Robyn: Sure. My research and work [are] primarily focused on early childhood and elementary. So, I'm going to focus there but then kind of expand up. Number one, let's play. That's the first thing that I want to walk into a classroom and see: I want to see the kids engaging with blocks, LEGOS, DUPLOS, and building with and without specific intentions. Not everything has to have a preconceived lesson. So, one of the activities I've been doing actually with teachers and professional development sessions lately is a presentation called “Whosits and Whatsits.” I have the teachers create whatsits that do thatsits; meaning, they create something that does something. I don't give them a prompt of what problem they're going to be solving or anything specific for them to build, but rather say, “Here are materials.” We give them large DUPLO blocks, magnet tiles and Magformers, different types of wooden, cardboard and foam blocks, PVC pipes, which are really interesting in the ways that teachers use them. And have them start thinking as though they're the children in the class, and they're trying to build something that takes space and can be used in different ways. Robyn: So, the session we did a couple of weeks ago, some teachers came up with … first, there was a swing that they had put a little frog in that they controlled with magnets. So, they had used the PVC pipe at the top that part of the swing connected over, and then were using the magnets to guide it back and forth without ever having to touch the swing. And I just thought, that was the coolest way for them to be using these materials in really playful, creative ways that could also engender them taking those lessons back into their classroom. I have also recently been reminded of the importance of modeling with fractions. So, are you familiar with the “Which One Doesn't Belong?” tasks? Mike: Absolutely love them. Robyn: Yes. There's also a website for fraction talks that children can look at visual representations of fractions and determine which one doesn't belong for some reason. That helps us to see the ways that children are thinking about the fractional spaces and then justifying their reason around them. With that, we can talk about the spatial positioning of the fractional pieces that are colored in. Or the ways that they're separated if those colored pieces are in different places on the figure that's being shown. They open up some nice spaces for us to talk about different concepts and use that language of spatial reasoning that is critical for teachers to engage in to show the ways that students can think about those things. Mike: So, I want to go back to this notion of play, and what I'm curious about is, why is situating this in play going to help these ideas around spatial reasoning come out as opposed to say, situating it in a more controlled structure? Robyn: Well, I think by situating spatial reasoning within play, we do allow teachers to respond in the moment rather than having these lesson plans that they are required to plan out from the beginning. A lot of the ideas within spatial reasoning, because it's a nebulous construct and it's learned through our everyday experiences and interactions with the world, they are harder to plan. And so, when children are engaged in play in the classroom, teachers can respond very naturally so that they're incorporating the mathematizing of the world into what the students are already doing. So, if you take, for example, one of my old teachers used to do a treasure hunt—great way to incorporate spatial reasoning with early childhood elementary classrooms—where she would set up a mapping task, is really what it was. But it was introducing the children to the school itself and navigating that environment, which is critical for spatial reasoning skills. Robyn: And they would play this gingerbread man-type game of, she would read the book and then everybody would be involved with this treasure hunt where the kiddos would start out in the classroom, and they would get a clue to help them navigate toward the cafeteria. When they got to the cafeteria, the gingerbread man would already be gone. He would've already run off. So, they would get their next clue to help them navigate to the playground, so on and so forth. They would go to the nurse's office, the principal, the library, all of the critical places that they would be going through on a daily basis or when they needed to within the school. And it reminds me that there was also a teacher I once interviewed who used orienteering skills with her students. Have you ever heard of orienteering? Mike: The connection I'm making is to something like geocaching, but I think you should help me understand it. Robyn: Yeah, that's really similar. So, it's this idea that children would find their way places. Path finding and way finding are also spatial reasoning skills that are applied within our real world. And so, while it may not be as scientific or sophisticated as doing geocaching, it has children with the idea of navigating in our real world, helps them start to learn cardinality and the different ways of thinking about traversing to a different location, which … these are all things that might better relate to social studies or technology, other STEM domains specifically, but that are undergirded by the spatial reasoning, which does have those mathematics connections. Mike: I think the first thing that occurred is, all of the directional language that could emerge from something like trying to find the gingerbread boy. And then the other piece that you made me think about just now is this opportunity to quantify distance in different ways. And I'm sure there are other things that you could draw out, especially in a play setting where the structure is a little bit looser and it gives you a little bit more space, as you said, to respond to kids rather than feeling like you have to impose the structure. Robyn: Yeah, absolutely. There's an ability when teachers are engaging in authentic ways with the students, that they're able to support language development, support ideation and creation, without necessarily having kids sit down and fill out a worksheet that says, “Where is the ball? The ball is sitting on top of the shelf.” Instead, we can be on the floor working with students and providing those directions of, “Oh, hey, I need you to get me those materials from the shelf on the other side of the room,” but thinking about, “How can I say that in a way that better supports children understanding the spatial reasoning that's occurring in our room?” So maybe it's, “Find the pencil inside the blue cup on top of the shelf that's behind the pencil sharpener,” getting really specific in the ways that we talk about things so that we're ingraining those ideas in such a way that it becomes part of the way that the kids communicate as well. Mike: You have me thinking that there's an intentionality in language choice that can create that, but then I would imagine as a teacher I could also revoice what students are saying and perhaps introduce language in that way as well. Robyn: Yeah, and now you have me thinking about a really fun routine number talks, of course. And if we do the idea of a dot talk instead of a number talk, thinking about the spatial structuring of the dots that we're seeing and the different ways that you can see those arrangements and describe the quantification of the arrangement. It's a nice way to introduce educators to spatial reasoning because it might be something that they're already doing in the classroom while also providing an avenue for children to see spatial structuring in a way that they're already accustomed to as well, based on the routines that they're receiving from the teacher.  Mike: I think what's really exciting about this, Robyn, is the more that we talk, the more two things jump out. I think one is, my language choices allow me to introduce these ideas in a way that I don't know that I'd thought about as a practitioner. Part two is that we can't really necessarily draw a distinction between work we're doing around numbers and quantity and spatial reasoning; that there are opportunities within our work around number quantity and within math content to inject the language of spatial reasoning and have it become a part of the experience for students. Robyn: Yeah, and that's important that I have conveyed that without explicitly saying it because that's the very work that I'm doing with teachers in their classrooms at this time. One, as you're talking about language, and I hate to do this, but I'm going to take us a little bit off topic for a moment. I keep seeing this idea on Twitter or whatever we call it at this point, that some people actually don't hear music in their heads. This idea is wild to me because I have songs playing in my head all the time. But at the same time, what if we think about the idea that some people don't also visualize things, they don't imagine those movements continuously that I just see. And so, as teachers, we really need to focus on that same idea that children need opportunities to practice what we think they should be able to hear but also practice what we think they should be able to see.  Robyn: I'm not a cognitive scientist. I can't see inside someone's head. But I am a teacher by trade, so I want to emphasize that teachers can do what's within their locus of control so that children can have opportunities to talk about those tasks. One that I recently saw was a lesson on clocks. So, while I was sitting there watching her teach, she was using a Judy Clock. She was having fun games with the kids to do a little competition where they could read the clock and tell her what time it was. But I was just starting to think about all of the ways that we could talk about the shorter and longer hands, the minute and hour hands, the ways that we could talk about them rotating around that center point. What shape does the hand make as it goes around that center point and what happens if it doesn't rotate fully? Now I'm going back to those fractional ideas from earlier with the “Which One Doesn't Belong?” tasks of having full shapes versus half shapes, and how we see those shapes in our real lives that we can then relate with visualized shapes that some children may or may not be able to see. Mike: You have me thinking about something. First of all, I'm so glad that you mentioned the role of visualization.  Robyn: Yeah. Mike: You had me thinking about a conversation I was having with a colleague a while ago, and we had read a text that we were discussing, and the point of conversation came up. I read this and there's a certain image that popped into my head. Robyn: Uh-hm. Mike: And the joke we were making is, “I'm pretty certain that the image that I saw in my head having read this text is not the same as what you saw.” What you said that really struck home for me is, I might be making some real assumptions about the pictures that kids see in their head and helping build those internal images, those mental movies. That's a part of our work as well. Robyn: Absolutely. Because I'm thinking about the way that we have prototypical shapes. So, a few years ago I was working with some assessments, and the children were supposed to be able to recognize an equilateral triangle—whether it was gravity-based or facing another orientation—and there were some children who automatically could see that the triangle was a triangle no matter which direction it was “pointing.” Whereas others only recognize it if a triangle, if it were gravity-based. And so, we need to be teaching the properties of the shapes beyond just that image recognition that oftentimes our younger students come out with. I tend to think of visualization and language as supporting one another with the idea that when we are talking, we're also writing a descriptive essay. Our words are what create the intended picture—can't say that it's always the picture that comes out. But the intended picture for the audience. What we're hopeful for in classrooms is that because we're sharing physical spaces and tangible experiences, that the language used around those experiences could create shared meaning. That's one of the most difficult pieces in talking about spatial reason or quite frankly, anything else, is that oftentimes our words may have different meanings depending on who the speaker and who the listener are. And so, navigating what those differences are can be quite challenging, which is why spatial reasoning is still so hard to define. Mike: Absolutely. My other follow-up is, if you were to offer people a way to get started, particularly on visualization, is there a kind of task that you imagine might move them along that pathway? Robyn: I think the first thing to do is really grasp an approximation. I'm not going to say figure out what spatial reasoning is, but just an approximation or a couple of the skills therein that you feel comfortable with. So, spatial reasoning is really the set of skills that undergirds almost all of our daily actions, but it also can be inserted into the lessons that teachers are already teaching. I think that we do have to acknowledge that spatial reasoning is hard to define, but the good news is that we do reason spatially all day every day. If I am in a classroom, I want to look first at the teaching that's happening, the routines that are already there, and see where some spatial reasoning might actually fit in. With our young classes, I like to think about calendar math. Every single kindergarten, first-grade classroom that you walk into, they're going to have that calendar on the wall. So how can you work into the routines that are occurring, that spatial language to describe the different components of the routine? Robyn: So, as a kiddo is counting on that hundreds chart, talking about the ways in which they're moving the pointer along the numbers … when they're counting by 10s, talk about the ways that they're moving down. When they're finding the patterns that are on the calendar, because all of those little calendar numbers for the day, they wind up having a pattern within them in most of the curricular kits. So, thinking about just the ways that we can use language therein. Now with older students, I think that offering that variety of models or manipulatives for them to use and then encourage them to translate from having a concrete manipulative into those more representational ideas, is great regardless of age or grade. So, students benefit from the modeling when they do diagramming of their models; that is, translating the 3-D model to 2-D, which is another component of spatial reasoning. And that gets me to this sticky point of, I'm not arguing against automaticity or being able to solve equations without physical or visual models. But I'm just acknowledging this idea that offering alternative ways for students to engage with content is really critical because we're no longer at a phase that we need our children to become computers. We have programs for that. We need children who are able to think and solve problems in novel ways because that's the direction that we're moving in problem-solving. Mike: That's fantastic. My final question before we close things up. If you were to make a recommendation for someone who's listening and they're intrigued and they want to keep learning, are there any particular resources that you'd offer people that they might be able to go to? Robyn: Yeah, absolutely. So, the first one that I like is the Learning Trajectories website. It's, uh, learning trajectories.org. It's produced by Doug Clements and Julie Sarama. There are wonderful tasks that are associated with spatial reasoning skills from very young children in the infants and toddler stages all the way up until 7 or 8 years old. So, that's a great place to go that will allow you to see how children are performing in different areas of spatial reasoning. There is also a book called “Taking Shape” by Cathy Bruce and colleagues that I believe was produced in 2016. And the grade levels might be a little bit different because it is on the Canadian school system, but it's for K–2 students, and that offers both the tasks and the spatial reasoning skills that are associated with them. For more of the research side, there's a book by Brent Davis and the Spatial Reasoning Study Group called “Spatial Reasoning in the Early Years,” and that volume has been one of my go-tos in understanding both the history of spatial reasoning in our schools and also ways to start thinking about spatializing school mathematics. Mike: One of the things that I really appreciate about this conversation is you've helped me make a lot more sense of spatial reasoning. But the other thing that you've done for me, at least, is see that there are ways that I can make choices with my planning, with my language … that I could pick up and do tomorrow. There's not a discreet separate bit that is about spatial reasoning. It's really an integrated set of ideas and concepts and skills that I can start to build upon right away whatever curriculum I have. Robyn: And that's the point. Often in mathematics, we think more explicitly about algebraic or numeric reasoning, but less frequently in classrooms about spatial reasoning. But spatial reasoning supports not only mathematics development, but other stem domains as well, and even skills that crossover into social studies and language arts as we're talking about mapping, as we're talking about language. So, as students have these experiences, they, too, can start to mathematize the world, see spatial connections as they go out to recess, as they go home from school, as they're walking through their neighborhoods, or just around the house. And it's ingrained ideas of measurement that we are looking at on a daily basis, the ways that we plan out our days and plan out our movements, whether it's really a plan or just our reactions to the world that support building these skills over time. And so, there are those really practical applications. But it also comes down to supporting overall mathematics development and then later STEM career interests, which is why I get excited about the work and want to be able to share it with more and more people. Mike: I think that's a great place to stop. For listeners, we're going to link all of the content that Robyn shared to our show notes. And, Robyn, I'll just say again, thank you so much for joining us. It's really been a pleasure talking with you. Robyn: Yes, absolutely. Thanks so much. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 16 – Strengthening Tasks Through Student Talk Guests: Dr. Amber Candela and Dr. Melissa Boston Mike Wallus: One of the goals I had in mind when we first began recording Rounding Up was to bring to life the best practices that we aspire to in math education and to offer entry points so that educators would feel comfortable trying them out in their classrooms. Today, we're talking with Drs. Amber Candela and Melissa Boston about powerful but practical strategies for supporting student talk in the elementary math classroom.  Welcome to the podcast, Amber and Melissa. We're really excited to be talking with you today. Amber Candela: Thank you for having us.  Melissa Boston: Yes, thank you. Mike: So we've done previous episodes on the importance of offering kids rich tasks, but one of the things that you two would likely argue is that rich tasks are necessary, but they're not necessarily sufficient, and that talk is actually what makes the learning experience really blossom. Is that a fair representation of where you all are at? Melissa: Yes. I think that sums it up very well. In our work, which we've built on great ideas from Smith and Stein, about tasks, and the importance of cognitively challenging tasks and work on the importance of talk in the classroom. Historically, it was often referred to as “talk moves.” We've taken up the term “discourse actions” to think about how do the actions a teacher takes around asking questions and positioning students in the classroom—and particularly these talk moves or discourse actions that we've named “linking” and “press”—how those support student learning while students are engaging with a challenging task. Mike: So I wonder if we could take each of the practices separately and talk through them and then talk a little bit about how they work in tandem. And Melissa, I'm wondering if you could start unpacking this whole practice of linking. How would you describe linking and the purpose it plays for someone who, the term is new for them? Melissa: I think as mathematics teachers, when we hear linking, we immediately think about the mathematics and linking representations or linking strategies. But we're using it very specifically here as a discourse action to refer to how a teacher links student talk in the classroom and the explicit moves a teacher makes to link students' ideas.  Sometimes a linking move is signaled by the teacher using a student's name, so referring to a strategy or an idea that a student might've offered. Sometimes linking might happen if a teacher revoices a student's idea and puts it back out there for the class to consider. The idea is in the way that we're using linking, that it's links within the learning community, so links between people in the classroom and the ideas offered by those people, of course. But the important thing here that we're looking for is how the links between people are established in the verbal, the explicit talk moves or discourse actions that the teacher's making. Mike: What might that sound like? Melissa: So that might sound like, “Oh, I noticed that Amber used a table. Amber, tell us how you used a table.” And then after Amber would explain her table, I might say, “Mike, can you tell me what this line of Amber's table means?” or “How is her table different from the table you created?” Mike: You're making me think about those two aspects, Melissa, this idea that there's mathematical value for the class, but there's also this connectivity that happens when you're doing linking. And I wonder how you think about the value that that has in a classroom.  Melissa: We definitely have talked about that in our work as well. I'm thinking about how a teacher can elevate a student's status in mathematics by using their name or using their idea, just marking or identifying something that the student said is mathematically important that's worthy of the class considering further. Creating these opportunities for student-to-student talk by asking students to compare their strategies or if they have something to add on to what another student said. Sometimes just asking them to repeat what another student said so that there's a different accountability for listening to your peers. If you can count on the teacher to revoice everything, you could tune out what your peers are saying, but if you might be asked to restate what one of your classmates had just said, now there's a bit more of an investment in really listening and understanding and making sense. Mike: Yeah, I really appreciate this idea that there's a way in which that conversation can elevate a student's ideas, but also to raise a student's status by naming their idea and positioning it as important. Melissa: I have a good example from a high school classroom where a student [...] was able to solve the contextual problem about systems of equations, so two equations, and it was important for the story when the two equations or the two lines intersected. And so one student was able to do that very symbolically. They created a graph, they solved the system of equations where another student said, “Oh, I see what you did. You found the difference in the cost per minute, and you also found the difference in the starting point, and then one had to catch up to the other.”  And so the way that the teacher kind of positioned those two strategies, one had used a sensemaking approach based really in the context. The other had used their knowledge of algebra. And by positioning them together, it was actually the student who had used the algebra had higher academic status, but the student who had reasoned through it had made this breakthrough that was really the aha moment for the class. Mike: That is super cool.  Amber, can we shift to press and ask you to talk a little bit about what press looks like? Amber: Absolutely. So how Melissa was talking about linking is holding students accountable to the community; press is more around holding students accountable to the mathematics.  And so the questions the teacher is going to ask is going to be more related specifically to the mathematics. So, “Can you explain your reasoning?” “How did you get that answer?” “What does this x mean?” “What does that intersection point mean?” And so the questions are more targeted at keeping the math conversation in the public space longer. Mike: I thought it was really helpful to just hear the example that Melissa shared. I'm wondering if there's an example that comes to mind that might shed some light on this. Amber: So when I'm in elementary classrooms and teachers are asking their kids about different problems, and kids will be like, “I got 2.” OK, “How did you get 2?” “What operation did you use?” “Why did you use addition when you could have used something else?”  So it's really pressing at the, “Yes, you got the answer, but how did you get the answer?” “How does it make sense to you?”, so that you're making the kids rather than the teacher justify the mathematics that's involved. And they're the ones validating their answers and saying, “Yes, this is why I did this because…” Mike: I think there was a point when I was listening to the two of you speak about this where, and forgive me if I paraphrase this a little bit, but you had an example where a teacher was interacting with a student and the student said something to the effect of, “I get it” or “I understand.” And the teacher came back and she said, “And what do you understand?” And it was really interesting because it threw the justification back to the student. Amber: Right. Really what the linking and press does, it keeps the math actionable longer to all of the peers in the room. So it's having this discussion out loud publicly. So if you didn't get the problem fully all the way, you can hear your peers through the press moves, talk about the mathematics, and then you can use the linking moves to think through, “Well, maybe if Mike didn't understand, if he revoices Melissa's comment, he has the opportunity to practice this mathematics speaking it.” And then you might be able to take that and be like, “Oh, wait, I think I know how to finish solving the problem now.” Mike: I think the part that I want to pull back and linger on a little bit is [that] part of the purpose of press is to keep the conversation about the mathematics in the space longer for kids to be able to have access to those ideas. I want y'all to unpack that just a little bit. Amber: Having linking and press at the end is holding the conversation longer in the classroom. And so the teacher is using the press moves to get at the mathematics so the kids can access it more. And then by linking, you're bringing in the community to that space and inviting them to add: “What do you agree [with]?” “Do you disagree?” “Can you revoice what someone said?” “Do you have any questions about what's happening?” Melissa: So when we talk about discourse actions, the initial discourse action would be the questions that the teacher asks. So there's a good task to start with. Students have worked on this task and produced some solution strategies. Now we're ready to discuss them. The teacher asks some questions so that students start to present or share their work and then it's after students' response [that] linking and press come in as these follow-up moves to do what Amber said: to have the mathematics stay in the public space longer, to pull more kids into the public space longer.  So we're hoping that by spending more time on the mathematics, and having more kids access the mathematics, that we're bringing more kids along for the ride with whatever mathematics it is that we're learning. Mike: You're putting language to something that I don't know that I had before, which is this idea that the longer we can keep the conversation about the ideas publicly bouncing around—there are some kids who may need to hear an idea or a strategy or a concept articulated in multiple different ways to piece together their understanding. Amber: And like Melissa was saying earlier, the thing that's great about linking is oftentimes in a classroom space, teachers ask a question, kids answer, the teacher moves on. The engagement does drop. But by keeping the conversation going longer, the linking piece of it, you might get called on to revoice, so you need to be actively paying attention to your peers because it's on the kids now. The math authority has been shared, so the kids are the ones also making sense of what's happening. But it's on me to listen to my peers because if I disagree, there's an expectation that I'll say that. Or if I agree or I might want to add on to what someone else is saying.  So oftentimes I feel like this pattern of teacher-student-teacher-student-teacher-student happens, and then what can start to happen is teacher-student-student-student-teacher. And so it kind of creates this space where it's not just back and forth, it kind of popcorns more around with the kids. Mike: You are starting to touch on something that I did want to talk about, though, because I think when I came into this conversation, what was in my head is, like, how this supports kids in terms of their mathematical thinking. And I think where you two have started to go is: What happens to kids who are in a classroom where link and press are a common practice? And what happens to classrooms where you see this being enacted on a consistent basis? What does it mean for kids? What changes about their mathematical learning experience? Melissa: You know, we observe a lot of classrooms, and it's really interesting when you see even primary grade students give an answer and immediately say, you know, “I think it's 5 because …,” and they provide their justification just as naturally as they provide their answer or they're listening to their peers and they're very eager to say, “I agree with you; I disagree with you, and here's why” or “I did something similar” or “Here's how my diagram is slightly different.”  So to hear children and students taking that up is really great. And it just—a big shift in the amount of time that you hear the teacher talking versus the amount of time you hear children talking and what you're able to take away as the teacher or the educator formatively about what they know and understand based on what you're hearing them say. And so [in] classrooms where this has become the norm, you see fewer instances where the teacher has to use linking and press because students are picking this up naturally. Mike: As we were sitting here and I was listening to y'all talk, Amber, the thing that I wanted to come back to is [that] I started reflecting on my own practice and how often, even if I was orchestrating or trying to sequence, it was teacher-student-teacher-student-teacher-student. It bounced back to me, and I'm really kind of intrigued by this idea, teacher-student-student-student-teacher—that the discourse, it's moving from a back and forth between one teacher, one student, rinse and repeat, and more students actually taking up the discourse. Am I getting that right? Amber: Yes. And I think really the thought is we always want to talk about the mathematics, but we also have to have something for the community. And that's why the linking is there because we also need to hold kids accountable to the community that they're in as much as we need to hold them accountable to the mathematics. Mike: So, Amber, I want to think about what does it look like to take this practice up? If you were going to give an educator a little nudge or maybe even just a starting point where teachers could take up linking and press, what might that look like? If you imagined kind of that first nudge or that first starting point that starts to build this practice? Amber: We have some checklists with sentence stems in [them], and I think it's taking those sentence stems and thinking about when I ask questions like, “How did you get that?” and “How do you know this about that answer?”, that's when you're asking about the mathematics. And then when you start to ask, “Do you agree with what so-and-so said? Can you revoice what they said in your own words?”, that's holding kids accountable to the community and just really thinking about the purpose of asking this question. Do I want to know about the math or do I want to build the conversation between the students? And then once you realize what you want that to be, you have the stem for the question that you want to ask. Mike: Same question, Melissa. Melissa: I think if you have the teacher who is using good tasks and asking those good initial questions that encourage thinking, reasoning, explanations, even starting by having them try out, once a student gives you a response, asking, “How do you know?” or “How did you get that?” and listening to what the student has to say. And then as the next follow-up, thinking about that linking move coming after that. So even a very formulaic approach where a student gives a response, you use a press move, hear what the student has to say, and then maybe put it back out to the class with a linking move. You know, “Would someone like to repeat what Amber just said?” or “Can someone restate that in their own words?” or whatever the linking move might be. Mike: So if these two practices are new to someone who's listening, are there any particular resources or recommendations that you'd share with someone who wants to keep learning? Amber: We absolutely have resources. We wrote an article for the NCTM's MTLT [Mathematics Teacher: Learning and Teaching PK-12] called “Discourse Actions to Promote Student Access .” And there are some vignettes in there that you can read through and then there [are] checklists with sentence stems for each of the linking and press moves. Melissa: Also, along with that article, we've used a lot of the resources from NCTM's Principles to Actions [Professional Learning] Toolkit.   that's online, and some of the resources are free and accessible to everyone. Amber: And if you wanted to dig in a bit more, we do have a book called Making Sense of Mathematics to Inform Instructional Quality. And that goes in-depth with all of our rubrics and has other scenarios and videos around the linking and press moves along with other parts of the rubrics that we were talking about earlier. Mike: That's awesome. We will link all of that in our show notes.  Thank you both so much for joining us. It was a real pleasure talking with you. Amber: Thanks for having us.  Melissa: Thank you. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org References and Resources: NCTM: https://pubs.nctm.org/view/journals/mtlt/113/4/article-p266.xml#:~:text=Discourse%20actions%20provide%20access%20to,up%20on%20contributions%20from%20students ERIC: https://eric.ed.gov/?id=EJ1275372 https://www.nctm.org/PtAToolkit/ https://www.nctm.org/uploadedFiles/Conferences_and_Professional_Development/Annual_Meetings/LosAngeles2022/Campaigns/12-21_PtA_Toolkit.pdf?utm_source=nctm&utm_medium=web&utm_campaign=LA2022&utm_content=PtA+Toolkit

Rounding Up Season 2 | Episode 15 – Making Sense of Story Problems Guest: Drs. Aina Appova and Julia Hagge Mike Wallus: Story problems are an important tool that educators use to bring mathematics to life for their students. That said, navigating the meaning and language found in story problems is a challenge for many students. Today we're talking with Drs. Aina Appova and Julia Hagge from [The] Ohio State University about strategies to help students engage with and make sense of story problems.  Mike: A note to our listeners. This podcast was recorded outside of our normal recording studio, so you may notice some sound quality differences from our regular podcast. Mike:   Welcome to the podcast, Aina and Julia. We're excited to be talking to both of you.  Aina Appova: Thank you so much for having us. We are very excited as well.  Julia Hagge: Yes, thank you. We're looking forward to talking with you today.   Mike: So, this is a conversation that I've been looking forward to for quite a while, partly because the nature of your collaboration is a little bit unique in ways that I think we'll get into. But I think it's fair to describe your work as multidisciplinary, given your fields of study.  Aina: Yes, I would say so. It's kind of a wonderful opportunity to work with a colleague who is in literacy research and helping teachers teach mathematics through reading story problems.  Mike: Well, I wonder if you can start by telling us the story of how you all came to work together. And describe the work you're doing around helping students make sense of word problems.  Aina: I think the work started with me working with fifth-grade teachers, for two years now, and the conversations have been around story problems. There's a lot of issues from teaching story problems that teachers are noticing. And so, this was a very interesting experience. One of the professional development sessions that we had, teachers were saying, “Can we talk about story problems? It's very difficult.” And so, we just looked at a story problem. And the story problem, it was actually a coordinate plane story problem. It included a balance beam, and you're supposed to read the story problem and locate where this balance beam would be. And I had no idea what the balance beam would be. So, when I read the story, I thought, “Oh, it must be from the remodeling that I did in my kitchen, and I had to put in a beam, which was structural.”  Aina: So, I'm assuming it's balancing the load. And even that didn't help me. I kept rereading the problem and thinking, “I'm not sure this is on the ceiling, but the teachers told me it's gymnastics.” And so even telling me that it was gymnastics didn't really help me because I couldn't think, in the moment, while I was already in a different context of having the beam, a load-bearing beam. It was very interesting that—and I know I'm an ELL, so English is not my first language—in thinking about a context that you're familiar with by reading a word or this term, “balance beam.” And even if people tell you, “Oh, it's related to gymnastics”—and I've never done gymnastics; I never had gymnastics in my class or in my school where I was. It didn't help. And that's where we started talking about underlying keywords that didn't really help either because it was a coordinate plane problem. So, I had to reach out to Julia and say, “I think there's something going on here that is related to reading comprehension. Can you help me?” And that's how this all started. ( chuckles ) Julia: Well, so Aina came to me regarding her experience. In fact, she sent me the math problem. She says, “Look at this.” And we talked about that. And then she shared frustration of the educators that she had been working with that despite teaching strategies that are promoted as part of instructional practice, like identifying mathematical keywords and then also reading strategies have been emphasized, like summarizing or asking questions while you're reading story problems. So, her teachers had been using strategies, mathematical and also reading, and their students were still struggling to make sense of and solve mathematical problems. Aina's experience with this word problem really opened up this thought about the words that are in mathematical story problems. And we came to realize that when we think about making sense of story problems, there are a lot of words that require schema. And schema is the background knowledge that we bring to the text that we interact with.  Julia: For example, I taught for years in Florida. And we would have students that had never experienced snow. So, as an educator, I would need to do read alouds and provide that schema for my students so that they had some understanding of snow. So, when we think about math story problems, all words matter—not just the mathematical terms, but also the words that require schema. And then when we think about English learners, the implications are especially profound because we know that, that vocabulary is one of the biggest challenges for English learners. So, when we consider schema-mediated vocabulary and story problems, this really becomes problematic. And so, Aina and I analyzed the story problems in the curriculum that Aina's teachers were using, and we had an amazing discovery.  Aina: Just the range of contexts that we came across from construction materials or nuts and bolts and MP3 players—that children don't really have anymore, a lot of them have a phone—to making smoothies and blenders, which some households may not have. In addition to that, we started looking at the words that are in the story problems. And like Julia said, there are actually mathematics teachers who are being trained on these strategies that come from literacy research. One of them was rereading the problem. And it didn't matter how many times I reread the problem or somebody reread it to me about the balance beam. I had no kind of understanding of what's going on in the problem. The second one is summarizing. And again, just because you summarized something that I don't understand or read it louder to me, it doesn't help, right? And I think the fundamental difference that we solve problems or the story problems … In the literacy, the purpose of reading a story is very different. In mathematics, the purpose of reading a story is to solve it, making sense of problems for the purpose of solving them. The three different categories of vocabulary we found from reading story problems and analyzing them is there's “technical,” there's “sub-technical” and there's “non-technical.” I was very good at recognizing technical words because that's the strategy that for mathematics teachers, we underline the parallelogram, we underline the integer, we underline the eight or the square root, even some of the keywords we teach, right? Total means some or more means addition.  Mike: So technical, they're the language that we would kind of normally associate with the mathematics  that are being addressed in the problem. Let's talk about sub-technical because I remember from our pre-podcast conversation, this is where some light bulbs really started to go off, and you all started to really think about the impact of sub-technical language.  Julia: Sub-technical includes words that have multiple meanings that intersect mathematically and other contexts. So, for example, “yard.” Yard can be a unit of measurement. However, I have a patio in my backyard. So, it's those words that have that duality. And then when we put that in the context of making sense of a story problem, it's understanding what is the context for that word and which meaning applies to that? Other examples of sub-technical would be table or volume. And so, it's important when making sense of a story problem to understand which meaning is being applied here. And then we have non-technical, which is words that are used in everyday language that are necessary for making sense of or solving problems. So, for example, “more.” More is more. So, more has that mathematical implication. However, it would be considered non-technical because it doesn't have dual meanings.  Julia: So, by categorizing vocabulary into these three different types, [that] helped us to be able to analyze the word problems. So, we worked together to categorize. And then Aina was really helpful in understanding which words were integral to solving those math problems. And what we discovered is that often words that made the difference in the mathematical process were falling within the sub-technical and non-technical. And that was really eye-opening for us. Mike: So, Aina, this is fascinating to me. And what I'm thinking about right now is the story that you told at the very beginning of this podcast, where you described your own experience with the word problem that contained the language “balance.” And I'm wondering if you applied the analysis that you all just described with technical and sub-technical and the non-technical, when you view your own experience with that story problem through that lens, what jumps out? What was happening for you that aligns or doesn't align with your analysis?  Aina: I think one of the things that was eye-opening to me is, we have been doing it wrong. That's how I felt. And the teachers felt the same way. They're saying, “Well, we always underline the math words because we assume those are the words that are confusing to them. And then we underline the words that would help them solve the problem.” So, it was a very good conversation with teachers to really, completely think about story problems differently. It's all about the context; it's all about the schema. And my teachers realize that I, as an adult who engages in mathematics regularly, have this issue with schema. I don't understand the context of the problem, so therefore I cannot move forward in solving it. And we started looking at math problems very differently from the language perspective, from the schema perspective, from the context perspective, rather than from underlining the technical and mathematical words first. That was very eye-opening to me.   Mike: How do you think their process or their perspective on the problems changed either when they were preparing to teach them or in the process of working with children?  Aina: I know the teachers reread a problem out loud and then typically ask for a volunteer to read the problem. And it was very interesting; some of the conversations were how different the reading is. When the teacher reads the problem, there is where you put the emotion, where the certain specific things in the problem are. Prosody?  Julia: Yes, prosody is reading with appropriate expression, intonation, phrasing.  Aina: So, when the teacher reads the problem, the prosody is present in that reading. When the child is reading the problems, it's very interesting how it sounds. It just sounds the word and the next word and the next word and the next word, right? So that was kind of a discussion, too. The next strategy the math teachers are being taught is summarizing. I guess discussing the problem and then summarizing the problem. So, we kind of went through that. And once they helped me to understand in gymnastics what it is, looking up the picture, what it looks like, how long it is, and where it typically is located and there's a mat next to it, that was very helpful. And then I could then summarize, or they could summarize, the problem. But even [the] summarizing piece is now me interpreting it and telling you how I understand the context and the mathematics in the problem by doing the summary. So, even that process is very different. And the teacher said that's very different. We never really experience that.  Mike: Julia, do you want to jump in?  Julia: And another area where math and reading intersect is the use of visualization. So, visualization is a reading strategy, and I've noticed that visualization has become a really strong strategy to teach for mathematics, as well. We encourage students to draw pictures as part of that solving process. However, if we go back to the gymnastics example, visualizing and drawing is not going to be helpful for that problem because you are needing a schema to be able to understand how a balance beam would situate within that context and whether that's relevant to solving that word problem. So, even though we are encouraging educators to use these strategies, when we think about schema-mediated vocabulary, we need to take that a step further to consider how schema comes into play and who has access to the schema needed, and who needs that additional support to be able to negotiate that schema-mediated vocabulary.  Mike: I was thinking the same thing, how we often take for granted that everyone has the same schema. The picture I see in my head when we talk about balance is the same as the picture you see in your head around balance. And that's the part where, when I think about some of those sub-technical words, we really have to kind of take a step back and say, “Is there the opportunity here for someone to be profoundly confused because their schema is different than mine?” And I keep thinking about that lived experience that you had where, in my head I can see a balance beam, but in your head you're seeing the structural beam that sits on the top of your ceiling or runs across the top of your ceiling.  Aina: Oh yeah. And at first, I thought the word “beam” typically, in my mind for some reason, is vertical.  Mike: Yeah. Aina: It's not horizontal. And then when I looked at the word balance, I thought, “Well, it could balance vertically.” And immediately what I think about is, you have a porch, then you see a lot of porches that balance the roof, and so they have the two beams …  Mike: Yes! Aina: … or sometimes more than that. So, at no point did I think about gymnastics. But that's because of my lack of experience in gymnastics, and my school didn't have the program. As a math person, you start thinking about it and you think, “If it's vertically, this doesn't make any sense because we're on a coordinate plane.” So, I started thinking about [it] mathematically and then I thought, “Oh, maybe they did renovations to the gymnasium, and they needed a balance beam.” So, I guess that's the beam that carries the load.  Aina: So, that's how I flipped, in my mind, the image of the beam to be horizontal. Then the teachers, when they told me it's gymnastics, that really threw me off, and it didn't help. And I totally agree with Julia. You know when we do mathematics with children, we tell them, “Can you draw me a picture?” Mike: Uh-hm. Aina: And what we mean is, “Can you draw me a mathematical picture to support your problem-solving or the strategies you used?” But the piece that was missing for me is an actual picture of what the balance beam is in gymnastics and how it's located, how long it is. So yeah, yeah, that was eye-opening to me.  Mike: It's almost like you put on a different pair of glasses that allow you to see the language of story problems differently, and how that was starting to play out with teachers. I wonder, could you talk about some of the things that they started to do when they were actually with kids in the moment that you looked at and you were like, “Gosh, this is actually accounting for some of the understanding we have about schema and the different types of words.”  Aina: So, the teacher would read a problem, which I think is a good strategy. But then it was very open-ended. “How do you understand what I just read to you? What's going on in the story problem? Turn to your partner, can you envision? Can you think of it? Do you have a picture in your mind?” So, we don't jump into mathematics anymore. We kind of talk about the context, the schema. “Can you position yourself in it? Do you understand what's going on? Can you retell the story to your partner the way you understand it?” And then, we talk about, “So how can we solve this problem? What do you think is happening?” based on their understanding. That really helped, I think, a lot of teachers also to see that sometimes interpretations lead to different solutions, and children pay attention to certain words that may take them to a different mathematical solution. It became really about how language affects our thinking, our schema, our image in the head, and then based on all of that, where do we go mathematically in terms of solving the problem?  Mike: So, there are two pieces that really stuck out for me in what you said. I want to come back to both of them. The first one was, you were describing that set of choices that teachers made about being really open-ended about asking kids, “How do you understand this? Talk to your neighbor about your understanding about this.” And it strikes me that the point you made earlier when you said context has really become an important part of some of the mathematics tasks and the problems we create. This is a strategy that has value not solely for multilingual learners, but really for all learners because context and schema matter a lot.  Aina: Yes. Mike: Yeah. And I think the other thing that really hits me, Aina, is when you said, “We don't immediately go to the mathematics, we actually try to help kids situate and make sense of the problem.” There's something about that that seems really obvious. When I think back to my own practice as a teacher, I often wonder how I was trying to quickly get kids into the mathematics without giving kids enough time to really make meaning of the situation or the context that we were going to delve into. Aina: Exactly. Mike, to go back to your question, what teachers can do, because it was such an eye-opening experience that, it's really about the language; don't jump into mathematics. The mathematics and the problem actually is situated around the schema, around the context. And so, children have to understand that first before they get into math. I have a couple of examples if you don't mind, just to kind of help the teachers who are listening to this podcast to have an idea of what we're talking about. One of the things that Julie and I were thinking about is, when you start with a story problem, you have three different categories of vocabulary. You have technical, sub-technical, non-technical. If you have a story problem, how do you parse it apart? OK, in the math story problems we teach to children, it's typically a number and operations.  Aina: Let's say we have a story problem like this: “Mrs. Tatum needs to share 3 grams of glitter equally among 8 art students. How many grams of glitter will each student get?” So, if the teacher is looking at this, technical would definitely be grams: 3, 8, and that is it. Sub-technical, we said “equally,” because equally has that kind of meaning here. It's very precise, it has to be exact amount. But a lot of children sometimes say, “Well that's equally interesting.” That means it's similarly or kind of, or like, but not exact. So, sub-technical might qualify as “equally.” Everything else in the story problem is non-technical: sharing and glitter, art students, each student, how much they would get. I want the teachers to go through and ask a few questions here that we have. So, for example, the teacher can think about starting with sub-technical and non-technical, right?  Aina: Do students understand the meaning of each of these words? Which of these could be confusing to them? And get them to think about the story, the context and the problem. And then see if they understand what the grams are, and 8 and 3. And what's happening. And what do those words mean in this context? Once you have done all this work with children, children are now in this context. They have situated themselves in this. “Oh, there's glitter, there's an art class, there's a teacher, they're going to do a project.” And so, they've discussed this context. Stay with it as a teacher and give them another problem that is the same context. Use as many words from the first problem as you can and change it up a little bit in terms of mathematical implication or mathematical solution. For example, I can change the same problem to be, “Mrs. Tatum needs to buy 3 grams of glitter for each of her 8 art students. How many grams of glitter does she need to buy?” So, the first problem was [a] division problem now becomes a multiplication problem. The context is the same. Children understand the context, especially children like myself, who are ELL, who took the time to process to learn new words, to understand new context, and now they're in this context. Let's use it. Let's now use it for the second piece. So, Mike, you've been talking about two things going on. There's a context, and then there's problem-solving or mathematical problem-solving. So, I believe posing the same question or kind of the same story problem with different mathematical implications gets at the second piece. So, first we make sense of the problem of the context schema. The second is, we make sense of that problem for the purpose of solving it.  Aina: And the purpose of solving it is where these two problems that sound so familiar and situate in the same context but have different mathematical implications for problem-solving. This is where the powerful piece, I think, is missing. If I give them a division problem, they can create a multiplication problem with the same Mrs. Tatum, the art students, the glitter. But what I'd like for them to do and what we've been discussing is how are these two problems similar?  Mike: Uh-hm. Aina: This kind of gets at children identifying some of the technical. So, the 3 is still there, the 8 is still there, you know, grams are still there. But then, “How are these two story problems different?” This is really schema-mediated vocabulary in the context where they now have to get into sub-technical and non-technical. “Oh, well there there's 3, but it's 3 per student. And this, there were 8 students, and they have to all share the 3 grams of glitter.”  Aina: So, children now get into this context and difference in context and how this is impacting the problem-solving strategies. I'd love for the teachers to then build on that and say, “How would you solve the first problem? What specifically is in the story problem [to] help you solve it, help you decide how to solve it, what strategies, what operations?” And do exactly the same thing for the second problem as well. “Would you solve it the same way? Are the two problems the same? Will they have the same solutions or different? How would you know? What tells you in the story? What helps you decide?” So, that really helps children to now become problem-solvers. The fun is the mathematical variations. So, for example, we can give them a third problem and say, “I have a challenge for you.” For example, “Mrs. Tatum needs to buy 3 grams of glitter for each of her 8 art students for a project, but she only has money today to buy 8 grams of glitter. How much more glitter does she need to still buy for her students to be able to complete their art project?” Again, it's art, it's glitter, it's 8 students, there's 3, the 8. I didn't change the numbers, I didn't change the context, but I did change the mathematical implication for their story problem. I think this is where Julia and I got very excited with how we can use schema-mediated vocabulary and schema in context to help children understand the story, but then really have mathematical discussions about solutions.  Mike: What's interesting about what you're saying is the practices that you all are advocating and describing in the podcast, to me, they strike me as good practice helping kids make meaning and understand and not jumping into the mathematics and recognizing how important that is. That feels like good practice, and it feels particularly important in light of what you're saying.  Julia: I agree. It's good practice. However, what we found when we reviewed literature, because one of the first steps that we took was what does the literature say? We found that focusing instructional practice on teaching children to look for key mathematical terms tends to lead to frequent errors. Mike: Yep. Julia: The mathematical vocabulary tends to be privileged when teaching children how to make sense of and solve word problems. We want to draw attention to the sub-technical and non-technical vocabulary, which we found to be influential in making sense of. And as in the examples Aina shared, it was the non-technical words that were the key players, if you will, in solving that problem.  Mike: I'm really glad you brought up that particular point about the challenges that come out of attempting to help kids mark certain keywords and their meanings. Because certainly, as a person who's worked in kindergarten, first grade, second grade, I have absolutely seen that happen. There was a point where I was doing that, and I thought I was doing something that was supporting kids, and I was consistently surprised that it was often like, that doesn't seem to be helping.  Julia: I also used that practice when I was teaching second grade. The first step was circle the keywords. And I would get frustrated because students would still be confused in the research that we found. When you focus on the keywords, which tend to be mathematical terms, then those other words that are integral to making sense of and solving the story problem get left behind. Mike: The question I wanted to ask both of you before we close is, are there practices that you would say like, “Here's a way that you can take this up in your classroom tomorrow and start to take steps that are supportive of children making sense of word problems”?  Julia: I think the first step is adding in that additional lens. So, when previewing story problems, consider what schema or background knowledge is required to understand this word, these words, and then what students would find additional schema helpful. So, thinking about your specific students, what students would benefit from additional schema and how can I support that schema construction?  Mike: Aina, how about for you?  Aina: Yeah, I have to say I agree with Julia. Schema seemed to be everything. If children don't understand the context and don't make sense of the problem, it's very hard to actually think about solving it. To build on that first step, I don't want teachers to stop there. I want teachers to then go one step further. Present a similar problem or problem that includes [the] same language, same words, as many as you can, maybe even same numbers, definitely same schema and context, but has a different mathematical implication for solving it. So maybe now it's a multiplication problem or addition problem. And really have children talk about how different or similar the problems are. What are the similarities, what are the differences, how their solutions are the same or different? Why that is. So really unpack that mathematical problem-solving piece. Now, after you have made sense of the context and the schema … as an ELL student myself, the more I talked as a child and was able to speak to others and explain my thinking and describe how I understand certain things and be able to ask questions, that was really, really helpful in learning English and then being successful with solving mathematical problems. I think it really opens up so many avenues and to just go beyond helping teachers teach mathematics.  Mike: I know you all have created a resource to help educators make sense of this. Can you talk about it, Julia?   Julia: Absolutely. Aina and I have created a PDF to explain and provide some background knowledge regarding the three types of vocabulary. And Aina has created some story problem examples that help to demonstrate the ways in which sub-technical and non-technical words can influence the mathematical process that's needed. So, this resource will be available for educators wanting to learn more about schema-mediated vocabulary in mathematical story problems.   Mike: That's fantastic. And for listeners, we're going to add this directly to our show notes. I think that's a great place for us to stop. Aina and Julia, I want to thank both of you so much for joining us. It has absolutely been a pleasure talking to both of you.  Aina: Thank you.  Julia: Thank you. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 14 – Three Resources to Support Multilingual Learners Guest: Dr. Erin Turner Mike Wallus: Many resources for supporting multilingual learners are included with curriculum materials. What's too often missing though is clear guidance for how to use them. In this episode, we're going to talk with Dr. Erin Turner about three resources that are often recommended for supporting multilingual learners. We'll unpack the purpose for each resource and offer a vision for how to put them to good use with your students.  Mike: Well, welcome to the podcast, Erin. We are excited to be chatting with you today.  Erin Turner: Thank you so much for inviting me.  Mike: So, for our listeners, the starting point for this episode was a conversation that you and I had not too long ago, and we were talking about the difference between having a set of resources which might come with a curriculum and having a sense of how to use them. And in this case, we were talking about resources designed to support multilingual learners. So, today we're going to talk through three resources that are often recommended for supporting multilingual learners, and we're going to really dig in and try to unpack the purpose and offer a vision for how to put them to use with students. What do you think? Are you ready to get started, Erin?  Erin: I am.  Mike: Well, one of the resources that often shows up in curriculum are what are often referred to as sentence frames or sentence stems. So, let's start by talking about what these resources are and what purpose they might serve for multilingual learners.  Erin: Great. So, a sentence stem, or sometimes it's called a sentence starter, this is a phrase that gives students a starting place for an explanation. So, often it includes three or four words that are the beginning part of a sentence, and it's followed by a blank that students can complete with their own ideas. And a sentence frame is really similar. A sentence frame just typically is a complete sentence that includes one or more blanks that again, students can fill in with their ideas. And in both cases, these resources are most effective for all students who are working on explaining their ideas, when they're flexible and open-ended. So, you always want to ensure that a sentence stem or a sentence frame has multiple possible ways that students could insert their own ideas, their own phrasing, their own solutions to complete the sentence. The goal is always for the sentence frame to be generative and to support students' production and use of language—and never to be constraining.  Erin: So, students shouldn't feel like there's one word or one answer or one correct or even intended way to complete the frame. It should always feel more open-ended and flexible and generative. For multilingual learners, one of the goals of sentence stems is that the tool puts into place for students some of the grammatical and linguistic structures that can get them started in their talk so that students don't have to worry so much about, “What do I say first?” or “What grammatical structures should I use?” And they can focus more on the content of the idea that they want to communicate. So, the sentence starter is just getting the child talking. It gives them the first three words that they can use to start explaining their idea, and then they can finish using their own insights, their own strategies, their own retellings of a solution, for example.  Mike: Can you share an example of a sentence frame or a sentence stem to help people understand them if this is new to folks?  Erin: Absolutely. So, let's say that we're doing number talks with young children, and in this particular number talk, children are adding two-digit numbers. And so, they're describing the different strategies that they might use to do either a mental math addition of two-digit numbers, or perhaps they've done a strategy on paper. You might think about the potential strategies that students would want to explain and think about sentence frames that would mirror or support the language that children might use. So, a frame that includes blanks might be something like, “I broke apart (blank) into (blank) and (blank).” If you think students are using 10s and 1s strategies, where they're decomposing numbers into 10s and 1s. Or if you think students might be working with open number lines and making jumps, you might offer a frame like, “I started at (blank), then I (blank),” which is a really flexible frame and could allow children to describe ways that they counted on on a number line or made jumps of a particular increment or something else. The idea again is for the sentence frame to be as flexible as possible. You can even have more flexible frames that imply a sequence of steps but don't necessarily frame a specific strategy. So, something like, “First I (blank), then I (blank)” or “I got my answer by (blank).” Those can be frames that children can use for all different kinds of operations or work with tools or representations.  Mike: OK, that sets up my next question. What I think is interesting about what you shared is there might be some created sentence frames or sentence stems that show up with the curricular materials I have, but as an educator, I could actually create my own sentence frames or sentence stems that align with either the strategies that my kids are investigating or would support some of the ideas that I'm trying to draw out in the work that we're doing. Am I making sense of that correctly?  Erin: Absolutely. So many curricula do include sample sentence frames, and they may support your students. But you can always create your own. And one place that I really like to start is by listening to the language that children are already using in the classroom because you want the sentence starters or the sentence frames to feel familiar to students. And by that, I mean you want them to be able to see their own ideas populating the sentence frames so that they can own the language and start to take it up as part of the repertoire of how they speak and communicate their ideas. So, if you have a practice in your classroom, for example, where children share ideas and maybe on chart paper or on the whiteboard you note down phrases from their explanations—perhaps labeled with their name so that we can keep track of who's sharing which idea—you could look across those notations and just start to notice the language that children are already using to explain their strategies and take that as a starting point for the sentence frames that you create. And that really honors children's contributions. It honors their natural ways of talking, and it makes it more likely that children will take up the frames as a tool or a resource.  Mike: Again, I just want to say, I'm so glad you mentioned this. In my mind, a sentence frame or a sentence stem was a tool that came to me with my curriculum materials, and I don't know that I understood that I have agency and that I could listen to kids' thinking and use that to help design my own sentence frames. One question that comes to mind is, do you have any guardrails or cautions in terms of creating them that would either support kids' language or that could inadvertently make it more challenging?  Erin: So, I'll start with some cautions. One way that I really like to think about sentence frames is that they are resources that we offer children, and I'm using “offer” here really strategically. They're designed to support children's use of language. And when they're not supportive, when children feel like it's harder to use the frame to explain their idea because the way they want to communicate something, the way they want to phrase something doesn't fit into the frame that we've offered, then it's not a useful support. And then it can become a frustrating experience for the child as the child's trying to morph or shape their ideas, which makes sense to them, into a structure that may not make sense. And so, I really think we want to take this idea of offering and not requiring frames really seriously.  Erin: The other caution that I would offer is that frames are not overly complex. And by that, I mean if we start to construct frames with multiple blanks where it becomes more about trying to figure out the teacher's intention and children are thinking, “What word would I put here? What should I insert into this blank?” Then we've lost the purpose. The purpose is to support generative language and to help children communicate their ideas, not to play guessing games with children where they're trying to figure out what we intend for them to fill in. This isn't necessarily a caution, but maybe just a strategy for thinking about whether or not sentence frames could be productive for students in your classroom— particularly for multilingual learners—is to think about multiple ways that they might complete the sentence stem or that they might fill in the sentence frame. And if as a teacher we can't readily come up with four or five different ways that they could populate that frame, chances are it's too constraining and it's not open-ended enough.  Erin: And you might want to take a step back toward a more open-ended or flexible frame. Because you want it to be something that the children can readily complete in varied ways using a range of ideas or strategies. So, something that I think can be really powerful about sentence frames is the way that they position students. For example, when we offer frames like, “I discovered that…” or “I knew my answer was reasonable because…” or “A connection I can make is… .” Those are all sentence starters. The language in those sentence starters communicates something really powerful to multilingual learners and to any student in our classroom. And that's that we assume as a teacher that they're capable of making connections, that they're capable of deciding for themselves if their answer is reasonable, that they're capable of making discoveries. So, the verbs we choose in our sentence frames are really important because of how they position children as competent, as mathematical thinkers, as people with mathematical agency.  Erin: So, sometimes we want to be really purposeful in the language that we choose because of the way that it positions students. Another kind of positioning to think about is that multilingual learners may have questions about things in math class. They may not have clarity about the meaning of a phrase or the meaning of a concept, and that's really true of all students. But we can use sentence frames to normalize those moments of uncertainty or struggle for students. So, at the end of a number talk or at the end of a strategy sharing session, we can offer a sentence frame like, “I had a question about…” or “Something I'm still not sure of is … .” And we can invite children to turn and talk to a partner and to finish that sentence frame. Erin: That's offering students language to talk about things that they might have questions about, that they might be uncertain about. And it's communicating to all kids that that's an important part of mathematics learning—that everyone has questions. It's not just particular students in the classroom. Everyone has moments of uncertainty. And so, I think it's really important that when we offer these frames to students in our classrooms, they're not positioned as something that some students might need, but they're positioned as tools and resources that all students can benefit from. We all can benefit from an example of a reflection. We all can learn new ways to talk about our ideas. We all can learn new ways to talk about our confusions, and that's not limited to the children that are learning the language of instruction. Otherwise, sentence frames become something that has low status in the classroom or is associated with students [who] might need extra help. And they aren't taken up by children if they're positioned in that way, at least not as effectively.  Mike: The comparison that comes to mind is the ways that in the past manipulatives have been positioned as something that's lower status, right? If you're using them, it means something. Typically, at least in the past, it was something not good. Whereas I hope as a field we've gotten to the place where we think about manipulatives as a tool for kids to help express their thinking and understand and make meaning, and that we're communicating that in our classrooms as well. So, I'm wondering if you can spend just a few minutes, Erin, talking about how an educator might introduce sentence frames or sentence stems and perhaps a little bit about the types of routines that keep them alive in the classroom.  Erin: Yes, thanks for this question. One thing that I found to be really flexible is to start with open-ended sentence frames or sentence stems that can be useful as an attachment or as an enhancement to a routine that children already know. So, just as an example, many teachers use an “I notice, I wonder”- or “We know, we wonder”-type of routine. Those naturally lend themselves to sentence starters. “I notice (blank), I wonder (blank).” Similarly, teachers may be already using a same and different routine in their classroom. You can add or layer a sentence frame onto that routine, and then that frame becomes a tool that can support students' communication in that routine. So, “These are the same because… .” “These are different because… .” And once students are comfortable and they're using sentence frames in those sorts of familiar routines, a next step can be introducing sentence frames that allow children to explain their own thinking or their own strategies.  Erin: And so, we can introduce sentence frames that map onto the strategies that children might use in number talks. We can introduce sentence frames that can support communication around problem-solving strategies. And those can be either really open-ended like, “First…, then I…”-type frames or frames that sort of reflect or represent particular strategies. In every case, it's really important that the teacher introduces the frame or the sentence starter in a whole group. And this can be done in a couple of ways. You can [chorally] read the frame so that all children have a chance to hear what it sounds like to say that frame. And as a teacher, you can model using the frame to describe a particular idea. One thing that I've seen teachers do really effectively is when children are sharing their strategy, teachers often revoice or restate children's strategies sometimes just to amplify it for the rest of the class or to clarify a particular idea.  Erin: As part of that revoicing, as teachers we can model using a sentence frame to describe the idea. So, we could say something like, “Oh, Julio just told us that he decomposed (blank) into two 10s and three 1s,” and we can reference the sentence frame on the board or in another visible place in the classroom so that children are connecting that mathematical idea to potential language that might help them communicate that idea. And that may or may not benefit Julio, the child [who] just shared. But it can benefit other children in the classroom [who] might have solved the problem or have thought about the problem in a similar way but may not yet be connecting their strategy with possible language to describe their strategy. So, by modeling those connections as a teacher, we can help children see how their own ideas might fit into some of these sentence frames. Erin: We also can pose sentence frames as tool to practice in a partner conversation. So, for example, if children are turning and talking during a number talk and they're sharing their strategy, we can invite children to practice using one of two sentence frames to explain their ideas to a partner. And after that turn and talk moment, we can have a couple of children in the class volunteer their possible ways to complete the sentence frame for the whole group. So, it just gives us examples of what a sentence frame might sound like in relation to an authentic activity. In this case, explaining our thinking about a number talk. And that sort of partner practice or partner rehearsal is really, really important because it gives children the chance to try out a new frame or a new sentence starter in a really low-stress context, just sharing their idea with one other peer, before they might try that out in a whole-class discussion.  Mike: That's really helpful, Erin. I think one of the things that jumps out for me is, when you initially started talking about this, you talked about attaching it to a routine that kids already have a sense of like, “I notice” or “I wonder” or “What's the same?” or “What's different?” And what strikes me is that those are routines that all kids participate in. So again, we're not positioning the resource or the tool of the sentence frame or the sentence starter as only for a particular group of children. They actually benefit all kids. It's positioned as a normal practice that makes sense for everybody to take up.  Erin: Absolutely. And I think we need to position them as ways to enhance things in classrooms for all students. And partner talk is another good example. We often send students off to talk with a partner and give them instructions like, “Go tell your partner how you solve the problem.” And many children aren't quite sure what that conversation looks like or sounds like, even children for whom English is their first language. And so, when we offer sentence frames to guide those interactions, we're offering a support or a potential support for all students. So, for partner talk, we often not only ask kids to explain their thinking, but we say things like, “Oh, and ask your partner questions.” “Find out more about your partner's ideas.” And that can be challenging for 7- and 8-year-olds. So, if we offer sentence frames that are in the form of questions, we can help scaffold those conversations.  Erin: So, things like, “Can you say more about?” or “I have a question about?” or “How did you know to?” If we want children asking each other questions, we need to often offer them supports or give them tools to support that conversation. And that helps them to learn from each other. It helps them to listen to each other, which we know benefits them in multiple ways. And I just want to share one final example about sentence frames that I think is so powerful. There are really different purposes for frames. They can be about reflection. They can be about asking questions of partners. We can use sentence frames to agree and disagree, to compare and contrast. Erin: One teacher that I've worked with uses sentence frames to guide end-of-lesson reflections. And after children have talked to a partner or shared ideas with a partner, she asked them to complete sentence frames that sound like this: “One thing I learned from my partner today is (blank).” Or “A new idea I got from my partner today was (blank).” And what I love about this is, it positions all kids as having valuable ideas, valuable contributions to offer the class. And if I'm in a partnership with a multilingual learner, I'm thinking deeply about what I learned from that partner, and I'm sharing with the teacher orally or in writing and sharing with the class what I learned from that child. And so, the sentence frame helps me because it gives me a support to think about that idea and to express that idea, but it really helps elevate other children in the classroom who might not always be seen in that way by their peers. So, I think there are just really powerful ways that, that we can use these tools.  Mike: I love that. I'm wondering if we can shift now to a different kind of resource, and this one might be a little less obvious. What we were talking about when we had this conversation earlier was the use of a repeated context across a series of lessons and the extent to which that, in itself, can actually be really supportive of multilingual learners. So, I'm wondering if we can talk about an example and share the ways that this might offer support to students?  Erin: Perfect. So, repeated contexts are wonderful because they offer a rich, really complex space for students to start thinking mathematically, for them to pose questions of their own, and for them to make mathematical observations and solve problems. And the benefit of a repeated context or a context that sort of returns over a sequence of lessons or even across a sequence of units, is that children can start to inhabit the story in the context. They start to learn who the characters are. They learn about the important features of the context, perhaps locations or objects in the context or relationships or key quantities. And every time that that context is reintroduced, the sense-making that they've done previously is a really powerful starting point for the new mathematical ideas that they can explore. And these repeated contexts are especially powerful when they're introduced with multiple supports. So, for multilingual learners, if we can introduce context with narrative stories, with pictures or images, with videos, with physical artifacts, whatever we can do to give children a sense of this, in most cases, imaginary worlds that we're creating, we support their sense-making.  Erin: And this is really different from curricula or programs that offer a new context with each word problem. Or perhaps with each page in a student book, there's a new context introduced. And for multilingual learners and really for all students, every time we introduce a new context, we have to make sense of what's happening in this story. What's happening in the situation? Who are the people? What does this new word mean that I haven't encountered before? And so, we limit our time to really think deeply about the mathematical ideas because we're repeating this space of sense-making around the context. And in classrooms we often don't have that time to unpack context. And so, what happens when we use new contexts every time is that we tend to fast track the sense-making, and children can start to develop all sorts of unproductive ways to dig into problems like looking for a particular word that they think means a particular operation because we just don't give them the time and space to really make sense of the story. And so, because we have this limited time in classrooms, when we can reintroduce context, it really offers that space to students.  Mike: Do you have an example that might help illustrate the point?  Erin: Absolutely. So, in second-grade curriculum that I have reviewed, there's a context around a character, “Jesse and [the] Beanstalk.” It's sort of an adaptation of the classic tale of “Jack and the Beanstalk.” And in this story, Jesse has beans that he gets from an interaction or a sale in a farmers market. And these beans, of course, grow into a giant beanstalk that has a friendly giant that lives at the top. And this beanstalk produces large, giant beans, which have all kinds of seeds inside of them. And this context is used over a series of units. It actually spans most of the school year to give children an opportunity to explore multiple mathematical ideas. So, they make representations of these giant beans with strips of paper, and they use cubes to measure the beans. So, they're looking at linear measurement concepts. They compare the length of different beans, so they're doing addition and subtraction to compare quantities. They find out how many seeds are inside of the beans, and they add those quantities together. So, they're doing all sorts of multi-digit operations, adding the beans. Erin: And then the context further develops into making bracelets with the seeds that are inside these bean pods, and they group these seeds in groups of 10. So, they have the chance to think about, “How many 10s can we make out of a larger quantity?” Later on, their bracelet-making business expands, and they have to think about how to package these seeds into 100s, 10s, and ones. So, it's a really rich context that develops over time. And children begin to learn about the people in the story, about the activities and the practices that they engage in, and they have the chance to ask their own questions about their story and to make their own connections, which is really powerful. As the story develops, you can see how children develop a sense of curiosity about what's happening, and they become invested in these stories, which really supports the mathematical work.  Mike: So, I want to walk back to our friend Jesse, and I'm glad to hear it's a friendly giant in this particular case. What you were making me think about as you were talking is the way that we introduce the context probably is really important. Could you shed some light on how you think about introducing a context?  Erin: So, asking children to share connections that they can make is really important. When we introduce context with different representations, it's really important to ask children to make connections as a place to start. So, we want to ask them what they already know about this context in particular, or similar context. What connections can they make to their own experiences? We want to ask them to share what they wonder about the context, what they're curious about, what they notice, what observations they can make. And when we have different representations like a story and a picture or a video or an artifact, we give children more possibilities for making those kinds of connections. One thing that we can also do to really support children's connections to the context is, as a context develops over time, we can create anchor charts or other written records with children that represent their perspectives on the key features of the context.  Erin: So, for example, if we go back to “Jesse and the Beanstalk,” after solving a couple of problems about “Jesse and the Beanstalk” and being introduced to that story, we can pause and talk with children about what they see as key aspects of this story. What are things they want to remember? When we come back to Jesse in a few weeks, who are the people [who] we want to remember in this story? What are some important quantities in this story? What are some other important features of this story? And this is not an anchor chart that we create ahead of time as teachers. It's really important that children own these ideas and that they get to start to identify the key quantities, the key features of the situation from their perspective, because then that can become a resource for their thinking later on. We don't have to re-explain the context completely every time. We can refer to these written records that we've co-created with children.  Mike: Well, let's close by talking about one more resource that educators will often find in their curriculum materials. Things like lists of academic vocabulary, or perhaps even cards with vocabulary words printed on them. I wonder how you think educators should understand the value of these particular resources.  Erin: These vocabulary cards can take the form of cards that can be inserted into a chart or even anchor charts themselves. And one thing that I think that's really important, especially when we're thinking about using this tool with multilingual learners, is that these include multiple representations of a concept. We always need to make sure that the cards include a picture or a diagram or a visual image of the term, in addition to an example of how the term can be used. So, that might be a phrase, it might be a symbolic representation of the term. It might be a whole sentence that uses the word to give children an idea of how to use the language in context, which is really important. And one thing that I've seen teachers do really effectively is to create large vocabulary cards with blank space, so that as these cards are introduced in the context of a lesson or activity when they would be relevant, children have the opportunity to share their own ideas about the term.  Erin: And that blank space on the card can be filled with connections that children make. So, children might know that term in another language. That can be added to the vocabulary card. Children might connect that term to another similar idea mathematically or a similar idea in daily life. So, they might know another meaning of the word. That can be added to this blank space so that it becomes a shared and collaboratively generated artifact and not just a static card on the wall of the classroom that is beautiful, but that children may not really use to support their sense-making. So, co-creating these cards with children I've seen to be really powerful, especially if we want them to be used by children and owned by children. And that leaving blank space can help with that.  Mike: So, you're taking this conversation to a place I hoped we might go, which is just to help paint a picture of what it might look like for a teacher to introduce this resource, but then also sustain it, how to bring it to life in the classroom. What does that look like? Or maybe what does that sound like, Erin?  Erin: So, I think when vocabulary card is first introduced, just like with many things in math classrooms, we want children to share what they already know. So, what does this word make you think of? Where do you see this word? Where have you heard this word? What are some other things we've done together in our mathematical work that relate to this word? You want children to share versions of the word in other languages. You want them to share real-world context, connections, anything that they can to connect to their experiences. And it's important that we introduce small sets of words at a time. So, if we're working on a unit on multiplication, we might have words related to “factor” and “multiple” and “products” that become additions to our word wall or to our anchor charts. And that we encourage children to use those words in particular activities in those units.  Erin: So, for example, if we're doing a number talk as part of a unit on multiplication, we might remind children of particular words that have been introduced in prior lessons and encourage them to try to take up those words in their explanations. “See if you can use the word ‘factor' today as you're sharing your strategy with a partner.” “See if you can use the word ‘product' today.” And then invite children to share examples of what that sounded like in their partner talk. Or what that looked like when they were writing about their explanation. And it's these constant invitations or repeated invitations that really make these words come to life in classrooms so that they don't just live on the wall. It's also really important that these words are highly visible and accessible for students. So, oftentimes teachers will display key vocabulary words alongside a whiteboard or underneath a whiteboard. Or children might have their own copy of a small set of keywords that they're working on to paste inside of their notebooks. If they're not highly accessible, it really limits children's opportunities to use them on a regular basis.  Erin: So, another way to introduce new vocabulary terms or to support students to use them in context is to connect specific words to a routine that is already in place in your classroom. So, one of my favorites for its potential to help kids use new vocabulary is a routine like, “Which One Doesn't Belong?” But same and different routines that many teachers use also work great for this reason. So, just as an example, if we're doing a “Which One Doesn't Belong?” routine, and the four images that we're using are geometric shapes, we might be able to come up with a list of vocabulary words that would help children describe their decisions about which of the shapes doesn't belong. And we could locate those words alongside our whiteboard or in a visible place in the classroom and just invite children as you're deciding which shape doesn't belong, as you're thinking about how you're going to explain your decision to your partner, think about how you could use some of these words. So, we could have words that describe different kinds of angles or other properties of shapes depending on what we're working on in the curriculum. But that's a way to show children the relevance of particular terms in a routine that they're familiar with and that they're engaged in in the classroom. And that's a way to keep these terms alive.  Mike: That's the thing that I really appreciate about what you just shared, Erin. If I'm autobiographical and I think back to my own practice, I recognize the value, and I aspired for these things to be useful. What you did just now is help paint a picture of what it looks like, not just to introduce the language or support the language, but also to keep these alive in classroom practice. I did have a question that occurred to me. Similar to sentence frames and sentence stems, is there any kind of caution that you would offer when people think about using these?  Erin: Definitely. For me, the most important caution is to not overemphasize formal mathematical vocabulary in classrooms, particularly for multilingual learners. Obviously, we want children to be developing mathematical language, and that's something we want for all children. But if we overemphasize the use of formal terminology, that can constrain communication for students who are developing the language. And we never want students' lack of familiarity or their lack of comfort with a particular vocabulary term to stop their communication or to hinder their communication. We would much rather have children explaining their ideas using all sorts of informal language and gestures and reference to physical models. The important thing is the idea and that children have the opportunity to communicate those ideas. And this formal mathematical language that might be represented in vocabulary cards or on anchor charts will come. It's part of the process, but they most importantly need opportunities to communicate their ideas.  Mike: Well, this has been a really enlightening conversation, Erin, and I'm wondering if before we go, if you have any particular recommendations for educators who are looking to build on what they heard today and continue to take up new ideas of how to support their multilingual learners?  Erin: There's a wonderful set of resources out of the Understanding Language Project at Stanford University, and they have a number of math language routines designed to support multilingual students. Some of them are related to introducing context, which we talked about today. They have a version of a “Three Reads” routine for introducing new contexts that people might find useful. But there's a whole collection of language routines on their website that teachers might find really useful. I always go to TODOS as one of my most meaningful resources for thinking deeply and critically about supporting multilingual learners.  Erin: So, I think that site and all of the books and the journals and the conferences that they develop should definitely be included. And many of the other colleagues that you've had on the podcast have wonderful resources to share, too. So, I think I would start with those two.  Mike: Well, thank you so much for joining us, Erin. It really has been a pleasure talking with you.  Erin: Oh, it's been a pleasure. Thank you again for inviting me.  Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 13 – Rough Draft Math Guest: Dr. Amanda Jansen Mike Wallus: What would happen if teachers consistently invited students to think of their ideas in math class as a rough draft? What impact might this have on students' participation, their learning experience, and their math identity? Those are the questions we'll explore today with Dr. Mandy Jansen, the author of “Rough Draft Math,” on this episode of Rounding Up.  Mike: Well, welcome to the podcast, Mandy. We are excited to be talking with you.  Mandy Jansen: Thanks, Mike. I'm happy to be here.  Mike: So, I'd like to start by asking you where the ideas involved in “Rough Draft Math” originated. What drove you and your collaborators to explore these ideas in the first place?  Mandy: So, I work in the state of Delaware. And there's an organization called the Delaware Math Coalition, and I was working in a teacher study group where we were all puzzling together—secondary math teachers—thinking about how we could create more productive classroom discussions. And so, by productive, one of the ways we thought about that was creating classrooms where students felt safe to take intellectual risks, to share their thinking when they weren't sure, just to elicit more student participation in the discussions. One way we went about that was, we were reading chapters from a book called “Exploring Talk in School” that was dedicated to the work of Doug Barnes. And one of the ideas in that book was, we could think about fostering classroom talk in a way that was more exploratory. Exploratory talk, where you learn through interaction. Students often experience classroom discussions as an opportunity to perform. "I want to show you what I know.” And that can kind of feel more like a final draft. And the teachers thought, “Well, we want students to share their thinking in ways that they're more open to continue to grow their thinking.” So, in contrast to final draft talk, maybe we want to call this rough draft talk because the idea of exploratory talk felt like, maybe kind of vague, maybe hard for students to understand. And so, the term “rough draft talk” emerged from the teachers trying to think of a way to frame this for students.  Mike: You're making me think about the different ways that people perceive a rough draft. So, for example, I can imagine that someone might think about a rough draft as something that needs to be corrected. But based on what you just said, I don't think that's how you and your collaborators thought about it, nor do I think that probably is the way that you framed it for kids. So how did you invite kids to think about a rough draft as you were introducing this idea?  Mandy: Yeah, so we thought that the term “rough draft” would be useful for students if they have ever thought about rough drafts in maybe language arts. And so, we thought, “Oh, let's introduce this to kids by asking, ‘Well, what do you know about rough drafts already? Let's think about what a rough draft is.'” And then we could ask them, “Why do you think this might be useful for math?” So, students will brainstorm, “Oh yeah, rough draft, that's like my first version” or “That's something I get the chance to correct and fix.” But also, sometimes kids would say, “Oh, rough drafts … like the bad version. It's the one that needs to be fixed.” And we wanted students to think about rough drafts more like, just your initial thinking, your first ideas; thinking that we think of as in progress that can be adjusted and improved. And we want to share that idea with students because sometimes people have the perception that math is, like, you're either right or you're wrong, as opposed to something that there's gradients of different levels of understanding associated with mathematical thinking. And we want math to be more than correct answers, but about what makes sense to you and why this makes sense. So, we wanted to shift that thinking from rough drafts being the bad version that you have to fix to be more like it's OK just to share your in-progress ideas, your initial thinking. And then you're going to have a chance to keep improving those ideas.  Mike: I'm really curious, when you shared that with kids, how did they react? Maybe at first, and then over time? Mandy: So, one thing that teachers have shared that's helpful is that during a class discussion where you might put out an idea for students to think about, and it's kind of silent, you get crickets. If teachers would say, “Well, remember it's OK to just share your rough drafts.” It's kind of like letting the pressure out. And they don't feel like, “Oh wait, I can't share unless I totally know I'm correct. Oh, I can just share my rough drafts?” And then the ideas sort of start popping out onto the floor like popcorn, and it really kind of opens up and frees people up. “I can just share whatever's on my mind.” So that's one thing that starts happening right away, and it's kind of magical that you could just say a few words and students would be like, “Oh, right, it's fine. I can just share whatever I'm thinking about.”  Mike: So, when we were preparing for this interview, you said something that has really stuck with me and that I've found myself thinking about ever since. And I'm going to paraphrase a little bit, but I think what you had said at that point in time was that a rough draft is something that you revise. And that leads into a second set of practices that we could take up for the benefit of our students. Can you talk a little bit about the ideas for revising rough drafts in a math classroom?  Mandy: Yes. I think when we think about rough drafts in math, it's important to interact with people thinking by first, assuming those initial ideas are going to have some merit, some strength. There's going to be value in those initial ideas. And then once those ideas are elicited, we have that initial thinking out on the floor. And so, then we want to think about, “How can we not only honor the strengths in those ideas, but we want to keep refining and improving?” So inviting revision or structuring revision opportunities is one way that we then can respond to students' thinking when they share their drafts. So, we want to workshop those drafts. We want to work to revise them. Maybe it's peer-to-peer workshops. Maybe it's whole-class situation where you may get out maybe an anonymous solution. Or a solution that you strategically selected. And then work to workshop that idea first on their strengths, what's making sense, what's working about this draft, and then how can we extend it? How can we correct it, sure. But grow it, improve it. Mandy: And promoting this idea that everyone's thinking can be revised. It's not just about your work needs to be corrected, and your work is fine. But if we're always trying to grow in our mathematical thinking, you could even drop the idea of correct and incorrect. But everyone can keep revising. You can develop a new strategy. You can think about connections between representations or connections between strategies. You can develop a new visual representation to represent what makes sense to you. And so, just really promoting this idea that our thinking can always keep growing. That's sort of how we feel when we teach something, right? Maybe we have a task that we've taught multiple times in a row, and every year that we teach it we may be surprised by a new strategy. We know how to solve the problem—but we don't have to necessarily just think about revising our work but revising our thinking about the ideas underlying that problem. So really promoting that sense of wonder, that sense of curiosity, and this idea that we can keep growing our thinking all the time.  Mike: Yeah, there's a few things that popped out when you were talking that I want to explore just a little bit. I think when we were initially planning this conversation, what intrigued me was the idea that this is a way to help loosen up that fear that kids sometimes feel when it does feel like there's a right or a wrong answer, and this is a performance. And so, I think I was attracted to the idea of a rough draft as a vehicle to build student participation. I wonder if you could talk a little bit about the impact on their mathematical thinking, not only the way that you've seen participation grow, but also the impact on the depth of kids' mathematical thinking as well.  Mandy: Yes, and also I think there's impact on students' identities and sense of self, too. So, if we first start with the mathematical thinking. If we're trying to work on revising—and one of the lenses we bring to revising, some people talk about lenses of revising as accuracy and precision. I think, “Sure.” But I also think about connectedness and building a larger network or web of how ideas relate to one another. So, I think it can change our view of what it means to know and do math, but also extending that thinking over time and seeing relationships. Like relationships between all the different aspects of rational number, right? Fractions, decimals, percents, and how these are all part of one larger set of ideas. So, I think that you can look at revision in a number of different grain sizes.  Mandy: You can revise your thinking about a specific problem. You can revise your thinking about a specific concept. You can revise your thinking across a network of concepts. So, there's lots of different dimensions that you could go down with revising. But then this idea that we can see all these relationships with math … then students start to wonder about what other relationships exist that they hadn't thought of and seen before. And I think it can also change the idea of, “What does it mean to be smart in math?” Because I think math is often treated as this right or wrong idea, and the smart people are the ones that get the right idea correct, quickly. But we could reframe smartness to be somebody who is willing to take risk and put their initial thinking out there. Or someone who's really good at seeing connections between people's thinking. Or someone who persists in continuing to try to revise. And just knowing math and being smart in math is so much more than this speed idea, and it can give lots of different ways to show people's competencies and to honor different strengths that students have.  Mike: Yeah, there are a few words that you said that keep resonating for me. One is this idea of connections. And the other word that I think popped into my head was “insights.” The idea that what's powerful is that these relationships, connections, patterns, that those are things that can be become clearer or that one could build insights around. And then, I'm really interested in this idea of shifting kids' understanding of what mathematics is away from answer-getting and speed into, “Do I really understand this interconnected bundle of relationships about how numbers work or how patterns play out?” It's really interesting to think about all of the ramifications of a process like rough draft work and how that could have an impact on multiple levels.  Mandy: I also think that it changes what the classroom space is in the first place. So, if the classroom space is now always looking for new connections, people are going to be spending more time thinking about, “Well, what do these symbols even mean?” As opposed to pushing the symbols around to get the answer that the book is looking for.  Mike: Amen. Mandy: And I think it's more fun. There are all kinds of possible ways to understand things. And then I also think it can improve the social dimension of the classroom, too. So, if there's lots of possible connections to notice or lots of different ways to relationships, then I can try to learn about someone else's thinking. And then I learn more about them. And they might try to learn about my thinking and learn more about me. And then we feel, like, this greater connection to one another by trying to see the world through their eyes. And so, if the classroom environment is a space where we're trying to constantly see through other people's eyes, but also let them try to see through our eyes, we're this community of people that is just constantly in awe of one another. Like, “Oh, I never thought to see things that way.” And so, people feel more appreciated and valued.  Mike: So, I'm wondering if we could spend a little bit of time trying to bring these ideas to life for folks who are listening. You already started to unpack what it might look like to initially introduce this idea, and you've led me to see the ways that a teacher might introduce or remind kids about the fact that we're thinking about this in terms of a rough draft. But I'm wondering if you can talk a little bit about, how have you seen educators bring these ideas to life? How have you seen them introduce rough draft thinking or sustain rough draft thinking? Are there any examples that you think might highlight some of the practices teachers could take up?  Mandy: Yeah, definitely. So, I think along the lines of, “How do we create that culture where drafting and revising is welcome in addition to asking students about rough drafts and why they might make sense of math?” Another approach that people have found valuable is talking with students about … instead of rules in the classroom, more like their rights. What are your rights as a learner in this space? And drawing from the work of an elementary teacher in Tucson, Arizona, Olga Torres, thinking about students having rights in the classroom, it's a democratic space. You have these rights to be confused, the right to say what makes sense to you, and represent your thinking in ways that make sense to you right now. If you honor these rights and name these rights, it really just changes students' roles in that space. And drafting and revising is just a part of that.  Mandy: So different culture-building experiences. And so, with the rights of a learner brainstorming new rights that students want to have, reflecting on how they saw those rights in action today, and setting goals for yourself about what rights you want to claim in that space. So then, in addition to culture building and sustaining that culture, it has to do—right, like Math Learning Center thinks about this all the time—like, rich tasks that students would work on. Where students have the opportunity to express their reasoning and maybe multiple strategies because that richness gives us so much to think about.  And drafts would a part of that. But also, there's something to revise if you're working on your reasoning or multiple strategies or multiple representations. So, the tasks that you work on make a difference in that space. And then of course, in that space, often we're inviting peer collaboration.  Mandy: So, those are kinds of things that a lot of teachers are trying to do already with productive practices. But I think the piece with rough draft math then, is “How are you going to integrate revising into that space?” So eliciting students' reasoning and strategies—but honoring that as a draft. But then, maybe if you're having a classroom discussion anyway, with the five practices where you're selecting and sequencing student strategies to build up to larger connections, at the end of that conversation, you can add in this moment where, “OK, we've had this discussion. Now write down individually or turn and talk. How did your thinking get revised after this discussion? What's a new idea you didn't have before? Or what is a strategy you want to try to remember?” So, adding in that revision moment after the class discussion you may have already wanted to have, helps students get more out of the discussion, helps them remember and honor how their thinking grew and changed, and giving them that opportunity to reflect on those conversations that maybe you're trying to already have anyway, gives you a little more value added to that discussion.  Mandy: It doesn't take that much time, but making sure you take a moment to journal about it or talk to a peer about it, to kind of integrate that more into your thought process. And we see revising happening with routines that teachers often use, like, math language routines such as stronger and clearer each time where you have the opportunity to share your draft with someone and try to understand their draft, and then make that draft stronger or clearer. Or people have talked about routines, like, there's this one called “My Favorite No,” where you get out of student strategy and talk about what's working and then why maybe a mistake is a productive thing to think about, try to make sense out of. But teachers have changed that to be “My Favorite Rough Draft.” So, then you're workshopping reasoning or a strategy, something like that. And so, I think sometimes teachers are doing things already that are in the spirit of this drafting, revising idea. But having the lens of rough drafts and revising can add a degree of intentionality to what you already value. And then making that explicit to students helps them engage in the process and hopefully get more out of it.  Mike: It strikes me that that piece that you were talking about where you're already likely doing things like sequencing student work to help tell a story, to help expose a connection. The power of that add-on where you ask the question, “How has your thinking shifted? How have you revised your thinking?” And doing the turn and talk or the reflection. It's kind of like a marking event, right? You're marking that one, it's normal, that your ideas are likely going to be refined or revised. And two, it sets a point in time for kids to say, “Oh yes, they have changed.” And you're helping them capture that moment and notice the changes that have already occurred even if they happened in their head.  Mandy: I think it can help you internalize those changes. I think it can also, like you said, kind of normalize and honor the fact that the thinking is continually growing and changing. I think we can also celebrate, “Oh my gosh, I hadn't thought about that before, and I want to kind of celebrate that moment.” And I think in terms of the social dimension of the classroom, you can honor and get excited about, “If I hadn't had the opportunity to hear from my friend in the room, I wouldn't have learned this.” And so, it helps us see how much we need one another, and they need us. We wouldn't understand as much as we're understanding if we weren't all together in this space on this day and this time working on this task. And so, I love experiences that help us both develop our mathematical understandings and also bond us to one another interpersonally.  Mike: So, one of the joys for me of doing this podcast is getting to talk about big ideas that I think can really impact students' learning experiences. One of the limitations is, we usually spend about 20 minutes or so talking about it, and we could talk about this for a long time, Mandy. I'm wondering, if I'm a person who's listening, and I'm really interested in continuing to learn about rough draft math, is there a particular resource or a set of resources that you might recommend for someone who wants to keep learning? Mandy: Thank you for asking. So, like you said, we can think about this for a long time, and I've been thinking about it for seven or eight years already, and I still keep growing in my thinking. I have a book called “Rough Draft Math: Revising to Learn” that came out in March 2020, which is not the best time for a book to come out, but that's when it came out. And it's been really enjoyable to connect with people about the ideas. And what I'm trying to do in that book is show that rough draft math is a set of ideas that people have applied in a lot of different ways. And I think of myself kind of as a curator, curating all the brilliant ideas that teachers have had if they think about rough drafts and revising a math class. And the book collects a set of those ideas together.  Mandy: But a lot of times, I don't know if you're like me, I end up buying a bunch of books and not necessarily reading them all. So, there are shorter pieces. There's an article in Mathematics Teaching in the Middle School that I co-wrote with three of the teachers in the Delaware Teacher Study Group, and that is at the end of the 2016 volume, and it's called “Rough-Draft Talk.” And that's only 1,800 words. That's a short read that you could read with a PLC or with a friend. And there's an even shorter piece in the NCTM Journal, MTLT, in the “Ear to the Ground” section. And I have a professional website that has a collection of free articles because I know those NCTM articles are behind a paywall. And so, I can share that. Maybe there's show notes where we can put a link and there's some pieces there.  Mike: Yes, absolutely. Well, I think that's probably a good place to stop. Thank you again for joining us, Mandy. It really has been a pleasure talking with you.  Mandy: Thank you so much, Mike. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 12 – Counting Guest: Dr. Kim Hartweg Mike Wallus: Counting is a process that involves a complex and interconnected set of concepts and skills. This means that for most children, the path to counting proficiency is not a linear process. Today we're talking with Dr. Kim Hartweg from Western Illinois University about the big ideas and skills that are a part of counting, and the ways educators can support their students on this important part of their math journey.  Mike: Well, hey, Kim, welcome to the podcast. We're excited to be talking with you about counting.   Kim Hartweg: Ah, thanks for having me. I'm excited, too.  Mike: So, I'm fascinated by all of the things that we're learning about how young kids count, or at least the way that they attend to quantities.  Kim: Yeah, it's exciting what all is taking place, with the research and everything going on with early childhood education, especially in regards to number and number sense. And I think back to an article I read about a 6-month-old baby who's in a crib and there's three pictures in this crib. One of them has two dots on it, another one has one dot, and then a third one has three dots. And a drum sounds, and it goes boom, boom, boom. And the 6-month-old baby turns their head and eyes and they look at the picture with three dots on it. And I just think that's exciting that even at that age they're recognizing that three dots [go] with three drum beats. So, it's just exciting.  Mike: So, you're actually taking us to a place that I was hoping we could go to, which is, there are some ideas and some concepts that we associate with counting. And I'm wondering if we could start the podcast by naming and unpacking a few of the really important ones.  Kim: OK, sure. I think of the fundamental counting principles, three different areas. And for me, the first one is that counting sequence, or just learning the language and that we count 1, 2, 3, 4, 5. However, in the English language, it's much more difficult [than] in other languages when we get beyond 10 because we have numbers like 11, 12, 13 that we never hear again. Like, you hear 21, 31, 41, but you don't hear 11. Again, it's the only time it's ever mentioned. So, I think it's harder for students to get that counting sequence for those who speak English.  Mike: I appreciate you saying that because I remember reading at one point that in certain Asian languages, the number 11, the translation is essentially 10 and 1, as opposed to for English speakers where it really is 11, which doesn't really follow the cadence of the number sequence that kids are learning: 1, 2, 3, 4, and so on.  Kim: Exactly. Yes.  Mike: It picks up again at 21, but this interim space where the teen numbers show up and we're first talking about a 10 and however many more, it's not a great thing about the English language that suddenly we decided to call those things that don't have that same cadence.   Kim: Yeah, after you get past 20, yes. And if you think of kids when they hear the number 16, a lot of times they'll say, “A 1 and a 6 or a 6 and a 1?” Because they hear 16, so you hear the 6 first. But like you said, in other languages, it's 10 six, 10 seven, 10 eight. So, it kind of fits more naturally with the way we talk and the language.   Mike: So, there's the language of the counting sequence. Let's talk about a couple of the other things.  Kim: OK. One-to-one correspondence is a key idea, and I think of this when I was first starting to teach undergraduate students about early math education. I had kids at the same age, so at a restaurant or wherever we happened to be, I'd get out the sugar packets and I would have them count. And at first when they're maybe 2 years old or so, and they're just learning the language, they may count those sugar packets as 1, 2, 3. There may be two packets. There may be five packets. But everything is 1, 2, 3, whether there's again, five packets or two packets. So, once they get that idea that each time they say a number word that it counts for an actual object and they can match them up, that's that idea of one-to-one correspondence to where they say a number and they either point or move the object so you can tell they're matching those up.   Mike: OK, let's talk about cardinality because this is one that I think when I first started teaching kindergarten, I took for granted how big of a leap this one is.   Kim: Yeah, that's interesting. So, once they can count out and you have five sugar packets and they count 1, 2, 3, 4, 5, and you ask how many are there, they should be able to say five. That's cardinality of number. If they have to count again, 1, 2, 3, 4, 5, then they don't have cardinality of number, where whatever number they count last is how many is in that set.   Mike: Which is kind of amazing actually. We're asking kids to decide that “I've figured out this idea that when I say a number name, I'm talking about an individual part of the count until I say the last one, and then I'm actually talking about the entire set.” That's a pretty big leap for kids to start to make sense of.   Kim: It is, and it's fun to watch because hear some of them say, “One, 2, 3, 4, 5. Five, there's five.” ( laughs ) So, they kind of get that idea. But yeah, that cardinality of number is a key principle and leads into the conservation of numbers.  Mike: Let's talk about conservation of number. What I'm loving about this conversation is the way that you're using these concrete examples from your own children, from sugar packets, to help us make sense of something that we might be seeing, but we might not have a name for.   Kim: Yeah, so the conservation of number, this is my favorite task when I have young kids around. I want to see if they can serve number or not. So, I might first do the sugar pack thing or whatever and see if they can tell you how many there are. But the real fun is, do they conserve that number? So, I think back to a friend of mine who brought her daughter over one time, and I had these toy matchbox cards on my table, and her name was Katie. And I said, “Katie, how many cars are there?” And she counted “One, 2, 3, 4, 5 … there's five toy cars.” And I moved them around and I said, “Now how many cars are there?” And she counted “One, 2, 3, 4, 5 … there's five toy cars.” So, she has cardinality of number. However, I kept moving those cars into different positions, never adding or taking any away.   Kim: That's all that were there the whole time. And after about seven or eight times, I said, “Now how many cars are there?” And her mom finally jumped in and said, “Katie, you've counted those already. There's five cars.” ( laughs ) And I said, “No, no, no. This is just whether she conserves number or not, it's a developmental-type thing.” But you know they conserve number when you ask them, “Well, now how many cars are there?” And they look at you and like, “Well, why would you ask that again? There's five.” ( chuckles ) So, then they can conserve number. It's real fun to do that with elementary students who are getting their number sense going and even before they enter school. However, there will be some that may not get that conservation of number until they're 5 or 6 years old.   Mike: Let's talk about something you named earlier. I've heard people pronounce this as (soobitizing) or (subitizing), but in any case, it's really an important idea for people, especially if you're teaching young children to make sense of this. Can you talk about what that means?   Kim: Yeah, so subitizing, I think that's interesting. We work so hard getting kids to count and learn the language and have one-to-one correspondence, and then be able to eventually conserve number. But then we want them to just be able to recognize a set of numbers without counting. And that's when they're really starting to develop some number sense. I think of dice. And if you roll a single [die], we want students to just know that when there's an arrangement of four dice, they know it's four without having to count 1, 2, 3, 4. So the subitizing idea, a lot of dice games, maybe some ten-frame cards, dot cards, lots of things like that can help students develop a little bit more of that subitizing, or recognizing a set of items without having to count those.   Mike: So, when I look at a set of three dots, I can just say that's three, as opposed to an earlier point where a child might actually say, “One, 2, 3 … that's three.”  Kim: Exactly. So, that would be subitizing—just instantly knowing what's there without having to count.   Mike: So, I wonder if we could unpack two other counting behaviors that sometimes pop up with kids when they're combining or separating quantities. And what I'm thinking about is the difference between the child who counts everything and the child who either counts on from a number or counts back from a number. And I'm wondering if you can talk about what these two behaviors can tell us about how kids are thinking about the numbers that they're operating on.   Kim: Yeah, it's so interesting when you have activities like a cup … and maybe you have eight counters and you put three under the cup and you say, “How many are here? Three.” And then you cover those up and you ask, “Well, how many are altogether?” There are some kids who don't have any trouble with counting on 4, 5, 6, 7, 8, but there's other kids who have to lift up the cup and start again at 1. So, they don't have that idea that there's three items under that cup whether you can see them or not. So, it's difficult for them to be able to count on, and we shouldn't as teachers force that on them until they're ready to do it. So, it's a hard concept for kids to get, but especially if they're not developmentally ready for it.   Mike: I think that's a really nice caution because I think you could accidentally potentially get kids to mimic a practice that you're trying to show them, but without understanding there's some real danger that you're just causing confusion.   Kim: And we want to give kids the idea that counting collections and things, it's a fun thing to do. And I know there may be teachers that have seashells or rocks or different types of collections they might count, and we want students to count those and then discuss how they counted them, arrange them. And I'm thinking of this little girl that I saw on a video where she was counting eight bears, and she arranged them first by color, then counted how many there were. And the teacher then went on to use that and make a problem-solving task for her, such as, “Well, how many green bears do you have?” And she would count them. “Well, what if you gave me those green bears? Do you know what you would have left?” And she said, “Well, I don't know. Let's try it.” And I love that because I think that's the kind of idea we want students to have. They're counting, and “I don't know, let's try it.” They're excited about it. They're not afraid to take chances, and we don't want them to think that “Oh, this is difficult to do.” It's just, “Hey, let's try it. Give it a try here.”   Mike: Well, I've heard people talking a lot about this idea of counting collections lately. It seems like we are almost rediscovering the value of a routine like that. I'm wondering if you could talk about the value that can come out of an experience of counting collections and help bring that idea to life for people.   Kim: The idea here is that we want students to get good at counting. And the research is showing that students who maybe don't show one-to-one correspondence when they count out, maybe eight counters, might show one-to-one correspondence when they count out 31 pennies, which seems like it shouldn't happen. But there's research out there that over 70 percent of them did better counting 31 pennies than they did with eight counters. So, I think what you count makes a big difference for kids—and to not hold them back, to not think that “OK, we've got to get one-to-one correspondence before we count this collection of 50 items.” I don't think that's the case. And the research is even showing that these ideas that we've talked about all develop concurrently. It's not a linear process. But this counting collections is kind of a big deal with that. And having students count, again, collections that they're interested in, writing number sentences about their collections, comparing what they counted with another partner, and then turning it into problem-solving questions where they're actually doing what happens if you lost five of yours. Or what happens if you combined your collection with somebody else? And turning it into where they're actually doing addition and subtraction, but not actually the formal process of that.   Mike: The other thing that you made me think about is, I would imagine you could also have kids finish counting a collection and then you could ask them to represent it either on paper or in some other way.   Kim: Exactly. And writing out those number sentences or even creating their own word problems so that they can ask a friend or a partner, it makes it fun. And it relates to what they've done. And let's face it, once you've taken that time to count those collections, you may as well get as much use out of it as you can. ( chuckles )  Mike: Kim, you're making me think of something that I don't know that I had words for when I was teaching kindergarten, which is, when I look back now, I was looking to see that kids could do a particular thing like one-to-one correspondence or that they had cardinality before I would give them access to a task like counting collections. And I think what you're making me think is that those things shouldn't be a gatekeeper; that they actually develop by doing those things. Am I making sense to you?   Kim: Yes. I always thought you had to have the language first. You had to be able to do one-to-one correspondence before you could get cardinality of number, and you needed cardinality of number before you could do conservation of number. But what the research is showing is, it develops concurrently with students; that it's not something that is a linear process by any means. So, when we have these activities, it's OK if they don't have one-to-one correspondence, and you're doing problem-solving tasks with counters. We need to be planning these activities so they're getting all of this, and they will develop it as it fits in the schema of what they're working on and thinking of in their minds.   Mike: So, I want to bring up a set of manipulatives that are actually attached to our bodies, particularly when it comes to counting. I'm thinking about fingers. And part of what's on my mind is, again, to go back to my practice, there was a point in time where I was really hung up on whether kids should make use of their fingers when they're counting or when they're operating on numbers. And I'm wondering if you could just offer some guidance around that.   Kim: Yes. I think again, it goes to that idea that we have these 10 fingers that are great manipulatives, that we shouldn't stop students from doing that. And I know there was a time when teachers would say, “Don't use your fingers, don't count on your fingers.” And I get the idea that we want students to start to subitize eventually and make combinations and not have to count on their fingers, but to stop them from doing it when they need that would be very detrimental to them. And I actually have a story. When I was supervising student teachers, one teacher was telling a student don't count with their fingers. And I saw them nodding their head, and I went over and I said, “What are you doing?” He said, “Well, I can't count my fingers, so I'm using my tongue, and I'm counting my teeth.” ( laughs ) So, coming up with a problem that way, still using a manipulative, but it wasn't their fingers.   Mike: That's pretty creative.   Kim: ( laughs ) Yeah.   Mike: Well, part of what strikes me about it, too, is our entire number system is based on 10s and ones, and we've got a set of them right in front of us, right? We're trying to get kids to make sense of shifting from units of one to units of 10 or maybe units of five. So, these tools that are attached to our bodies are actually pretty helpful because they're really the basis for our number system in a lot of ways.   Kim: Yes, exactly. And being able to come up with even using your fingers to answer questions … I'm thinking, we want students to subitize. So, even something to where there's a dot card that a teacher flashes for 3 seconds, and it's in the formation of maybe a five on a [die]. And you could have students hold up how many there are. And you could do that five or 10 times, with dot flashes. Or you could hold up one more than what you see on the [die]. So, they only see it for 5 seconds and the number's five, but they hold up six. So just uses of fingers to kind of make those connections can be very helpful.   Mike: So, before we go, you mentioned that you work with pre-service teachers, folks who are getting ready to go into the field and work with elementary children in the area of mathematics. I'm wondering if there are any particular resources that really help your students and perhaps teachers who are already in the field just make sense of counting and number to really understand some of the ideas that we've been talking about today. Do you have anything in particular that you would recommend to teachers?   Kim: Yeah, I'll just mention a few that we use a lot of. We do the two-color counters a lot where one side is yellow and one is red. But we do a lot of dot cards, where again, there are arrangements of dots on a card that you could just flash to a student kind of like I've already explained. There's lots of resources on the National Council of Teachers of Mathematics website. That has ten-frame activities. And if you haven't used rekenreks before, I think those are pretty amazing as well—along with hundreds charts. And just being able to have students create some of their own manipulatives and their own numbers makes a huge difference for kids.   Mike: I think that's a great place to close the conversation. Thank you so much for joining us, Kim. It's really been a pleasure chatting with you.  Kim: Hey, thanks so much. It's been fun, Mike. Thanks for asking me. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 11 – Translanguaging Guest: Tatyana Kleyn Mike Wallus: Over the past two years, we've done several episodes on supporting multilingual learners in math classrooms. Today we're going back to this topic to talk about “translanguaging,” an asset-focused approach that invites students to bring their full language repertoire into the classroom. We'll talk with Tatyana Kleyn about what translanguaging looks like and how all teachers can integrate this practice into their classrooms.  Mike: Well, welcome to the podcast, Tatyana. We're excited to be talking with you today.  Tatyana Kleyn: Thank you. This is very exciting.  Mike: So, your background with the topic of multilingual learners and translanguaging, it's not only academic. It's also personal. I'm wondering if you might share a bit of your own background as a starting point for this conversation.  Tatyana: Yes, absolutely. I think for many of us in education, we don't randomly end up teaching in the areas that we're teaching in or doing the work that we're doing. So, I always like to share my story so people know why I'm doing this work and where I'm coming from. So, my personal story, I work a lot at the intersection of language migration and education, and those are all three aspects that have been critical in bringing me here. So, I was actually born in what was the Soviet Union many, many years ago, and my family immigrated to the United States as political refugees, and I was just 5½ years old. So, I actually never went to school in the Soviet Union. Russian was my home language, and I quickly started speaking English, but my literacy was not quick at all, and it was quite painful because I never learned to read in my home language. I never had that foundation.  Tatyana: So, when I was learning to read in English, it wasn't meaning making, it was just making sounds. It was kind of painful. I once heard somebody say, “For some people, reading is like this escape and this pure joy, and for other people it's like cleaning the toilet. You get in and you get out.” And I was like, “That's me. I'm the toilet cleaner.” ( laughs ) So, that was how reading was for me. I always left my home language at the door when I came into school, and I wanted it that way because I, as a young child, got this strong message that English was the language that mattered in this country. So, for example, instead of going by Tatyana, I went by Tanya. So, I always kind of kept this secret that I spoke this other language. I had this other culture, and it wasn't until sixth grade where my sixth-grade teacher, Ms. Chang, invited my mom to speak about our immigration history.  Tatyana: And I don't know why, but I thought that was so embarrassing. I think in middle school, it's not really cool to have your parents around. So, I was like, “Oh my God, this is going to be horrible.” But then I realized my peers were really interested—and in a good way—and I was like, “Wait, this is a good thing?” So, I started thinking, “OK, we should be proud of who we are and let just people be who they are.” And when you let people be who they are, they thrive in math, in science, in social studies, instead of trying so hard to be someone they're not, and then focusing on that instead of everything else that they should be focusing on as students.  Mike: So, there's a lot there. And I think I want to dig into what you talked about over the course of the interview. I want to zero in a little bit on translanguaging though, because for me, at least until quite recently, this idea of translanguaging was really a new concept, a new idea for me, and I'm going to guess that that's the case for a lot of the people who are listening to this as well. So, just to begin, would you talk briefly about what translanguaging is and your sense of the impact that it can have on learners?  Tatyana: Sure. Well, I'm so glad to be talking about translanguaging in this space specifically, because often when we talk about translanguaging, it's in bilingual education or English as a second language or is a new language, and it's important in those settings, right? But it's important in all settings. So, I think you're not the only one, especially if we're talking about math educators or general elementary educators, it's like, “Oh, translanguaging, I haven't heard of that,” right? So, it is not something brand new, but it is a concept that Ofelia García and some of her colleagues really brought forth to the field in the early 2000s … around 2009. And what it does is instead of saying English should be the center of everything, and everyone who doesn't just speak English is peripheral. It's saying, “Instead of putting English at the center, let's put our students' home language practices at the center. And what would that look like?” So, that wouldn't mean everything has to be in English. It wouldn't mean the teacher's language practices are front and center, and the students have to adapt to that. But it's about centering the students and then the teacher adapting to the languages and the language practices that the students bring. Teachers are there to have students use all the language at their resource—whatever language it is, whatever variety it is. And all those resources will help them learn. The more you can use, when we're talking about math, well, if we're teaching a concept and there are manipulatives there that will help students use them, why should we hide them? Why not bring them in and say, “OK, use this.” And once you have that concept, we can now scaffold and take things away little by little until you have it on your own. And the same thing with sometimes learning English.  Tatyana: We should allow students to learn English as a new language using their home language resources. But one thing I will say is we should never take away their home language practices from the classroom. Even when they're fully bilingual, fully biliterate, it's still about, “How can we use these resources? How can they use that in their classroom?” Because we know in the world, speaking English is not enough. We're becoming more globalized, so let's have our students grow their language practices. And then students are allowed and proud of the language practices they bring. They teach their language practices to their peers, to their teachers. So, it's really hard to say it all in a couple of minutes, but I think the essence of translanguaging is centering students' language practices and then using that as a resource for them to learn and to grow, to learn languages and to learn content as well.  Mike: How do you think that shifts the experience for a child?  Tatyana: Well, if I think about my own experiences, you don't have to leave who you are at the door. We are not saying, “Home language is here, school language is there, and neither shall the two meet.” We're saying, “Language, and in the sense that it's a verb.” And when you can be your whole self, it allows you to have a stronger sense of who you are in order to really grow and learn and be proud of who you are. And I think that's a big part of it. I think when kids are bashful about who they are, thinking who they are isn't good enough, that has ripple effects in so many ways for them. So, I think we have to bring a lens of critical consciousness into these kind of spaces and make sure that our immigrant-origin students, their language practices, are centered through a translanguaging lens.  Mike: It strikes me that it matters a lot how we as educators—internally, in the way that we think and externally, in the things that we do and the things that we say—how we position the child's home language, whether we think of it as an asset that is something to draw upon or a deficit or a barrier, that the way that we're thinking about it makes a really big difference in the child's experience.  Tatyana: Yes, absolutely. Ofelia García, Kate Seltzer and Susana Johnson talk about a translanguaging stance. So, translanguaging is not just a practice or a pedagogy like, “Oh, let me switch this up, or let me say this in this language.” Yes, that's helpful, but it's how you approach who students are and what they bring. So, if you don't come from a stance of valuing multilingualism, it's not really going to cut it, right? It's something, but it's really about the stance. So, something that's really important is to change the culture of classrooms. So, just because you tell somebody like, “Oh, you can say this in your home language, or you can read this book side by side in Spanish and in English if it'll help you understand it.” Some students may not want to because they will think their peers will look down on them for doing it, or they'll think it means they're not smart enough. So, it's really about centering multilingualism in your classroom and celebrating it. And then as that stance changes the culture of the classroom, I can see students just saying, “Ah, no, no, no, I'm good in English.” Even though they may not fully feel comfortable in English yet, but because of the perception of what it means to be bilinguals.  Mike: I'm thinking even about the example that you shared earlier where you said that an educator might say, “You can read this in Spanish side by side with English if you need to or if you want to.” But even that language of you can implies that, potentially, this is a remedy for a deficit as opposed to the ability to read in multiple languages as a huge asset. And it makes me think even our language choices sometimes will be a tell to kids about how we think about them as a learner and how we think about their language.  Tatyana: That's so true, and how do we reframe that? “Let's read this in two languages. Who wants to try a new language?” Making this something exciting as opposed to framing it in a deficit way. So that's something that's so important that you picked up on. Yeah.  Mike: Well, I think we're probably at the point in the conversation where there's a lot of folks who are monolingual who might be listening and they're thinking to themselves, “This stance that we're talking about is something that I want to step into.” And now they're wondering what might it actually look like to put this into practice? Can we talk about what it would look like, particularly for someone who might be monolingual to both step into the stance and then also step into the practice a bit?  Tatyana: Yes. I think the stance is really doing some internal reflection, questioning about what do I believe about multilingualism? What do I believe about people who come here, to come to the United States? In New York City, about half of our multilingual learners are U.S. born. So, it's not just immigrant students, but their parents, or they're often children of immigrants. So, really looking closely and saying, “How am I including respecting, valuing the languages of students regardless of where they come from?” And then, I think for the practice, it's about letting go of some control. As teachers, we are kind of control freaks. I can just speak for myself. ( laughs ) I like to know everything that's going on.  Mike: I will add myself to that list, Tatyana.  Tatyana: It's a long list. It's a long list. ( laughs ) But I think first of all, as educators, we have a sense when a kid is on task, and you can tell when a kid is not on task. You may not know exactly what they're saying. So, I think it's letting go of that control and letting the students, for example, when you are giving directions … I think one of the most dangerous things we do is we give directions in English when we have multilingual students in our classrooms, and we assume they understood it. If you don't understand the directions, the next 40 minutes will be a waste of time because you will have no idea what's happening. So, what does that mean? It means perhaps putting the directions into Google Translate and having it translate the different languages of your students. Will it be perfect? No. But will it be better than just being in English? A million times yes, right?  Tatyana: Sometimes it's about putting students in same-language groups. If there are enough—two or three or four students that speak the same home language—and having them discuss something in their home language or multilingually before actually starting to do the work to make sure they're all on the same page. Sometimes it can mean if asking students if they do come from other countries, sometimes I'm thinking of math, math is done differently in different countries. So, we teach one approach, but what is another approach? Let's share that. Instead of having kids think like, “Oh, I came here, now this is the bad way. Or when I go home and I ask my family to help me, they're telling me all wrong.” No, again, these are the strengths of the families, and let's put them side by side and see how they go together.  Tatyana: And I think what it's ultimately about is thinking about your classroom, not as a monolingual classroom, but as a multilingual classroom. And really taking stock of who are your students? Where are they and their families coming from, and what languages do they speak? And really centering that. Sometimes you may have students that may not tell you because they may feel like it's shameful to share that we speak a language that maybe other people haven't heard of. I'm thinking of indigenous languages from Honduras, like Garífuna, Miskito, right? Of course, Spanish, everyone knows that. But really excavating the languages of the students, the home language practices, and then thinking about giving them opportunities to translate if they need to translate. I'm not saying everything should be translated. I think word problems, having problems side by side, is really important. Because sometimes what students know is they know the math terms in English, but the other terms, they may not know those yet.  Tatyana: And I'll give you one really powerful example. This is a million years ago, but it stays with me from my dissertation. It was in a Haitian Creole bilingual classroom. They were taking a standardized test, and the word problem was where it was like three gumballs, two gumballs, this color, what are the probability of a blue gumball coming out of this gumball machine? And this student just got stuck on gumball machine because in Haiti people sell gum, not machines, and it was irrelevant to the whole problem. So, language matters, but culture matters, too, right? So, giving students the opportunity to see things side by side and thinking about, “Are there any things here that might trip them up that I could explain to them?” So, I think it's starting small. It's taking risks. It's letting go of control and centering the students.  Mike: So, from one recovering control freak to another, there are a couple of things that I'm thinking about. One is expanding a little bit on this idea of having two kids who might speak to one another in their home language, even if you are a monolingual speaker and you speak English and you don't necessarily have access to the language that they're using. Can you talk a little bit about that practice and how you see it and any guidance that you might offer around that?  Tatyana: Yeah, I mean, it may not work the first time or the second time because kids may feel a little bit shy to do that. So maybe it's, “I want to try out something new in our class. I really am trying to make this a multilingual class. Who speaks another language here? Let's try … I am going to put you in a group and you're going to talk about this, and let's come back. And how did you feel? How was it for you? Let me tell you how I felt about it.” And it may be trying over a couple times because kids have learned that in most school settings, English is a language you should be using. And to the extent that some have been told not to speak any other language, I think it's just about setting it up and, “Oh, you two spoke, which language? Wow, can you teach us how to say this math term in this language?”  Tatyana: “Oh, wow, isn't this interesting? This is a cognate, which means it sounds the same as the English word. And let's see if this language and this language, if the word means the same thing,” getting everyone involved in centering this multilingualism. And language is fun. We can play with language, we can put language side by side. So, then if you're labeling or if you have a math word wall, why not put key terms in all the languages that the students speak in the class and then they could teach each other those languages? So, I think you have to start little. You have to expect some resistance. But over time, if you keep pushing away at this, I think it will be good for not only your multilingual students, but all your students to say like, “Oh, wait a minute, there's all these languages in the world, but they're not just in the world. They're right here by my friend to the left and my friend to the right” and open up that space.  Mike: So, I want to ask another question. What I'm thinking about is participation. And we've done an episode in the past around not privileging verbal communication as the only way that kids can communicate their ideas. We were speaking to someone who, their focus really was elementary years mathematics, but specifically, with multilingual learners. And the point that they were making was, kids gestures, the way that they use their hands, the way that they move manipulatives, their drawings, all of those things are sources of communication that we don't have to only say, “Kids understand things if they can articulate it in a particular way.” That there are other things that they do that are legitimate forms of participation. The thing that was in my head was, it seems really reasonable to say that if you have kids who could share an explanation or a strategy that they've come up with or a solution to a problem in their home language in front of the group, that would be perfectly legitimate. Having them actually explain their thinking in their home language is accomplishing the goal that we're after, which is can you justify your mathematical thinking? I guess I just wanted to check in and say, “Does that actually seem like a reasonable logic to follow that that's actually a productive practice for a teacher, but also a productive practice for a kid to engage in?”  Tatyana: That makes a lot of sense. So, I would say for every lesson you, you may have a math objective, you may have a language objective, and you may have both. If your objective is to get kids to understand a concept in math or to explain something in math, who cares what language they do it in? It's about learning math. And if you're only allowing them to do it in a language that they are still developing in, they will always be about English and not about math. So, how do you take that away? You allow them to use all their linguistic resources. And we can have students explain something in their home language. There are now many apps where we could just record that, and it will translate it into English. If you are not a speaker of the language that the student speaks, you can have a peer then summarize what they said in English as well. So, there's different ways to do it. So yes, I think it's about thinking about the objectives or the objective of the lesson. And if you're really focusing on math, the language is really irrelevant. It's about explaining or showing what they know in math, and they can do that in any language. Or even without spoken language, but in written language artistically with symbols, et cetera.  Mike: Well, and what you made me think, too, is for that peer, it's actually a great opportunity for them to engage with the reasoning of someone else and try to make meaning of it. So, there's a double bonus in it for that practice.  Tatyana: Exactly. I think sometimes students don't really like listening to each other. They think they only need to listen to the teacher. So, I think this really has them listen to each other. And then sometimes summarizing or synthesizing is a really hard skill, and then doing it in another language is a whole other level. So, we're really pushing kids in those ways as well. So, there's many advantages to this approach.  Mike: Yeah, absolutely. We have talked a lot about the importance of having kids engage with the thinking of other children as opposed to having the teacher be positioned as the only source of mathematical knowledge. So, the more that we talk about it, the more that I can see there's a lot of value culturally for a mathematics classroom in terms of showing that kids thinking matters, but also supporting that language development as well.  Tatyana: Yes, and doing it is hard. As I said, none of this is easy, but it's so important. And I think when you start creating a multilingual classroom, it just has a different feel to it. And I think students can grow so much in their math, understanding it and in so many other ways.  Mike: Absolutely. Well, before we close the interview, I invite you to share resources that you would recommend for an educator who's listening who wants to step into the stance of translanguaging, the practice of translanguaging, anything that you would offer that could help people continue learning.  Tatyana: I have one hub of all things translanguaging, so this will make it easy for all the listeners. So, it is the CUNY New York State Initiative on Emergent Bilinguals. And let me just give you the website. It's C-U-N-Y [hyphen] N-Y-S-I-E-B.org. And I'll say that again. C-U-N-Y, N-Y-S-I-E-B.org, cuny-nysieb.org. That's the CUNY New York State Initiative on Emergent Bilinguals. And because it's such a mouthful, we just say “CUNY NYSIEB,” as you could tell by my own, trying to get it straight. You can find translanguaging resources such as guides. You can find webinars, you can find research, you can find books. Literally everything you would want around translanguaging is there in one website. Of course, there's more out there in the world. But I think that's a great starting point. There's so many great resources just to start with there. And then just start small. Small changes sometimes have big impacts on student learning and students' perceptions of how teachers view them and their families.  Mike: Thank you so much for joining us, Tatiana. It's really been a pleasure talking with you.  Tatyana: Yes, it's been wonderful. Thank you so much. And we will just all try to let go a little bit of our control little by little. Both: ( laugh)  Tatyana: Because at the end of the day, we really don't control very much at all. ( laughs )  Mike: Agreed. ( chuckles ) Thank you.  Tatyana: Thank you. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 10 – Place Value Guest: Dr. Eric Sisofo Mike Wallus: If you ask an educator to share some of the most important ideas in elementary mathematics, I'm willing to bet that most would include place value on that list. But what does it mean to understand place value really? And what types of language practices and tools support students as they build their understanding? Today we're digging deep into the topic of place value with Dr. Eric Sisofo from the University of Delaware.  Mike: Welcome to the podcast, Eric. We're glad to have you with us.  Eric Sisofo: Thanks for having me, Mike. Really excited to be here with you today.  Mike: I'm pretty excited to talk about place value. One of the things that's interesting is part of your work is preparing pre-service students to become classroom elementary teachers. And one of the things that I was thinking about is what do you want educators preparing to teach to understand about place value as they're getting ready to enter the field?  Eric: Yeah, that's a really great question. In our math content courses at the University of Delaware, we focus on three big ideas about place value with our novice teachers. The first big idea is that place value is based on the idea of grouping a total amount of stuff or bundling a total amount of stuff into different size units. So, as you know, we use groups of ones, tens, hundreds, thousands and so on, not just ones in our base 10 system to count or measure a total amount of stuff. And we write a numeral using the digit 0 through 9 to represent the amount of stuff that we measured. So interestingly, our novice teachers come to us with a really good understanding of this idea for whole numbers, but it's not as obvious to them for decimal quantities. So, we spend a lot of time with our novice teachers helping them think conceptually about the different groupings, or bundlings, that they're using to measure a decimal amount of stuff. In particular, getting them used to using units of size: one-tenth, one-hundredth, one-thousandth, and so on. So, that's one big idea that really shines through whether you're dealing with whole numbers or decimal numbers, is that place value is all about grouping, or bundling, a total amount of stuff with very specific, different-size units.  Eric: The second big idea we'd help our novice teachers make sense of at UD is that there's a relationship between different place value units. In particular, we want our novice teachers to realize that there's this 10 times relationship between place value units. And this relationship holds true for whole numbers and decimal numbers. So, 10 of one type of grouping will make one of the next larger-sized grouping in our decimal system. And that relationship holds true for all place value units in our place value system. So, there might be some kindergarten and first-grade teachers listening who try to help their students realize that 10 ones are needed to make one 10. And some second- and third-grade teachers who try to help their students see that 10 tens are needed to make 100. And 10 hundreds are needed to make 1,000, and so on. In fourth and fifth grade, we kind of extend that idea to decimal amounts. So, helping our students realize that 10 of these one-tenths will create a one. Or 10 of the one-hundredths are needed to make one-tenth, and so on and so on for smaller and smaller place value units. So, that's the second big idea. Eric: And the third big idea that we explicitly discuss with our pre-service teachers is that there's a big difference between the face value of a digit and the place value of a digit. So, as you know, there are only 10 digits in our base 10 place value system. And we can reuse those digits in different places, and they take on a different value. So, for example, for the number 444, the same digit, 4, shows up three different times in the numeral. So, the face value is four. It's the same each digit in the numeral, but each four represents a different place value or a different grouping or an amount of stuff. So, for 444, the 4 in the hundreds place means that you have four groupings of size 100, the four in the tens place means you have four groupings of size 10, and the four in the ones place means you have four groupings of size one.  Eric: So, this happens with decimal numbers, too. With our novice teachers, we spend a lot of time trying to get them to name those units and not just say, for example, 3.4 miles when they're talking about a numeral. We wouldn't want them to say 3.4. We instead want them to say three and four-tenths, or three ones and four-tenths miles. So, saying the numeral 3.4 focuses mostly just on the face value of those digits and removes some of the mathematics that's embedded in the numeral. So, instead of saying the numerals three ones and four-tenths or three and four-tenths really requires you to think about the face value and the place value of each digit. So those are the three big ideas that we discuss often with our novice teachers at the University of Delaware, and we hope that this helps them develop their conceptual understanding of those ideas so that they're better prepared to help their future students make sense of those same ideas. Mike: You said a lot there, Eric. I'm really struck by the point two where you talk about the relationship between units, and I think what's hitting me is that I don't know that when I was a child learning mathematics—but even when I was an adult getting started teaching mathematics—that I really thought about relationships. I think about things like add a zero, or even the language of point-something. And how in some ways some of the procedures or the tricks that we've used have actually obscured the relationship as opposed to shining a light on it. Does that make sense?  Eric: I think the same was true when I was growing up. That math was often taught to be a bunch of procedures or memorized kinds of things that my teacher taught me that I didn't really understand the meaning behind what I was doing. And so, mathematics became more of just doing what I was told and memorizing things and not really understanding the reasoning why I was doing it. Talking about relationships between things I think helps kids develop number sense. And so, when you talk about how 10 tenths are required to make 1 one, and knowing that that's how many of those one-tenths are needed to make 1 one, and that same pattern happens for every unit connected to the next larger unit, seeing that in decimal numbers helps kids develop number sense about place value. And then when they start to need to operate on those numerals or on those numbers, if they need to add two decimal numbers together and they get more than 10 tenths when they add down the columns or something like that in a procedure—if you're doing it vertically. If they have more of a conceptual understanding of the relationship, maybe they'll say, “Oh, I have more than 10 tenths, so 10 of those tenths will allow me to get 1 one, and I'll leave the others in the tens place,” or something like that. So, it helps you to make sense of the regrouping that's going on and develop number sense so that when you operate and solve problems with these numbers, you actually understand the reasoning behind what you're doing as opposed to just memorizing a bunch of rules or steps. Mike: Yeah. I will also say, just as an aside, I taught kindergarten and first grade for a long time and just that idea of 10 ones and 1 ten, simultaneously, is such a big deal. And I think that idea of being able to say this unit is comprised of these equal-sized units, how challenging that can be for educators to help build that understanding. But how rich and how worthwhile the payoff is when kids do understand that level of equivalence between different sets of units. Eric: Absolutely, and it starts at a young age with children. And getting them to visualize those connections and that equivalence that a 10, 1 ten, can be broken up into these 10 ones or 10 ones can create 1 ten, and seeing that visually multiple times in lots of different situations really does pay off because that pattern will continue to show up throughout the grades. When you're going into second, third grade, like I said before, you've got to realize that 10 of these things we call tens, then we'll make a new unit called 100. Or 10 of these 100s will then make a unit that is called a thousand. And a thousand is equivalent to 10 hundreds. So, these ideas are really critical pieces of students understanding about place value when they go ahead and try to add or subtract with these using different strategies or the standard algorithm, they're able to break numbers up, or decompose, numbers into pieces that make sense to them. And their understanding of the mathematical relationships or ideas can just continue to grow and flourish.  Mike: I'm going to stay on this for one more question, Eric, and then I think you're already headed to the place where I want to go next. What you're making me think about is this work with kids not as, “How do I get an answer today?” But “What role is my helping kids understand these place value relationships going to play in their long-term success?”  Eric: Yeah, that's a great point. And learning mathematical ideas, it just doesn't happen in one lesson or in one week. When you have a complex idea like place value that … it spans over multiple years. And what kindergarten and first-grade teachers are teaching them with respect to the relationship, or the equivalence, between 10 ones and 1 ten is setting the foundation, setting the stage for the students to start to make sense of a similar idea that happens in second grade. And then another similar idea that happens in third grade where they continue to think about this 10 times relationship between units, but just with larger and larger groupings. And then when you get to fourth, fifth, sixth, seventh grade, you're talking about smaller units, units smaller than 1, and seeing that if we're using a decimal place value system, that there's still these relationships that occur. And that 10 times relationship holds true. And so, if we're going to help students make sense of those ideas in fourth and fifth grade with decimal units, we need to start laying that groundwork and helping them make sense of those relationships in the earlier grades as well.  Mike: That's a great segue because I suspect there are probably educators who are listening who are curious about the types of learning activities that they could put into place that would help build that deeper understanding of place value. And I'm curious, when you think about learning activities that you think really do help build that understanding, what are some of the things that come to mind for you?  Eric: Well, I'll talk about some specific activities in response to this, and thankfully there are some really high-quality instructional materials and math curricula out there that suggest some specific activities for teachers to use to help students make sense of place value. I personally think there are lots of cool instructional routines nowadays that teachers can use to help students make sense of place value ideas, too. Actually, some of the math curricula embed these instructional routines within their lesson plans. But what I love about the instructional routines is that they're fairly easy to implement. They usually don't take that much time, and as long as you do them fairly consistently with your students, they can have real benefits for the children's thinking over time. So, one of the instructional routines that could really help students develop place value ideas in the younger grades is something called “counting collections.”  Eric: And with counting collections, students are asked to just count a collection of objects. It could be beans or paper clips or straws or unifix cubes, whatever you have available in your classroom. And when counting, students are encouraged to make different bundles that help them keep track of the total more efficiently than if they were just counting by ones. So, let's say we asked our first- or second-grade class to count a collection of 36 unifix cubes or something like that. And when counting, students can put every group of 10 cubes into a cup or make stacks of 10 cubes by connecting them together to represent every grouping of 10. And so, if they continue to make stacks of 10 unifix cubes as they count the total of 36, they'll get three stacks of 10 cubes or three cups of 10 cubes and six singletons. And then teachers can have students represent their count in a place value table where the columns are labeled with tens and ones. So, they would put a 3 in the tens column and a 6 in the ones column to show why the numeral 36 represents the total. So, giving students multiple opportunities to make the connection between counting an amount of stuff and using groupings of tens and ones, writing that numeral that corresponds to that quantity in a place value table, let's say, and using words like 3 tens and 6 ones will hopefully help students over time to make sense of that idea. Mike: You're bringing me back to that language you used at the beginning, Eric, where you talked about face value versus place value. What strikes me is that counting collections task, where kids are literally counting physical objects, grouping them into, in the case you used tens, you actually have a physical representation that they've created themself that helps them think about, “OK, here's the face value. Where do you see this particular chunk of that and what place value does it hold?” That's a lovely, super simple, as you said, but really powerful way to kind of take all those big ideas—like 10 times as many, grouping, place value versus face value—and really touch all of those big ideas for kids in a short amount of time.  Eric: Absolutely. What's nice is that this instructional routine, counting collections, can be used with older students, too. So, when you're discussing decimal quantities let's say, you just have to make it very clear what represents one. So, suppose we were in a fourth- or fifth-grade class, and we still wanted students to count 36 unifix cubes, but we make it very clear that every cup of 10 cubes, or every stack of 10 cubes, represents, let's say, 1 pound. Then every stack of 10 cubes represents 1 pound. So, every cube would represent just one-tenth of a pound. Then as the students count the 36 unifix cubes, they would still get three stacks of 10 cubes, but this time each stack represents one. And they would get six singleton cubes where each singleton cube represents one-tenth of a pound. So, if you have students represent this quantity in a place value table labeled ones and tenths, they still get 3 in the ones place this time and 6 in the tenths place. So over time, students will learn that the face value of a digit tells you how many of a particular-size grouping you need, and the place value tells you the size of the grouping needed to make the total quantity. Mike: That totally makes sense. Eric: I guess another instructional routine that I really like is called “choral counting.” And with coral counting, teachers ask students to count together as a class starting from a particular number and jumping either forward or backward by a particular amount. So, for example, suppose we ask students to start at 5 and count by tens together. The teacher would record their counting on the board in several rows. And so, as the students count together, saying “5 15, 25, 35,” and so on, the teacher's writing these numerals across the board. He or she puts 10 numbers in a row. That means that when the students get to 105, the teacher starts a new row beginning at 105 and records all the way to 195, and then the third row would start at 205 and go all the way to 295. And after a few rows are recorded on the board, teachers could ask students to look for any patterns that they see in the numerals on the board and to see if those patterns can help them predict what number might come in the next row. Eric: So, students might notice that 10 is being added across from one number to the next going across, or 100 is being added down the columns. Or 10 tens are needed to make a hundred. And having students notice those patterns and discuss how they see those patterns and then share their reasoning for how they can use that pattern to predict what's going to happen further down in the rows could be really helpful for them, too. Again, this can be used with decimal numbers and even fractional numbers. So, this is something that I think can also be really helpful, and it's done in a fun and engaging way. It seems like a puzzle. And I know patterns are a big part of mathematics and coral counting is just a neat way to incorporate those ideas. Eric: Yeah, I've seen people do things like counting by unit fractions, too, and in this case counting by tenths, right? One-tenths, two-tenths, three-tenths, and so on. And then there's a point where the teacher might start a new column and you could make a strategic choice to say, “I'm going to start a new column when we get to ten-tenths.” Or you could do it at five-tenths. But regardless, one of the things that's lovely is choral counting can really help kids see structure in a way that counting out loud, if it doesn't have the, kind of, written component of building it along rows and columns, it's harder to discern that. You might hear it in the language, but choral accounting really helps kids see that structure in a way that, from my experience at least, is really powerful for them. Eric: And like you said, the teacher, strategically, chooses when to make the new row happen to help students, kind of, see particular patterns or groupings. And like you said, you could do it with fractions, too. So even unit fractions: zero, one-seventh, two-sevenths, three-sevenths, four-sevenths all the way to six-sevenths. And then you might start a new row at seven-sevenths, which is the same as 1. And so, kind of realize that, “Oh, I get a new 1 when I regroup 7 of these sevenths together.” And so, with decimal numbers, I need 10 of the one-tenths to get to 1. And so, if you help kids, kind of, realize that these numerals that we write down correspond with units and smaller amounts of stuff, and you need a certain amount of those units to make the next-sized unit or something like that, like I said, it can go a long way even into fractional or decimal kinds of quantities. Mike: I think you're taking this conversation in a place I was hoping it would go, Eric, because to be autobiographical, one thing that I think is an advance in the field from the time when I was learning mathematics as a child is, rather than having just a procedure with no visual or manipulative support, we have made progress using a set of manipulative tools. And at the same time, there's definitely nuance to how manipulatives might support kids' understanding of place value and also ways where, if we're not careful, it might actually just replace the algorithm that we had with a different algorithm that just happens to be shaped like cubes. What I wanted to unpack with you is what's the best-case use for manipulatives? What can manipulatives do to help kids think about place value? And is there any place where you would imagine asking teachers to approach with caution? Eric: Well, yeah. To start off, I'll just begin by saying that I really believe manipulatives can play a critical role in developing an understanding of a lot of mathematical ideas, including place value. And there's been a lot of research about how concrete materials can help students visualize amounts of stuff and visualize relationships among different amounts of stuff. And in particular, research has suggested that the CRA progression, have you heard of CRA before?  Mike: Let me check. Concrete, Representational and Abstract. Am I right? Eric: That's right. So, because “C,” the concrete representation, is first in this progression, this means that we should first give students opportunities to represent an amount of stuff with concrete manipulatives before having them draw pictures or write the amount with a numeral. To help kindergarten and first-grade students begin to develop understandings of our base 10 place value system, I think it's super important to maybe use unifix cubes to make stacks of 10 cubes. We could use bundles of 10 straws wrapped up with a rubber band and singleton straws. We could use cups of 10 beans and singleton beans … basically use any concrete manipulative that allows us to easily group stuff into tens and ones and give students multiple opportunities to understand that grouping of tens and ones are important to count by. And I think at the same time, making connections between the concrete representation, the “C” in CRA, and the abstract representation, the “A,” which is the symbol or the numeral we write down, is so important. Eric: So, using place value tables, like I was saying before, and writing the symbols in the place value table that corresponds with the grouping that children used with the actual stuff that they counted will help them over time make sense that we use these groupings of tens and ones to count or measure stuff. And then in second grade, you can start using base 10 blocks to do the same type of thing, but for maybe groupings of hundreds, thousands, and beyond. And then in fourth and fifth grade, base 10 blocks are really good for tenths and hundredths and ones, and so on like that. But for each of these, making connections between the concrete stuff and the abstract symbols that we use to represent that stuff. So, one of the main values that concrete manipulatives bring to the table, I think, is that they allow students to represent some fairly abstract mathematical ideas with actual stuff that you can see and manipulate with your hands. Eric: And it allows students to get visual images in their heads of what the numerals and the symbols mean. And so, it brings meaning to the mathematics. Additionally, I think concrete manipulatives can be used to help students really make sense of the meaning of the four operations, too, by performing actions on the concrete stuff. So, for example, if we're modeling the meaning of addition, we can use concrete manipulatives to represent the two or more numerals as amounts of stuff and show the addition by actually combining all the stuff together and then figuring out, “Well, how much is this stuff altogether?” And then if we're going to represent this with a base 10 numeral, we got to break all the stuff into groupings that base 10 numerals use. So, ones, tens, hundreds if needed, tenths, hundredths, thousandths. And one thing that you said that maybe we need to be cautious about is we don't want those manipulatives to always be a crutch for students, I don't think. So, we need to help students make the transition between those concrete manipulatives and abstract symbols by making connections, looking at similarities, looking at differences. Eric: I guess another concern that educators should be aware of is that you want to be strategic, again, which manipulatives you think would match the students' development in terms of their mathematical thinking? So, for example, I probably wouldn't use base 10 blocks in kindergarten or first grade, to be honest. When students are just learning about tens and ones, because the long in a base 10 block is already put together for them. The 10-unit cubes are already formed into a long. So, some of the cognitive work is already done for them in the base 10 blocks, and so you're kind of removing some of the thinking. And so that's why I would choose unifix cubes over base 10 blocks, or I would choose straws to, kind of, represent this relationship between ones and tens in those early grades before I start using base 10 blocks. So, those are two things that I think we have to be thoughtful about when we're using manipulatives. Mike: My wife and I have this conversation very often, and it's fascinating to me. I think about what happens in my head when a multi-edition problem gets posed. So, say it was 13 plus 46, right? In my head, I start to decompose those numbers into place value chunks, and in some cases I'll round them to compensate. Or in some cases I'll almost visualize a number line, and I'll add those chunks to get to landmarks. And she'll say to me, “I see the standard algorithm with those two things lined up.” And I just think to myself, “How big of a gift we're actually giving kids, giving them these tools that can then transfer.” Eventually they become these representations that happen in their heads and how much more they have in their toolbox when it comes to thinking about operating than many of us did who grew up learning just a set of algorithms. Eric: Yeah, and like you said, decomposing numerals or numbers into place value parts is huge because the standard algorithm does the same thing. When you're doing the standard addition algorithm in vertical form, you're still adding things up, and you're breaking the two numbers up by place value. It's just that you're doing it in a very specific way. You're starting with the smallest unit first, and you add those up, and if you get more than 10 of that particular unit, then you put a little 1 at the top to represent, “Oh, I get one of the next size unit because 10 of one unit makes one of the next size.” And so, it's interesting how the standard algorithm kind of flows from some of these more informal strategies that you were talking about—decomposing or compensating or rounding these numbers and other strategies that you were talking about—really, I think help students understand, and manipulatives, too, help students understand that you can break these numbers up into pieces where you can figure out how close this amount of stuff is to another amount of stuff and round it up or round it down and then compensate based off of that. And that helps prepare students to make sense of those standard algorithms when we go ahead and teach those. Mike: And I think you put your finger on the thing. I suspect that some people would be listening to this and they might think, “Boy, Mike really doesn't like the standard algorithm.” What I would say is, “The concern I have is that oftentimes the way that we've introduced the algorithm obscures the place value ideas that we really want kids to have so that they're actually making sense of it.” So, I think we need to give kids options as opposed to giving them one way to do it, and perhaps doing it in a way that obscures the mathematics. Eric: And I'm not against the standard algorithm at all. We teach the standard algorithms at the University of Delaware to our novice teachers and try to help them make sense of those standard algorithms in ways that talk about those big ideas that we've been discussing throughout the podcast. And talking about the place values of the units, talking about how when you get 10 of a particular unit, it makes one of the next-size unit. And thinking about how the standard algorithm can be taught in a more conceptual way as opposed to a procedural, memorized kind of set of steps. And I think that's how it sounds like you were taught the standard algorithm, and I know I was taught that, too. But giving them the foundation with making sense of the mathematical relationships between place value units in the early grades and continuing that throughout, will help students make sense of those standard algorithms much more efficiently and soundly. Mike: Yeah, absolutely. One of the pieces that you started to talk about earlier is how do you help bring meaning to both place value and, ultimately, things like standard algorithms. I'm thinking about the role of language, meaning the language that we use when we talk in our classrooms, when we talk about numbers and quantities. And I'm wondering if you have any thoughts about the ways that educators can use language to support students understanding of place value? Eric: Oh, yeah. That's a huge part of our teaching. How we as teachers talk about mathematics and how we ask our students to communicate their thinking, I think is a critical piece of their learning. As I was saying earlier, instead of saying 3.4, but expecting students to say three and four-tenths, can help them make sense of the meaning of each digit and the total value of the numeral as opposed to just saying 3.4. Another area of mathematics where we tend to focus on the face value of digits, like I was saying before, rather than the place value, is when we teach the standard algorithms. So, it kind of connects again. I believe it's really important that students and teachers alike should think about and use the place value words of the digits when they communicate their reasoning. So, if we're adding 36 plus 48 using the standard addition algorithm and vertical format, we start at the right and say, “Well, 6 plus 8 equals 14, put the 4 carry the 1 … but what does that little 1 represent, is what we want to talk about or have our students make sense of. And it's actually the 10 ones that we regrouped into 1 ten. Eric: So, we need to say that that equivalence happened or that regrouping or that exchange happened, and talk about how that little 1 that's carried over is actually the 1 ten that we got and not just call it a 1 that we carry over. So, continuing with the standard algorithm for 36 plus 48, going over to the tens column, we usually often just say, “Three plus 4 plus the 1 gives us 8,” and we put down the 8 and get the answer of 84. But what does the 3 and the 4 and the 1 really represent? “Oh, they're all tens.” So, we might say that we're combining 3 tens, or 30, with 4 tens, or 40. And the other 10 that we got from the regrouping to get 8 tens, or 80, as opposed to just calling it 8. Eric: So, talking about the digits in this way and using the place value meaning, and talking about the regrouping, all of this is really bringing meaning to what's actually happening mathematically. That's a big part of it. I guess to add onto that, when I was talking about the standard algorithm, I didn't use the words “add” or “plus,” I was saying “put together,” “combine,” to talk about the actual action of what we're doing with those two amounts of stuff. Even that language is, I think, really important. That kind of emphasizes the action that we're taking when we're using the plus symbol to put two things together. And also, I didn't say “carry.” Instead, I said, we want to “regroup” or “exchange” these 10 ones for 1 ten. So, I'm a big believer in using language that tries to precisely describe the mathematical ideas accurately because I just have seen over and over again how this language can benefit students' understanding of the ideas, too. Mike: I think what strikes me, too, is that the kinds of suggestions you're talking about in terms of describing the units, the quantities, the actions, these are things that I hope folks feel like they could turn around and use tomorrow and have an immediate impact on their kids. Eric: I hope so, too. That would be fantastic. Mike: Well, before we close the interview, I wanted to ask you, for many teachers thinking about things like place value or any big idea that they're teaching, often is kind of on the job learning and you're learning along with your kids, at least initially. So, I wanted to step back and ask if you had any recommendations for an educator who's listening to the podcast. If there are articles, books, things, online, particular resources that you think would help an educator build that understanding or think about how to build that understanding with their students? Eric: Yeah. One is to listen to podcasts about mathematics teaching and learning like this one. There's a little plug for you, Mike. Both: (laugh) Eric: I guess … Mike: I'll take it. Eric: Yeah! Another way that comes to mind is if your school uses a math curriculum that aims to help students make sense of ideas, often the curriculum materials have some mathematical background pages that teachers can read to really deepen their understanding of the mathematics. There's some really good math curricula out there now that can be really educative for teachers. I think teachers also can learn from each other. I believe teachers should collaborate with each other, talk about teaching specific lessons with each other, and through their discussions, teachers can learn from one another about the mathematics that they teach and different ways that they can try to help their students make sense of some of those ideas. Another thing that I would suggest is to become a member of an organization like NCTM, the National Council of Teachers of Mathematics. I know NCTM has some awesome resources for practitioners to help teachers continue to learn about mathematical ideas and different ways to teach particular ideas to kids. And you can attend a regional or national conference with some of these organizations. Eric: I know I've been to several of them, and I always learn some really great ideas about teaching place value or fractions or early algebraic thinking. Whatever it is, there's so many neat ideas that you can learn from others. I've been teaching math for so many years. What's cool is that I'm still learning about math and how to teach math in effective ways, and I keep learning every day, which is really one of the fun things about teaching as a profession. You just keep learning. So, I guess one thing I would suggest is to keep plugging away. Stay positive as you work through any struggles you might experience, and just know that we all wrestle with parts of teaching mathematics especially. So, stay curious and keep working to make sense of those concepts that you want your students to make sense of so that they can be problem-solvers and thinkers and sensemakers. Mike: I think it's a great place to leave it. Eric, thank you so much for joining us. It's really been a pleasure talking to you. Eric: Thanks, Mike. It's been a pleasure. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org  

Rounding Up Season 2 | Episode 9 – Instructional & Assessment Practices Guest: Dr. Kim Morrow-Leong Mike Wallus: What are the habits of mind that educators can adopt to be more responsive to our students' thinking? And how can we turn these habits of mind into practical steps that we can take on a regular basis? Dr. Kim Morrow-Leong has some thoughts on this topic. Today, Kim joins the podcast, and we'll talk with her about three mental shifts that can profoundly impact educators instructional and assessment practices.  Mike: Kim Morrow-Leong, welcome to the podcast. We're excited to have you. Kim Morrow-Leong: Thank you, Mike. It's nice to be here. Mike: I'm really excited to talk about the shifts educators can make to foster responsive interpretations of student thinking. This is an idea that for me has been near and dear for a long time, and it's fun to be able to have this conversation with you because I think there are some things we're going to get into that are shifts in how people think. But they're also practical. You introduced the shift that you proposed with a series of questions that you suggested that teachers might ask themselves or ask their colleagues, and the first question that you posed was, “What is right?” And I'm wondering what do you mean when you suggest that teachers might ask themselves or their colleagues this question when they're interpreting student thinking? Kim: So, I'm going to rephrase your question a little bit and change the emphasis to say, “What is right?” And the reason I want to change the emphasis of that is because we often talk about what is wrong, and so rather than talking about what is wrong, let's talk about what's right. When we look at student work, it's a picture. It's a snapshot of where they are at that particular moment. And the greater honesty that we can bring to that situation to understand what their thinking is, the better off we're going to be. So, there's a lot of talk lately about asset-based instruction, asset-based assessment, and I think it's a great initiative and it really gets us thinking about how we can think about what students are good at and what they bring to the table or what they bring to the classroom culture. But we don't often talk a lot about how we do that, how we break the mold. Because many of our metaphors and our language about learning are linear, and they indicate that students are moving from somewhere to achieve a goal somewhere down the path, somewhere down the line. Kim: How do you switch that around? Well, rather than looking at what they're missing and what part of the path they haven't achieved yet, we can look at where they are at the moment because that reflects everything they've learned up to that moment. So, one of the ways we can do this is to unpack our standards a little more carefully, and I think a lot of people are very good at looking at what the skills are and what our students need to be able to do by the end of the year. But a lot of what's behind a standard are concepts. What are some big ideas that must be in place for students to be successful with the skills? So, I'm going to give a very specific example. This one happens to be about a fourth-grade question that we've asked before in a district I used to work at. The task is to sketch as many rectangles as you can that are 48 square units. Kim: There's some skills behind this, but understanding what the concepts are is going to give us a little more insight into student thinking. So, one of the skills is to understand that there are many ways to make 48: to take two factors and multiply them together and only two factors, and to make a product of 48 or to get the area. But a concept behind that is that 48 is the product of two numbers. It's what happens when you multiply one dimension by the other dimension. It's not the measure of one of the dimensions. That's a huge conceptual idea for students to sort out what area is and what perimeter is, and we want to look for evidence of what they understand about the differences between what the answer to an area problem is and what the answer to, for example, a perimeter problem is. Another concept is that area indicates that a space is covered by squares. Kim: The other big concept here is that this particular question is going to have more than one answer. You're going to have 48 as a product, but you could have six times eight and four times 12 and many others. So that's a lot of things going into this one, admittedly very rich, task for students to take in. One of the things I've been thinking a lot about lately is this idea of a listening stance. So, a listening stance describes what you're listening for. It describes how you're listening. Are you listening for the right answer? Are you listening to understand students' thinking? Are you listening to respond or are you listening to hear more—and asking for more information from your student or really from any listener? So, one of the ways we could think about that, and perhaps this sounds familiar to you, is you could have what we call an evaluative listening stance. Kim: An evaluative listening stance is listening for the right answer. As you listen to what students say, you're listening for the student who gives you the answer that you're looking for. So, here's an example of something you might see. Perhaps a student covers their space and has dimensions for the rectangle of seven times six, and they tell you that this is a space that has an area of 48 square units. There's something right about that. They are really close. Because you can look at their paper and you can see squares on their paper and they're arranged in an array and you can see the dimensions on this side and the dimensions on that side, and you can see that there's almost 48 square units. I know we all can see what's wrong about that answer, but that's not what we're thinking about right now. We're thinking about what's right. And what's right is they covered that space with an area that is something by six. This is a great place to start with this student to figure out where they got that answer. If you're listening evaluatively, that's a wrong answer and there's nowhere else to go. So, when we look at what is right in student work, we're looking for the starting point. We're looking for what they know so that we can begin there and make a plan to move forward with them. You can't change where students are unless you meet them where they are and help them move forward. Mike: So, the second question that you posed was, “Can you cite evidence for what you're saying?” So again, talk us through what you're asking, when you ask teachers to pose this question to themselves or to their colleagues. Kim: Think about ways that you might be listening to a student's answer and very quickly say, “Oh, they got it,” and you move on. And you grab the next student's paper or the next student comes up to your desk and you take their work and you say, “Tell me what you're thinking.” And they tell you something. You say, “That's good,” and you move to the next one. Sometimes you can take the time to linger and listen and ask for more and ask for more and ask for more information. Teachers are very good at gathering information, at a glance. We can look at a stack of papers and in 30 seconds get a good snapshot of what's happening in that classroom. But in that efficiency we lose some details. We lose information about specifics, about what students understand, that we can only get by digging in and asking more questions. Kim: Someone once told me that every time a student gives an answer, you should follow it with, “How do you know?” And somebody raised their hand and said, “Well, what if it's the right answer?” And the presenter said, “Oh, you still ask it. As a matter of fact, that's the best one to ask. When you ask, ‘How do you know?' you don't know what you're going to hear, you have no idea what's going to happen.” And sometimes those are the most delightful surprises, is to hear some fantastical creative way to solve a problem that you never would've thought about. Unless you ask, you won't hear these wonderful things. Sometimes you find out that a correct answer has some flawed reasoning behind it. Maybe it's reasoning that only works for that particular problem, but it won't work for something else in the future. You definitely want to know that information so that you can help that student rethink their reasoning so that the next time it always works. Kim: Sometimes you find out the wrong answers are accidents. They're just a wrong computation. Everything was perfect up until the last moment and they said three times two is five, and then they have a wrong answer. If you don't ask more either in writing or verbally, you have incorrect information about that student's progress, their understandings, their conceptual development and even their skills. That kind of thing happens to everyone because we're human. By asking for more information, you're really getting at what is important in terms of student errors and what is not important, what is just easily fixable. I worked with a group of teachers once to create some open-ended tasks that require extended answers, and we sat down one time to create rubrics. And we did this with student work, so we laid them all out and someone held up a paper and said, “This is it!” Kim: “This student gets it.” And so, we all took a copy of this work and we looked at it. And we were trying to figure out what exactly does this answer communicate that makes sense to us? That seems to be an exemplar. And so, what we did was we focused on exactly what the students said. We focused on the evidence in front of us. This one was placing decimal numbers on a number line. We noted that the representation was accurate, that the position of the point on the number line was correct. We noticed that the label on the point matched the numbers in the problem, so that made sense. But then all of a sudden somebody said, “Well, wait a minute. There's an answer here, but I don't know how this answer got here.” Something happened, and there's no evidence on the page that this student added this or subtracted this, but magically the right answer was there. And it really drove home for this group—and for me, it really stuck with me—the idea that you can see a correct answer but not know the thinking behind it. Kim: And so, we learned from that point on to always focus on the evidence in front of us and to make declarative statements about what we saw, what we observed, and to hold off on making inferences. We saved our inferences for the end. After I had this experience with the rubric grading and with this group of teachers and coaches, I read something about over attribution and under attribution. And it really resonated with me. Over attribution is when you make the claim that a student understands something when there really isn't enough evidence to make that statement. It doesn't mean that's true or not true, it means that you don't have enough information in front of you. You don't have enough evidence to make that statement. You over attribute what it is they understand based on what's in front of you. Similarly, you get under attribution. You have a student who brings to you a drawing or a sketch or a representation of some sort that you don't understand because you've never seen anybody solve a problem this way before. Kim: You might come to the assumption that this student doesn't understand the math task at hand. That could be under attribution. It could be that you have never seen this before and you have not yet made sense of it. And so, focusing on evidence really gets us to stop short of making broad, general claims about what students understand, making broad inferences about what we see. It asks us to cite evidence to be grounded in what the student actually put on the paper. For some students, this is challenging because they mechanically have difficulties putting things on paper. But we call a student up to our desk and say, “Can you tell me more about what you've done here? I'm not following your logic.” And that's really the solution is to ask more questions. I know, you can't do this all the time. But you can do it once in a while, and you can check yourself if you are assigning too much credit for understanding to a student without evidence. And you can also check yourself and say, “Hmm, am I not asking enough questions of this student? Is there something here that I don't understand that I need to ask more about?” Mike: This is really an interesting point because what I'm finding myself thinking about is my own practice. What I feel like you're offering is this caution, which says, “You may have a set of cumulative experiences with children that have led you to a set of beliefs about their understanding or how they come to understanding. But if we're not careful—and even sometimes even if we are careful—we can bring that in a way that's actually less helpful, less productive,” right? It's important to look at things and actually say, “What's the evidence?” Rather than, “What's the body of my memory of this child's previous work?” It's not to say that that might not have value, but at this particular point in time, “What's the evidence that I see in front of me?” Kim: That's a good point, and it reminds me of a practice that we used to have when we got together and assessed these open-ended tasks. The first thing we would do is we put them all in the middle of the table and we would not look at our own students' work. That's a good strategy if you work with a team of people, to use these extended assessments or extended tasks to understand student thinking, is to share the load. You put them all out there. And the other thing we would do is we would take the papers, turn them over and put a Post-it note on the back. And we would take our own notes on what we saw, the evidence that we saw. We put them on a Post-it note, turn them over and then stick the Post-it note to the back of the work. There are benefits to looking at work fresh without any preconceived notions that you bring to this work. There are other times when you want all that background knowledge. My suggestion is that you try it differently, that you look at students' work for students you don't know and that you not share what you're seeing with your colleagues immediately, is that you hold your opinions on a Post-it to yourself, and then you can share it afterwards. You can bring the whole conversation to the whole table and look at the data in front of you and discuss it as a team afterwards. But to take your initial look as an individual with an unknown student. Mike: Hmm. I'm going to jump to the third shift that you suggest, which is less of a question and more of a challenge. You talk about the idea of moving from anticipating to targeting a learning trajectory, and I'm wondering if you could talk about what that means and why you think it's important. Kim: Earlier we talked about how important it is to understand and unpack our standards that we're teaching so that we know what to look for. And I think the thing that's often missed, particularly in standards in the older grades, is that there are a lot of developmental steps between, for example, a third-grade standard and a fourth-grade standard. There are skills and concepts that need to grow and develop, but we don't talk about those as much as perhaps we should. Each one of those conceptual ideas we talked about with the area problem we discussed may come at different times. It may not come during the unit where you are teaching area versus perimeter versus multiplication. That student may not come to all of those conceptual understandings or acquire all of the skills they need at the same time, even though we are diligently teaching it at the same time. Kim: So, it helps to look at third grade to understand, what are these pieces that make up this particular skill? What are the pieces that make up the standard that you're trying to unpack and to understand? So, the third shift in our thinking is to let go of the standard as our goal, but to break apart the standard into manageable pieces that are trackable because really our standards mean by the end of the year. They don't mean by December, they mean by the end of the year. So that gives you the opportunity to make choices. What are you going to do with the information you gather? You've asked what is right about student work. You've gathered evidence about what they understand. What are you going to do with that information? That perhaps is the hardest part. There's something out there called a learning trajectory that you've mentioned. Kim: A learning trajectory comes out of people who really dig in and understand student thinking on a fine-grain level, how students will learn … developmentally, what are some ideas they will develop before they develop other ideas? That's the nature of a learning trajectory. And sometimes those are reflected in our standards. The way that kindergartners are asked to rote count before they're asked to really understand one-to-one correspondence. We only expect one-to-one correspondence up to 20 in kindergarten, but we expect counting up to a hundred because we acknowledge that that doesn't come at the same time. So, a learning trajectory to some degree is built into your standards. But as we talked about earlier, there are pieces and parts that aren't outlined in your standards. One of the things we know about students and their interactions with grids and arrays is that a student might be able to recognize an array that is six by eight, but they may not yet be able to draw it. Kim: The spatial structuring that's required to create a certain number of lines going vertically and a certain number of lines going horizontally may not be in place. At the same time, they are reading a arrays and understanding what they mean. So, the skill of structuring the space around you takes time. The task where we ask them to draw these arrays is asking something that some kids may not yet be able to do, to draw these grids out. If we know that we can give them practice making arrays, we can give them tools to make arrays, we can give them blocks to make arrays, and we can scaffold this and help them move forward. What we don't want to assume is that a student who cannot yet make a six by eight array can't do any of it because that's not true. There's parts they can do. So, our job as teachers is to look at what they do, look carefully at the evidence of what they do, and then make a plan. Use all of that skill and experience that's on our teams. Even if you're a new teacher, all those people on your teams know a lot more than they're letting on, and then you can make a plan to move forward and help that student make these smaller steps so they can reach the standard. Mike: When we talked earlier, one of the things that you really shifted for me was some of the language that I found myself using. So, I know I have been in the habit of using the word “misconception” when we're talking about student work. And the part of the conversation that we had that really has never left me is this idea of, what do we actually mean when we say “misconception”? Because I found that the more I reflected on it, I used that language to describe a whole array of things that kids were doing, and not all of them were what I think a misconception actually is. Can you just talk about this language of misconception and how we use it and perhaps what we might use instead to be a little bit more precise? Kim: I have stopped using the word misconception myself. Students understand what they understand. It's our job to figure out what they do understand. And if it's not at that mature level we need it to be for them to understand the concept, what disequilibrium do I need to introduce to them? I'm borrowing from Piaget there. You have to introduce some sort of challenge so that they have the opportunity to restructure what it is they understand. They need to take their current conception, change it with new learning to become a new conception. That's our teaching opportunity right there. That's where I have to start. Mike: Before we close, I have to say one of the big takeaways from this conversation is the extent to which the language that I use, and I mean literally what I say to myself internally or what I say to my colleagues when we're interpreting student work or student thinking, that that language has major implications for my instruction and that the language that surrounds my assessing, my interpreting and my planning habits really matters. Kim: It does. You are what you practice. You are what you put forth into the world. And to see a truly student-centered point of view requires a degree of empathy that we have to learn. Mike: So, before you go, Kim, I'm wondering, can you share two or three resources that have really shaped your thinking on the interpretation of student learning? Kim: Yes, I could. And one of them is the book, “Children's Mathematics.” There's a lot of information in this book, and if you've ever engaged with the work of cognitively guided instruction, you're familiar with the work in this book. There's plenty of content knowledge, there's plenty of pedagogical content knowledge in this book. But the message that I think is the most important is that everything they learn, they learn by listening. They listen to what students were saying. And the second piece is called “Warning Signs!” And this one is one of my favorites. And in this book, they give three warning signs that you as a teacher are taking over students' learning. And one example that comes to mind for me is you take the pencil from the student. It's such a simple thing that we would just take it into quickly get something out, but to them, they expressed that that's a warning sign that you're about to take over their thinking. So, I highly recommend that one. And there's another one that I always recommend. It was published in Mathematics Teaching in the Middle School. It's called “Never Say Anything a Kid Can Say!” That's a classic. I highly recommend it if you've never read it. Mike: Kim Morrow-Leong, thank you for joining us. It's really been a pleasure. Kim: Mike, thank you for having me. This has been delightful. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 8 – It's a Story, Not a Checklist! Guest: Dr. John Staley Mike Wallus: There's something magical about getting lost in a great story. Whether you're reading a book, watching a movie, or listening to a friend, stories impart meaning, and they capture our imagination. Dr. John Staley thinks a lot about stories. On this episode of Rounding Up, we'll talk with John about the ways that he thinks that the concept of story can impact our approach to the content we teach and the practices we engage in to support our students.  Well, John, welcome to the podcast. We're really excited to talk with you today. John Staley: I'm glad to be here. Thank you for the invitation, and thank you for having me. Mike: So when we spoke earlier this year, you were sharing a story with me that I think really sets up the whole interview. And it was the story of how you and your kids had engaged with the themes and the ideas that lived in the Harry Potter universe. And I'm wondering if you could just start by sharing that story again, this time with the audience. John: OK. When I was preparing to present for a set of students over at Towson University and talking to them about the importance of teaching and it being a story. So the story of Harry Potter really began for me with our family—my wife, Karen, and our three children—back in '97 when the first book came out. Our son Jonathan was nine at that time and being a reader and us being a reading family, we came together. He would read some, myself and my wife would read some, and our daughter Alexis was five, our daughter Mariah was three. So we began reading Harry Potter. And so that really began our journey into Harry Potter. Then when the movies came out, of course we went to see the movies and watch some of those on TV, and then sometimes we listened to the audio books. And then as our children grew, because Harry Potter took, what, 10 years to develop the actual book series itself, he's 19 now, finally reading the final book. By then our three-year-old has picked them up and she's begun reading them and we're reading. So we're through the cycle of reading with them.  But what they actually did with Harry Potter, when you think about it, is really branch it out from just books to more than books. And that right there had me thinking. I was going in to talk to teachers about the importance of the story in the mathematics classroom and what you do there. So that's how Harry Potter came into the math world for me, [chuckles] I guess you can say. Mike: There's a ton about this that I think is going to become clear as we talk a little bit more.  One of the things that really struck me was how this experience shaped your thinking about the ways that educators can understand their role when it comes to math content and also instructional practice and then creating equitable systems and structures. I'm wondering if we can start with the way that you think this experience can inform an educator's understanding for content. So in this case, the concepts and ideas in mathematics. Can you talk about that, John? John: Yeah, let's really talk about the idea of what happens in a math classroom being a story. The teaching and learning of mathematics is a story that, what we want to do is connect lesson to lesson and chapter to chapter and year to year.  So when you think about students' stories, and let's start pre-K. When students start coming in pre-K and learning pre-K math, and they're engaging in the work they do in math with counting and cardinality initially, and as they grow across the years, especially in elementary, and they're getting the foundation, it's still about a story. And so how do we help the topics that we're taught, the grade level content become a story? And so that's the connection to Harry Potter for me, and that's what helped me elevate and think about Harry Potter because when you think about what Harry Potter and the whole series did, they've got the written books. So that's one mode of learning for people for engaging in Harry Potter.  Then they went from written books to audiobooks, and then they went from audiobooks to movies. And so some of them start to overlap, right? So you got written books, you got audiobooks, you got movies—three modes of input for a learner or for an audience or for me, the individual interested in Harry Potter, that could be interested in it. And then they went to additional podcasts, Harry Potter and the Sacred Text and things like that. And then they went to this one big place called Universal Studios where they have Harry Potter World. That's immersive. That I can step in; I can put on the robes; I can put the wand in my hand. I can ride on, I can taste, so my senses can really come to play because I'm interactive and engaged in this story. When you take that into the math classroom, how do we help that story come to life for our students? Let's talk one grade. So it feels like the content that I'm learning in a grade, especially around number, around algebraic thinking, around geometry, and around measurement and data. Those topics are connected within the grade, how they connect across the grade and how it grows. So the parallel to Harry Potter's story—there's, what, seven books there? And so you have seven books, and they start off with this little young guy called Harry, and he's age 11. By the time the story ends, he's seven years later, 18 years old. So just think about what he has learned across the years and how what they did there at Hogwarts and the educators and all that kind of stuff has some consistency to it. Common courses across grade levels, thinking, in my mind, common sets of core ideas in math: number, algebra, thinking, geometry, measurement of data. They grow across each year. We just keep adding on.  So think about number. You're thinking with base ten. You then think about how fractions show up as numbers, and you're thinking about operations with whole numbers, base ten, and fractions. You think about decimals and then in some cases going into, depending if you're K–8 or K–5, you might even think about how this plays into integers. But you think about how that's all connected going across and the idea of, “What's the story that I need to tell you so that you understand how math is a story that's connected?” It's not these individual little pieces that don't connect to each other, but they connect somehow in some manner and build off of each other. Mike: So there are a couple of things I want to pick up on here that are interesting. When you first started talking about this, one of the things that jumped out for me is this idea that there's a story, but we're not necessarily constrained to a particular medium. The story was first articulated via book, but there are all of these ways that you can engage with the story. And you talked about the immersive experience that led to a level of engagement. John: Mm-hmm. Mike: And I think that is helping me make sense of this analogy—that there's not necessarily one mode of building students' understanding. We actually need to think about multiple modes. Am I picking up on that right? John: That's exactly right. So what do I put in my tool kit as an educator that allows me to help tap into my students' strengths, to help them understand the content that they need to understand that I'm presenting that day, that week, that month, that I'm helping build their learning around? And in the sense of thinking about the different ways Harry Potter can come at you—with movies, with audio, with video—I think about that from the math perspective. What do I need to have in my tool kit when it comes to my instructional practices, the types of routines I establish in the classroom?  Just think about the idea of the mathematical tools you might use. How do the tools that you use play themselves out across the years? So students working with the different manipulatives that they might be using, the different mathematical tools, a tool that they use in first grade, where does that tool go in second grade, third grade, fourth grade, as they continue to work with whole numbers, especially with doing operations, with whatever the tool might be? Then what do you use with fractions? What tools do you use with decimals? We need to think about what we bring into the classroom to help our students understand the story of the mathematics that they're learning and see it as a story. Is my student in a more concrete stage? Do they need to touch it, feel it, move it around? Are they okay visually? They need to see it now, they're at that stage. They're more representational so they can work with it in a different manner or they're more abstract. Hmm. Oh, OK. And so how do we help put all of that into the setting? And how are we prepared as classroom teachers to have the instructional practices to meet a diverse set of students that are sitting in our classrooms? Mike: You know, the other thing you're making me think about, John, is this idea of concepts and content as a story. And what I'm struck by is how different that is than the way I was taught to think about what I was doing in my classroom, where it felt more like a checklist or a list of things that I was tracking. And oftentimes those things felt disconnected even within the span of a year.  But I have to admit, I didn't find myself thinking a lot about what was happening to grade levels beyond mine or really thinking about how what I was doing around building kindergartners' understanding of the structure of number or ten-ness. John: Mm-hmm. Mike: How that was going to play out in, say, fifth grade or high school or what have you. You're really causing me to think how different it is to think about this work we're doing as story rather than a discrete set of things that are kind of within a grade level. John: When you say that, it also gets me thinking of how we quite often see our content as being this mile-wide set of content that we have to teach for a grade level. And what I would offer in the space is that when you think about the big ideas of what you really need to teach this year, let's just work with number. Number base ten, or, if you're in the upper elementary, number base ten and fractions. If you think about the big ideas that you want students to walk away with that year, those big ideas continue to cycle around, and those are the ones that you're going to spend a chunk of your time on. Those are the ones you're going to keep bringing back. Those are the ones you're going to keep exposing students to in multiple ways to have them make sense of what they're doing. And the key part of all of that is the understanding, the importance of the vertical nature as to what is it I want all of my students sitting in my classroom to know and be able to do, have confidence in, have their sense of agency. Like, “Man, I can show you. I can do it, I can do it.” What do we want them to walk away with that year? So that idea of the vertical nature of it, and understanding your learning progressions, and understanding how number grows for students across the years is important. Why do I build student understanding with a number line early? So that when we get the fractions, they can see fractions as numbers. So later on when we get the decimals, they can see decimals as numbers, and I can work with it. So the vertical nature of where the math is going, the learning progression that sits behind it, helps us tell the story so that students, when they begin and you are thinking about their prior knowledge, activate that prior knowledge and build it, but build it as part of the story.  The story piece also helps us think about how we elevate and value our students in the classroom themselves. So that idea of seeing our students as little beings, little people, really, versus just us teaching content. When you think about the story of Harry Potter, I believe he survived across his time at Hogwarts because of relationships. Our students make it through the math journey from year to year to year to year because of relationships. And where they have strong relationships from year to year to year to year, their journey is a whole lot better. Mike: Let's make a small shift in our conversation and talk a little bit about this idea of instructional practice. John: OK.  Mike: I'm wondering how this lived experience with your family around the Harry Potter universe, how you think that would inform the way that an educator would think about their own practice? John: I think about it in this way. As I think about myself being in the classroom—and I taught middle school, then high school—I'm always thinking about what's in my tool kit. I think about the tools that I use and the various manipulatives, the various visual representations that I need to have at my fingertips. So part of what my question would be, and I think about it, is what are those instructional strategies that I will be using and how do I fine-tune those? What are my practices I'm using in my routines to help it feel like, “OK, I'm entering into a story”?  Harry Potter, when you look at those books, across the books, they had some instructional routines happening, some things that happen every single year. You knew there was going to be a quidditch match. You knew they were going to have some kind of holiday type of gathering or party or something like that. You knew there was going to be some kind of competition that happened within each book that really, that competition required them to apply the knowledge and skills from their various courses that they learned. They had a set of core courses that they took, and so it wasn't like in each individual course that they really got to apply. They did in some cases, they would try it out, they'd mess up and somebody's nose would get big, ears would get big, you know, change a different color. But really, when they went into some of those competitions, that's when the collection of what they were learning from their different courses, that's when the collection of the content. So how do we think about providing space for students to show what they know in new settings, new types of problems? Especially in elementary, maybe it's science application type problems, maybe they're doing something with their social studies and they're learning a little bit about that. As an educator, I'm also thinking about, “Where am I when it comes to my procedural, the conceptual development, and the ability to think through and apply the applications?” And so I say that part because I have to think about students coming in, and how do I really build this? How do I strike this balance of conceptual and procedural? When do I go conceptual? When do I go procedural? How do I value both of them? How do I elevate that? And how do I come to understand it myself? Because quite often the default becomes procedural when my confidence as a teacher is not real deep with building it conceptually. I'm not comfortable, maybe, or I don't have the set of questions that go around the lesson and everything. So I've got to really think through how I go about building that out. Mike: That is interesting, John, because I think you put your finger on something. I know there have been points in time during my career when I was teaching even young children where we'd get to a particular idea or concept, and my perception was, “Something's going on here and the kids aren't getting it.” But what you're causing me to think is often in those moments, the thing that had changed is that I didn't have a depth of understanding of what I was trying to do. Not to say that I didn't understand the concept myself or the mathematics, but I didn't have the right questions to draw out the big ideas, or I didn't have a sense of, “How might students initially think about this and how might their thinking progress over time?” So you're making me think about this idea that if I'm having that moment where I'm feeling frustrated, kids aren't understanding, it might be a point in time where I need to think to myself, “OK, where am I in this? How much of this is me wanting to think back and say, what are the big ideas that I'm trying to accomplish? What are the questions that I might need to ask?” And those might be things that I can discover through reflection or trying to make more sense of the mathematics or the concept. But it also might be an opportunity for me to say, “What do my colleagues know? Are there ways that my colleagues are thinking about this that I can draw on rather than feeling like I'm on an island by myself?” John: You just said the key point there. I would encourage you to get connected to someone somehow. As you go through this journey together, there are other teachers out there that are walking through what they're walking through, teaching the grade level content. And that's when you are able to talk deeply about math. Mike: The other thing you're making me think about is that you're suggesting that educators just step back from whether kids are succeeding or partially succeeding or struggling with a task and really step back and saying, like, “OK, what's the larger set of mathematics that we're trying to build here? What are the big ideas?” And then analyzing what's happening through that lens rather than trying to think about, “How do I get kids to success on this particular thing?” Does that make sense? Tell me more about what you're thinking. John: So when I think about that one little thing, I have to step back and ask myself the question, “How and where does that one thing fit in the whole story of the unit?" The whole story of the grade level. And when I say the grade level, I'm thinking about those big ideas that sit into the big content domains, the big idea number. How does this one thing fit into that content domain? Mike: That was lovely. And it really does help me have a clearer picture of the way in which concepts and ideas mirror the structures of stories in that, like, there are threads and connections that I can draw on from my previous experience to understand what's happening now. You're starting to go there.  So let's just talk about where you see parallels to equitable systems and structures in the experience that you had with Harry Potter when you were in that world with your family. John: First, let's think about this idea of grouping structures. And so when you think about the idea of groups and the way groups are used within the classroom, and you think about the equitable nature of homogeneous, heterogeneous, random groupings, truly really thinking about that collectively. And I say collectively in this sense, when you think about the parallel to the Harry Potter story, they had a grouping structure in place. They had a random sorting. Now who knows how random it was sometimes, right? But they had a random sorting the minute the students stepped into the school. And they got put into one of the four houses. But even though they had that random sorting then, and they had the houses structured, those groups, those students still had opportunities as they did a variety of things—other than the quidditch tournaments and some other tournaments—they had the opportunity where as a collection of students coming from the various houses, if they didn't come together, they might not have survived that challenge, that competition, whatever it was. So the idea of grouping and grouping structures and how we as educators need to think about, “What is it really doing for our students when we put them in fixed groups? And how is that not of a benefit to our students? And how can we really go about using the more random grouping?”  One of the books that I'm reading is Building Thinking Classrooms [in Mathematics: Grades K–12: 14 Teaching Practices for Enhancing Learning]. And so I'm reading Peter [Liljedahl]'s book and I'm thinking through it in the chapter when he talks about grouping. I think I read that chapter and highlighted and tapped every single page in it multiple times because it really made me think about what's really happening for our students when we think about grouping. So one structure and one part to think about is, “What's happening when we think we're doing our grouping that's not really getting students engaged in the lesson, keeping them engaged, and benefiting them from learning?” Another part, and I don't know if this is a part of equitable systems and structures or just when I think about equity work: One of the courses that they had to take at Hogwarts was about the history of wizarding. I bring that up in this space because they learned about the history of what went on with wizards and what went on with people. And to me, in my mindset, that's setting up and showing the importance of us sharing the history and bringing the history of our students—their culture, their backgrounds, in some cases their lived experiences—into the classroom. So that's us connecting with our students' culture and being culturally responsive and bringing that into the classroom. So as far as an equitable structure, the question I would ask you to think about is, “Do my students see themselves in my mathematics classroom?”  And I say it that way versus “in the mathematics,” because some people will look at the problems in the math book and say, “Oh, I don't see them there. I don't see, oh, their names, their culture, their type of foods.” Some of those things aren't in the written work in front of you. But what I would offer is the ability for me as the educator to use visuals in my classroom, the ability for me to connect with the families in my classroom and learn some of their stories, learn some of their backgrounds—not necessarily learn their stories, but learn about them and bring that in to the space—that's for me to do. I don't need a textbook series that will do that for me. And as a matter of fact, I'm not sure if a textbook series can do that for you, for all the students that you have in your classroom or for the variety of students that you have in your classroom, when we think about their backgrounds, their culture, where they might come from. So thinking about that idea of cultural responsiveness, and really, if you think about the parallel in the Harry Potter series, the history of wizarding and the interaction, when you think about the interaction piece between wizards and what they call Muggles, right? That's the interactions between our students, learning about other students, learning about other cultures, learning about diverse voices. That's teaching students how to engage with and understand others and learn about others and come to value that others have voice also. Mike: I was just thinking, John, if I were to critique Hogwarts, I do wonder about the houses. Because in my head, there is a single story that the reader comes to think about anyone who is in Harry's house versus, say, like Slytherin house. John: Yes. Mike: And it flattens anyone who's in Slytherin house into bad guys, right? John: Mm-hmm. Mike: And so it makes me think there's that element of grouping where as an educator, I might tell a single story about a particular group, especially if that group is fixed and it doesn't change. But there's also, like, what does that do internally to the student who's in that group? What does that signal to them about their own identity? Does that make sense? John: That does make sense. And so when you think about the idea of grouping there at Hogwarts, and you think about these four fixed groups, because they were living in these houses, and once you got in that house, I don't think anybody moved houses. Think about the impact on students. If you put them in a group and they stay in that group and they never change groups, you will have students who realize that the way you did your groups and the way you named your groups and the way they see others in other groups getting more, doing different, and things like that. That's a nice caution to say the labels we put on our groups. Our kids come to internalize them and they come to, in some cases, live up to the level of expectations that we set for “just that group.” So if you're using fixed groups or thinking about fixed groups, really I'd offer that you really get into some of the research around groups and think, “What does it do for students?” And not only what does it do for students in your grade, but how does that play out for students across grades? If that student was in the group that you identified as the “low group” in grade 2, [exhales] what group did they show up in grade 3? How did that play with their mindset? Because you might not have said those words in front of students, but our students pick up on being in a fixed group and watching and seeing what their peers can do and what their peers can't do, what their group members can do and what their group can't do. As our students grow from grades 2 to 3, 4, 5, that really has an impact. There's somewhere between grade 3 and 5 where students' confidence starts to really shake. And I wonder how much of it is because of the grouping and types of grouping that is being used in the classroom that has me in a group of, “Oh, I am a strong doer [of mathematics]” or, “Oh, I'm not a good doer of mathematics.” And that, how much of that just starts to resonate with students, and they start to pick that up and carry that with them, an unexpected consequence because we thought we were doing a good thing when we put 'em in this group. Because I can pull them together, small group them, this and that. I can target what I need to do with them in that moment. Yeah, target what you need to do in that moment, but mix them up in groups. Mike: Just to go back and touch on the point that you started with. Building Thinking Classrooms has a lot to say about that particular topic among others, and it's definitely a book that, for my money, has really caused me to think about a lot of the practices that I used to engage in because I believed that they were the right thing to do. It's a powerful read. For anyone who hasn't read that yet, I would absolutely recommend it. John: And one last structure that I think we can speak to. I've already spoken to supports for students, but the idea of a coherent curriculum is I think an equitable structure that systems put in place that we need to put in place that you need to have in place for your students. And when I say a coherent curriculum, I'm thinking not just your one grade, but how does that grow across the grades? It's something for me, the teacher, to say, “I need to do it my way, this way…”. But it's more to say, “Here's the role I play in their pre-K to 12 journey.” Here's the chapter I'm going to read to them this year to help them get their deep understanding of whichever chapter it was, whichever book it happened to be of.  In the case of the parallel of Harry Potter, here's the chapter I'm doing. I'm the third grade chapter, I'm the fourth grade chapter, I'm the fifth grade chapter. And the idea of that coherent curriculum allows the handoff to the next and the entry from the prior to be smoother. Many of the curriculums, when you look at them, a K–5 curriculum series will have those coherent pieces designed in it—similar types of tools, similar types of manipulatives, similar types of question prompts, similar types of routines—and that helps students build their confidence as they grow from year to year. And so to that point, it's about this idea of really thinking about how a coherent curriculum helps support equity because you know your students are getting the benefit of a teacher who is building from their prior knowledge because they've paid attention to what came before in this curriculum series and preparing them for where they're going. And that's quite often what the power of a coherent curriculum will do.  The parallel in the Harry Potter series, they had about five to seven core courses they had to take. I think about the development of those courses. Boom. If I think about those courses as a strand of becoming a wizard, [laughs] how did I grow from year to year to year to year in those strands that I was moving across? Mike: Okay, I have two thoughts. One, I fully expect that when this podcast comes out, there's going to be a large bump in whoever is tracking the sale of the Harry Potter series on Amazon or wherever it is.  John: [laughs] Mike: But the other question I wanted to ask you is what are some books outside of the Harry Potter universe that you feel like you'd recommend to an educator who's wanting to think about their practice in terms of content or instructional practices or the ways that they build equitable structure? John: When I think about the works around equitable structure, I think about The Impact of Identity and K–8 Mathematics: Rethinking Equity-Based Practices by Julia Aguirre, Karen Mayfield-Ingram, and Danny Martin as being one to help step back and think about how am I thinking about what I do and how it shows up in the classroom with my students.  Another book that I just finished reading: Humanizing Disability in Mathematics Education[: Forging New Paths]. And my reason for reading it was I continue to think about what else can we do to help our students who are identified, who receive special education services? Why do we see so many of our students who sit in an inclusive environment—they're in the classroom on a regular basis; they don't have an IEP that has a math disability listed or anything along those lines—but they significantly underperform or they don't perform as well as their peers that don't receive special education services. So that's a book that got me just thinking and reading in that space.  Another book that I'm reading now, or rereading, and I'll probably reread this one at least once a year, is Motivated[: Designing Mathematics Classrooms Where Students Want to Join In] by Ilana [Seidel] Horn. And the reason for this one is the book itself, when you read it, is written with middle schools' case stories. Part of what this book is tackling is what happens to students as they transition into middle school. And the reason why I mentioned this, especially if you're elementary, is somewhere between third grade and fifth grade, that process of students' self-confidence decreasing their beliefs in themselves as doers of math starts to fall apart. They start to take the chips in the armor. And so this book, Motivated itself, really does not speak to this idea of intrinsic motivation. “Oh, my students are motivated.” It speaks to this idea of by the time the students get to a certain age, that upper fifth grade, sixth grade timeframe, what shifts is their K, 1, 2, 3, “I'm doing everything to please my teacher.” By [grades] 4 or 5, I'm realizing, “I need to be able to show up for my peers. I need to be able to look like I can do for my peers.” And so if I can't, I'm backing out. I'm not sharing, I'm not volunteering, I'm not “engaging.”  So that's why I bring it into this elementary space because it talks about five pieces of a motivational framework that you can really push in on, and not that you push in on all five at one time. [chuckles] But you pick one, like meaningfulness, and you push in on that one, and you really go at, “How do I make the mathematics more meaningful for my students, and what does it look like? How do I create that safe space for them?” That's what you got to think about. Mike: Thanks. That's a great place to stop. John Staley, thank you so much for joining us. It's really been a pleasure. John: Thank you for having me. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. ©2023 The Math Learning Center - www.mathlearningcenter.org

Rounding Up Season 2 | Episode 7 – Making Fractions More Meaningful Guest: Dr. Susan Empson Mike Wallus: For quite a few adults, fractions were a stumbling block in their education that caused many to lose their footing and begin to doubt their ability to make sense of math. But this doesn't have to be the case for our students. Today on the podcast, we're talking with Dr. Susan Empson about big ideas and fractions and how we can make them more meaningful for our students. Welcome to the podcast. Susan. Thanks for joining us. Susan Empson: Oh, it's so great to be here. Thank you for having me. Mike: So, your book was a real turning point for me as an educator, and one of the things that it did for me at least, it exposed how little that I actually understood about the meaning of fractions. And I say this because I don't think that I'm alone in saying that my own elementary school experience was mostly procedural. So rather than attempting to move kids quickly to procedures, what types of experiences can help children build a more meaningful understanding of fractions? Susan: Great question. Before I get started, I just want to acknowledge my collaborators because I've had many people that I've worked with. There's Linda Levi, co-author of the book, and then my current research partner, Vicki Jacobs. And of course, we wouldn't know anything without many classroom teachers we've worked with in the current and past graduate students. In terms of the types of experiences that can help children build more meaningful experiences of fractions, the main thing we would say is to offer opportunities that allow children to use what they already understand about fractions to solve and discuss story problems. Children's understandings are often informal and early on, for example, may consist mainly partitioning things in half. What I mean by informal is that understandings emerge in situations out of school. So, for example, many children have siblings and have experienced situations where they have had to share, let's say three cookies or slices of pizza between two children. In these kinds of situations, children appreciate the need for equal shares, and they also develop strategies for creating them. So, as children solve and discuss story problems in school, their understandings grow. The important point is that story problems can provide a bridge between children's existing understandings and new understandings of fractions by allowing children to draw on these informal experiences. Generally, we recommend lots of experiences with story problems before moving on to symbolic work to give children plenty of opportunity to develop meaningful fractions. And we also recommend using story problems throughout fraction instruction. Teachers can use different types of story problems and adjust the numbers in those problems to address a range of fraction content. There are also ideas that we think are foundational to understanding fractions, and they're all ideas that can be elicited and developed as children engage in solving and discussing story problems.  Susan: So, one idea is that the size of a piece is determined by its relationship to the whole. What I mean is that it's not necessarily the number of pieces into which a whole is partitioned that determines the size of a piece. Instead, it's how many times the piece fits into the whole. So, in their problem-solving, children create these amounts and eventually name them and symbolize them as unit fractions. That's any fraction with 1 in the numerator. Mike: You know, one of the things that stands out for me in that initial description that you offered, is this idea of kids don't just make meaning of fractions at school, that their informal lived experiences are really an asset that we can draw on to help make sense of what a fraction is or how to think about it. Susan: That's a wonderful way to say it. And absolutely, the more teachers get to know the children in their classrooms and the kinds of experiences those children might have outside of school, the more of that can be incorporated into experiences like solving story problems in school. Mike: Well, let's dig into this a little bit. Let's talk a little bit about the kinds of story problems or the structure that actually provides an entry point and can build understanding of fractions for students. Can you talk a bit about that, Susan? Susan: Yes. So, I'll describe a couple types of story problems that we have found especially useful to elicit and develop children's fraction understandings. So first, equal sharing story problems are a powerful type of story problem that can be used at the beginning of and even throughout instruction. These problems involve sharing multiple things among multiple sharers. So, for example, four friends equally sharing 10 oranges. How much orange would each friend get? Problems like this one allow children to create fractional amounts by drawing things, partitioning those things, and then attaching fraction names and symbols. So, let's [talk] a little bit about how a child might solve the oranges problem. A child might begin by drawing four friends and then distributing whole oranges one by one until each friend has two whole oranges. Now, there are two oranges left and not enough to give each friend another whole orange. So, they have to think about how to partition the remaining oranges. Susan: They might partition each orange in half and give one more piece to each friend, or they might partition each of the remaining oranges into fourths and give two pieces to each friend. Finally, they have to think about how to describe how much each friend gets in terms of the wholes and the pieces. They might simply draw the amount, they might shade it in, or they might attach number names to it. I also want to point out that a problem about four friends equally sharing 10 oranges can be solved by children with no formal understanding of fraction names and symbols because there are no fractions in the story problem. The fractions emerge in children's strategies and are represented by the pieces in the answer. The important thing here is that children are engaged in creating pieces and considering how the pieces are related to the wholes or other pieces. The names and symbols can be attached gradually. Mike: So, the question that I wanted to ask is how to deal with this idea of how you name those fractional amounts, because the process that you described to me, what's powerful about it is that I can directly model the situation. I can make sense of partitioning. I think one of the things that I've always wondered about is, do you have a recommendation for how to navigate that naming process? I've got one of something, but it's not really one whole orange. So how do I name that? Susan: That's a great question. Children often know some of the informal names for fractions, and they might understand halves or even fourths. Initially, they may call everything a half or everything a piece or just count everything as one. And so, what teachers can do is have conversations with children about the pieces they've created and how the pieces relate to the whole. A question that we've found to be very helpful is, how many of those pieces fit into the whole? Mike: Got it. Susan: Not a question about how many pieces are there in the whole, but how many of the one piece fit into the whole. Because it then focuses children on thinking about the relationship between the piece and the whole rather than simply counting pieces. Mike: Let's talk about the other problem type that was kind of front and center in your thinking. Susan: Yes. So, another type of story problem that can be used early in fraction instruction involves what we think of as special multiplication and division story problems that have a whole number of groups and a unit fraction amount in each group. So, what do I mean by that? For example, let's say there are six friends and they each will get one-third of a sub sandwich for lunch. So, there's a whole number of groups—that's the six friends—and there's a unit fraction amount in each group that's the one-third of a sandwich that they each get. And then the question is how many sandwiches will be needed for the friends? So, a problem like this one essentially engages children in reasoning about six groups of one-third. And again, as with the equal sharing problem about oranges, they can solve it by drawing out things. They might draw each one-third of a sandwich, and then they have to consider how to combine those to make whole sandwiches. An important idea that children work on with this problem then is that three groups of one-third of a sandwich can be combined to make one whole sandwich. There are other interesting types of story problems, but teachers have found these two types, in particular, effective in developing children's understandings of some of the big ideas and fractions. Mike: I wonder if you have educators who hear you talk about the second type of problem and are a little bit surprised because they perceive it to be multiplication. Susan: Yes, it is surprising. And the key is not that you teach all of multiplying and dividing fractions before adding and subtracting fractions, but that you use these problem types with special number combinations. So, a whole number of groups, for example, the six groups unit fractions in each group—because those are the earliest fractions children understand. And I think maybe one way to think about it is that fractions come out of multiplying and dividing, kind of in the way that whole numbers come out of adding and counting. And the key is to provide situations story problems that have number combinations in them that children are able to work with. Mike: That totally makes sense. Can you say more about the importance of attending to the number combinations? Susan: Yes. Well, I think that the number combinations that you might choose would be the ones that are able to connect with the fraction understandings that children already have. So, for example, if you're working with kindergartners, they might have a sense of what one half is. So, you might choose equal sharing problems that are about sharing things among two children. So, for example, three cookies among two children. You could even, once children are able to name the halves, they create in a problem like that, you can even pose problems that are about five children who each get half of a sandwich, how many sandwiches is that? But those are all numbers that are chosen to allow children to use what they understand about fractions. And then as their understandings grow and their repertoire of fractions also grows, you can increase the difficulty of the numbers. So, at the other end, let's think about fifth grade and posing equal sharing problems. If we take that problem about four friends sharing 10 oranges, we could change the number just a little bit to make it a lot harder to, four friends sharing 10 and a half oranges, and then fifth-graders would be solving a problem that's about finding a fraction of a fraction, sharing the half orange among the four children. Mike: Let me take what you've shared and ask a follow-up question that came to me as you were talking. It strikes me that the design, the number choices that we use in problems matter, but so does the space that the teacher provides for students to develop strategies and also the way that the teacher engages with students around their strategy. Could you talk a little bit about that, Susan? Susan: Yes. We think it's important for children to have space to solve problems, fraction story problems, in ways that make sense to them and also space to share their thinking. So, just as teachers might do with whole number problem-solving in terms of teacher questioning in these spaces, the important thing is for the teacher to be aware of and to appreciate the details of children's thinking. The idea is not to fix children's thinking with questioning, but to understand it or explore it. So, one space that we have found to be rich for this kind of questioning is circulating. So, that's the time when as children solve problems, the teacher circulates and has conversations with individual children about their strategies. So, follow-up questions that focus on the details of children's strategies help children to both articulate their strategies and to reflect on them and help teachers to understand what children's strategies are. We've also found that obvious questions are sometimes underappreciated. So, for example, questions about what this child understands about what's happening in a story problem, what the child has done so far in a partial strategy, even questions about marks on a child's paper; shapes or tallies that you as a teacher may not be quite sure about, asking what they mean to the child. “What are those? Why did you make those? How did they connect with the problem?” So, in some it benefits children to have the time to articulate the details of what they've done, and it benefits the teacher because they learn about children's understandings. Mike: You're making me think about something that I don't know that I had words for before, which is I wonder if, as a field, we have made some progress about giving kids the space that you're talking about with whole number operations, especially with addition and subtraction. And you're also making me wonder if we still have a ways to go about not trying to simply funnel kids to, even if it's not algorithms, answer-getting strategies with rational numbers. I'm wondering if that strikes a chord for you or if that feels off base. Susan: It feels totally on base to me. I think that it is as beneficial, perhaps even more beneficial for children to engage in solving story problems and teachers to have these conversations with them about their strategies. I actually think that fractions provide certain challenges that whole numbers may not, and the kinds of questioning that I'm talking about really depend on the details of what children have done. And so, teachers need to be comfortable with and familiar with children's strategies and how they think about fractions as they solve these problems. And then that understanding, that familiarity, lays the groundwork for teachers to have these conversations. The questions that I'm talking about can't really be planned in advance. Teachers need to be responsive to what the child is doing and saying in the moment. And so that also just adds to the challenge. Mike: I'm wondering if you think that there are ways that educators can draw on the work that students have done composing and decomposing whole numbers to support their understanding of fractions? Susan: Yes. We see lots of parallels just as children's understandings of whole numbers develop. They're able to use these understandings to solve multi-digit operations problems by composing and decomposing numbers. So, for example, to take an easy addition, to add 37 plus eight, a child might say, “I don't know what that is, but I do know how to get from 37 to 40 with three.” So, they take three from the eight, add it to the 37 get to 40, and then once at 40 they might say, “I know that 40 plus five more is 45.” So, in other words, they decompose the eight in a way that helps them use what they understand about decade numbers. Operations with fractions work similarly, but children often do not think about the similarities because they don't understand fractions or numbers to, versus two numbers one on top of the other. Susan: If children understand that fractions can be composed and decomposed just as whole numbers can be composed and decomposed, then they can use these understandings to add, subtract, multiply, and divide fractions. For example, to add one and four-fifths plus three-fifths, a child might say, “I know how to get up to two from one in four-fifths. I need one more fifth, and then I have two more fifths still to add from the three-fifths. So, it's two and two-fifths.” So, in other words, just as they decompose the eight into three and five to add eight to 37, they decompose the three-fifths into one-fifth and two-fifths to add it to one and four-fifths. Mike: I could imagine a problem like one and a half plus five-eighths. I could say, “Well, I know I need to get a half up. Five-eighths is really four-eighths and one-eighths, and four-eighths is a half.” Susan: Yep. Mike: “So, I'm actually going from one and a half plus four-eighths. OK. That gets me to two, and then I've got one more eighth left. So, it's two and an eighth.” Susan: Nice. Yeah, that's exactly the kind of reasoning this approach can encourage. Mike: Well, I have a final question for you, Susan. “Extending Children's Mathematics” came out in 2011, and I'm wondering what you've learned since the book came out. So, are there ideas that you feel like have really been affirmed or refined, and what are some of the questions about the ways that students make meaning of fractions that you're exploring right now? Susan: Well, I think, for one, I have a continued appreciation for the power of equal sharing problems. You can use them to elicit children's informal understandings of fractions early in instruction. You can use them to address a range of fraction understandings, and they can be adapted for a variety of fraction content. So, for example, building meaning for fractions, operating with fractions, concepts of equivalence. Vicki and I are currently writing up results from a big research project focused on teachers' responsiveness to children's fraction thinking during instruction. And right now, we're in the process of analyzing data on third-, fourth-, and fifth-grade children's strategies for equal sharing problems. We specifically focused on over 1,500 drawing-based strategies used by children in a written assessment at the end of the school year. We've been surprised both by the variety of details in these strategies—so, for example, how children represent items, how they decide to distribute pieces to people—and also by the percentages of children using these drawing-based strategies. For each of grades three, four, and five, over 50 percent of children use the drawing-based strategy. There are also, of course, other kinds of strategies that don't depend on drawings that children use, but by far the majority of children were using these strategies. Mike: That's interesting because I think it implies that we perhaps need to recognize that children actually benefit from time using those strategies as a starting point for making sense of the problems that they're solving. Susan: I think it speaks to the length of time and the number of experiences that children need to really build meaning for fractions that they can then use in more symbolic work. I'll mention two other things that we've learned for which we actually have articles in the NCTM publication MTLT, which is “Mathematics Teacher: Learning and Teaching in PK–I2.” So first, we've renewed appreciation for the importance of unit fractions and story problems to elicit and develop big ideas. Another idea is that unit fractions are building blocks of other fractions. So, for example, if children solve the oranges problem by partitioning both of the extra oranges into fourths, then they have to combine the pieces in their answer. One-fourth from each of two oranges makes two-fourths of an orange. Another idea is that one whole can be seen as the same amount as a grouping of same-sized unit fractions. So, those unit fractions can all come from the same hole or different wholes, for example, to solve the problem about six friends who will each get one-third of a sub sandwich. A child has to group the one-third sandwiches to make whole sandwiches. Understanding that the same sandwich can be seen in these two ways, both as three one-third sandwiches or as one whole sandwich, provides a foundation for flexibility and reasoning. For those in the audience who are familiar with CGI, this idea is just like the IDM base ten, that 1 ten is the same amount as ten 1s, or what we describe in shorthand as 10 as a unit. And we also have an article in MTLT. It's about the use of follow-up equations to capture and focus on fraction ideas in children's thinking for their story problems. So basically, teachers listen carefully as children solve problems and explain their thinking to identify ideas that can be represented with the equations. Susan: So, for example, a child solving the sub-sandwiches problem might draw a sandwich partitioned into thirds and say they know that one sandwich can serve three friends because there are three one-thirds in the sandwich. That idea for the child might be drawn, it might be verbally stated. A follow-up equation to capture this idea might be something like one equals one-third plus one-third plus blank, with the question for the child, “Could you finish this equation or make it a true equation?” So, follow-up equation[s] often make ideas about unit fractions explicit and put them into symbolic form for children. And then at the same time, the fractions in the equations are meaningful to children because they are linked to their own meaning-making for a story problem. And so, while follow-up equations are not exactly a question, they are something that teachers can engage children with in the moment as a way to kind of put some symbols onto what they are saying, help children to reflect on what they're saying or what they've drawn, in ways that point towards the use of symbols. Mike: That really makes sense. Susan: So, they could be encouraged to shade in the piece and count the total number of pieces into which an orange is cut. However, we have found that a better question is, how many of this size piece fit into the whole? Because it focuses children on the relationship between the piece and the whole, and not on only counting pieces. Mike: Oh, that was wonderful. Thank you so much for joining us, Susan. It's really been a pleasure talking with you. Susan: Thank you. It's been my pleasure. I've really enjoyed this conversation. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 6 – Multiplicative Thinking Guest: Dr. Anderson Norton   Mike Wallus: One of the most important shifts in students' thinking during their elementary years is also one of the least talked about. I'm talking about the shift from additive to multiplicative thinking. If you're not sure what I'm talking about, I suspect you're not alone. Today we talk with Dr. Anderson Norton about this important but underappreciated shift.  Mike: Welcome to the podcast, Andy. I'm excited to talk with you about additive and multiplicative thinking. Andy Norton: Oh, thank you. Thanks for inviting me. I love talking about that. Mike: So, I want to start with a basic question. When we're talking about additive and multiplicative thinking, are we just talking about strategies or operations that students would carry out to find a sum or a product of a problem? Or are we talking about something larger? Andy: Yeah, definitely something larger, and it doesn't come down to strategies. Students can solve multiplication tasks, what to us look like multiplication tasks, using additive reasoning. And they often do, I think, they get through a lot of elementary school using, for example, repeated edition. If I gave a task like what is four times five? Then they might just say that's five and five and five and five, which is fine. They're solving a multiplication problem, but their method for solving it is repeated addition, so it's basically additive reasoning. But it starts to catch up to them in later grades where that kind of additive reasoning requires them to do more and more sophisticated or complicated strategies that maybe their teachers can teach them, but it starts to add up, especially when they get to fractions or algebra. Mike: So, let's dig into this a little bit deeper. How would you describe the difference between additive and multiplicative thinking? And I'm wondering if there's an example of the differences in how a student might approach a task or a problem that could maybe highlight that distinction. Andy: The main distinction is with additive reasoning, you're working within one level of unit. So, for example, if I want to know, and going back to that four times five example, really what I'm doing is I'm working with ones. So, I say I have five ones and five ones and five ones and five ones, and that's 20 ones. But in a multiplication problem, you're really transforming across units. If I want to understand four times five as a multiplication problem, what I'm saying is, “If I measure a quantity with a unit of five, the measure is four,” just to make it a little more concrete. Suppose my unit of measure is like a stick that's 5 feet long, and then I say, “OK, I measured this length, and it was four of these sticks. So, it's four of these 5-foot sticks. But I want to know what it is in just feet.” So, I've changed my unit. I'm saying, “I measured this thing in one unit, this stick length, but I want to understand its measure in a different unit, a unit of ones.” So, you're transforming between this one kind of unit into another kind of unit, and it's a five-to-one transformation. So, I'm not just doing five plus five plus five plus five, I'm saying every one of that stick length contains 5 feet, five of these 1-foot measures. And so, it's a transformation from one unit into another, one unit for measuring into a different unit for measuring. Mike: I mean, that's a really big shift, and I'm glad that you were able to describe that with a practical example, that someone could listen to this and visualize. I think understanding that for me clarifies the importance of not thinking about this in terms of just procedural steps that kids would take to either add or multiply; that really there's a transformation in how kids are thinking about what's happening rather than just the steps that they're following. Andy: Yeah, that's right. And a lot of times as teachers or even as researchers studying children, we're frustrated like the kids are when they're solving tasks, when they're struggling. And so we try to give them those procedures. We might give them a visual model, we might give them an array model for multiplication, which can solve a lot of problems. You just sort of think about things going vertically and things going horizontally, and then you're looking at an area or a number of intersections. So, that makes it possible for them to solve these individual tasks. And there's a lot of pressure on teachers to cover curriculum. So, we feel like we have to support them by giving them these strategies. But in the end, it just becomes more and more of these complicated strategies without really necessitating the need for something we might call a “productive struggle”; that is, where students can actually start to go through developmental changes by allowing them to struggle so that they actually develop these kinds of multiplicative structures instead of just giving them a bunch of strategies for dealing with that one task at a time. Mike: I'm wondering if you might share some examples of what multiplicative thinking might look like or sound like in different scenarios. For example, with whole numbers, with fractions or decimals …  Andy: Uh-hm. Mike: … and perhaps even in a context like measurement. What might an educator who was listening or observing students' work, what might they see that would indicate to them that multiplicative reasoning or multiplicative thinking was something that was happening for the student? Andy: So, it really is that sort of transformation of units. Like imagine, I know something is nine-fifths, and nine-fifths doesn't make a whole lot of sense unless I can think about it as nine units of one-fifth. We have to think about it as a measure like it's nine of one-fifth. And then I have to somehow compare that to, OK, it's nine of this one unit, this one-fifth unit, but what is it of a whole unit? A unit of one? So, having an estimate for how big nine-fifths is, yes, it's nine units of one-fifth. But at the same time, I want to know how big that is relative to a one. So, there's this multiplicative nature kind of built into tasks like that, and it's one explanation for why students struggle so much with improper fractions. Mike: So, I'm going to put my teacher hat on for a second because what you've got me thinking is, what are the types of tasks or experiences or even questions that an educator could put in front of students that would nudge them to make this shift without potentially pushing them to a place where they're not quite ready to go yet?  Andy: Hmm. Mike: Could you talk a little bit about what types of tasks or experiences or questions might help provide a little bit of that nudge? Andy: Yeah, that's a really good question, because it goes back to this idea that students are already solving the kinds of tasks that should involve multiplicative reasoning, but they might be using additive strategies to do it. Those strategies get more and more complicated, and we as teachers facilitate students just, sort of, doing something more procedural instead of really struggling with the issue. And what the issue should be is opportunities to work with multiple levels of units and then to reflect on their activity and working with them. So, for example, one task I like to give students is, I'll cut out a piece of construction paper and I'll hand it to the student, and I'll have hidden what I'm going to label a whole, and I'll have hidden what I'm going to label to be the unit fraction that might be appropriate for measuring this thing I gave them. So, I'll give them this piece of construction paper and I'll say, “Hey, this is five-sevenths of my whole.” Now what I've given them as a rectangular strip of paper without any partitions in it, I've hidden the whole from which I created this five-sevenths. I've hidden one-seventh, and I've put them away, maybe inside of envelopes. So, it becomes like a game. Can you guess what I have in this envelope? I just gave you five-sevenths. Can you guess, what is this five of? What is the unit that this is five of and what is the whole this five-sevenths fraction is? So, it's getting them thinking about two different levels of units at once. They've been given this one measurement, but they don't know the unit in which it's measured, and they don't even have visually present for them what the whole unit would be. Andy: So, what they might do, is they might engage in partitioning activity. Sometimes they might partition what I give them into seven equal parts instead of five because I told them five sevenths and five sevenths to them, that means partition it a seventh. Well, that could lead to problems, and if they see that their unit is smaller than the one I have hidden, they might have to reason through what went wrong, “Why might have you have gotten a different answer than I did?” So, it's those kinds of activities—of partitioning or iterating a unit, measuring out with a unit, and then reflecting on that activity—that give them a basis for starting to coordinate these units at higher and higher levels and, therefore, in line with Amy Hackenberg's framing, develop multiplicative concepts. Mike: I think that example is really helpful. I was picturing it in my head, and I could see the opportunities that that affords for, kind of, pressing on some of those big ideas. One of the things that you made me think about is the idea of manipulatives, or even if we broaden it out a little bit, visual models. Because the question I was going to ask is, “What role might a visual model or a manipulative play in supporting a shift from additive to multiplicative thinking?” I'm curious about how you would respond to that initially. And then I think I have a follow-up question for you as well. Andy: OK. I can think of two important roles for visual models—or at least two for manipulatives—and at least one works with visual models as well. But before answering that, the bigger answer is, no one manipulative is going to be the silver bullet. It's how we use them. We can use manipulatives in ways where students are just following our procedures. We can use visual models where students are just doing what we tell them to do and reading off the answer on paper. That really isn't qualitatively any different than when we just teach them an algorithm. They don't know what they're doing. They get the answer, they read it off the paper. You could consider that to be a visual model, what they're doing on their paper or even a manipulative, they're just following a procedure. What manipulatives should afford is opportunities for students to manipulate. They should be able to carry out their mental actions. So, maybe when they're trying to partition something and then iterate it, or they're thinking about different units. That's too much for them to keep in mind in their visual imagination. So, a visual model or a manipulative gives them a way to carry those actions out to see how they work with each other, to notice the effects of those actions.  Andy: So, if the manipulative is used truly as a manipulative, then it's an opportunity for them to carry out their mental actions to coordinate them with a physical material and to see what happens. And visual models could be similar, gives them a way to sort of carry out their mental actions, maybe a little more abstractly because they're just using representations rather than the actual manipulative, but maybe gives them a way to keep track of what would happen if I partitioned this into three parts and then took one of those parts and partitioned into five. How would that compare to the whole? So, it's their actions that have to be afforded by the manipulative or the visual model. And to decide what is an appropriate manipulative or an appropriate task, we need to think about, “OK, what can they already do without it?” And I'm trying to push them to do the next thing where it helps them coordinate at a level they can't just do in their imagination, and then to reflect on that activity by looking at what they wrote or looking at what they did. So, it's always that: Carrying out actions in slightly more powerful ways than they could do in their mind. That's sort of the sense in which mathematics builds on itself. After they've reflected on what they've done and they've seen the results, now maybe that's something that they can take as an object, as something that's just there for them in imagination so they can do the next thing, adding complexity. Mike: OK. So, I take it back. I don't think I have a follow-up question because you answered it in that one. What I was kind of going to dig into is the thing that you said, which is, there's a larger question about the role that a manipulative plays, and I think that your description of a manipulative should be there to manipulate …  Andy: Uh-hm. Mike: … to help kids carry out the mental action and make meaning of that. I think that piece to me is one that I really needed clarified, just to think about my own teaching and the role the manipulatives are going to play when I'm using them to support student thinking. Andy: And I'll just add one thing, not to use too many fractions examples, but that is where most of my empirical research has been, was working with elementary and middle-school children with fractions. But I have to make these decisions based on the child. So, sometimes I'll use these cuisenaire rods, the old fraction rods, the colored fraction rods. Sometimes I'll use those with students because then it sort of simplifies the idea. They don't have to wonder whether a piece fits in exactly a certain number of times. The rods are made to fit exactly. And maybe I'm not as concerned about them cutting a construction paper into equal parts or whatever. So, the rods are already formed. But other times I feel like they might be relying too much on the rods, where they start to see the brown rod as a four. They're not even really comparing the red rod, which fits into it twice. They're just, “Oh, the red is a two, the brown is a four. I know it's in there twice because two and two is four.” So, you start to think about them whole numbers. And so sometimes I'll use the rods because I want them to manipulate them in certain ways, and then other times I'll switch to the construction paper to sort of productively frustrate this idea that they're just going to work with whole numbers. I actually want them to create parts and to see the measurements and actually measure things out. So, it all depends on what kind of mental action I want them to carry out that would determine what manipulative as well. Because manipulatives have certain affordances and certain constraints. So, sometimes cuisenaire rods have the affordances I want, and other times they have constraints that I want to go beyond with, say, construction paper. Mike: Absolutely. So, there's kind of a running theme that started to develop on the podcast. And one of the themes that comes to mind is this idea that it's important for us to think about what's happening with our students thinking as a progression rather than a checklist. What strikes me about this conversation is this shift from additive to multiplicative thinking has really major implications for our students beyond simple calculation. And I'm wondering if you could just afford us a view of, why does this shift in thinking matter for our students both in elementary school, and then also when they move beyond elementary school into middle and high school? Could you just talk about the ramifications of that shift and why it matters so much that we're not just building a set of procedures, we're building growth in the way that kids are thinking? Andy: Yeah. So, one big idea that comes up starting in middle school—but becomes more and more important as they move into algebra and calculus, any kind of engineering problem—is a rate of change. So, a rate of change is describing a relationship between units. It's like, take a simple example of speed. It's taking units of distance and units of time and transforming them into a third level of unit that is speed. So, it's that intensive relationship that's defining a new unit. When I talk about units coordination, I'm not usually talking about physical units like distance, time and speed. I'm just talking about different numerical units that students might have to coordinate. But to get really practical when we talk about the sciences, units coordinations have to happen all the time. So, students are able to be successful with their additive reasoning up to a point, and I would argue that point is probably around where they first see improper fractions. ( chuckles ) They're able to work with them up to a point, and then after that, things [are] going to be less and less sensible if they're just relying on these additive sort of strategies that each have a separate rule for a different task instead of being able to think more generally in terms of multiplicative relationships. Mike: Well, I will say from a former K–12 math curriculum director, thank you for making a very persuasive case for why it's important to help kids build multiplicative thinking. You certainly hit on some of the things that can be pitfalls for kids who are still thinking in an additive way when they start to move into upper elementary, middle school and beyond. Before we go, Andy, I suspect that this idea of shifting from additive to multiplicative thinking, that it's probably a new idea for our listeners. And you've hinted a bit about some of the folks who have been powerful in the field in terms of articulating some of these ideas. I'm wondering if there are any particular resources that you'd recommend for someone who wants to keep learning about this topic? Andy: Yeah. So, there are a bunch of us developing ideas and trying to even create resources that teachers can pick up and use. Selfishly, I'll mention one called “Developing Fractions Knowledge,” used by the U.S. Math Recovery Council in their professional development programs for teacher-leaders across the country. That book is probably, at least as far as fractions, that book is maybe the most comprehensive. But then beyond that, there are some research articles that people can access, even going in Google Scholar and looking up units, coordination and multiplicative reasoning, maybe put in Steffe's name for good measure, S-T-E-F-F-E. You'll find a lot of papers there. Some of them have been written in teacher journals as well, like journals published by the National Council of Teachers of Mathematics, like Teaching Children Mathematics materials that are specifically designed for teachers. Mike: Andy, thank you so much for joining us. It's really been a pleasure talking with you. Andy: OK. Yeah, thank you. This was fun. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 5 – Horizontal Enrichment Guest: Tisha Jones Mike Wallus: At their best, programs with titles such as “gifted and talented” seek to provide enrichment to a subset of learners. That said, these initiatives sometimes have unintended consequences, sending messages about which students are, or are not, capable doers of mathematics. What if there was a way educators could offer problems that extend grade-level learning to each and every student? Today we'll explore the concept of horizontal enrichment with Tisha Jones, MLC's senior manager of assessment.  Mike: Well, thanks for joining us, Tisha. I am excited to explore this idea of horizontal enrichment. Tisha Jones: I am excited to be here and talk about it. Mike: So, we're using the term “horizontal enrichment,” and I think we should define the term and talk about, what do we mean when we say that? Tisha: When we're talking about horizontal enrichment, we are looking at how do we enrich the curriculum, but on grade level. So, not trying to accelerate into the next grade level. But how do we help them go deeper with the content that is at their developmental level currently? Mike: That's really interesting because when I was teaching, I would've said enrichment and acceleration are exactly the same thing, which, I think, leads me to the next question, which is: What are the features of a task that might be designed with horizontal enrichment in mind? Tisha: So, I like to think about horizontal enrichment as an opportunity to engage the practice standards. So, how do we help kids do more of the things that we think being a [mathematician] actually is? So, how can we get them more invested in problem-solving? How can we get them using tools? How can we get them thinking creatively in math and not just procedurally. And, of course, we try to do that on a daily basis in math, but when we're enriching, we want to give them tasks that raise the ceiling of their thinking, where they can approach things in lots of different ways and push their thinking in ways that maybe they haven't, where they can apply the concepts that they're using to solve interesting and novel problems. Mike: I think that's really helpful because you're really clarifying for me, one way that we could “enrich” kids would be to teach them procedures that they might learn in a grade or several grades that are of beyond where they're at right now. But what you're suggesting is that enrichment really looks like problem-solving and novelty and creativity. And we can do that with grade-level ideas. Am I making sense of that correctly? Tisha: Absolutely, and I get excited because I also think that it's fun working a problem where the path is not clear-cut to get to the answer and try some things out and see what happens and look at how can I learn from what I did to make new decisions to try to get to where I'm going? To me, that's bringing in the joy of doing math. Mike: So, this is interesting. I think that maybe the best way to unpack these ideas might be to look at a specific task. So, I'm wondering, is there a specific task that you could help us take a look at more closely? Tisha: Absolutely. So, we're going to take a look at a task from third grade, and it comes out of Concept Quests, which is a supplemental resource that's published by Math Learning Center, and this task is called “The Lasagna Task.” So, I'm just going to read it and then we can talk about what is it asking kids to do. So, it says, “You need to assume that you like lasagna and would like as much lasagna as possible. For each of the ‘Would you rather…?' scenarios below, justify your reasoning with equations, pictures, or both.” So, that's the setup for the kids. And then there's three “Would you rather…?” scenarios. So, the first is, “Would you rather: a.) share three lasagnas between two families or share four lasagnas between three families? b.) Would you rather share four lasagnas between six families or share three lasagnas between four families?” And the last one is, “c.) Would you rather share five lasagnas between three families or share six lasagnas between four families?” Mike: Ahh, this is so great. There's so much to unpack here to step back and try to analyze this. What are some things that you would want us to notice about the way this task is set up for kids? Tisha: So, there's a few things. The first thing is, I love that there's this progression of questions, of scenarios. I think what's also really important is, when you're looking at this on the page, there's no front-loading here. No, “Well, let me tell you about how to do this.” This is just, “I'm going to give you this problem, and I'm going to ask you to just take a stab at it, give it a shot.” So, what we want kids to do is start to learn, how do you approach a problem? What is your first step? What things do you do to make sense of what it's asking? Do you draw a picture? Do you start with numbers? Do you try to find important information? How do you even get started on a problem? And that's so important, right? That's a huge part of the process of problem-solving. And when we front-load for kids, we take away their opportunities to work on those skills. Mike: So, there's a couple things that jump out for me when I've been reading the text of what you were reading aloud to the group. One bit is this language at the end where it says, “For each of the ‘Would you rather…?' scenarios below, justify your reasoning with equations, pictures, or both.” And that language just pops out for me. I'm wondering if you could talk a little bit about the choice of that language in the way that this is set up for kids. Tisha: Ahh, I love that language. So, I think this is amazing for kids because as a teacher, we've all had kids that come up to us and they hand us their paper and they say, “Is this right?” And when we ask them to justify their response, I think we're putting the responsibility back on them to be able to come up to me and say, “I think this is right because of this.” So now, who is owning what they did? The kids are owning what they did, right? And they're owning it because they've gone through this process of trying to prove it not just to somebody else but to themselves. If you're justifying it, you should be able to go back through and say, “Well, because I did this and this is this and because I did this next step and this is how this worked out, this is why I know my answer is correct.” And I love that kids can own their own answers and their own work to be able to determine whether it makes sense or not. Mike: I'm going to read a part of this again because I just think it's worth lingering on and spending a little bit of time thinking about how this question structure impacts kids or has the potential to impact kids. So, I'm going to read it again for the audience: “Would you rather: a.) share three lasagnas between two families or share four lasagnas between three families?” So, listeners, just pause for a second and think about the mathematics in that question, and then also think about what mathematics might come out of it. What is it about the structure of that question that creates space for kids to solve problems, encounter novelty, and make decisions? Well, Tisha, since we can't hear their answer, I would love it if you could share a little bit of your thinking. What is it about the design that you think creates those conditions for kids? Tisha: So, while there is an implied operation, it's not necessarily an obvious operation, right? I think that it is something that easily lends itself to drawing a picture, which, I think, when students start modeling the scenario, they now have … that opens up all kinds of creativity, right? They're going to model in the way that they're seeing it in their head. They're not focused on trying to divide this number by that number. They may not even, at first, realize that they're working with fractions. But by the end of it, because it's something that they can model, there's still a lot of room for them to be able to find success on this task, which I think is really important. Mike: It seems like there's also opportunities for teachers to engage with kids because there's a fair number of assumptions that live inside of this question structure, right? Like three lasagnas for two families, four lasagnas for three families, but we haven't talked about how large those families are, how many people are in each family. Tisha: How much lasagna there is ( chuckles ). Mike: Yeah! Right? Tisha: Absolutely. So, I think it's also fair to say that maybe a kid would decide that the four lasagnas between three families, those are going to need to be bigger pans of lasagna. So, how are they bringing in their world experience with feeding people and having to make these decisions? There's nothing in here that says that the lasagnas have to be the same size or that the families have to be the same size. So, as they're justifying the way that they would go as a teacher, I'm looking for: Is their justification, a sound justification? Mike: Well, the thing that I started to think about, too, is, if you did introduce the variable that, “Oh, this family has three members and this family has, say, 12. Well, how many lasagnas would you need in order to give an equal share to the family with 12 versus the family with three?” There's a lot of ways as a teacher that I can continue to adapt and play with the ideas and really press kids to examine their own assumptions and their own logic. Tisha: Absolutely, yeah. So, I think that's a really great point, too, is that, there's a lot of room to even extend these problems further. Would your answer change if you knew that one family was a family of six people, so you can even push their thinking even further than what's just on the paper. Mike: I keep going back to this notion of justification. And we've talked about the structure of the problems as a way to differentiate for kids, to really press them on justification. But the other side of the coin is, as an educator, [it] really gives me a chance to understand my students' thinking and then continue to make moves or offer tasks that either shine a light on the blind spots that they have or extend some of the ideas in interesting and productive ways. Tisha: Yes, I would agree with that. Mike: So, I want to play with a couple more questions, Tisha. One of the ones that we touched on right at the beginning was this idea that a task can be characterized as enriching and challenging, and yet it can still be at a student's grade level. And I think that really stands out for me, and I suspect it probably might be a challenging idea for educators to get their heads around, especially if you've been a teacher, and for the majority of your career, acceleration and enrichment have meant the same thing. Can you unpack this just a little bit for the audience, this idea of enrichment? Tisha: So, I like to think about enrichment as, how do we help our students think more deeply? There's so much room within a school year for a particular concept, for example. Like, let's say with fractions. There's a lot of room for students to think about things in ways they haven't thought about or ways that maybe we don't ask them to think about things in the curriculum; that, if we don't give them the opportunity, they're not going to, right? With enrichment, it's like we're giving them more opportunities to apply what they're learning about concepts. The other thing that I think is really important about enrichment is that it isn't just for the kids that may be characterize as being your high-level students. Because enrichment is still important. Problem-solving is still important for all kids. No matter where they are computationally, we want to make sure that all kids are getting opportunities to be problem-solvers, to apply their thinking in ways that work for them and not just the ways that we're asking them to through our curriculum. Acceleration, I think, often applies when kids are just well beyond grade level—but enrichment is really for every single kid. Mike: Yeah, I think you answered, at least partly, the question that I was going to pose next, which was a question about access. Because at least with Concepts Quests, which is the MLC supplemental resource, we would describe this as a tool that should be made available to all students, not a particularly small subset of students. And I'm wondering if you can talk a little bit more about the case for that. Tisha: So, if we go back to our lasagna problem, once our kids have had opportunities to read it and make sense out of it, at that point, I truly believe that there is an entry point in these problems for any kid. These are not dependent on computation. So, a student can draw pictures. I believe that all of my students that I've had throughout my years of teaching were capable of drawing a picture to model a problem. Then, I really believe that a good problem can have an entry point for every student. Mike: The other thing that you're really making me think about is, how much we've equated the idea of enrichment, acceleration. We've fused those ideas, and we've really associated it with procedure and calculation versus problem-solving and thinking creatively. Tisha: I think that happens a lot. I think that's a lot of how people think about math. You know, it's who can do it fast, who can get there? But what I think our goal is, is to create students who are not just able to be calculators, but who are able to apply their understandings of multiplication, addition, subtraction, division. They can apply them to novel problems. Mike: Yeah, and the real world isn't designed with a set of “Free set, here's what you should do, repeat directions.” Tisha: ( laughs ) I would love some of those. Where can I find them? Mike and Tisha: ( laugh) Mike: This has been fascinating, and I think we could and probably should do more work on Rounding Up talking about these versions of enrichment that are available for all kids. And I have a suspicion that this conversation is going to cause a lot of folks to reassess, reevaluate, and reflect on how they've understood the idea of enrichment. I'm wondering if we can help those folks out. If I'm an educator who's really interested in exploring the idea of horizontal enrichment in more detail, where might I get started? Or, perhaps, where are there some resources out there that might contain the types of problems that you introduced us to today? Tisha: Well, of course, I have to say Concept Quests. We've put a lot of work into creating some really great tasks. But some other places where you can find tasks that are engaging and help kids to think more deeply are “Open Middle” and “NRICH” and “YouCubed” are just a few resources that I can think of off the top of my head. Mike: Ahh, those are great ones. Tisha, thank you so much for joining us. It's really been a pleasure to have this conversation. Tisha: This has been so fun. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 4 – Joy in the Elementary Math Classroom Guest: Amy Parks, Ph.D. Mike Wallus: Teaching is a complex and challenging job. It's also one where educators experience moments of deep joy and satisfaction. What might it look like to build a culture of joy in an elementary mathematics classroom? Michigan State professor Amy Parks has some ideas. Today on the podcast, we explore ways educators can construct joyful experiences for their youngest mathematics learners.  Mike: Well, welcome to the podcast, Amy. I'm so excited to be talking with you about joy in the elementary mathematics classroom. Amy Parks: I'm so happy to be here. Mike: So, your article in MTLT was titled, “Creating Joy in PK–Grade 2 Mathematics Classrooms.” And early on you draw a distinction between math classrooms where students are experiencing joy and those that are fun. And you quote Desmond Tutu and the Dalai Lama, who say, “Being joyful is not just about having more fun, we're talking about a more empathic, more empowered, more spiritual state of mind that's totally engaged with the world.” That really is powerful. So, I'm wondering if you could tell me about the difference between classrooms that foster joy and those that are just more fun. Amy: Yeah, I was very struck by that quote when I read it the first time in “The Book of Joy.” And I think one of the reasons that book is powerful for me is that the two people writing it didn't have these super easy lives, right? Particularly the Archbishop Desmond Tutu was imprisoned in the country that was openly hostile to him, and yet he was still really committed to approaching his work and the world with joy. And so, I often think if he could do that, then surely the rest of us can get up and do that. And it also tied into something I often see in elementary classrooms, which is this focus on activities that are fun, like sugary cereal, right? They're immediately attractive, but they don't stick with us and maybe they're not really good for us. I often think the prototypical example is, like, analyses of packets of M&Ms. When I think about the intellectual energy that has gone into counting and sorting and defining colors of M&Ms, it makes me a little sad, given all the big questions that are out there that even really young kids can engage with. And so, yes, I want children to be playful and to laugh and to engage with materials they enjoy. But also, I think there is this quieter kind of joy that comes from making mathematical connections and understanding the world in new ways and grasping the thinking and ideas of others. And so, when I'm pointing toward joy, that's part of what I'm trying to point toward. Mike: So, I want to dig into this a little bit more because one of your first recommendations for sparking joy is this idea that we need to make some room for play. And my guess is that that raises many questions for elementary educators, like “What do you mean by play?” and “What role does the teacher play in play?” Can you talk a little bit about this recommendation, Amy? Amy: Yeah. So, when I have more time than that very short article to talk about, one of the things that I like to bring out to teachers is that we can think of play in sort of three broad buckets. So, one is “free play,” and this is an area where the teacher may not have a lot of roles except to sort of define health and safety limits. So certainly, recess is a place of free play. But there are places at recess where children are encountering mathematical ideas, right? There are walking in straight lines and they're balancing on things and they're seeing whether they all have the same amount of materials and toys. So, those are all mathematical contexts that we can, as teachers later bring in and highlight in places where they can engage. But they're not places where teachers are setting learning goals and reinforcing things. And particularly in the lower grades, we might see also free play opportunities in the classroom. Amy: You know, many kindergarten classrooms have opportunities for free play during the school day. So, while kids are playing in the kitchen for example, or doing puzzles, they may be again encountering mathematical ideas and teachers certainly can capitalize on that. But they're not directing or shaping the play. And then there are these two other categories where the teacher's role is maybe more present. So, one I would call “guided play.” And this is a case where the teacher and the children are really handing responsibility back and forth. So, the teacher might set up a relatively open-ended task like pattern block puzzles or a commercial game that gets at counting or something like that. And so, the teacher has an intended mathematical goal. She has set some limits to keep children focused on that in some way. But the task is in the hands of the kids. They're playing together, they're negotiating roles, they have that more central responsibility. And the learning goals may be a little bit broader and more open because of that. Because since you're not centrally involved, you can't be so specific. Amy: And then the last kind of play I talk with teachers about are “playful lessons.” Children might not have as much choice in the activity that they do. They might not be able to stop and start it or move in certain ways, but teachers are intentionally bringing aspects of play into the mathematics lesson. And that could be by using engaging materials. It could be by creating places for creativity. It could be by creating spaces for social collaboration. It could be just by inviting children to use their bodies in ways that are comfortable to them instead of being really constrained. But the mathematical task might be much more specific and “Build this cube and identify the vertices on it.” So, the task is constrained, but because they're using materials, because they can do it in different ways, there's this playful aspect to it. So, I like to encourage teachers to sort of think those three buckets of play and where kids are getting access to them during the day. Mike: Yeah, I think that's really helpful. Because I did teach kindergarten for a long time, and so I think my definition of play was really the first one that you were talking about, which is free play. But hearing you talk about the other two definitions actually helps open space up for me. I feel like with that broader definition, it helps me consider the choices that I've got in front of me. Amy: Yeah, and if you talk [to]—or read even—mathematicians, they will often talk about playing with ideas. So, there is a part of play that is inherently mathematical, the part that is about experimenting and investigating and trying things out and recognizing that you might be wrong and getting this engagement from others. So, I think sometimes even mathematics lessons that look relatively traditional can also have this playful spirit if we bring that to it. Mike: I would love to talk to you a little bit about the way that choice can be a key component in sparking joy. So, what are some of the options that teachers have at their disposal to offer choice to learners in their classrooms? Amy: Yeah, I think that this is something that's often overlooked. And I think that for kids in school right now, they often have so few choices. Their experiences are often so constrained by adults. And simply by allowing children to choose when they can, we can make experiences more joyful for them. So, one easy thing is who or whether children will work with other people. So yes, there are all kinds of benefits to group tasks and social interactions, but also lots of children are introverts. And being in a small room for six hours a day with 25 other people can be exhausting. And so, simply giving the children the choice to say, “I'm going to do this one on my own,” can be a huge relief to some children. Other children, like, need to talk—just like other adults—talk to others to know what they're thinking. And so, they need these groups. Amy: And then I think also teachers can get really involved in choosing the magic right group, but often there is no magic right group as we know because we're constantly rejuggling these groups because they didn't work in the magic way we thought. And so just letting kids pick their groups, because then they have responsibility for that interaction. And it's not that they never have difficult social interactions, but they've chosen to be with this person and they have to work through it. So that's one. The other thing is letting children choose physically where they work. Some children lie on the floor while they work, or some children stand up at their seat. Allowing some choice in freedom of movement doesn't mean allowing total chaos. And I think even pretty young children can be taught that they can move within limits in the classroom. And I think if children get to stop expending so much energy trying to control their bodies in the ways adults find helpful, they can engage more fully in the academics of the day. Amy: And then, like, choices of materials. So, we can make different things available to kids as they engage with mathematics, choices of problems. They may choose to do some and not others. Lower grades like using centers. If we have multiple centers that all get at the same mathematical idea, maybe it doesn't actually matter whether all kids get to all of them, right? As long as they're engaging with making units of 10, however they're doing that, can work for us. So, I think in general, the more often we can give children choices about anything, the better off all of us are. Mike: I think that last bit is really interesting. I just want to pause for a second on it. Because what you've got me thinking is, if I have options available and they're all really addressing some of the same mathematical goals or a range of goals that I have in my class, this idea that I can release control and invite kids to make choices, that seems like a really practical first step that a teacher could take to think about, “What are the options? What are the goals that they meet?” And then, “To what degree can I offer those as choices?” Amy: Yeah, and in a really basic way, right? Sometimes we might have a game that works with kids on making tens, and then other times we might have a project or even a worksheet. And different kids may be drawn to those different things. There are some kids for whom games might be really exciting, but there are some kids for whom games might be really stressful, and they would just rather do something else. And that's fine because the point isn't actually playing the game, right? Mike: I think that's really interesting. I could get so caught up as a teacher sometimes trying to get the mechanics of getting kids out to places and getting kids started and making sure that kids were doing the thing that I would sometimes lose track of, “My point in doing this is to have kids think about structuring 10 or making sense of fractions.” That's a lovely reminder. I really appreciate that. I think that this is a really nice turning point because this question about choice actually plays into one of the other recommendations you had regarding time on task. So, I would love to have you unpack your thinking on this topic, Amy. Amy: Yeah. Well, you talked about being autobiographical, and this is definitely autobiographical for me because I am very on task. I like to get things done. I like to check things off my list. And that was definitely a force for me when I was teaching. And I think it was something that, one, caused anxiety for me and my kids, and two, limited our opportunities to engage in more playful ways and more joyful ways to follow curiosities because I was so worried about that. And honestly, when it came home to me was when I started teaching university students because I think it is a little harder to clap your hands at 19-year-olds and tell them to get back to work than to do it with 7-year-olds. And what I realized was if I step back and I let my students talk about “The Bachelor” for a minute, they would have the conversation and then they would move on to the mathematical task, and I actually didn't need to intervene. And me intervening would've shifted the emotional tone of the class in a way that would not have been productive for learning, right? Amy: They would've become resentful or maybe felt self-conscious. And now I have this thing in the way as opposed to just letting them have that break. And I think if we pay attention as adults to how we are in staff meetings or how we are in professional development, we recognize we have a lot of informal conversations around the work we do, and that those informal conversations are not distractions. They're actually, like, building the relationships that let us do the work. And it is similarly true for children. And then I think another thing to remember about particularly young children is language learning, social relationships, all of those are things they actually need to develop. That's part of our work as teachers is to help them grow in those things. And so, giving them the opportunity to build those relationships is, in fact, part of our work. Mike: I think that's really interesting because I found myself, as you were talking, thinking through my own day, when I log into Zoom to talk to someone across the country. We don't immediately start just working through our agenda. We exchange pleasantries, we tell a joke or two, we talk about what's going on in our world, and we can have an incredibly productive chunk of time. But there are these pieces of social reality that kind of bind us together as people, right? When I'm talking to my friend Nataki in North Carolina, I'm asking her about her son. That might take two minutes out of 55. We've still done a tremendous amount of work and thought deeply about the kind of professional learning we want to provide to teachers. But there's the reality that if we didn't do that, how are we connected? If we're partnering to do this work, there's something about being connected to the other person that we can't schedule out of the experience of working together. Does that make sense? Amy: Yeah, a hundred percent. And it's true in classroom settings, too. I was thinking the “Batman” movie, the Ben Affleck one was filmed in Detroit, and they happened to be filming right outside the building where I was teaching. And at some point, one of my adult students looked out the window and was, like, there's Ben Affleck. And of course, all my students got up and went to the window. I could have as the teacher been, like, “OK, sit down. We're doing whatever we're doing.” But their minds were all going to be on Ben Affleck out the window. And so instead, we stopped and we watched the movie for a little bit, and that became an experience we came back to as a class over and over in the semester. “Remember when that happened?” And so, yeah, that pressure to be productive I think often interferes with the relationship building that does support good work among adult colleagues and among kids in classrooms. And I would also connect it to the opening conversation on play. Mike: So, before we close the interview, I'm wondering if you have any recommendations for someone who wants to continue learning about how they could design opportunities for joy in their classrooms. Are there any resources that you would point a listener to? Amy: I mean, I have a book on play in early mathematics, and that would certainly be a place that someone could start. But, you know, the other thing that I might do is just look at some of the great materials that are out there, both like physical things like Legos and magnet tiles, which often if you don't have at your school, you can get through thrift stores and things. And just bringing them into classrooms and seeing what kids do with them. Oh, the other thing that I always recommend is looking at some of the resources on “soft starts.” And if you just Google this, you'll see videos and articles. And this is often a really, like, nonthreatening way for teachers who are interested in this but haven't done a lot of play in their classrooms, to begin.  Amy: And the idea is instead of immediately starting with a worksheet or whatever, that you bring in some kind of toy or tool, and maybe children can make some choices about whether they're going to paint or they're going to work on a puzzle, and you just take 15 minutes and that's how you begin the day. And people who have done this, so many people have said it's just been such a lovely culture shift in their classroom, and it also means that children are coming in a little late. It's fine. They can just come in and join, and then everyone's ready to go 15 minutes later, and you really haven't given up that much of your day. So, I think that can be a really, a really smooth entry into this if you're interested. Mike: Well, I want to thank you so much for joining us, Amy. It really has been a pleasure talking with you. Amy: Oh, you, too. It was so fun. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 3 – Student Engagement Guest: Dr. Meghan Shaughnessy Mike Wallus: When we say students are engaged in a discussion or a task, what do we really mean? There are observable behaviors that we often code as engaged, but those are just the things that we can see or hear. What does engagement really mean, particularly for students who may not verbally participate on a regular basis? Today on the podcast, we're talking with Dr. Meghan Shaughnessy about the meaning of engagement and a set of strategies teachers can use to extend opportunities for participation to each and every student. Mike: Welcome to the podcast, Meghan. We are super excited to have you joining us. Meghan: I'm excited to be here. Mike: So, I want to start with a question that I think in the past I would've thought had an obvious answer. So, what does or what can participation look like? Meghan: So, I think in answering that question, I want to start with thinking about one of the ways that teachers get feedback on participation in their classroom is through administrator observation. And oftentimes those observations are focused on students making whole-group verbal contributions and discussions, particularly with a focus on students sharing their own ideas. Administrators are often looking at how quiet the space is and how engaged students appear to be, which is often determined by looking at students' body language and whether or not that language matches what is often seen as listening body language, such as having your head up, facing the speaker, et cetera. And as I say all of this, I would also say that defining participation in this way for discussions is both a limited and a problematic view of participation. I say limited in the sense that not all participation is going to be verbal, and it certainly won't always include sharing new ideas. Meghan: So, to give a concrete example, a student might participate by revoicing another student's strategy, which could be really important, providing other students a second chance to hear that strategy. A second example is that a student might create a representation of a strategy being shared verbally by a classmate. And this nonverbal move of creating a representation could be really useful for the class in developing collective understanding of the strategy. The traditional view is problematic, too, in the sense that it assumes that students are not participating when they don't display particular behaviors. To turn to a more equitable approach to conceptualizing and supporting participation, I and my colleagues would argue that this includes learning children's thinking body language, including a focus on written pair talk, and supporting contributions. In other words, moving beyond just having students share their own ideas, having students share what they learned from our classmate. Mike: Yeah. I want to dig into this a little bit more. Because this idea that my read on a child's behavior influences my understanding of what's happening, but also my practice, is really interesting to me. You've really had me thinking a lot about the way that a teacher's read on a student's engagement or participation, it has a lot to do with the cultural script for how adults and children are expected to interact, or at least what we've learned about that in our own lived experiences. I'm wondering if you could just talk a little bit about that.  Meghan: Yeah. One way to start answering that question might be to ask everyone to take a minute to think about how you participate in a discussion. Do you use the sort of listening behaviors that teachers are told matter? Are you always sharing new ideas when you participate in a discussion? You also might want to imagine sitting down with a group of your colleagues and asking them to think about when they engage in a discussion outside of class, what does it look and feel like? Are there lots of people talking at once or people talking one at a time? Is everyone that's participating in the discussion sharing new ideas, or are they participating in other sorts of ways? And further, you might imagine asking those colleagues about their discussions outside of class as a child. What did those discussions look and feel like? One of the challenges of being teachers is that we bring our own experiences and sometimes we don't reflect on what children are experiencing. Children's experiences don't necessarily match our own, and we need to be thinking about changing our expectations or explicitly teaching what it means to participate in particular sorts of ways. Yet another layer of challenge here is a tendency to make assumptions about how students from particular cultural groups engage in discussions. You only know what you know. And teachers need opportunities to learn from their students about how they engage in discussions inside and outside of math class, and to be able to think about the connections and disconnections and the opportunities to leverage. Mike: So, you really have me deconstructing some of the norms that were unspoken in my own childhood about being a learner, being a good student. And what you have me thinking is, some of those were voiced, some of those were unvoiced, but I'm really reflecting on how that showed up in the way that I read kids. So, I want to ask you to even go a little bit deeper. Can you share some examples of where our read on the meaning of behaviors might lead to an inaccurate understanding of students' cognitive engagement or the contributions that they might make to discourse? Meghan: Yeah. Some of it can be thinking about sort of traditional behavior reads in a traditional sense. Oftentimes, when children have their heads down or their eyes closed or they're not looking at the speaker, the child is seen as not engaging or participating. But if we think about it, people have lots of different thinking postures, and for some people having their heads down or closing their eyes is actually the way in which they're thinking deeply about the ideas that are being shared in the discussion. And so, engagement might look for them. They may be carefully tracking and thinking about the ideas, but the way that that gets expressed may not be the way that we traditionally think about what engagement should look like in classrooms. Mike: It feels like there's two pieces to this question about reading behavior and interpretation. One piece that you talked about there was just this idea that we need to have conversations with children. The other piece that I kept thinking about is, how might an educator interrogate their own cultural script around participation? Are there questions that educators could ask themselves or practices that they might engage in with colleagues that would help them take these things that are subconscious and unspoken and maybe raise them up? So, if you have an awareness of them, it's easy to recognize how that's influencing your read or your instructional moves. Meghan: Yeah, I think there are kind of two pieces to this. So, one goes back to the idea that I shared about the importance of recognizing our own experiences in school as a student and our experiences out of school, both as a child and as an adult in discussions and trying to think about what are we bringing to our work as a teacher that we might need to interrogate because it may be different than the experiences of children? And at the same time, we need to be having conversations with children about what it looks like to participate in discussions in different sorts of spaces so that we can learn more about what children's experiences are outside of school. The big idea is to recognize that children's experiences are often very different from our own, and we have to be careful at the same time not to make assumptions that all children from particular communities experience participation and discussion in the same way. This can be highly variable. Mike: I think what's really interesting about the work that you and your colleagues have done is, there's an element of it that's really about taking a step back and recognizing these ideas like cultural scripts that we have about participation and really trying to interrogate our own understandings that we've come to, and then how do we interact with kids. But on the other hand, you all have some really practical strategies and suggestions for educators on how they can use an expanded understanding of participation to create more opportunity for kids. So, I'm wondering if we can talk a little bit about some of those things. Meghan: Absolutely. So, I have a set of four different strategies that my colleagues and I have been working on over time. So, I'm going to start by talking about task selection. Sometimes students' cultural backgrounds and experiences in schools may be at odds, particularly around the work of critiquing the ideas of others. And this can in particular be a challenge when the critiquing is about critiquing the teacher's ideas. So, it leads to this question of, “How can we support students in learning to critique in ways that don't dismiss their own culture and experience?” So, our practical solution to working in this space is that we've used written critique tasks. So, when working with students, we'll show a fictitious person's response to a mathematics task and ask students to do three sorts of things. So, one is to describe the student's strategy in their own words. A second thing is to think about and write down the questions that they have about the student's strategy. And then the third piece is for students to think about and record what suggestions they have for the student and how they would convince the student to use those suggestions. Meghan: So, how does this support participation? Well, it can explicitly support the work of critiquing. It's written, and it allows students to think carefully rather than needing to think on the spot. And thirdly, the student is not a classmate, which can reduce the feeling of confrontation that some students feel when engaging in critique. So, one thing that I want to name with this particular strategy around task selection and using a written critique task, is that we've recognized that the way that critiquing is often worked on in mathematics classrooms may be at odds with some students' experiences with critique outside of school. And so, we're not trying to say that students shouldn't be supported in learning to critique mathematical ideas. That's an important part of mathematical work. But rather we're trying to design a structure that's going to not dismiss students' experiences outside of school, but at the same time give them experiences with the mathematical work of critiquing. Mike: Yeah, the questions themselves are powerful, but it seems like the choice to use a fictitious person is really critical to this task design. Meghan: Absolutely. And as a teacher, too, it really does give us a little bit more control in terms of what is the critique that's going to unfold in that particular classroom. Mike: It strikes me that they're able to engage in the task of critique without that feeling of conflict. Meghan: Absolutely. It really opens up space for students to engage in that critiquing work and takes a lot of that pressure off of them. Mike: Let's talk about the second idea. Meghan: Alright. So, the second strategy is to use a deliberate turn and talk. In discussions, some students are ready to share their ideas right away, but other students need a chance to practice verbalizing the ideas that they're about to share. Sometimes students' ideas are not completely formed, and they need to learn how others hear the ideas to refine their arguments. Further, in multilingual classrooms, sometimes students need opportunities to refine their thinking in their home language, and importantly, they also need opportunities to develop academic language in their home language. So, in a deliberate turn and talk, a teacher deliberately pairs students to share their thinking with a partner, and the partner asks clarifying questions. The pairs might be made based on knowledge of students' home language use, their mathematical understandings, or some other important thing the teacher is thinking about as they engage in that pairing. So how might using deliberately paired turn and talks broaden participation in a discussion? Meghan: Well, first, all students are being asked to participate and have the opportunity to refine their own mathematical argument and consider someone else's ideas. In a whole-class discussion, it's not the case that every student is likely to have that opportunity. So, turn and talks provide that opportunity. Second, turn and talks can support a broader range of students in feeling ready and willing to share their thinking in a whole group. Third, these pairs can also set up students who are not yet comfortable sharing their own ideas in whole group to be able to share someone else's idea. So, a way for them to still share ideas in whole group, even though it's not necessarily their own idea that's being shared. Mike: So, what I'm thinking about is, if you and I were engaged in a deliberate turn and talk, what might it look like if I'm a student, you're a student and we've engaged in the norms of the deliberate turn and talk as you described them? Let's just walk through that for a second. What would it look like? Meghan: So, in a pair turn and talk, it really has the structure of partner A, sharing their thinking, and then partner B being responsible for asking questions about the ideas that they just heard in order to further their own understanding of partner's ideas, but also to provide partner A with some feedback about the ways in which they've been expressing their ideas. So, that's pretty different than what often happens in classrooms where kids are invited to share in a discussion and they actually haven't tried verbalizing it yet, right? And they have no way of thinking about, or limited ways of thinking about, how other people might hear those ideas that they're about to share. Mike: I think the other thing that pops up to me is that another scenario that often occurs in turn and talk is it's really turn and tell. Because one person is essentially sharing their thinking and the norms aren't necessarily that they respond, it's just that they share in kind, right? So, this idea that you're actually engaging with someone's idea feels like an important piece of what it looks like to do a deliberate turn and talk versus some of the other iterations that we've just been describing. Meghan: Absolutely. Mike: Well, I'm excited to hear about the third strategy. Meghan: Alright. Our third strategy focuses on supporting participation through connection-making. So, when you think about a typical discussion in a classroom, opportunities for individual students to make explicit connections between ideas shared, are often pretty limited—or at least their opportunities to verbalize or to record in some other way. Often, only one or two students are able to share the connections. And so, a question for us has been how can we provide opportunities for students who are not yet ready to share those connections in whole group or might not have the opportunity? When you think about the fact that 28 students are not going to be able to share connections on a given day to be able to engage in the making of those connections. So, we have two different structures that we have been exploring. The first structure is really a pair share. Students are paired, if possible, with a student who used a different strategy, who has a different solution. Meghan: Each partner explains their strategy, and then together they look for connections between their thinking. So again, this moves beyond the traditional turn and talk because in addition to sharing your thinking, there's a task that the partners are doing about thinking about the connections between those two strategies. A second sort of structure is really using a stop and jot. In this instance, the teacher selects one strategy for students to be thinking about making a connection to, and then each student jots a connection between their strategy or solution and the strategy that the teacher has selected. And they do this in their notebook or in some other written form in the classroom. And so, these two different structures can support participation by having all students have an opportunity to share their own thinking, either verbally with a partner or by recording it in written form. And all students at the same time are having an opportunity to make connections in the classroom. Mike: I think what's interesting about that is to compare that one with the initial idea around critique. In this particular case, I'm going to make a guess that part of the reason that in this one you might actually use students from the classroom versus a fictitious student, is that connecting versus critiquing our two really different kind of social practices. Is that sensible? Meghan: That is sensible. And I would argue that if you're going to be engaging in critique work just to say it, that part of critiquing actually is recognizing, too, what is similar and different about strategies. Mike: Gotcha. Meghan: Right? So, there is that piece in addition to put that out there. Mike: Gotcha. Let's talk about the fourth one. Meghan: Alright. So, the fourth strategy really focuses on broadening participation in the conclusion of a discussion. So, as we all know in a discussion, students hear lots of different ideas, but they don't all get to share their thinking in a discussion, nor do they all get to share what they are thinking at the end of the discussion. But we also know that students need space to consolidate their own thinking and the questions that they have about the ideas that have been shared. At the same time, teachers need access to students' thinking to plan for the next day, particularly when a discussion is not finished at the end of a given math lesson. With all of this, the challenge is that time is often tight at the end of a discussion. So, one structure that we've used has been a note to self. And in a note to self, students write a note to themselves about how they are currently thinking about a particular sort of problem at the end of a discussion. And a note to self allows students to take stock of where they are with respect to particular ideas, similar to a stop and jot. It can create a record of thinking that can be accessed on a subsequent day by students. If those notes yourself are recorded in a notebook. Again, support students and tracking on their own questions and how their thinking is changing over time, and it can provide the teacher with a window into all students' thinking. Mike: Can you talk about the experience of watching the note to self and just seeing the impact that it had? Meghan: So, it was day one of our mathematics program, and we had done a discussion around an unequally partitioned rectangle task, and students were being asked to figure out what fraction of the hole was shaded. And there clearly wasn't enough time that day to really explore all the different sorts of ideas. And so, Darius Robinson, who was one of the co-teachers, invited students to share some of their initial ideas about the task. And the way that Darius then ended up deciding to conclude things that day was saying to students, “I think we're going to do this thing that I'm going to call a note to self.” And he invited the students to open up their notebooks and to record how they were thinking about the different ideas that had gotten shared thus far in the discussion. There was some modeling of what that might look like, something along the lines of, “I agree with … because,” but it really opened up that space then for students to begin to record how they were thinking about otherwise ideas in math class. So, how might using a note to self-broaden participation in a discussion? Well, first of all, students have the opportunity to participate. All students are being asked to write a note to themselves. It creates space for students to engage with others' ideas that doesn't necessarily require talk, right? So, this is an opportunity to privilege other ways of participating, and it also allows for thinking and processing time for all students. Mike: I think the other piece that jumps out for me is this idea that it's normal and to be expected that you're going to have some unfinished thinking or understanding at the end of a particular lesson or what have you, right? That partial understanding or growing understanding is a norm. That's the other thing that really jumps out about this practice is it allows kids to say, “This is where I am now,” with the understanding that they have room to grow or they have room to continue refining their thinking. I really love that about that. Meghan: I think it's so important, right? And oftentimes, we read curriculum materials, we read through a lesson for a particular day and get the sense that everything is going to be tied off with a bow at the end of the lesson, and that we're expecting everybody to have a particular sort of understanding at the end of Section 3.5. But as we all know, that's not the reality in classrooms, right? Sometimes discussions take longer because there are really rich ideas that are being shared, and it's just not feasible to get to a particular place of consensus on a particular day. So, it is for teachers to have access to where students are. But at the same time to feel empowered, to be able to say, “I'm going to pick this up the next day, and that doesn't need to be finished on Monday, but that these ideas that we're working on Monday can flow nicely into Tuesday. And as students, your responsibility is to think about, ‘How are you thinking about the task right now?' Jot some notes so when we come back to it tomorrow, we can pick that up together.” Mike: Well, I think that's the other lovely piece about it, too, is that they're engaging in that self-reflection, but they've got an artifact of sorts that they can come back to and say, “Oh yeah, that's where I was, or that's how I was thinking about it.” That allows for a smoother re-engagement with this or that idea. Meghan: Absolutely. And you can add on the pieces of notation that students might choose to do the next day as well, where they might choose to annotate their notes with notes that said, “Yesterday I was thinking this, but now I think this” as a way to further record the ideas that thinking changes over time. Mike: So, I think before we close this interview, I want to say to you that I watched you do your presentation in Los Angeles at NCTM, and it was really eye-opening for me, and I found myself stuck on this for some time. And I suspect that there are people who are going to listen to this podcast who are going to think the same thing. So, what I want to ask you is, if someone's a listener, and this is a new set of ideas for them, do you have any recommendations for where they might go to kind of deepen their understanding of these ideas we've been talking about? Meghan: Sure. I want to give three different sorts of suggestions. So, one suggestion is to look at the fabulous books that have been put together by Amy Lucenta and Grace Kelemanic, who are the authors of “Routines for Reasoning and Thinking for Teaching.” And I would argue that many of the routines that they have developed and that they share in those resources are ones that are really supportive of thinking about, “How do you broaden participation in mathematics discourse?” A second resource that someone might be interested in exploring is a research article that was written in 2017 by Cathy O'Connor, Sarah Michaels, Suzanne Chapin, and Alan (G.) Harbaugh that focuses on the silent and the vocal participation in learning in whole-class discussion, where they carefully looked at learning outcomes for students who were vocally expressing ideas and discussion as well as the silent participants in the discussion, and really found that there was no difference in the learning outcomes for those two groups of students. And so that's important, I think, for us to think about as teachers. At the same time, I want to be clear in acknowledging that all of what we do as teachers needs to be in relation to the learning goals that we have for students. So, sometimes our learning goals are that we want students to be able to share ideas and discussions. And if that's the case, then we actually do need to make sure that we build in opportunities for students to share their ideas verbally in addition to participating in other sorts of ways. Mike: I'm really glad you said that because what I hear you saying is, “This isn't a binary. We're not talking about … Meghan: Correct. Mike: … verbal participation and other forms of participation and saying you have to choose.” I think what I hear you saying is, “If you've only thought about participation from a verbal perspective, these are ways that you can broaden access and also access your students' thinking at the same time.” Meghan: Absolutely. The third thing to share, which has been a theme across this podcast, has really been the importance of learning from our students and talking with the children with whom we're working about their experiences, participating in discussions both in school and outside of school. Mike: Megan, thank you so much for joining us. It really was a pleasure. Meghan: Thank you, Mike, for the opportunity to really share all of these ideas that my colleagues and I have been working on. I want to acknowledge my colleagues, Nicole Garcia, Aileen Kennison, and Darius Robinson, who all played really important roles in developing the ideas that I shared with you today. Mike: Fabulous. Thank you so much. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 2 – Empathy Interviews Guest: Dr. Kara Imm Mike Wallus: If there were a list of social skills we hope to foster in children, empathy is likely close to the top. Empathy matters. It helps us understand how others are feeling so we can respond appropriately, and it can help teachers understand the way their students are experiencing school. Today on a podcast, we talk with Dr. Kara Imm about a practice referred to as an empathy interview. We'll discuss the ways empathy interviews can help educators understand their students' lived experience with mathematics and make productive adaptations to instructional practice. Mike: Well, welcome to the podcast, Kara. We're excited to have you join us. Kara Imm: Thanks, Mike. Happy to be here. Mike: So, I have to confess that the language of an empathy interview was new to me when I started reading about this, and I'm wondering if you could just take a moment and unpack, what is an empathy interview, for folks who are new to the idea? Kara: Yeah, sure. I think I came to understand empathy interviews in my work with design thinking as a former teacher, classroom teacher, and now teacher-educator. I've always thought of myself as a designer. So, when I came to understand that there was this whole field around design thinking, I got very intrigued. And the central feature of design thinking is that designers, who are essentially thinking about creating new products, services, interactions, ways of being for someone else, have to start with empathy because we have to get out of our own minds and our own experiences and make sure we're not making assumptions about somebody else's lived experience. So, an empathy interview, as I know it now, is first and foremost a conversation. It's meant to be as natural a conversation as possible. When I do empathy interviews, I have a set of questions in mind, but I often abandon those questions and follow the child in front of me or the teacher, depending on who I'm interviewing. Kara: And the goal of an empathy interview is to elicit stories; really granular, important stories, the kind of stories that we tell ourselves that get reiterated and retold, and the kinds of stories that cumulatively make up our identities. So, I'm not trying to get a resumé, I'm not interested in the facts of the person, the biography of the person. I'm interested in the stories people tell about themselves. And in my context, the stories that kids tell themselves about their own learning and their own relationship to school, their classrooms, and to mathematics. I'm also trying to elicit emotions. So, designers are particularly listening for what they might call unmet needs, where as a designer we would then use the empathy interview to think about the unmet needs of this particular person and think about designing something uniquely and specifically for them—with the idea that if I designed something for them, it would probably have utility and purpose for other people who are experiencing that thing. So, what happened more recently is that I started to think, “Could empathy interviews change teachers' relationship to their students? Could it change leaders' relationships to the teachers?” And so far, we're learning that it's a different kind of conversation, and it's helping people move out of deficit thinking around children and really asking important questions about, what does it mean to be a kid in a math class? Mike: There's some language that you've used that really stands out for me. And I'm wondering if you could talk a little bit more about it. You said “the stories that we tell about ourselves”; or, maybe paraphrased, the stories that kids tell themselves. And then you had this other bit of language that I'd like to come back to: “the cumulative impact of those stories on our identity.” Can you unpack those terms of phrase you used and talk a little bit about them specifically, as you said, when it comes to children and how they think about their identity with relation to mathematics? Kara: Sure. I love that kind of phrase, “the story we tell ourselves.” That's been a pivotal phrase for me. I think stories kind of define and refine our existence. Stories capture this relationship between who we are and who we want to become. But when I'm thinking about stories in this way, I imagine as an interviewer that I'm trying to paint a portrait of a child, typically. And so, I'm trying to interact with this child in such a way that I can elicit these stories, painting a unique picture of this kid, not only as a learner but also as a human. What inevitably happens when you do these interviews is that I'm interested in their experience in math class. When I listen to kids, they have internalized, “I'm good at math, and here's why” or “I'm bad at math, and here's why. I just know it.” But when you dig a little bit deeper, the stories they tell are a little more nuanced, and they kind of live in the space of gray. And I'm interested in that space, not the space of testing and measurement that would land you in a particular identity as meant for math or not meant for math. Mike: I think what I was going to suggest is, why don't we listen to a few, because you shared a couple clips before we got ready for the interview, and I was fascinated by the approach that you had in chatting with these children and just how much information I could glean from even a minute or two of the interview slices that you shared. Why don't we start and get to know a few of these kiddos and see what we can learn together. Kara: Sounds great. Mike: We've got a clip that I'm going to invite you to set it up and give us as much context as you want to, and then we'll play the clip and then we can talk a little bit about it. I would love to start with our friend Leanna. Kara: Great. Leanna is a third-grader. She goes to an all-girls school. I've worked in Leanna's school over multiple years. I know her teacher well. I'm a part of that community. Leanna was kind of a new mathematician to me. Earlier in the day I had been in Leanna's classroom, and the interview starts with a moment that really struck me, which I won't say much more about. And I invited Leanna to join me after school so we could talk about this particular moment. And I really wanted to know how she made sense of what happened. So, I think we'll leave it at that and we'll listen to what happened. Mike: Alright, let's give it a listen. Leanna: Hi, I'm Leanna, and I'm 8 years old. Kara: Hi, Leanna. Today when I was in your class, something interesting happened where I think the kids said to me, and they said, “Do you know we have a math genius in our class?” Do you remember that moment? Leanna: Yeah. Kara: Tell me what happened in that moment. Leanna: Um, they said, “We have a math genius in our class.” And then they all started pointing at me. Kara: And what was that like for you? Leanna: It was … like, maybe, like, it was nice, but also it was kind of like, all the pressure was on me. Kara: Yeah, I was wondering about that. Why do you think the girls today—I mean, I'm a visitor, right?—why do you think they use the word “math genius”? And why did they choose you? What do you think they think of you? Leanna: A mathematician … Kara: Yeah. Leanna: … because I go to this thing every Wednesday. They ask me what I want to be when I grow up, and I always say a mathematician. So, they think that I am a math genius. Kara: Gotcha. Do you think all the girls in your class know that you want to be a mathematician when you grow up? But do they mean something else? They didn't say, “We have a mathematician in our class.” They said, “We have a math genius.” Leanna: Maybe. Kara: Are you a math genius? Do think, what does that even mean? Leanna: Like, I'm really good at math. Kara: Yeah. Do you think that's a true statement? Leanna: Yeah, a little bit. Kara: A little bit? Do you love math? Leanna: Yeah. Kara: Yeah. Have you always loved math? Leanna: Yeah. Kara: And so, it might be true that, like, is a math genius the same as a mathematician? Leanna: No. Kara: OK. Can you say how they're different? Leanna: Like, a mathematician is, like … Like, when you're a math genius, you don't always want to be a mathematician when you grow up. A math genius is when you just are really good at math, but, like, a mathematician is when you really, like, want to be when you grow up. Kara: Yeah. Mike: That was fascinating to listen to. So, my first inclination is to say, as you were making meaning of what Leanna was sharing, what were some of the things that were going on for you? Kara: Yeah, I was thinking about how math has this kind of unearned status, this measure of success in our culture that in this interview, Leanna is kind of pointing to. I was thinking about the mixed emotions she has being positioned as a math genius. It called into mind the model minority myth in which folks of Asian descent and Asian Americans are often positioned as stereotypically being good at math. And people say, “Well, this is such a lovely and respectful stereotype, who cares if it's not true?” But she later in the interview talks about the pressure of living up to this notion of math genius and what means. I think about her status in the classroom and how she has the agency to both take up this idea of math genius, and does she have the agency to also nuance it or reject it? And how that might play out in her classroom? So yeah, those are all the things that kind of come to mind as I listen to her. Mike: I think you're hitting on some of the themes that jumped out for me; this sense that kids who are participating in particular activities have been positioned, either by their participation or by their kids' perceptions of what participation means. And I thought the most interesting part was when she said, “Well, it's nice”—but there was a long pause there. And then she talked about this sense of pressure. What it's making me think about as a practitioner is that there are perhaps ways that as a teacher, if I'm aware of that, that might change something small, some things big about the way that I choose to engage with Leanna in the classroom; that I choose to help her navigate that space that she finds herself in. There's a lot for me there as a practitioner in that small clip that helps me really see her, understand her, and think about ways that I can support her. Kara: Yeah. And, like, from a design perspective, I huddled with her teacher later in the day, and we talked about this interview, and we thought about what would it mean to design or redesign a space where Leanna could feel really proud of who she was as a mathematician, but she didn't feel the kind of pressure that this math genius moniker is affording her. And so, ultimately, I want these interviews to be conducted by teachers so that, as you said, practitioners might show up differently for kids or think about what we might need to think more deeply about or design for kids like her. She's certainly not the only one. Mike: Yeah, absolutely. And I think part of what's hitting me in the face is that the term “empathy interview” really is taking on new meaning, even listening to this first one. Because feeling the feelings that she's sharing with us, feeling what it would be like to be in those shoes, I've had kiddos in my class who have been identified or whose folks have chosen to have them participate in programming. And I have to confess that I don't know that I thought as much about what that positioning meant to them or what it meant about how kids would perceive them. I was just struck by how, in so many subtle ways doing an interview like this, might really shift the way that I showed up for a child. Kara: Yeah, I think so. Mike: Well, let's listen to another one. Kara: OK. Maybe Matthew, should we meet Matthew? Mike: I think we should meet Matthew. Kara: Yeah. Mike: Do you want to set up Matthew and give us a sense of what we might need to know about the context? Kara: Absolutely. Matthew is a fifth-grader who describes, in my conversation with him, several years of what he calls “not good” years in math. And he doesn't enjoy mathematics. He doesn't think he's good at it. He has internalized, he's really blamed himself and taken most of the responsibility for those “bad“ years of learning. When I meet him, he's a fifth-grader, and he has written a mathography at the invitation of his classroom teacher. This is a practice that's part of this school. And in his mathography as a fifth-grader, he uses the word “evolving,” and he tells the story of how he's evolving as a mathematician. That alone is pretty profound and beautiful that he has the kind of insight to describe this kind of journey with mathematics. And he really just describes a fourth-grade teacher who fundamentally changed his relationship to mathematics, his sense of himself, and how he thinks about learning. Mike: Let's give it a listen. Kara: Maybe we'll end, Matthew, with: If people were thinking about you as—and maybe there's other Matthews in their class, right—what kinds of things would've helped you back in kindergarten, first and second grade to just feel like math was for you? It took you until fourth grade, right … Matthew: Yeah. Kara: … until you really had any positive emotions about math? I'm wondering what could we have done for younger Matthew? Matthew: Probably, I think I should have paid a lot more attention. Kara: But what if it wasn't about you? What if it's the room and the materials and the teacher and the class? Matthew: I think it was mostly just me, except for some years it was really, really confusing. Kara: OK. Matthew: And when … you didn't really want in third grade or second grade, you didn't want to be the kid that's always, like, “Hey, can you help me with this?” or something. So that would be embarrassing for some people. Kara: OK. You just made air quotes right, when you did embarrassing? Matthew: Yeah. Kara: Was it embarrassing to ask for help? Matthew: It wasn't embarrassing to ask for help, and now I know that. But I would always not ask for help, and I think that's a big reason why I wasn't that good at math. Kara: Got it. So, you knew in some of these math lessons that it was not making sense? Matthew: It made no sense. Kara: It made no sense. Matthew: And then I was, like, so I was in my head, “I think I should ask, but I also don't want to embarrass myself.” Kara: Hmm. Matthew: But also, it's really not that embarrassing. Kara: OK, but you didn't know that at the time. At the time it was like, “Ooh, we don't ask for help.” Matthew: Yeah. Kara: OK. And did that include asking another kid for help? You didn't ask anybody for help? Matthew: Um, only one of my friends that I knew for a really long time … Kara: Hmm. Matthew: He helped me. So, I kind of got past the first stage, but then if he was absent on those days or something, then I'd kind of just be sitting at my desk with a blank sheet. Kara: Wow, so it sounds like you didn't even know how to get started some days. Matthew: Yeah, some days I was kind of just, like, “I'm not even going to try.” Kara: “I'm not” … OK. Matthew: But now I'm, like, “It's not that big of a deal if I get an answer wrong.” Kara: Yeah, that's true. Right? Matthew: “I have a blank sheet. That is a big deal. That's a problem.” Kara: So having a blank sheet, nothing written down, that is a bigger problem for you than, like, “Oh, whoops, I got the answer wrong. No big deal.” Matthew: I'd rather just get the answer wrong because handing in a blank sheet would be, that would probably be more embarrassing. Mike: Oh, my goodness. There is a lot in a little bit of space of time. Kara: Yeah. These interviews, Mike, are so rich, and I offer them to this space and to teachers with such care and with such a deep sense of responsibility 'cause I feel like these stories are so personal. So, I'm really mindful of, can I use this story in the space of Matthew for a greater purpose? Here, I feel like Matthew is speaking to all the kind of socio-mathematical norms in classrooms. And I didn't know Matthew until this year, but I would guess that a kid like Matthew, who is so quiet and so polite and so respectful, might've flown under the radar for many years. He wasn't asking for help, but he was also not making trouble. It makes me wonder, “How would we redesign a class so that he could know earlier on that asking for help—and that this notion that in this class, mathematics—is meant to make sense, and when it doesn't make sense, we owe it to ourselves and each other to help it make sense?” I think it's an invitation to all of us to think about, “What does it mean to ask for help?” And how he wants deep down mathematics to make sense. And I agree with him, that should be just a norm for all of us. Mike: I go back to the language that you used at the beginning, particularly listening to Matthew talk, “the stories that we tell ourselves.” The story that he had told himself about what it meant to ask for help or what that meant about him as a person or as a mathematician. Kara: Yeah. I mean, I am trained as a kind of qualitative researcher. So as part of my dissertation work, I did all kinds of gathering data through interviews and then analyzing them. And one of the ways that is important to me is thinking about kind of narrative analysis. So, when Matthew tells us the things that were in his head, he tells you the voice that his head is saying back to him. Kids will do that. Similarly, later in the interview I said, “What would you say to those kids, those kids who might find it?” And what I was interested in is getting him to articulate in his own voice what he might say to those children. So, when I think about stories, I think about when do we speak in a first person? When do we describe the voices that are in our heads? When do we quote our teachers and our mothers and our cousins? And how that's a powerful form of storytelling, those voices. Mike: Well, I want to listen to one more, and I'm particularly excited about this one. This is Nia. I want to listen to Nia and have you set her up. And then I think what I want to do after this is talk about impact and how these empathy interviews have the potential to shift practice for educators or even school for that matter. So, let's talk about Nia and then let's talk about that. Kara: You got it. Nia is in this really giant classroom of almost 40 kids, fifth-graders, and it's co-taught. It's purposely designed as this really collaborative space, and she uses the word “collaboration,” but she also describes how that's a really noisy environment. On occasion, there's a teacher who she describes pulling her into a quieter space so that she can concentrate. And so, I think that's an important backstory for her just in terms of her as a learner. I ask her a lot of questions about how she thinks about herself as a mathematician, and I think that's the clip we're going to listen to. Mike: Alright, let's listen in. Nia: No, I haven't heard it, but … Kara: OK. I wonder what people mean by that, “I'm not a math person.” Nia: I'm guessing, “I don't do math for fun.” Kara: “I don't do math for fun.” Do you do math for fun? Nia: Yes. Kara: You do? Like, what's your for-fun math? Nia: Me and my grandma, when we were in the car, we were writing in the car. We had this pink notebook, and we get pen or a pencil, and she writes down equations for me in the backseat, and I do them and she times me, and we see how many questions I could get right in, like, 50 seconds. Kara: Oh, my gosh. What's an example of a question your grandma would give you? Nia: Like, they were just practice questions, like, three times five, five times eight. Well, I don't really do fives because I already know them. Mike: So, we only played a real tiny snippet of Nia. But I think one of the things that's really sticking out is just how dense these interviews are with information about how kids think or the stories that they've told themselves. What strikes you about what we heard or what struck you as you were having this conversation with Nia at that particular point in time? Kara: For me, these interviews are about both storytelling and about identity building. And there's that dangerous thinking about two types of people, math people and non-math people. I encounter adults and children who have heard of that phrase. And so, I sometimes offer it in the interview to find out what sense do kids make of that? Kids have told me, “That doesn't make sense.” And other kids have said, “No, no, my mom says that. My mom says she's not a math person.” So, she, I'm playing into it to see what she says. And I love her interpretation that a math person is someone who does math for fun. And truthfully, Mike, I don't know a lot of kids who describe doing math for fun. And so, what I loved about that she, A: She a described a math person's probably a person who, gosh, enjoys it, gets some joy or pleasure from doing mathematics. Kara: But then the granularity of the story she offers, which is the specific pink notebook that she and her grandmother are passing back and forth in the backseat of the car, tell you about mathematics as a thing that she shares a way of relating to her grandmother. It's been ritualized, and really all they're doing if you listen to it is, her grandmother's kind of quizzing her on multiplication facts. But it's such a different relationship to multiplication facts because she's in relationship to her grandmother. They have this beautiful ongoing ritual. And quite honestly, she's using it as an example to tell us that's the fun part for her. So, she just reminds us that mathematics is this human endeavor, and for her, this one ritual is a way in which she relates and connects to her grandmother, which is pretty cool. Mike: So, I want to shift a little bit and talk about a couple of different things: the types of questions that you ask, some of the norms that you have in mind when you're going through the process, and then what struck me about listening to these is you're not trying to convince the kids who you're interviewing of anything about their current thinking or their feelings or trying to shift their perspective on their experience. And I'm just wondering if you can think about how you would describe the role you're playing when you're conducting the interview. 'Cause it seems that that's pretty important. Kara: Yeah. I think the role I'm playing is a deep listener. And I'm trying to create space. And I'm trying to make a very, very, very safe environment for kids to feel like it's OK to tell me a variety of stories about who they are. That's my role. I am not their classroom teacher in these interviews. And so, these interviews probably look and sound differently when the relationship between the interviewer and the interviewee is about teachers and students and/or has a different kind of power differential. I get to be this frequent visitor to their classroom, and so I just get to listen deeply. The tone that I want to convey, the tone that I want teachers to take up is just this fascination with who they are and a deep curiosity about their experience. And I'm positioned in these interviews as not knowing a lot about these children. Kara: And so, I'm actually beautifully positioned to do what I want teachers to do, which is imagine you didn't know so much. Imagine you didn't have the child's cumulative file. Imagine you didn't know what they were like last year. Imagine you didn't know all that, and you had to ask. And so, when I enter these interviews, I just imagine, “I don't know.” And when I'm not sure, I ask another smaller question. So I'll say, “Can you say more about that?” or “I'm not sure if you and I share the same meaning.” The kinds of questions I ask kids—and I think because I've been doing this work for a while, I have a couple questions that I start with and after that I trust myself to follow the lead of the children in front of me—I often say to kids, “Thank you for sitting down and having a conversation with me today. I'm interested in hearing kids' stories about math and their math journey, and somebody in your life told me you have a particularly interesting story.” And then I'll say to kids sometimes, “Where do you want to start in the story?” And I'll try to give kids agency to say, “Oh, well, we have to go back to kindergarten” or “I guess we should start now in high school” or kids will direct me where they think are the salient moments in their own mathematical journey. Mike: And when they're sharing that story, what are the types of questions that you might ask along the way to try to get to clarity or to understanding? Kara: Great question. I'm trying to elicit deep emotion. I'm trying to have kids explain why they're telling me particular stories, like, what was significant about that. Kids are interesting. Some kids in these interviews just talk a lot. And other kids, I've had to really pepper them with questions and that has felt a little kind of invasive, like, this isn't actually the kind of natural conversation that I was hoping for. Sometimes I'll ask, “What is it like for you or how do you think about a particular thing?” I ask about things like math community, I ask about math partners. I ask about, “How do you know you're good at math and do you trust those ways of knowing?” I kind of create spaces where we could have alternative narratives. Although you're absolutely right, that I'm not trying to lead children to a particular point of view. I'm kind of interested in how they make sense. Mike: One of the things that, you used a line earlier where you said something about humanizing mathematics, and I think what's striking me is that statement you made: “What if you didn't have their cumulative report card?” You didn't have the data that tells one story, but not necessarily their story. And that really is hitting me, and I'm even feeling a little bit autobiographical. I was a kid who was a lot like Matthew, who, at a certain point, I just stopped raising my hand because I thought it meant something about me, and I didn't want people to see that. And I'm just struck by the impact of one, having someone ask you about that story as the learner, but also how much an educator could take from that and bring to the relationship they had with that child while they were working on mathematics together. Kara: You said a lot there, and you actually connect to how I think about empathy interviews in my practice now. I got to work with Rochelle Gutiérrez this summer, and that's where I learned deeply about her framework, rehumanizing mathematics. When I do these empathy interviews, I'm living in this part of her framework that's about the body and emotions. Sometimes kids in the empathy interview, their body will communicate one thing and their language will communicate something else. And so, that's an interesting moment for me to notice how body and motions even are associated with the doing of mathematics. And the other place where empathy interviews live for me is in the work of “Street Data,” Jamila Dugan and Shane Safir's book, that really call into question this idea that what is measurable and what is quantifiable is really all that matters, and they invite us to flip the data dashboard. Kara: In mathematics, this is so important 'cause we have all these standardized tests that tell children about who they are mathematically and who they're about to become. And they're so limiting, and they don't tell the full story. So, when they talk about “Street Data,” they actually write about empathy interviews as a way in which to be humanizing. Data can be liberatory, data can be healing. I feel that when I'm doing these interviews, I have this very tangible example of what they mean because it is often the case that at the end of the interview—and I think you might've had this experience just listening to the interview—there's something really beautiful about having a person be that interested in your story and how that might be restorative and might make you feel like, “There's still possibility for me. This isn't the last story.” Mike: Absolutely. I think you named it for me, which is, the act of telling the story to a person, particularly someone who, like a teacher, might be able to support me being seen in that moment, actually might restore my capacity to feel like, “I could do this” or “My fate as a mathematician is not sealed.” Or I think what I'm taking away from this is, empathy interviews are powerful tools for educators in the sense that we can understand our students at a much deeper level, but it's not just that. It's the experience of being seen through an empathy interview that can also have a profound impact on a child. Kara: Yes, absolutely. I'm part of a collaboration out of University of California where we have thought about the intersection of disability and mathematics, and really thinking about how using the tools of design thinking, particularly the empathy interview can be really transformative. And what the teachers in our studies have told us is that just doing these empathy interviews—and we're not talking about interviewing all the kids that you teach. We're talking about interviewing a select group of kids with real intention about, “Who's a kid who has been marginalized?” And/or “Who's a kid who I don't really know that much about and/or I don't really have a relationship with?” Or “Who's a kid who I suspect doesn't feel seen by me or doesn't feel, like, a deep sense of belonging in our work together?” Teachers report that just doing a few of these interviews starts to change their relationship to those kids. Kara: Not a huge surprise. It helped them to name some of the assumptions they made about kids, and it helped them to be in a space of not knowing around kids. I think the other thing it does for teachers that we know is that they describe to do an empathy interview well requires a lot of restraint, restraint in a couple of ways. One, I'm not fixing, I'm not offering advice. I'm also not getting feedback on my teaching. And I also think it's hard for teachers not to insert themselves into the interview with our own narratives. I really try to make sure I'm listening deeply and I'm painting a portrait of this kid, and I'm empathetic in the sense I care deeply and I'm deeply listening, which I think is a sign of respect, but the kids don't need to know about my experience in the interview. That's not the purpose. Mike: We could keep going for quite a long time. I'm going to make a guess that this podcast is going to have a pretty strong on a lot of folks who are out in the field listening. Kara: Hmm. Mike: If someone was interested in learning more about empathy interviews and wanted to explore or understand more about them, do you have any particular recommendations for where someone might go to continue learning? Kara: Yes, and I wish I had more, but I will take that as an invitation that maybe I need to do a little bit more writing about this work. I think the “Street Data” is an interesting place where the co-authors do reference empathy interviews, and I do think that they have a few videos online that you could see. I think Jamila Dugan has an empathy interview that you could watch and study. People can write me and/or follow me. I'm working on an article right now. My colleagues in California and I have a blog called “Designing4Inclusion,” “4” being the number four, and we've started to document the work of empathy and how it shows up in teachers' practice there. Mike: Well, I want to thank you so much for joining us, Kara. It has really been a pleasure talking with you. Kara: Thank you, Mike. I was really happy to be invited. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 2 | Episode 1 – Practical Ways to Build Strengths-based Math Classrooms Guest: Beth Kobett Mike Wallus: What if it were possible to capture all of the words teachers said or thought about students and put them in word clouds that hovered over each student throughout the day? What impact might the words in the clouds have on students' learning experience? This is the question that Beth Kobett and Karen Karp pose to start their book about strengths-based teaching and learning. Today on the podcast, we're talking about practices that support strengths-based teaching and learning and ways educators can implement them in their classrooms.  Mike: Hey, Beth, welcome to the podcast. Beth Kobett: Thank you so much. I'm so excited to be here, Mike. Mike: So, there's a paragraph at the start of the book that you wrote with Karen Karp. You said: ‘As teachers of mathematics, we've been taught that our role is to diagnose, eradicate, and erase students' misconceptions. We've been taught to focus on the challenges in students' work rather than recognizing the knowledge and expertise that exist within the learner.' This really stopped me in my tracks, and it had me thinking about how I viewed my role as a classroom teacher and how I saw my students' work. I think I just want to start with the question, ‘Why start there, Beth?' Beth: Well, I think it has a lot to do with our identity as teachers, that we are fixers and changers and that students come to us, and we have to do something. And we have to change them and make sure that they learn a body of knowledge, which is absolutely important. But within that, if we dig a little bit deeper, is this notion of fixing this idea that, ‘Oh my goodness, they don't know this.' And we have to really attend to the ways in which we talk about it, right? For example, ‘My students aren't ready. My students don't know this.' And what we began noticing was all this deficit language for what was really very normal. When you show up in second grade, guess what? There's lots of things you know, and lots of things you're going to learn. And that's absolutely the job of a teacher and a student to navigate. So, that really helped us think about the ways in which we were entering into conversations with all kinds of people; teachers, families, leadership, and so on, so that we could attend to that. And it would help us think about our teaching in different ways. Mike: So, let's help listeners build a counter-narrative. How would you describe what it means to take a strengths-based approach to teaching and learning? And what might that mean in someone's daily practice? Beth: So, we can look at it globally or instructionally. Like, I'm getting ready to teach this particular lesson in this class. And the counter-narrative is, ‘What do they know? What have they been showing me?' So, for example, I'm getting ready to teach place value to second-graders, and I want to think about all the things that they've already done that I know that they've done. They've been grouping and counting and probably making lots of collections of 10 and so on. And so, I want to think about drawing on their experiences, A. Or B, going in and providing an experience that will reactivate all those prior experiences that they've had and enable students to say, ‘Oh yeah, I've done this before. I've made sets or groups of 10 before.' So, let's talk about what that is, what the names of it, why it's so important, and let's identify tasks that will just really engage them in ways that help them understand that they do bring a lot of knowledge into it. And sometimes we say things so well intentioned, like, ‘This is going to be hard, and you probably haven't thought about this yet.' And so, we sort of set everybody on edge in ways that set it's going to be hard, which means, ‘That's bad.' It's going to be hard, which means, ‘You don't know this yet.' Well, why don't we turn that on its edge and say, ‘You've done lots of things that are going to help you understand this and make sense of this. And that's what our job is right now, is to make sense of what we're doing.' Mike: There's a lot there. One of the things that I think is jumping out for me is this idea is multifaceted. And part of what we're asking ourselves is, ‘What do kids know?' But the other piece that I want to just kind of shine a flashlight on, is there's also this idea of what experiences have they had—either in their home life or in their learning life at school—that can connect to this content or these ideas that you're trying to pull out? That, to me, actually feels like another way to think about this. Like, ‘Oh my gosh, we've done partitioning, we've done grouping,' and all of those experiences. If we can connect back to them, it can actually build up a kid's sense of, like, ‘Oh, OK.' Beth: I love that. And I love the way that you just described that. It's almost like positioning the student to make those connections, to be ready to do that, to be thinking about that and providing a task or a lesson that allows them to say, ‘Oh!' You know, fractions are a perfect example. I mean, we all love to use food, but do we talk about sharing? Do we talk about when we've divided something up? Have we talked about, ‘Hey, you both have to use the same piece of paper, and I need to make sure that you each have an equal space.' I've seen that many times in a classroom. Just tweak that a little bit. Talk about when you did that, you actually were thinking about equal parts. So, helping students … we don't need to make all those connections all the time because they're there for students and children naturally make connections. That's their job ( chuckles ). It really is their job, and they want to do that. Mike: So, the other bit that I want to pick up on is the subtle way that language plays into this. And one example that really stood out for me was when you examined the word ‘misconception.' So, talk about this particular bit of language and how you might tweak it or reframe it when it comes to student learning. Beth: Well, thank you for bringing this up. This is a conversation that I am having consistently right now. Because this idea of misconception positions the student. ‘You're wrong, you don't understand something.' And again, let's go back to that again, ‘I've got to fix it.' But what if learning is pretty natural and normal to, for example, think about Piaget's conservation ideas, the idea that a young child can or can't conserve based on how the arrangement. So, you put in a, you know, five counters out, they count them and then you move them, spread them out and say, ‘Are they the same, more or less?' We wouldn't say that that's a misconception of a child because it's developmental. It's where they are in their trajectory of learning. And so, we are using the word misconception for lots of things that are just natural, the natural part of learning. And we're assuming that the student has created a misunderstanding along the way when that misunderstanding or that that idea of that learning is very, very normal. Beth: Place value is a perfect example of it. Fractions are, too. Let's say they're trying to order fractions on a number line, and they're just looking at the largest value wherever it falls, numerator, denominator, I'm just throwing it down. You know, those are big numbers. So, those are going to go at the end of a number line. But what if we said, ‘Just get some fraction pieces out'? That's not a misconception 'cause that's normal. I'm using what I've already learned about value of number, and I'm throwing it down on a number line ( chuckles ). Um, so it changes the way we think about how we're going to design our instruction when we think about what's the natural way that students do that. So, we also call it fragile understanding. So, fragile understanding is when it's a little bit tentative. Like, ‘I have it, but I don't have it.' That's another part, a natural part of learning. When you're first learning something new, you kind of have it, then you've got to try it again, and it takes a while for it to become something you're comfortable doing or knowing. Mike: So, this is fascinating because you're making me think about this, kind of, challenge that we sometimes find ourselves facing in the field where, at the end of a lesson or a unit, there's this idea that if kids don't have what we would consider mastery, then there's a deficit that exists. And I think what you're making me think is that framing this as either developing understanding or fragile understanding is a lot more productive in that it helps us imagine what pieces have students started to understand and where might we go next? Or like, what might we build on that they've started to understand as opposed to just seeing partial understanding or fragile understanding from a deficit perspective. Beth: Right. I love this point because I think when we think about mastery, it's all or nothing. But that's not learning either. Maybe on an exam or on a test or on assessment, yes, you have it or you don't have it. You've mastered or you haven't. But again, if we looked at it developmentally that ‘I have some partial understanding or I have it and … I'm inconsistent in that,' that's OK. I could also think, ‘Well, should I have a task that will keep bringing this up for students so that they can continue to build that rich understanding and move along the trajectory toward what we think of as mastery, which means that I know it now, and I'm never going to have to learn it again?' I don't know that all things we call mastery are actually mastered at that time. We say they are. Mike: So, I want to pick up on what you said here because in the book there's something about the role of tasks in strengths-based teaching and learning. And specifically, you talk about ‘the cumulative impact that day-to-day tasks have on what students think mathematics is and how hard and how long they should have to work on ideas so that they make sense.' That kind of blows me away. Beth: Well, I want to know more about why it blows you away. Mike: It blows me away because there's two pieces of the language. One is that the cumulative impact has an effect on what students actually think mathematics is. And I think there's a lot there that I would love to hear you talk about. And then also this second part, it has a cumulative impact on how hard and how long kids believe that they should have to work on ideas in order to have them be sensible. Beth: OK, thank you so much for talking about that a little bit more. So, there's two ways to think about that. One is, and I've done this with teams of teachers, and that's bring in a week's worth of tasks that you designed and taught for two weeks. And I call this a ‘task autopsy.' It's a really good way because you've done it. So, bring it in and then let's talk about, do you have mostly conceptual ideas? How much time do students get to think about it? Or are students mimicking a procedure or even a solution strategy that you want them to use or a model? Because if most of the time students are mimicking or repeating or modeling in the way that you've asked them, then they're not necessarily reasoning. And they're building this idea that math means that ‘You tell me what I'm supposed to do, I do it, yay, I did it.' And then we move on to the next thing.  Beth: And I think that sometimes we have to really do some self-talk about this. I show what I value and what I believe in those decisions that I'm making on a daily basis. And even if I say, ‘It's so important for you to reason, it's so important for you to make sense of it.' If all the tasks are, ‘You do this and repeat what I've shown you,' then students are going to take away from that, that's what math is. And we know this because we ask students, ‘What is math?' Math is, ‘When the teacher shows me what to do, and I do it, and I make my teacher happy.' And they say lots of things about teacher pleasing because they want to do what they've been asked to d,o and they want to repeat it and they want to do well, right? Or do they say, ‘Yeah, it's problem-solving. It's solving a problem, it's thinking hard. Sometimes my brain hurts. I talk to other students about what I'm solving. We share our ideas.' We know that students come away with big impressions about what math means based on the daily work of the math class. Mike: So, I want to take the second part up now because you also talk about what I would call ‘normalizing productive struggle' for kids when they're engaged in problems. What does that mean and what might it sound like for an educator on a day-to-day basis? Beth: So, I happened to be in a classroom yesterday. It was a fifth-grade classroom, and the teacher has been really working on normalizing productive struggle. And it was fabulous. I just happened to stop in, and she stopped everything to say, ‘We want to have this conversation in front of you.' And I said, ‘All right, go for it.' And the question was, ‘What does productive struggle feel like to you and why is it important?' That's what she asked her fifth-graders. And they said, ‘It feels hard at first. And uh, amazing at the end of it. Like, you can't feel amazing unless you've had productive struggle.' We're taking away that opportunity to feel so joyous about the mathematics that we're learning because we got to the other side. And some of the students said, ‘It doesn't feel so good in the beginning, but I know I have to remember what it's going to feel like if I keep going.' I was blown away. I mean, they were like little adults in there having this really thoughtful conversation. And I asked her what … she said, ‘We have to stop and have this conversation a lot. We need to acknowledge what it feels like because we're kind of conditioned when we don't feel good that somebody needs to fix it.' Mike: Yeah, I think what hits me is there's kind of multiple layers we consider as a practitioner. One layer is, do I actually believe in productive struggle? And then part two is, what does that look like, sound like? And I think what I heard from you is, part of it is asking kids to engage with you in thinking about productive struggle, that giving them the opportunity to voice it and think about it is part of normalizing it. Beth: It's also saying, ‘You might be feeling this way right now. If you're feeling like this,' like for example, teaching a task and students are working on a task trying to figure out how to solve it and, and it's starting to get a little noisy and hands start coming up, stopping the class for a second and saying, ‘If you're feeling this way, that's an OK way to feel,' right? ‘And here's some things we might be thinking about. What are some strategies'—like re-sort-of focusing them on how to get out of that instead of me fixing it—like, ‘What are some strategies you could think about? Let's talk about that and then go back to this.' So, it's the teacher acknowledging. It's allowing the students to talk about it. It's allowing everybody … it's not just making students be in productive struggle, or another piece of that is ‘just try harder.' That's not real helpful.  Like, OK, ‘I just need you to try harder because I'm making you productively struggle.' I don't know if anyone has had someone tell them that, but I used to run races and when someone said, ‘Try harder' to me, I'm like, ‘I'm trying as hard as I can.' That isn't that helpful. So, it's really about being very explicit about why it's important. Getting students to the other side of it should be the No. 1 goal. And then addressing it. ‘OK, you experienced productive struggle, now you did it. How do you feel now? Why is it worth it?' Mike: I think what you're talking about feels like things that educators can put into practice really clearly, right? So, there's the fron- end conversation maybe about normalizing. But there's the backend conversation where you come back to kids and say, ‘How do you feel once this has happened? It feels amazing.' This is why productive struggle is so important because you can't get to this amazingness unless you're actually engaged in this challenge, unless it feels hard on the front end. And helping them kind of recalibrate what the experience is going to feel like. Beth: Exactly. And another example of this is this idea of … so I had a pre-service teacher teaching a task. She got to teach it twice. She taught it in the morning. Students experienced struggle and were puffed up and running around, so engaged when they solved it. Beyond proud. ‘Can we get the principal in here? Who needs to see this, that we did this?' And then she got some feedback to reduce the level of productive struggle for the second class based on expectations about the students. And she said the engagement, everything went down. Everything went down, including the level of productive struggle went way down. And so, the excitement and joy went way down, too. And so, she did her little mini-research experiment there. Mike: So, I want to stay on this topic of what it looks like to enact these practices. And there are a couple practices in the book that really jumped out at me that I'd like to just take one at a time. So, I want to start with this idea of giving kids what you would call a ‘walk-back option.' What's a walk-back option? Beth: So, a walk-back option is this opportunity once you've had this conversation—or maybe one-on-one, or it could be class conversation—and a walk-back option is to go look at your work. Is there something else that you'd like to change about it? One of the things that we want to be thinking about in mathematics is that solutions and pathways and models and strategies are all sort of in flux. They're there, but they're not all finished all the time. And after having some conversation or time to reason, is there something that you'd like to think about changing? And really building in some of that mathematical reflection. Mike: I love that. I want to shift and talk about this next piece, too, which is ‘rough-draft thinking.' So, the language feels really powerful, but I want to get your take on, what does that mean and how might a teacher use the idea of rough-draft thinking in a classroom? Beth: So rough-draft thinking is really Mandy Jansen's work that we brought into the strengths work because we saw it as an opportunity to help lift up the strengths that students are exhibiting during rough-draft thinking. So, rough-draft thinking is this idea that most of the time ( chuckles ), our conversations in math as we're thinking through a process is rough, right? We're not sure. We might be making a conjecture here and there. We want to test an idea. So, it's rough, it's not finished and complete. And we want to be able to give students an opportunity to do that talking, that thinking and that reasoning while it is rough, because it builds reasoning, it builds opportunities for students to make those amazing connections. You know, just imagine you're thinking through something, and it clicks for you. That's what we want students to be able to do. So, that's rough-draft thinking and that's what it looks like in the math classroom. It's just lots of student talk and lots of students acknowledging that ‘I don't know if I have this right yet, but here's what I'm thinking. Or I have an idea, can I share this idea?' I watched a pre-service teacher do a number talk and a student said, ‘I don't know if this is going to work all the time, but can I share my idea?' Yes, that's rough-draft thinking. ‘Let's hear it. And wow, how brave of you and your strength and risk-taking. Uh, come over here and share it with us.' Mike: Part of what I'm attracted to is even using that language in a classroom with kids, to some degree it reduces the stakes that we traditionally associate with sharing your thinking in mathematics. And it normalizes this idea that you just described, which is, like, reasoning is in flux, and this is my reasoning at this point in time. That just feels like it really changes the game for kids. Beth: What you hear is very authentic thinking and very real thinking. And it's amazing because even very young children—young children are very at doing this. But then as you move, students start to feel like their thinking has to be polished before it's shared. And then that gives other students who may be on some other developmental trajectory in their understanding, so much more afraid to share their rough-draft thinking or their thoughts or their ideas because they think it has to be at the polished stage. It's very interesting how this sort of idea has developed that you can't share something that you think in math because it's got to be right and completed. And everything's got to be perfect. And before it gets shared, because, ‘Wait, we might confuse other people.' But students respond really beautifully to this. Mike: So, the last strategy that I want to highlight is this one of a ‘math amendment.' I love the language again. So same question, how does this work? What does it look like? Beth: OK, so how it works is that you have done some sharing in the class. So, for example, you may have already shared some solutions to a task. Students have been given a task they're sharing, they may be sharing a pair-to-pair share or a group-to-group share, something like that. It could be whole class sharing. And then you say, ‘Hmm, you've heard lots of good ideas today, lots of interesting thinking and different strategies. If you'd like to provide a math amendment, which is a change to your solution in addition, something else that you'd like to do to strengthen it, you can go ahead and do that and you can do it in that lesson right there.' Or what's really, what we're finding is really powerful, is to bring it back the next day or even a few days later, which connects us back to this idea of what you were saying, which is, ‘Is this mastered? Where am I on the developmental trajectory?' So, I'm just strengthening my understanding, and I'm also hearing … I'm understanding the point of hearing other people's ideas is to go and try them out and use them. And we're really allowing that. So, this is take, this has been amazing, the math amendments that we're seeing students do, taking someone else's idea or a strategy and then just expanding on their own work. And it's very similar to, like, a writing piece, right? Writing. You get a writing piece and you polish and you polish. You don't do this with every math task that you solve or problem that you solve, but you choose and select to do that. Mike: Totally makes sense. So, before we go, I have the question for you. You know, for me this was a new idea. And I have to confess that it has caused me to do a lot of reflection on language that I used when I was in the classroom. I can look back now and say there are some things that I think really aligned well with thinking about kids' assets. And I can also say there are points where, gosh, I wish I could wind the clock back because there are some practices that I would do differently. I suspect there's probably a lot of people where this is a new idea that we're talking about today. What are some of the resources that you'd recommend to folks who want to keep learning about strengths-based or asset-based teaching and learning? Beth: So, if they're interested, there's several … so strengths-based or asset-based is really the first step in building equity. And TODOS, they use the asset-based thinking, which is mathematics for all organization. And it's a wonderful organization that does have an equity tool that would be really helpful. Mike: Beth, it has been such a pleasure talking to you. Thank you for joining us.  Beth: Thank you so much. I appreciate it. It was a good time. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 1 | Episode 20 – Work Places Guest: Lori Bluemel Mike Wallus: When I meet someone new at a gathering and tell them that I work in math education, one of the most common responses I hear is, “I was never good at math in school.” When I probe a bit further, this belief often originated in the person's experience memorizing basic facts. How can we build students' fluency with facts, encourage flexible thinking, and foster students' confidence? That's the topic we'll explore in this episode of Rounding Up.  Mike: One of the challenges that we face in education can be letting go of a practice—even if the results are questionable—when the alternative is unclear. In elementary math, this challenge often arises around building computational fluency. We know that speed tests, drill and kill, and worksheets, those are all ineffective practices. And even worse, they can impact students' math identity. So, today we're going to spend some time unpacking an alternative, a component of the Bridges in Mathematics curriculum called Work Places. We're doing this not to promote the curriculum, but to articulate an alternative vision for ways that students can develop computational fluency. To do that, we're joined by Lori Bluemel, a curriculum consultant for The Math Learning Center.  Mike: Lori, welcome to the podcast. It's great to have you with us. Lori: Thank you. It's good to be here. Mike: Well, let's just start with a basic question: If I'm a listener who's new to the Bridge's curriculum, can you describe what a Work Place is? Lori: The simple answer would be that it's math activities or games that are directly focusing on the skills or the ideas and concepts that students are working on during Problems & Investigations. The best aspect, or the feature about Work Places, is that teachers have an opportunity to be like a fly on the wall as they're listening into their students and learning about what strategies they're using and the thinking process that they're going through. Mike: How do you think practicing using a Work Place differs from the version of practice that children have done in the past? What changes for the child or for the learner? Lori: Well, I always felt like a piece of paper was pretty static. There wasn't a lot of interaction. You could run through it so quickly and be finished with it without really doing a lot of thinking and processing—and with absolutely no talking. Whereas during Work Places, you're discussing what you're doing. You're talking to your partner. You're listening to your partner. You're hearing about what they're doing and the different methods or strategies that they're using. And [there's] nothing at all static about it because you're actively working together to work through this game or this activity. Mike: That is so fascinating. It makes me think of a book that I was reading recently about thinking classrooms, and one of the things that they noted was, there's data that suggests that the more talk that's happening in a classroom, the more learning that's actually happening. It really connects me to what you just said about Work Places. Lori: Yeah, and I feel like that's the big difference between Work Places and doing a worksheet on your own. You can do it completely isolated without any outside interaction, whereas Work Places, it's very interactive, very collaborative. Mike: Yeah. So, as a former classroom teacher who used Work Places on a daily basis, how did you set up norms and routines to make them successful for students? Lori: Well, I actually went through several different methods, or routines, before I landed on one that really worked well for me. One that worked best for me is, at the beginning of the year when we first started doing Work Places, I would take that very first Work Place time, and we would just have a class meeting and talk about what we're doing in Work Places. Why would we even have Work Places? We would create an anchor chart, and we'd have one side that would say “Students.” The other side would say “Teachers.” And then we would talk about the expectations. And the students would come up with those. Then we would talk about me as the teacher, what do they think I should be doing? And again, that would come up with all different ideas. And then we always came back to that final thought of, “We need to be having fun.” Mike: Hmm.  Lori: Math needs to be fun during Work Places. And then we would start in, and students would go to Work Places. They would choose their partner, and then they would get started. And that first few times we did Work Places, I always just kind of watched and listened and walked around. And if I felt like things needed to be slightly different, maybe they weren't talking about math or they weren't really playing the Work Place, then we would call a class meeting. And everyone would freeze, and we'd go to our meeting spot, and we would talk about what I saw. And we would also talk about what was going well and what they personally could do to improve. And then we'd go back to Work Places and try it again. Needless to say, a lot of times those first few times at Work Places they didn't play the games a lot because we were setting up expectations. But in the long run, it made Work Places run very smoothly throughout the rest of the year. Mike: Yeah. The word that comes to mind as I listen to you talk, Lori, is investment.  Lori: Um-hm.  Mike: Investing the time to help set the norms, set the routines, give kids a vision of what things look like, and the payoff is productive math talk. Lori: Exactly. And that was definitely the payoff. They needed reminders on occasion, but for the most part, they really understood what was expected. Mike: I think it's fascinating that you talked about your role and asked the kids to talk about that. I would love if you could say more about why you asked them to think about your role when it came to Work Places. Lori: I wanted them to realize that I was there to help them. But at the same time, I was there to help their peers as well. So, if I was working with a small group, I wanted them to understand that they might need to go to another resource to help them answer a question. They needed to make sure that I was giving my attention to the, the small group or the individual that I was working with at that time. So, by talking about what was expected from me, my hope was that they would understand that there were times when they might have to wait a minute, or they might go to another resource to find an answer to their question, or to help them with the situation that they were in. And that seemed to be the case. I think I alleviated a lot of those interruptions just by talking about expectations. Mike: So, I want to return to something that you said earlier, Lori, 'cause I think it's really important. I can imagine that there might be some folks who are listening who are wondering, “What exactly is the teacher doing while students are engaged in Work Places?”  Lori: Um-hm.  Mike: And I wanted to give you an opportunity to really help us understand how you thought about what your main focus was during that time. So, children are out, they're engaged with the Work Places. How do you think about what you want to do with that time? Lori: OK. So, I often look at the needs of my students and, and think about “What have I seen during Problems & Investigations? What have I seen during Work Places previously? And where do I focus my time?” And then I kind of gravitate towards those students that I want to listen in on. So, I want to again, be like that fly on the wall and just listen to them, maybe ask a few questions, some clarifying questions about what they're doing, get an idea of what strategies or the thinking that they're going through as they're processing the problem. And then from there, I can start focusing on small groups, maybe adjust the Work Place so that they can develop that skill at a deeper level. It helps me during that time to really facilitate my students' practice; help students make the most of their practice time so that as they're going through the Work Place, it's not just a set of rules and procedures that they're following. That they're really thinking about what they're doing and being strategic with those skills as well. So that's my opportunity to really help and focus in on my small groups and provide the support that students need. Or maybe I want them to advance their skills, go a little bit deeper so that they are working at a little bit different level. Mike: You know, I'm really interested in this idea that Work Places present an opportunity to listen to students' thinking in real time. I'm wondering if you can talk about an experience where you were able to tuck in with a small group and listen to their thinking and use what you learned to inform your teaching. Lori: ( chuckles ) One experience kind of stands out to me more than others just because it helped me understand that I need to not assume that my students are thinking about, or thinking in a specific way. So, there was one student, they were playing the Work Place game in grade 3, Loops & Groups, and she had spun a six and rolled, I think, a six as well. So, her problem was to solve six times six. And this student had actually been in front of the class just a few days before, and several times actually when I had worked with her, had solved a problem similar to this by thinking of it as three times six and three times six, which is a great strategy. But what I really wanted this student to develop was some flexibility.  Lori: So, I asked her to explain her thinking, and I fully expected her to solve it: “Oh, yeah. I thought of it as three times six and three times six. And when I add those two together, I get 36.” And she totally shocked me. ( laughs ) She said, “Oh, I, I thought of it as five times six, and I know what five times six is. That's 30. And if I just add one more set of six, I get 36. So, she had already developed another strategy, which was not what I was expecting. With that, her partner was a little bit confused and said, “I don't understand how you could do that.” So, I asked this little girl if she could use tile maybe to explain her thinking to her friends. So, we got out the tile. She set it up and she explained this thinking to her partner. And her partner was still a little bit unsure, not really sure she could use that with her own thinking. But what it did was, in the future, just days later, that partner started trying that particular strategy. So, it taught me several things. First of all, don't assume. You don't always know what students are thinking. And also, students are their peers' best teachers. It really encouraged her partner to try that method just a few days later. Mike: We kind of zoomed really in on a pair of children and, and kind of the impact. The other thing that it makes me think is, by doing the fly on the wall, you as a teacher get a better sense of kind of the themes around thinking that are happening across the classroom. Lori: Yeah. You definitely do get that, that perspective. And I think the questioning that you use also will help draw that out. Asking students to explain their thinking: “How did you solve the problem? How could you check your work? Is there a different strategy that you could use that would help you make sure that the answer you came up with, the first strategy you used, was correct?” Those kinds of questions always seem to really help students kind of pull out that thinking and be able to explain what they were doing. Mike: Lori, thank you so much for joining us today. It has really been a pleasure to have you on the podcast and to be able to talk about this. Lori: You bet. Thank you for having me. It was fun. Mike: I want to thank all of you who've listened in during the first season of Rounding Up. We're going on a short break this summer, but we'll be back for Season 2 in September. Before we go, we're wondering what topics you'd like us to explore, what guests you'd like to hear from, and what questions you'd like us to take up in Season 2. This week's episode includes a link you can use to share your ideas with us. Let us know what you're thinking about, and we'll use your ideas to inform the topics we consider in Season 2.  Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 1 | Episode 19 – Building a Broader Definition of Participation Guest: Juanita Silva Mike Wallus: Participation is an important part of learning to make sense of mathematics. But stop and ask yourself, “What counts as participation?” In this episode, we'll talk with Dr. Juanita Silva from Texas State University about an expanded definition of participation and what it might mean for how we engage with and value our students' thinking.  Mike: Welcome, Juanita. Thanks for joining us on the podcast. Juanita Silva: Hi. Thank you for inviting me. I'm excited to talk about this topic. Mike: I think I'd like to start by asking you to just talk about the meaning of participation. What is it and what forms can participation take in an elementary math classroom? Juanita: Well, there's a mixture of nonverbal and verbal communication. And you can add in there gestures [as a] form of communication, not just in an interconnected space, but also thinking about students' respect. And it's not just bidirectional, but there's a lot of things that are kind of added in that space. Mike: So, it strikes me that when I was a classroom teacher, when I look back, I probably overemphasized verbal communication when I was assessing my students' understanding of math concepts. And I have a feeling that I'm not alone in that. And I'm wondering if you could talk about the way that we've traditionally thought about participation and how that might have impacted student learning? Juanita: Yes, this is a great question. In thinking about, “What does this look like, how to participate in the classroom?” Mostly teachers think about this as whole group discussions or in small group discussions. And I emphasize the word “their” discussions, where students can share verbally how they thought about the problem. So, for example, if a student is solving a fraction word problem, the teacher may ask, “OK, so how did you solve this problem? Can you share your strategy with the class? What does that look like?” And so, the student sometimes will say, “If I'm solving a fraction word problem about four parts or four chocolate bars, then I can cut those leftovers into four parts.” So that's usually what we think of, as in our teaching and practice in elementary schooling. We think of that as verbal communication and verbal participation, but there are others. ( laughs ) Mike: Let's talk about that. I think part of what you have pushed me to think about is that a student's verbal communication of their thinking, it really only offers a partial window into their actual thinking. What I'd like to do is just talk about what it might look like to consciously value participation that's nonverbal in an elementary classroom. Like, what are the norms and the routines that a teacher could use to value nonverbal communication, maybe in a one-to-one conversation in a small group or even in a whole group discussion? Juanita: Yes. So, I can share a little bit for each one of those. For example, in a one-to-one environment, the teacher and student can more effectively actually communicate ideas if the teacher attends to that child's thinking in nonverbal ways as well. So, for instance, I've had a student before in the past where he would love to explain his thinking using unifix cubes and to share his thinking on a multiplication problem that was about three sets of cookies. And those sets were in groups of seven. So, there were seven cookies in each bag. And I asked him, “Well, how would you share? Could you explain your thinking to me?” And so, he showed me three sets of seven unifix cubes, and he pointed to each of the seven linking cubes and then wrote on his paper, the number sentence, “seven plus seven plus seven is 21.” And when I asked him if the seven represented the cookies, he simply nodded yes and pointed to his paper, saying and writing the words “21 total.” Juanita: So, I didn't ask him to further explain anything else to me verbally because I had completely understood how he thought of the problem. And in this example, I'm showing that a student's gestures and a student's explanation on a piece of paper should be valued enough. And we don't necessarily need to engage in a verbal communication of mathematical ideas because this honors his ways of thinking. But at the same time, I could clearly understand how this child thought of the problem. So, I think that's one way to think about how we can privilege a nonverbal communication in a one-to-one setting. Mike: That's really helpful. I think that part of the example that you shared that jumps out for me is attending to the ways that a child might be using manipulative tools as well, right?  Juanita: Correct.  Mike: So, it was kind of this interaction of the student's written work, their manipulative tools, the way that they gestured to indicate their thinking … that gave you a picture of how this child was thinking. And you didn't really need to go further than that. You had an understanding as an educator that would help you think about what you might do next with that child. Juanita: Absolutely. And that is one of the tools that I find to be super useful, is to not just have students explain their thinking, but also just listen to their nonverbal cues. And so, paying attention to those and also valuing those is extremely important in our practice. I can share one of my favorites, which is a small group example. And this one is kind of foundational to think of the practice when we're teaching in our elementary math classrooms. It's not just that interactions between student and teacher, but the interactions between students and students can be very powerful. So, that's why this is one of my favorite examples. I had two students at one point in my practice. And this was Marco and José, and they were in fourth grade. They were having a hard time communicating verbally with one another, and José was trying to convince Marco of his strategy to split the leftovers of an equal-sharing problem into three parts instead of halves. Juanita: But his verbal communication of these ideas were not clear to Marco. And José explains to Marco, “You have to cut it into halves.” And Marco would say, “Yes, that is what I did.” Like, frustrated, as if, like, “You have to cut this into halves.” And José would say, and Marco was like, “Yes, that's exactly what I did.” So, this exchange of verbal communication was not really helping both of them showcase how they were trying to communicate. So, then José started to insist, and he said, “No, look.” And then he showed Marco his strategy on his paper. And in his paper, he had split the bar into three parts. And then Marco looked at José and said, “Ah, OK.” Had José not shown this strategy on his paper, then Marco would have never really understood what he meant by “You have to cut it into halves.” And so, I share this example because it really showcases that sometimes what we're trying to say and communicate might come across differently verbally, but we mean something else when we showcase it nonverbally. So, in this instance, José was trying to explain that, but he couldn't figure out how to tell that to Marco. And so, in this instance, I feel like it really showcases the power of the nonverbal communication among students. Mike: I think what's fascinating about that is, conceptually the strategy was right there. It was kind of like, “I'm going to equally partition into three parts.” The issue at hand was the language choice. I'm essentially referring to this equal partition as a half, this second equal partition as a half, and this third equal partition as a half. That's a question of helping figure out what is the language that we might use to describe those partitions. But if we step back and say, “Mathematically, does the child actually understand the idea of equal partitioning?” Yes. And then it seems as though it becomes a second question about how do you work with children to actually say what we call this, or the way that we name fractions is—that's a different question, as opposed to, “Do you understand equal partitioning, conceptually?” Juanita: Yeah. So, you're pointing at something that I've found in my research in the past. Oftentimes students will use the word half. And verbally explaining, use the word to mean that they're trying to equally partition a piece of a bar. They'll say, “Well, I cut it into halves.” And then when we look at the document, they're pointing to the lines, the partition lines, that are within the bar. And that's what they're referring to. So, we know that they don't necessarily mean that the part itself is a half, but that the partition is what they're indicating. It means that it's a half. And it's this idea that it's behind … languages really attained to this development over time, where students really think about their prior experiences, as in, “I've cut items before. And those cuts before have been halves.” And so, that particular prior knowledge can transfer into new knowledge. And so, there's this disjuncture, or there's this complexity, within the language communication and those actions. And that's why it's important not just to value the verbal communication—but also nonverbals—because they might mean something else. Mike: Well, part of what you're making me think about, too, is in practice, particularly the way that you described that, Juanita, was this idea that my prior knowledge, my lived experience led me to call the partitions “half.” And the mathematical piece of that is, like, “I understand equal partitioning. The language that I use to describe partitioning is the language of half.” So, my wondering for you is, what would it look like to value the child's partitioning and value the fact that they used this idea of partitioning when they were thinking about halves—and then also build on that to help them have the language of, “We call this type of a partition a third or a fourth,” or what have you. Juanita: So, this is one of those conundrums that I've talked to and discussed with other colleagues, and we talk about how sometimes they're just not ready for it. And so, when we are trying, and that's the other thing, right? Honoring what they say and taking it as they're saying it. And sometimes it's OK not to correct that. So, because we as the teachers have that, you know, we're honoring their thinking as it is, and eventually that language will develop. It eventually will become where they're no longer calling the partitions halves, and they're calling them appropriately, and they're using the part instead. So, it takes time for the student to really understand that connection. So, if we just say it and we tell them, it doesn't necessarily mean it's going to transfer and that they're going to pick up on that. So, I often try not to tell them, and I just let them explain how they're thinking and how they're saying. Juanita: And if I honor their nonverbal ways, then I definitely can see what they mean by halves, that they're not necessarily thinking of the part, they're thinking of the partition itself. And so, that is a very important, nuanced, mathematical evolution in their knowledge. And that sometimes, we as teachers try and say, “Oh, well, we should just tell him how it is.” Or how we should develop the appropriate language. And in some instances, it might be OK. But I think most often I would defer not to do something like that because like I said, I still can access their mathematical thinking even if they don't have that language yet. ( chuckles ) Mike: That's super helpful. I think we could probably do a podcast … Juanita: On that alone? ( laughs ) Mike: The nuances of thinking about that decision. But I want to ask you before we close about whole group. Let's talk a little bit about whole group and what it looks like to value nonverbal communication in a whole group setting. Tell me your thinking. Juanita: Yeah, so this one is a fascinating one that I've recently come across in my own work. And I have to say, it takes a lot of effort on the part of the teacher to enact these things in the classroom, but it is possible. And so, I'll share an example of what I came across in my practice. So, if this was a bilingual classroom, and the teacher was asking students to participate silently and in written form to attend to each other's mathematical ideas, and they had examples. They had to solve a multiplication area problem individually, and then the teacher would post the student's solutions on a large poster paper and then ask all of the students to go around the room with a sticky note offering comments to each of their peer solutions. And so, what we found was just fascinating because the students were able to really dive deep into the students' solutions. Juanita: So, they were more deeply involved in those mathematical ideas with … when you took out the verbal communication. We had an instance where a student was like, “Well, you solved it this way, and I noticed that you had these little pencil marks on each of those squares.” And the student was saying, “Did you count 25 or did you count 26? I think you missed one.” And so, the gestures and the marks, the pencil marks on the piece of paper, that's how detailed the students were kind of attending to each other's thinking. So, they were students that were offering ideas to other students' solutions. So, they were saying, “Well, what if you thought about it this way?” And they would write their explanation of that strategy of how they would solve it instead of how the student actually did it. And so, it was just fantastical. We were just amazed by how much richness there was to their explanations. Had the teacher done this particular activity verbally, then I wonder how many students would have actually participated. Right? So that was one of our bigger or larger questions, was noticing how many students participated in the level and the depth of their justifications for each other, versus had the teacher done this verbally with the students and had them communicate in a whole group discussion. How many students would've been able to do this? So, it is just fascinating. ( chuckles ) Mike: You touched on some of the things that were coming to mind as I heard you describe this practice, and I'd love your take on it. One of the things that strikes me about this strategy of posting solutions and then asking kids to use Post-it Notes to capture the comments or capture the noticings: Does it have the potential to break down some of the status dynamics that might show up in a classroom if you're having this conversation verbally? What I mean by that is, kids recognize that when someone speaks who they've perceived as, like, “Well, that person understands it, so I'm going to privilege their ideas.” That kind of goes away, or at least it's minimized, in the structure that you described. Juanita: That is correct. So, I do a lot of writing on also thinking about culturally sustaining pedagogies in our teaching of practice of math. And some of the things that we find, is that a lot of the students that do participate verbally tend to be white monolinguals. And that oftentimes the teacher or other students privilege their knowledge over the student of color. And so being able to participate in nonverbal ways in this manner really showcases that everybody's knowledge can be privileged. And so, those kind of dynamics within the classroom go away. And so, it really highlights that everybody is valued equally, and that everybody can contribute to these ideas, and that everybody has a voice. That's one of the reasons why this particular piece is just dear to my heart, is because it really showcases to teachers that this can be done in the classroom. Mike: Yeah, I've said this oftentimes on the podcast. I find myself wanting to step back into my classroom role and try this protocol out. It just feels really powerful. Let me go back to something that I wanted to clarify. So, as we've talked about practices that value nonverbal communication, a question that I've been forming and that I suspect other people might be wondering about is, I don't think you're saying that teachers have to either choose to value verbal or nonverbal communication. Juanita: Yes, that is correct. So, I often do both. ( laughs ) It's a mixture of both. Students will communicate verbally to some extent in the same strategy and nonverbally at the same time. And valuing all forms of communication is most important. In my practice as a bilingual teacher and teaching bilingual students, I've also understood that language can't be the sole focus. And the nonverbal cues also highlighted in that communication are just as important as the language, as the bilingualism, when we're communicating ideas. And so, as teachers, there's a law that we also have to pay attention to. So, it's not just that it's nonverbal or verbal communication, but it's also how we approach the teaching, right? Because we as teachers can definitely take over students' thinking and not necessarily pay attention to what they're actually saying. So, only valuing verbal communication would be detrimental to the student. Juanita: So, it has to be a little bit of both and a mixture of everything. I've had students [who] have tried to show me in gestures alone with no written comments on a piece of paper, and that sometimes can work. I've had instances where students can gesture with their hands and say they're pointing, and they're using both hands as, “This is how many I mean, and this is how I'm partitioning with my fingers. I'm doing three partitions, and I'm using three fingers, and I'm showing you three iterations of that with closing and opening my fists.” And so, there's just so much that kids can do with their body. And they're communicating ideas not just in a formal written format, but also using gestures. So, there's lots of ways that students can communicate, and I think teachers should pay attention to all of those ways. Mike: Yeah. The connection that I'm making is, we've done several podcasts, and I've been thinking a lot about this idea of strengths-based, or asset-based, instruction. And I think what you're saying really connects to that because my interpretation is, gestures, nonverbal communication, using manipulative tools, things that kids have either written or drawn, those are all assets that I need to pay attention to in addition to the things that they might use language to describe. Juanita: That's right. That's right. So, everything. ( laughs ) The whole student. ( laughs ) Mike: Well, I suspect you've given our listeners a lot to think about. For folks who want to keep learning about the practices that value nonverbal communication, what research or resources would you suggest? Juanita: Yeah, so I have two articles, one that's particular to bilingual pre-service teachers, and another one that I just explained within a whole group discussion. That's an article, titled, “Attending to others' mathematical ideas: a semiotic alternative to logocentrism in bilingual classrooms.” So, I can give you both links and you can share those along with the podcast. Mike: That sounds fantastic. We'll put a link to that up when we publish the podcast. I just want to thank you, Juanita. It was lovely to have you with us. I've learned a lot, and I sure appreciate you joining us. Juanita: Thank you. Well, thank you for having me. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 1 | Episode 18 – Why Progressions Matter Guest: Graham Fletcher Mike Wallus: Many educators were first introduced to the content that they teach as a series of items on a checklist. What impact might that way of thinking have on a teacher's approach to instruction? And what if there were another way to understand the mathematics that our students are learning? In this podcast, we talk with Graham Fletcher about seeing mathematics as a progression and how this shift could have a profound impact on teaching and learning.  Mike: Welcome to the podcast, Graham. We're glad to have you with us. Graham Fletcher: Yeah, really excited to just kind of play around, uh, in this space with you here talking about math and supporting teachers so that they can, in turn, support kids. Mike: You bet. So, just as a starting point, we're talking about progressions, and we're talking about some of the work that you've done, building progression videos. I have, maybe, what is kind of a weird opening question: How would you define the term “progression” so that we're all starting with the same understanding? Graham: So, when I think about progression, I think a lot of the times as teachers we can become, like, hyper focused on one grade level. And within that one grade level there can be a progression of where things are learned in a sequential order. It's probably not as linear as we'd like it to be, but I think that little micro progression, or sequence, of learning that we see in one grade level, we start thinking about what that might look like over a grade band, over like K–2 or even K–5. So, there's things that happen within certain grade levels, and that's kind of where progressions happen. How do we move kids through this understanding of learning? And it's that progression of understanding that we tend to want to move kids through, where everything's kind of connected. And that's really where I see progressions. Mike: So, I think you're kind of leading into my second question, which is—I love the work that you've put together on your website. I'm unabashedly going to say that this is a great place for teachers to go. But part of what strikes me is that there are a lot of things that you could have done to support elementary math educators and yet you chose to invest time to build this series of videos that unpack the ideas that underlie processes, like counting or addition and subtraction or fractions. Like, why that? Why was that a thing where you're like, “I should invest some time in putting this together.” Graham: So, I guess we're all teachers at heart, and so I start thinking about how I'm in a place of privilege where I've had an opportunity to work with some really amazing educators that I've stood on their shoulders over the years. And I think about all the times that I've been able to huddle up in a classroom at the end of the day and just listen to those people who are brilliant and really understand those progressions and the smaller nuances of what it is to just understand student thinking and how to keep moving it forward. So, I started thinking about, “Well, what does this look like in one grade level?” But then, when I was starting to think about that whole idea, the big piece for me is: Not every teacher has a person that they can sit next to. And so, if I've had the opportunity to sit down and make sense of these things where, like, on a Friday night (laughs) maybe I'm sitting down with some math books, which most people don't choose to do, I enjoy doing that. Graham: And so, if I've had the opportunity to do that, and I'm able to make these connections, I start thinking about those other teachers who, teachers that teach 75 subjects 54 days a week, right? And we want them to focus solely on math. So, maybe just sharing some of that knowledge to kind of lessen the burden of understanding that content. So, giving them like a 60,000-foot view of what those progressions could look like. And then them saying, “OK, well, wait a minute. Maybe I can do a deeper dive,” where we're giving them those [aha moments] that they might want or need to kind of do that deeper dive. And the big piece for it was, there's always talk about progressions. There's always talk about, “This is the content that you need to know,” content after content after content. But very seldom is it ever in a coherent, consumable manner. So, when I start thinking about teachers, we don't have that time to sit down and give hours and hours and hours to the work. So really, just what is a consumable amount of time to where teachers won't be overwhelmed? And I think that's why I tried to keep them at about 5 to 6 minutes; to where you can go kind of light that fire to go and continue building your own capacity. So, that's kind of where it was. My North Star: just building capacity and supporting teachers in their own growth. For sure. Mike: You know, it's interesting, 'cause when I was a classroom teacher, the lion's share of my time was kindergarten and first grade, with a little bit of time in second grade. So, I was thinking about that when I was watching these because I watched some of the ones for younger kids and I was like, “This makes a ton of sense to me.” But I really kind of perked up when I started watching the ones for kids in the intermediate grades. And I think for me it was kind of like, “Ah, these ideas that I was working on in K and 1, so often, I wasn't quite sure what seeds was I planting or how would those seeds grow in the long term—not just next year, but in the long term. I wonder if that's part of what you think about comes out of a teacher's experience with these. Graham: Yeah, I definitely think so. I think finding that scalability in reasoning and relationships is key for students, and it's key for teachers as well. So, for instance, when we start thinking about, in kindergarten, where kids are sitting and they're practicing counting and they're counting by singular units; singular units of 1, where it's 1, 2, 3. Well, then when we start making that connection into third grade, where kids are counting by fractions instead of going ahead and saying, like, “One-fourth, two-fourth, three-fourths,” really focusing on that iteration of the unit, that rote counting where it's one one-fourth, two one-fourths, three one-fourths. And then, even that singular unit that we're talking about in kindergarten, which now is in fractions in third grade, well that begins to connect in sixth grade when we start talking about unit rate, when we start getting into ratios and proportions. So, that scalability of counting is massive. So, that's just one little example of taking something and seeing how it progresses throughout the grade level. And making those connections explicit becomes really powerful because I know, just in my own experiences, in talking with teachers as well, is when they start making those connections. Bingo, right? So, now when you're looking at students, it's like, “OK, they're able to count by unit fractions. Well, what now happens if we start grouping fractions together and units and we start counting by two-thirds?” So, now you start moving from counting strategies to additive strategies and then additive strategies to multiplicative, and seeing how it all kind of grows together. That scalability is what I'm really after a lot of the time, which falls in line with that idea of teaching through progressions. Mike: Yeah, I think one of the things that's really hitting me about this, too, is that understanding children's mathematical thinking as a progression is really a different experience than thinking about math as a set of procedures or skills that kids need to leave second grade with. It feels really different. I wonder if you could talk about that. Graham: Yeah, absolutely. So, working with Tracy Zager—good friend of mine—we've done a lot of work around fact fluency here over the last three, four years, per se. And one of the biggest things that we have spent a lot of time just grappling and chewing on, is when we have students in second grade and they move to third grade, how do we move students from additive thinking, which is adding of singular units, to multiplicative thinking? So, seeing groups of groups of groups. And so, I think when we start thinking about third grade teachers, I'll go ahead and throw myself under the bus here. Like, as a third-grade teacher, when we start thinking about that idea of multiplication, it becomes skip counting and repeated addition. But then no kids ever really move from skip counting and repeated addition to knowing their multiplication facts. Like, I could sit there and do jumping jacks in class, but kids aren't going to know their facts. Graham: So, then what I would do is, I would jump to having kids try to memorize their facts. And just because kids can memorize their facts doesn't mean that they can reason multiplicatively and seeing those groups of groups. So, I think, thinking of that, what [are] those big jumps in the progression from grade level to grade level? That's probably one of the ones for me that really stands out that I know I struggled for. And we always look back and say, “What are the things I wish I knew back then that I know now?” And I think that jump from additive thinking to multiplicative thinking is a really big jump that is often overlooked, which is now why we have kids struggling in fourth and fifth grade and middle school. 'Cause they're still stuck in additive, but we want them to think multiplicatively and proportionally. But yeah, that's one of those big jumps in terms of a progression that we want kids to make. Mike: Yeah, this is a great transition because I think, like, what we've been exploring is, how if I understand what I'm helping kids think about in the context of a larger story rather than a set of discreet things that I need to check a box on, that has impact on my practice. But I almost wanted to ask you, just on a day-to-day basis, what's your sense of, if I'm a teacher who's absorbed this sense of progression either across my grade level or across a larger band of time, how do you think that changes the way someone approaches teaching? Or maybe the way that they set up tasks with students? Graham: Well, I start thinking about learning objectives as they're handed down, and standards. And a lot of the time standards can become, or learning objectives can become, more of a checklist. And so not necessarily looking at these ideas of learning as a checklist, but how do they connect between the grade levels? And so, I think it's important as much as on the day-to-day practice that we're really down in the trenches and we're doing the work and we're making sure that we're meeting those learning objectives, I think it becomes really important that we provide ourselves that space and grace to zoom back out to that 60,000-foot view and say, “Wait a minute, how are all of these connected?” And I think that's a really big piece that maybe we don't always do when we start thinking, even planning, on a day-to-day or a week or a unit. “Where am I going to be able to zoom out and maybe connect some big ideas around an understanding or around a piece of learning?” And I think it can become cumbersome when we start looking at those learning objectives and they're so granular. But I think when we can zoom out and make connections between them, it lessens a little bit of the burden from having to go ahead. “Well, there's just so much to teach, trying to make those connections.” There is a lot to teach, don't get me wrong here. But I think going ahead and making those connections just lessens that burden for us a little bit. Mike: It's interesting, because I think part of what is coming to mind for me is this ability to zoom out and zoom back in and be able to say, “In what way is this relatively granular learning objective or learning goal serving to advance this larger set of ideas that I want kids to understand about, say, additive thinking as they're making a shift to multiplicative thinking?” And the other connection I'm making is, in what way can I ask a question in this moment that's going to actually advance that larger goal rather than—again, guilty as charged—rather than what I've done often in the past, which is how can I help them just complete the task or get this particular thing right? And if by them getting it right in the moment, I failed to advance their thinking, that's a place where I'd want to take it back. Does that make sense to you? Graham: Yeah, absolutely. I think about tasks and really about when I first would start to use problem-based lessons or three act tasks and start thinking about those lessons. Normally it would be, like, “OK, I just taught the task for no rhyme or reason just to see if kids could get the right answer.” And so, for me, the big piece with that is a shift in my own craft, is looking at that task placement. And so, thinking of, “Are you a teacher who learns math to solve problems or are you a teacher who solves problems to learn math?” A little play on words there. And I think by default, many of us were taught to learn math to go ahead and solve the problems. But when I start thinking about this idea of using tasks and why we use tasks, it's to use … well, to quote Dan Meyer, talking about this headache and aspirin analogy where you have a problem that's your headache, and then from that problem, the math serves the headache, that's the aspirin that you need. Graham: So, when we talk about zooming back out, instead of saving the really good tasks for the end of the unit, what would it look like if we put it on day one of a unit? Knowing that the goal on day one isn't for kids to get the right answer, but it's for us to just pull the veil back and see, “Hey, where are my students thinking?” And what I've realized is that when we don't front-end load or pre-teach things, students will usually fall back to the strategy that they feel safe enough. And if you have a student who, say we're in fourth grade and we're playing with two- by two-digit multiplication, if you have a student on day one of a unit who's doing draw all, count all, great, right? That's what they're doing on day one? But if they're still using that same strategy at the end of the unit, that falls back on me. Graham: Like, what have I done to be intentional enough about moving that student's thinking forward? So, even in the moment when students might not be getting the right answer, it might be wrong answer, but it might be the right thinking. And I think at that moment I need to zoom back out and say, “They don't have the answer yet, but I've still got three or four weeks to get there.” So, now that I know what students are thinking, how can I be intentional? How can I be purposeful about asking the right questions, presenting the right activities and tasks to continue to move that student's thinking forward to the end goal? The end goal isn't on day one of a unit. So yeah, I think that's such a great question because I think a lot of the times we feel as if we fall short or we failed as a teacher if kids aren't getting the right answer. But so often there's beautiful thinking that's happening, it just might not have the right answer. So yeah, big, big change in my practice. Mike: We've been talking about the use of the progression videos that you've built, and I think in my mind I've imagined myself as a classroom teacher, as the consumer. And I think that's a really powerful way to use those. My wondering is, if you have any thoughts about how someone who might be an instructional coach or an instructional leader in a building or a district, if you could wave a magic wand, how you wish folks who have that type of role might take and use the things that you've built? Graham: I can share how I've used them in the past. I don't know, I'm sure there's coaches out there that are probably using the progression videos way better than I'm using them. But many times, I've found that when we start looking at individual standards, it's standards out of context. And granted, the progression videos, if I could go back and redo them, I would love to embed much more context into those progression videos. It would definitely lengthen them, which kind of defeats the original purpose of keeping them short and compact. So, now when we show those videos, what's nice is it's not really a coach in that moment talking with the teachers. The coach can now, after the video, say, “Hey, what was new to you? What was something that, that maybe you didn't recognize?” And also, like, “What are you doing well?” There's so much goodness that's already happening. Graham: I think as coaches, we have to be really mindful, like, there's great things that [are] happening with teachers, let's support and lift up those great things that are already happening with our teachers that we're supporting, just like teachers do with students as well. So, I think showing the videos and asking, “Hey, what's the same, what are you comfortable with? What doesn't sit well with you?” Thinking about kindergarten teachers when they see five frames, it's like, “Whoa, wait a minute. I've never really thought about using five frames.” So, just different ways of thinking it to kind of be a catalyst for the conversation, just a launch point. Mike: Totally makes sense. So, I suspect there are some folks who are going to be listening to this who are, like, “Oh my goodness, I want to go check these things out right now. Or I want to think about sharing them with my teammates that I'm working with on a daily basis.” Walk me through how to find these and any kind of advice that you might have for people as they start to initially poke around and look at what's there. Graham: Well, you can jump on my website, gfletchy.com, with my full name, Graham Fletcher. Just one of those things that we kind of went with growing up. I was called “Fletchy” as a kid. So yeah, at gfletchy.com you can look on progression videos, and then right there you'll see five of them. But as you start poking around, I'm going to harness my inner Brené Brown here and just say, “Vulnerability is the birthplace of professional growth.” And so, no one is ever going to get a new idea and go ahead and try it and then it be successful right on that get-go. So, when you poke around there, give things a try. I love reaching out on Twitter, sharing on Twitter, and just kind of growing in that space. Find a colleague. Or if you are a coach, one of the things I love doing is when coaches ask for ideas, go muck about, find a good task, and then muck about in a third-grade classroom with that task and make yourself vulnerable around the teachers you're supporting. Graham: And that really helps build and solidify that relationship where, “Hey, we're in this together and I'm trying to fumble through this just like you, let's kind of work here together. Give me feedback and, and in the end, I think kids win.” I'm a firm believer that all of us are smarter than one of us. And so, I love finding new things, testing new things with a friend, and trying not to lock myself in a silo. So, that would kind of be it in terms of poking around there. Yeah, find an idea and go share it with a friend and see how it works and keep on tweaking and revising. Mike: I love that. Graham, thank you so much for joining us. It's really been a pleasure. Graham: Yeah, it's been great. I appreciate it. And thanks for the opportunity. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 1 | Episode 17 – Asset-Based Approach to Assessment Guest: Tisha Jones Mike Wallus: When you look at the results of your students work, what types of things are you attending to? Many of us were trained to look for the ways that students were not understanding concepts or ideas. But what if we flipped that practice on its head and focused on the things students did understand? Today on the podcast, we're talking with Tisha Jones, senior adviser for content development at The Math Learning Center, about building an asset-based approach to assessment.  Mike: Tisha, first of all, thanks for joining us. We're thrilled to have you with us. Tisha Jones: I'm really excited to be here. Mike: I have a sense that for a lot of people, the idea of asset-based assessment is something that we might need to unpack to offer, kind of, a basic set of operating principles or a definition. So, my first question is, how would you describe asset-based assessment? What would that mean for a practitioner? Tisha: I think the first part of it is thinking just about assessment. Assessment is a huge part of every school that is in this country. So, there are formative assessments, which are ongoing assessments that teachers are doing while students are considered “in the process of learning”—although we know that students really are never not in the process of learning. And then there are also summative assessments, when we want to see if they have demonstrated proficiency or mastery of the concepts that they've been learning throughout that unit. But when we're thinking about assessments, oftentimes the idea of assessment is that we are looking for what students don't know. And asset-based assessment means that we're taking this idea and we're flipping it, and we're saying, “Let's start by looking at what students are showing us that they do know.” And we're trying to really focus on the things that our students are showing us that they're able to do. Mike: So, that's a lot. And I think one of many of the things that's going on for me is that that's a pretty profound mind shift, I think, for a lot of folks in the field; not because they necessarily want to look at their students as a set of deficits, but because most of the training that a lot of us got actually was focused on “What are the deficits?” Tisha: Most of the training when we're talking about kids casually, or with our colleagues or administrators, we're often worried about, “Well, our kids don't know this. Our kids are struggling here.” And that really becomes the way that we see our students, right? And our kids are so much more than that, right? And our kids are coming to us with knowledge, and we can forget that when we're only focused on what they don't know. Mike: There's a great quote that you're making me think about. It's from the 14th century, and the person has said, essentially, “The language that we use becomes the world that we live in.” And I think that's a little bit of where you're going, is that deficit-focused language kind of lives in the DNA of a lot of either the training that we've had or the structures of schools. And so, flipping this is a mind shift, and I think it's really exciting that we're talking about this. I have two things on my mind. I think one is, let's talk about the assessments themselves first. So, if I want to start thinking about using my assessments in an asset-based way, if we just think about the assessments themselves, be they formative or summative, tell me about what you think an educator might do with the assessments that they're using, whether they're coming from a curriculum or whether they're some that they're designing on their own. How should I think about the assessment materials that I have, and are there ways that I should imagine shifting them? Tisha: That's a great question. I think that when you're looking at your assessments, you may or may not need to change them. They might be fine the way that they are. But the way to know is when you see the opportunities kids have to give their answers, what is that going to tell you about what they understand? So, if you have, for example, a problem that is computation, if you have a problem that has just asked the kids for an answer, or if you have a problem that's multiple choice, what are you learning about their thinking, about their understanding from what they put on the paper? Now, I'm not saying don't ever use those questions. They have their purpose. But that is really what I am asking you to do, is to think about “What is their purpose? What is the intention behind the questions on the assessment?” So, are there ways for you to open up the assessment to give kids more ways of showing what they do understand as opposed to limiting them to saying, “You must show something in this way” or “You're either right or you're wrong”? Mike: Yeah, that really hits home for me. And I think one of the operating principles that I'm hearing is, regardless of what assessment tools you're using, creating space for kids to show you how they're thinking is really a starting, foundational, kind of, centerpiece for asset-based assessment. Tisha: Absolutely. And I want to also add that I'm talking a lot about paper and pencil because we think about assessments as paper and pencil. But assessment's also not just paper and pencil. Assessment, especially formative assessment, it's your conversations that you have with kids in class. As far as I am concerned, there is no better way to know what a kid's thinking than to talk to them. Talk to your kids as much as you absolutely, possibly can. Ask them so many questions. Mike: Well, you're bringing me to the second piece about the assessments themselves. One piece is, create space, regardless of whether it's a question in a conversation or whether it's a question in a paper-pencil assessment or what have you, for them to show their thinking. The other thing that it makes me think is, part of my work as an educator is to look at the questions and say, “What are the big ideas that I'm really looking for? And what is it that I'm hoping that I can understand about children's thinking with each of these questions that I'm asking?”  Tisha: Yes. Mike: Beyond just right and wrong. Tisha: Yes, this is hard work. But this, to me, is not extra work. When you think about a gap, sometimes that can feel very disheartening. It can feel like, “I can't close it. My kids don't know this. They're never going to get it.” It almost just drains the joy of teaching out. This is the job, and this is the part that I am hoping we can all get excited about. I am excited to know what my kids understand. I feel like that gives me a better entryway to being a better teacher for them. If we can start to shift how we think about assessing our students to looking for what they know, to me, that feels very different. It feels different for your kids, and it feels different for you. It's much more fun to walk into a classroom thinking about what my kids know than what they don't. Mike: Yeah. And I think you're hinting at the next place that I wanted to go, which is, there's the assessments themselves and both how I use them and how I make space for kids to show their thinking. And then there's “How do I approach the things that kids are showing me in their assessments?” And I think that feels like another one of these mind-shift pieces where, what kept coming to mind for me is, if you and I and a colleague or two were sitting together at a table and we were teaching third grade and we had a set of student work in front of us, part of what I'm thinking about is what would a conversation sound like if we were really taking an asset-based perspective on looking at our students' work? What questions might we ask? What kind of a process might we use to, kind of, really focus on assets as opposed to focusing on deficits and gaps? Tisha: So, as we're looking at the work, I think the best place to start is, if we're talking as colleagues, “What do you see that the kids know? What are they doing well?” Whether you're talking about one kid or whether you're talking about a group of kids or your class collectively, “What are they doing well?” And for me, even just sitting here across from you saying this, that feels like a much brighter place to start. I'm like, “OK, I'm into this conversation about what my kids know,” and I would then start to say, “OK, and how can we build on what they know?” Mike: Ooh, I love that. Keep talking about that. Tisha: So, if we're looking at say, fractions, and we're kind of at the beginning, we could come in and we could say, “Oh, our kids are just not getting it. They don't know anything about fractions.” And that feels very defeating. But if you start with, “OK, well, I can see that they can partition into half, great. OK, so can we get them to fourths? Can we get them to eighths? How about thirds? All right. Can they get it on a rectangle? Can they get it on a circle? Can they get it in this context? Can they get it if it's a sharing situation?” Right? Now, we're brainstorming all of these questions of what can they do next. Mike: And those are actionable things, right? Like … Tisha: Right.  Mike: … in addition to saying, “This is what kids are doing,” thinking about “What I can build from” actually leads to action, it leads me to a path of instruction, and that does feel really different. Tisha: So, if we are here and we take the perspective that our kids don't get fractions, then that could bleed into our instruction in a different way. So, instead of now thinking about what we can do next and how we can keep building them up, we may be thinking about how do we need to water things down? How do I need to make things easier? And we want to make sure that we are not taking away rich mathematical opportunities from our students because our perspective is that they're not able, they have deficits. We want to instead think about “How do we build them up? How do we still make sure that they're getting these rich mathematical problems and opportunities in class and being able to grow them in that way?” Mike: Love that. So, one of the things that really just jumped out, and I want to come back to this because I think the language is so darn important: This idea that an asset-based perspective leads to thinking about instruction as “building upon.” That just seems like such a practical, simple thing. But boy, shifting your mindset and approaching it the way you described it, Tisha, that really does feel profoundly different than a lot of the data conversations that I've sat in over the years. Tisha: At that point, we should be stopping to think, “What do they need next?” But it's hard to make that [determination] based on saying, “Well, they don't know this.” It's much easier to think about what they need next if you're looking for what they do know. And you can say, “Oh, I can make some connections to that and move them maybe even just a little bit to a little bit further, help them take another step.” Mike: It strikes me that what I don't hear you saying is, “We can't acknowledge that there's sometimes going to be a difference between what kids understand and our ultimate goals for them.” That can still be true, but we're looking at their starting point as the starting point and the next steps, rather than just only saying, like, “The gap is this wide.” And even using the language of “gap” is challenging, right?  Tisha: Absolutely.  Mike: Because we're trying to say, like, “Our job is to build, not just to measure.” Tisha: Well, and when you think about talking about a gap, it almost feels like it's the kids' fault.  Mike: Uh-hm.  Tisha: But right now, in our conversation, we are talking about where the responsibility is.  Mike: Oh! Yeah! Tisha: And the responsibility is on me to keep thinking about “How do I help this kid grow?”  Mike: Uh-hm. Tisha: “How do I keep helping this kid grow in their math understanding?” It is not uncommon in elementary schools to group or classify kids based on their abilities. And coming from the best place, right? Like, we're all wanting to help our students. I believe that everybody wants to help their students grow.  Mike: This conversation has really got me thinking a lot, and I suspect that anyone who's listening is in the same place. I'm curious, if I'm a person who's new to this conversation, if these ideas are new, I'm wondering if you have any recommendations about where someone could go to keep learning, be it, uh, a book, a website, something along those lines that could keep me thinking about this and exploring these ideas? Tisha: A good place to start is a book called “The Impact of Identity in K–8 Mathematics: Rethinking Equity-Based Practices.” And that is an NCTM publication. Mike: I love that one. It's fantastic. In fact, I've read it myself. We'll put a link to that in the podcast notes. Tisha: That would be great. I think that it's a great resource for thinking about assessment and just equity-based practices in general. Mike: Fabulous. Tisha, it was lovely having you on. Thank you so much. Tisha: Oh, it's been so much fun. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up  Season 1 | Episode 16 – Math Talk in Kindergarten & Beyond  Guest: Dr. Hala Ghousseini  Mike Wallus: Kindergarten is a joyful, exciting, and challenging grade level to teach. It's also a time when  educators can develop a set of productive norms and routines around discourse that can have long lasting effects on students. On today's podcast, we talk with Dr. Hala Ghousseini, a professor at the  University of Wisconsin, about building a solid foundation from math talk in kindergarten and beyond.   Mike: Welcome, Hala. We're really excited to have you on the podcast today talking about math talk in  kindergarten.  Hala Ghousseini: Thank you very much for having me. This is exciting. I love this topic, and the chance  to really talk about this with you is great.  Mike: Well, I feel the same way. I spent eight of my 17 years teaching kindergarten, so I've been  dreaming about a podcast like this for a long time.  Hala: ( laughs ) I can imagine the magic of kindergarten just because it's a time where people think that  they know what to expect, but literally you don't know what to expect with children in kindergarten.  Mike: You started to hint at the first thing that I hope to talk about. I would love to talk about norms. This  feels so important because the norms and the culture that we set in kindergarten, from my perspective,  those might be some of the first messages students receive about what's valued in a mathematics  classroom. And I'm wondering if you could talk just a bit about the norms that you think are important. I  mean, perhaps what it looks like to support them in kindergarten.  Hala: Absolutely. And I just want to situate a little bit some of the things that I have been studying and  thinking about. When I think of math in kindergarten, it very much exists within the learning altogether  that happens in kindergarten; whether it's social-emotional skills, whether they're learning about other  subject areas. So, when I think about the norms, I think often of them as embedded within the fabric of  what's happening in kindergarten. In the research that we've done, we've seen it happening at two  levels. One in relation to what we would call ‘norms related to what's conceptual,' or what [people  might] call more like the disciplinary aspects of norms. So, some of the things that we've seen is, first of  all, centered on children's thinking. The idea that first as an individual in class, that I'm a contributor to  everyone's understanding. So, the way that is typically continuously communicated by the teacher, in the  sense that it's important to share our thinking. And it's important to share it, not just because I'm the  teacher and I asked you to do it, but because it's going to contribute to everyone else's learning.  Hala: My learning as the teacher, others learning in the classroom. And we've seen examples from  teachers where often, as they're asking students to get ready to go into their small groups, they would  always say, ‘Remember, it's important to show our thinking and our work because we want to help  someone else learn it.' You want to help the class understand this idea better. And even with the use of  representations, resources, those were all really in the service of helping someone make their thinking  explicit so that someone else is going to understand it or use it or build on it. So, I'll give you another  example. The idea of saying, ‘Remember, we want to listen now to Hala share her thinking because we want to think how we make sense of it, what Hala is helping us think about. So, those were the typical  expressions or things that teachers would say in building these norms in the classroom.  Hala: The other norm, when it comes to the social aspects of the norm, was really this explicit work on  the sense of the collective as an intellectual community. The idea that we are in this together. It's not  about me and you as the teacher, but it's about the us. What do we make of it? How do we really flag  certain things that may help the group process and think about something? And those were also done  constantly across the times we've spent in these classrooms, in the way teachers would really point to  something that may help us as a group later. ‘Hey, look at this, this might help us later in the way we're  going to work on certain ideas together.'  Mike: Well, I do want to ask you about something else that really struck me when I was reading the  article. So, you and your co-authors talked a great deal about orienting students to and then encouraging  the use of resources to communicate their thinking. That really hit me as a person who used to teach  these young kiddos. Can you talk a little bit about what this looks like?  Hala: Yes. This drew our attention, given where kindergartners are in their language development. They  bring a lot of language from home that actually is going to be essential to build on in explaining the  reasoning, talking about their thinking, reacting to someone else's thinking. So, we started thinking  about the way students' thinking, the way their language that they bring with them, becomes a resource  that they could use. So, encouraging them that ‘Yes, that is one way you can explain your thinking,' so  that really they find that language that is going to give them an entry point into the collective as an  intellectual community. The second thing in relation to resources, also availing in the classroom. We've  noticed these teachers that—besides the fact that you have, like, a number line or a hundredth chart  displayed on the board or even the physical tools that usually typically students play with—how those  become things that the teacher points to and says, ‘Wow, you know what you're doing.'  Hala: This might help us think about this idea. So, let's remember that what struck us was that, when  students were explaining their thinking, we rarely saw a student asking for permission to go and use  something to come and support their thinking. We saw that they were really going to things and bringing them. So that was a norm in that class. That kind of intersects with the idea of normative ways of  working. You can just go and reach it. You don't have to get that teacher's permission to do it. I think one  more thing I'll say about resources. We've noticed the teacher, typically if a student used a particular  resource that supported them in their thinking, when they're sharing, they make sure to actually  highlight it, lift it up in what the student is saying so that others see that those resources could be  contributions to supporting the reasoning in this class.  Mike: So, boy, there's a lot there. I think the first thing that really hits me is this idea that part of the  culture that you want to establish, is that the resources are available and it's contingent on the teacher  saying, ‘Yes, you can go get that right now.'  Hala: Absolutely. And it's a way of socializing the students to be aware of what's in their classroom that is  actually part of what's supporting their learning. You know, there is a thing that I always work in when  I'm working with teachers, this idea that, you know, children are sense makers. And we tend to think of  children as sense makers beyond just mathematics. Of course they are, but also they're sense makers as  learners in general. So, we treat them as sense makers in the way as teachers. We owe it to them to  explain to them why, for example, we're asking them to do something. And we say, ‘So, I want you to  show your work—not just to please me, because this contributes to the collective work in this way.' And we reinforce this message continuously. Similarly, the idea of what's in our class, like, when we see, for  example, base ten blocks. I have a few things in this corner. The idea that these are there to also support  our learning. So, we treat them as sense makers in the sense, these are all shared tools for our  classrooms. So, that's kind of how we think about it in relation to the orienting to resources.  Mike: I want to check my own understanding. I was struck by the way that you talked about the way that  the teacher positions the materials. It seems like a pitfall, I know that I have fallen into at different points  in time is: Using the materials to set a conversation up in a way where children might come away  thinking, ‘Oh, that's the way to do it,' which is very different from, I think the way I heard you describe it. It was more like, this is a tool that can help us think about for future reference. I just wanted to call that  out because I thought I heard that, but I wasn't exactly sure if I was interpreting that accurately.  Hala: Thank you for mentioning that. I think what you're really referring to is what often happens,  especially when we use some manipulatives, let's say, or resources or tools. Where the idea becomes  that the tool equates what it means to do or to reason, like, as if the idea is within the tool and/or the  representation, uh, et cetera. And I think the idea that there is a lot of choice. So, one of the things for  example, that we are currently studying is in kindergarten classrooms, the nature of the use of multiple  representations. There's one question, ‘How often can students come up with their own  representations?' They invent the representations. How often can they go on their own to draw on  certain tools to represent an idea? Those say something when it's actually coming from the student, where you can follow up with questions and say, ‘So, tell me why you use this? Like how do you see it in  this one?' And that's the work that we saw teachers do often, is that they're orienting the resources but  then they're orienting to resources as supporting reasoning.   Hala: And there is the question of why, pressing students. There is a nice example that I always love to  think about, especially with kindergarteners using multiple representations and their own choices. Of  course, students come to class with various fluency in academic language, vocabulary, et cetera. So, there was an instance where the teacher was asking the students, ‘If we've been in school for 129 days, in how many days like that number 29 is going to, we are going to get another 10?' And they were  working with bundling sticks and other things. They focused on the number 9 as nine ones. And how  many more ones till we get another 10? Then the teacher asks the class, ‘Well, is there another way we  can think about how many more days till we get to another 10?'  Hala: ‘Can we use the number 29 altogether?' And a student raises her hand, we call her Gloria, and  actually points to the number line above the whiteboard and says, ‘One twenty-nine, 130.' And the  teacher says, ‘What do you mean by those two?' That literally points to it: 129, 130. So, what the teacher  does, she presses Gloria to explain more and says, ‘Tell us a little bit more. What do you mean by 129  and 130?' Then Gloria actually sees that just looking at the number line as a representation—we call it a  language proxy—to help her really explain her thinking, according to Gloria, wasn't enough for her. She  actually goes back to the hundreds chart. She points at 29, makes a hub, and says, ‘One jump and we get  to 30.' So, we see this is just as a small example of where the student is really using their agency in  deciding on the representation, and the teacher then helps the class try to see the connection that  Gloria was trying to make between this representation. We think this is important for not only this grade  level, but whenever we use multiple representations. The power of multiple representations is in helping  the students see the conceptual connection between them. So, that's where I would caution all of us  when we are doing this, to try to make sure we are focusing on the conceptual piece that the  representation is allowing us to see. Mike: I think part of what you had me thinking about is The Math Learning Center and Bridges. We have  kind of hung our hat on this idea that visual representations are a powerful tool. But the caution that I  always feel is, if those visual representations just turn into another version of an algorithm that's more  like geometric or visually laid out, then we are not advancing the kind of classroom culture or discourse  or thinking that we want, right? That it really is to expose the big ideas. And I think that's what I take, particularly from that example is, the visual actually served as, like, a tool that helped them find the  language to describe the concept rather than just as, like, a here's how you do it. Does that make sense?  Hala: Exactly. I think the tool here is a way for them … the difference is that they're using it not to apply  the reasoning, it's not an application. That's kind of where I see it. Don't just come and show me how  like, like base ten blocks can represent a number. Base ten blocks are used as a way to support a  mathematical idea, not just to apply, like, to show you and show you how something looks like on a hundreds chart. Actually going back to the hundreds chart, to the hub between 29 and 30, was in the  service of really explaining what they meant by 130, 129, 100, there is a hub. That's what they were  talking about in class that when you, you're counting by ones, you're actually now, you got no more 9,  10—9 ones—you actually have one more. And now you could bundle it, and it's your extra 10. So, it's all  couched in the history of working with these representations, like how these students experienced the  work as to not just, ‘Hey, come, let's represent the numbers.' Or there was more talk about, like, those  key ideas that the students were talking about.  Mike: What you're making me think about is that there's an overall pattern that I want to explore in the  context of kindergarten, which is that, as a field, in my mind, one of the things that I wonder about is  whether we have almost explicitly thought about communicating our thinking as something that  happens in the verbal realm. And the more that I've been in the profession is, that we need to broaden  that, particularly when we're talking about young children in pre-K and kindergarten. And I'm wondering, in your mind, what broadening out communication might look like, particularly in kindergarten?  Hala: That's a great question. And I would link it again, like, whenever I think about the norms, the  resources, I see them literally as a triangle with other things working together. Especially critical at this  young age is verbal and non-verbal communication; or really, assets for the students to express their  thinking and communicate with others. And that's where, in a way, the resources become the mediators  of this, with non-verbal—we call them language proxies—is that they become ways of helping the  communication without necessarily waiting for that correct vocabulary or the specific language. And I  think the more we honor various ways of participating and contributing to the learning of the collective,  the more students are going to be able to make improvements, and to make connections, and to show  us what they know, rather than thinking it's too difficult for them to do something maybe because they  don't have that particular, specialized language that someone is looking for.  Hala: We actually think of kindergartners in the way they're really acquiring this new—not only the  verbal language, so that they become more proficient in it—the academic language. And actually, if you  come to think of it, every student in math class, in a way, is a language learner, especially the idea of  what does it mean to explain one's reasoning? And when we are thinking about certain ways that  schools go, they want to follow, for example, the Common Core standards and what they expect in terms  of providing evidence, supporting it. That's actually a language learning process. And there is actually the  literature about supporting bilingual students and multilingual students in classrooms, helps us a lot  think about how we could support learners in the early childhood span. And most recently I was reading  an opinion piece by Tim Boals at the WIDA at the University of Wisconsin. I just actually highlighted a few things in what he said in his opinion piece, which is basically about what it takes to make sure that  multilingual students encounter opportunities to learn.  Hala: So, in a parallel way, it makes me think what it takes for opportunities for early childhood learners  and kindergartners to learn. I just highlighted a few elements that might be one of the resources I share  with you in the end, in case someone is interested in them; about what school programs could do to  ensure that multilingual learners have opportunities to learn. One of them is actually the idea that  always encourage the can-do kind of stance, that you can do it. It's not too difficult for you, like, even in  the choice of tasks. How this guides us for kindergartners is tha,t let's not just give tasks that allow  kindergartners even to skip count on a number line. Actually using tasks where they can reason and  think about why something is true, would be something they can do. So, thinking about not what they  can't do because they're restricted with what they know with numbers, et cetera, it's actually what they  can do.  Hala: So, the idea of designing tasks that leverages what they know, that they could really show you the  way they're reading a situation, what they know about the situation, and really leverage the resources  they have to explain their thinking. My favorite in terms of what he lists in terms of opportunities for  multilingual learners, is this idea of building academic identities, where he says that ‘this is much more  than merely teaching content knowledge and skills. It's about learning to communicate and think like  people who work in those academic or vocational areas.' That's all of this can do. And opening  possibilities for reasoning helps our kindergartners develop really mathematical identities early on that  we know are going to impact their opportunities to learn later. And that's what research shows.  Mike: So, in the third part of your article, you talk about the idea of narration. And I'm wondering if you  could explain narration in this context and then talk a little bit about why it's particularly helpful for young learners?  Hala: So, let me explain what we meant by it in that article. It's literally when, because students may not  have that facility to explain their thinking articulately, elaborately, it's when the teacher actually supports  them by recapping what they said to the class. And on top of it, building on it and setting it up for further   articulation or investigation. So, we try to distinguish here, that's why we're trying to revisit the word  ‘narration' because, we don't think of it just as revoicing. We think of it as a way where the teacher is  highlighting something the student did and, often, we see it in exchange. It's highlighted not only in  terms of the verbatim words that they used or the actions that they took. Highlighting why this is really  helping in the task that we are working on together, and then follows it. It positions it in a way where, now this is what Gloria did.  Hala: So, really it positions the student in a way where other students are now listening, are trying to see  what the student is doing and saying, and then it sets the stage for further focus or deeper conceptual  exploration of particular ideas. So, an example of that would be when Gloria went from 129 to 130 and  went down to the hundreds chart and said, ‘You know, there is a hop from 29 to 30.' So, the teacher may  say, ‘OK, here's what Gloria said so far. She picked those two numbers, she saw that they follow each  other. Actually we're going to get to 130. Then she went down to the hundreds chart to really focus on  that jump of one from 29 to 30.' And then she would immediately go on with a question to the group.  ‘Now what do we do?' I think that makes it more ambitious than just simply revoicing or appropriating  something that the student said, or trying to put words that they may not have used. I think positioning  it for further and deeper conceptual work takes us a bit away from that. Mike: That's really helpful. You started to address the question that I was going to ask next, which is  what's the sweet spot for what you described in the article as narration? It struck me, at least as I was  reading it, that over narrating, if we were defining it as kind of revoicing for kids, might impact kids in  ways that are not productive. But what I hear you saying is, narration is much more than revoicing.  Hala: Absolutely. And that sweet spot that I think you are getting at is really knowing when do you do it  and when do you hold off. In the sense, I don't think there is a rule, but it all goes to the teacher's ability  to know: ‘Is there a shared language here that the students can access through what a student said?' So,  knowing your students in terms of, is this something that I need to further articulate so that now they  could engage productively with someone's idea? And if it's not, then actually it's just highlighting; pulling  from what a student says, the valuable pieces that you think are going to be important for the continued  work of the class, rather than, literally, a student says something, you say verbatim, and then you ask  more questions. It's really tracking what seems to be important for the development of everyone's  thinking, that collective as an intellectual community that's working together.  Mike: That's really helpful. And I think what I heard are simultaneous things that are happening. One is  attending to the ideas that you want to position as important. And the other thing that really jumps is this idea that we're also positioning the child as the author of the ideas.  Hala: Yes. And you know, in later grades we've seen teachers being able to do this in grades 1 and 2, is  often—especially when we are working early on to build that classroom talk community, that math talk  community—is encouraging students as listeners to someone to say, ‘Did you hear something that you  think is important for the way we are really working on this problem in what Mike said? So, let's listen.  Was there something you have a question about, you're not certain about?' Also, distributing the work of the narration, if we want to call it that way, so it's distributed. It's not just about me, but now the class  is listening and trying to pull what's important and worthy of focusing on.  Mike: I love that. Particularly that idea that you can in fact distribute the idea of narration to the class, and it doesn't just live with the teacher. It also advances that broader cultural goal that you have, which  is that the students are actually sense makers, which is the thing from the very beginning of this  conversation.  Hala: Again, it goes back to the way I think about all the practices that we've talked about, to be very  interconnected. It's not like we know you set up norms, you put them on a chart. You know, norms are  reinforced, are renegotiated with your students through the work that you do. And there's a lot of  socializing that you're doing while you're working on content. It reinforces certain ideas, it reintroduces  certain ideas for others to see how they're able to access them and be part of them. So yes, I agree with  you. They're all connected in that way.  Mike: Well, Hala, before we close the podcast, I'm wondering if you could share some resources with  listeners who might be encountering some of the ideas we're talking about for the first time. Is there  anything that you might suggest for a listener who just wants to keep thinking about this and perhaps  learn more?  Hala: So, if they're interested in thinking a little bit more about representations, there is a recent article that I published with Dr. Eric Siy, who is currently at Boston University, in relation to what multiple  representations mean. And how different they are from just using different representations. Mike: Yep. We could absolutely put a link to that on the podcast notes.  Hala: Yeah. And I find the work of Dr. Amy Parks at Michigan State University. You know, she has this  book called ‘Exploring Mathematics Through Play in the Early Childhood Classroom.' [It] has wonderful  pieces that really could support this work in relation to the idea of reasoning in kindergarten, discourse  in kindergarten. And it could happen during play. It doesn't have to happen necessarily only during  academic tasks that are, like, problem-solving situations or worth problems.  Mike: We could absolutely add a link to that. And I think that's probably another great podcast that we  should do relatively soon.  Hala: Yes, I find you really connecting wonderful, cohesive dots together here, which I think is really  going to be helpful to the listener.  Mike: Well, I want to thank you so much for joining us, Hala. It's really been a pleasure talking with you.  Hala: Thank you very much. And it's been a great opportunity to talk about these ideas with you, and the  questions are on target in terms of the things that we have to pay attention to.  Mike: Oh, thank you so much.   Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence  and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org   

  Rounding Up  Season 1 | Episode 15 – Productive Ways to Build Fluency with Basic Facts Guest: Dr. Jennifer Bay-Williams  Mike Wallus: Ensuring students master their basic facts remains a shared goal among parents and educators. That said, many educators wonder what should replace the memorization drills that cause so much harm to their students' math identities. Today on the podcast, Jenny Bay-Williams talks about how to meet that goal and shares a set of productive practices that also support student reasoning and sense making.  Mike: Welcome to the podcast, Jenny. We are excited to have you.  Jennifer Bay-Williams: Well, thank you for inviting me. I'm thrilled to be here and excited to be talking about basic facts.  Mike: Awesome. Let's jump in. So, your recommendations start with an emphasis on reasoning. I wonder if we could start by just having you talk about the ‘why' behind your recommendation and a little bit about what an emphasis on reasoning looks like in an elementary classroom when you're thinking about basic facts.  Jenny: All right, well, I'm going to start with a little bit of a snarky response: that the non-reasoning approach doesn't work.  Mike and Jenny: ( laugh )  Jenny: OK. So, one reason to move to reasoning is that memorization doesn't work. Drill doesn't work for most people. But the reason to focus on reasoning with basic facts beyond that fact, is that the reasoning strategies grow to strategies that can be used beyond basic facts. So, if you take something like the making 10 idea—that nine plus six, you can move one over and you have 10 plus five—is a beautiful strategy for a 99 plus 35. So, you teach the reasoning upfront from the beginning, and it sets students up for success later on.  Mike: That absolutely makes sense. So, you talk about the difference between telling a strategy and explicit instruction. And I raised this because I suspect that some people might struggle to think about how those are different. Could you describe what explicit instruction looks like and maybe share an example with listeners?  Jenny: Absolutely. First of all, I like to use the whole phrase: ‘explicit strategy instruction.' So, what you're trying to do is have that strategy be explicit, noticeable, visible. So, for example, if you're going to do the making 10 strategy we just talked about, you might have two ten-frames. One of them is filled with nine counters, and one of them is filled with six counters. And students can see that moving one counter over is the same quantity. So, they're seeing this flexibility that you can move numbers around, and you end up with the same sum. So, you're just making that idea explicit and then helping them generalize. You change the problems up and then they come back and they're like, ‘Oh, hey, we can always move some over to make a 10 or a 20 or a 30' or whatever you're working on. And so, I feel like, in using the counters, or they could be stacking unifix cubes or things like that. That's the explicit instruction. Jenny: It's concrete. And then, if you need to be even more explicit, you ask students in the end to summarize the pattern that they noticed across the three or four problems that they solved. ‘Oh, that you take the bigger number, and then you go ahead and complete a 10 to make it easier to add.' And then, that's how you're really bringing those ideas out into the community to talk about. For multiplication, I'm just going to contrast. Let's say we're doing add a group strategy with multiplication. If you were going to do direct instruction, and you're doing six times eight, you might say, ‘All right, so when you see a six,' then a direct instruction would be like, ‘Take that first number and just assume it's a five.' So then, ‘Five eights is how much? Write that down.' That's direct instruction. You're like, ‘Here, do this step here, do this step here, do this step.'  Jenny: The explicit strategy instruction would have, for example—I like eight boxes of crowns because they oftentimes come in eight. So, but they'd have five boxes of crowns and then one more box of crowns. So, they could see you've got five boxes of crowns. They know that fact is 40, they—if they're working on their sixes, they should know their fives. And so, then what would one more group be about? So, just helping them see that with multiplication through visuals, you're adding on one group, not one more, but one group. So, they see that through the visuals that they're doing or through arrays or things like that. So, it's about them seeing the number of relationships and not being told what the steps are.  Mike: And it strikes me, too, Jenny, that the role of the teacher in those two scenarios is pretty different.  Jenny: Very different. Because the teacher is working very hard ( chuckles ) with the explicit strategy instruction to have the visuals that really highlight the strategy. Maybe it's the colors of the dots or the exact ten-frames they've picked and have they filled them or whether they choose to use the unifix cubes and how they're going to color them and things like that. So, they're doing a lot of thinking to make that pattern noticeable, visible. As opposed to just saying, ‘Do this first, do that second, do that third.'  Mike: I love the way that you said that you're doing a lot of thinking and work as a teacher to make a pattern noticeable. That's powerful, and it really is a stark contrast to, ‘Let me just tell you what to do.' I'd love to shift a little bit and ask you about another piece of your work. So, you advocate for teaching facts in an order that stresses relationships rather than simply teaching them in order. I'm wondering if you can tell me a little bit more about how relationships-based instruction has an impact on student thinking.  Jenny: So, we want every student to enact the reasoning strategies. So, I'm going to go back to addition, for example. And I'm going to switch over to the strategy that I call pretend-to-10, also called use 10 or compensation. But if you're going to set them up for using that strategy, [there are] a lot of steps to think through. So, if you're doing nine plus five, then in the pretend-to-10 strategy, you just pretend that nine is a 10. So now you've got 10 plus five and then you've got to compensate in the end. You've got to fix your answer because it's one too much. And so, you've got to come back one. That's some thinking. Those are some steps. So, what you want is to have the students automatic with certain things so that they're set up for that task. So, for that strategy, they need to be able to add a number onto 10 without much thought.  Jenny: Otherwise, the strategy is not useful. The strategy is useful when they already know 10 plus five. So, you teach them this, you teach them that relationship, you know 10 and some more, and then they know that nine's one less than 10. That relationship is hugely important, knowing nine is one less than 10. Um, and so then they know their answer has to be one less. Nine's one less than 10. So, nine plus a number is one less than 10 plus the number. Huge idea. And there's been a lot of research done in kindergarten on students understanding things like seven's one more than six, seven's one less than eight. And they're predictive studies looking at student achievement in first grade, second grade, third grade. And students, it turns out that one of the biggest predictors of success, is students understanding those number relationships. That one more, one less, um, two more, two less. Hugely important in doing the number sense. So that's what the relationship piece is, is sequencing facts so that what is going to be needed for the next thing they're going to do, the thinking that's going to be needed, is there for them. And then build on those relationships to learn the next strategy.  Mike: I mean, it strikes me that there's a little bit of a twofer in that one. The first is this idea that what you're doing is purposely setting up a future idea, right? It's kind of like saying, ‘I'm going to build this prior knowledge about ten-ness, and then I'm going to have kids think about the relationship between 10 and nine.' So, like, the care in this work is actually really understanding those relationships and how you're going to leverage them. The other thing that really jumps out from what you said, this has long-term implications for students thinking. It's not just fact acquisition, it's what you said, research shows that this has implications for how kids are thinking further down the road. Am I understanding that right?  Jenny: That's absolutely correct. So just that strategy alone. Let's say they're adding 29 plus 39. And they're like, ‘Oh hey, both of those numbers are right next to the next benchmark. So instead of 29 plus 39, I'm going to add 30 plus 40, 70. And I got, I went up two, so I'm going to come back down two. And I know that two less than a benchmark's going to land on an eight to that.' Again, it's coming back to this relationship of how far apart numbers are, what's right there within a set of 10, helps then to generalize within 10s or within 100s. And by the way, how about fractions?  Mike: Hmm. Talk about that.  Jenny: ( laughs ) It generalizes to fractions. So, let's take that same idea of adding. Let's just say it's like, two and seven-eighths plus two and seven-eighths. So, if we just pretended those were both threes because they're both super close to three, then you'd have six, and then you added on two-eighths too much. So, you come back two-eighths, or a fourth, and you have your answer. You don't have to do the regrouping with fractions and all the mess that really gets bogged down. And it's a much more efficient method that, again, you set students up for when they understand these number relationships. When you get into fractions, you're thinking about, like, how close are you to the next whole number maybe, instead of to the next 10s number.  Mike: It strikes me that if you have a group of teachers who have a common understanding of this approach to facts, and everyone's kind of playing the long game and thinking about how what they're doing is going to support what's next, it just creates a system that's much more intentional in helping kids not only acquire the facts, but build a set of ways of thinking.  Jenny: Mike, that's exactly it. I mean, here we are, we're trying to make up for lost time. We never have enough time in the classroom. We want an efficient way to make sure our kids get the most learning in. And so, to me that is about investing early in the fact strategies. Because then actually when you get up to those other things that you're adding or subtracting or multiplying or whatever you're doing, you benefit from the fact that you took time early to learn those strategies. Because those strategies are now very useful for all this other math that you're doing. And then students are more successful in making good choices about how they're going to solve those problems that are, oftentimes—especially when, I like to mention fractions and decimals at least once in a basic facts talk because we get back, by the time we get into fractions and decimals—we're back to just sometimes only showing one way. The sort of standard algorithm way. When, in fact, those basic facts strategies absolutely apply to almost-always-more-efficient strategies for working with fractions and decimals.  Mike: I want to shift a little bit. One of the things that was really helpful for me in growing my understanding is, the way that you talk about a set of facts that you would describe as ‘foundational' facts and another set of facts that you would describe as ‘derived' facts. And I'm wondering if you can unpack what those two subsets are and how they're related to one another.  Jenny: Yeah. So, the foundational facts are ones where automaticity is needed in order to enact a strategy. So, to me, the foundational fact strategies are, they're names. Like the doubling strategy or double and double again, some people call it. Or add a group for multiplication, and the addition ones of making 10s and pretend-to-10 strategies. And in those strategies, you can solve lots of different facts. But there's too much going on ( laughs ) in your brain if you don't have automaticity with the facts you need. So, for example, if you have your six facts, and you're trying to get your six facts down. And you already know your fives, like, automaticity with your fives. Then that becomes a useful way to get your sixes. So, if you have six times eight, and you know five times eight is 40, then you're like, ‘I got one more 8, 48.'  Jenny: That's an added group strategy. But if you're not automatic with your fives, this is how this sounds when you're interviewing a child. They're going to use add a group strategy, but they don't know their fives. So, then they're like, ‘Let's see, five times eight is 5, 10, 15, 20, 25, 30, 40. Now, what was I doing?' Like, they can't finish it because they were skip-counting with their fives. They lose track of what they're doing, is my point. So, the key is that they just know those facts that they need in order to use a strategy. And that, going back to, like, the pretend-to-10, they got to know 10-and-some-more facts to be successful. They have to know nine's one less than 10 to be successful. So, that's the idea is, if they reach automaticity with the foundational fact sets, then their brain is freed up to go through those reasoning strategies.  Mike: That totally makes sense. I want to shift a little bit now. One of the things that I really appreciated about the article was that you made what I think is a very strong, unambiguous case for ending many of the past practices used for fact acquisition—worksheets and timed tests, in particular. This can be a tough sell because this is often what is associated with elementary mathematics, and families kind of expect this kind of practice. How would you help an educator explain the shift away from these practices to folks who are out in the larger community? What is it that we might help say to folks to help them understand this shift?  Jenny: That's a great question, and the real answer is it depends, again, on audience. So, who is your audience? Even if the audience is parents, what do those parents prioritize and want for their children? So, I feel like [there are] lots of reasons to do it, but to really speak to what matters to them. So, I'm going to give a very generic answer here. But for everyone, they want their child to be successful. So, I feel that that opportunity to show, to give a problem like 29 plus 29, and ask how parents might add that problem. And if they think 30 plus 30 and subtract two to get to the answer, whatever, then that gives this case to say, ‘Well this is how we're going to work on basic facts. We're building up so that your child is ready to use these strategies. We're going to start right with the basic facts, learning these strategies. These really matter.' Jenny: And the example I gave could be whatever fits with the level of their kid. So, it could be like 302 minus 299. It's a classic one where you don't want your child to implement an algorithm there, you want them to notice those numbers are three apart. And so, there's this work that begins early. So, I think that's part of it. I think another part of it is helping people just reflect on their own learning experiences. What were your learning experiences with basic facts? And even if they liked the speed drills, they oftentimes recognize that it was not well-liked by most people. And also, then they really didn't learn strategies. So, I feel like we have to be showing that we're not taking something away, we're adding something in. They are going to become automatic with their facts. They're not going to forget them because we're not doing this memorizing that leads to a lot of forgetting. And bonus, they're going to have these strategies that are super useful going forward. So, to me, those are some of the really strong speaking points. I like to play a game and then just stop and pause for a minute and just say, ‘Did you see how hard it was for me to get you quiet? Do you see how much fun you were having?' And then I just hold up a worksheet ( laughs ). I'm like, ‘And how about this?' You know, again, that emotional connection to the experience and the outcomes.  Mike: That is wonderful. Since you brought it up, let's talk about replacements for worksheets and timed tests.  Jenny: Um-hm.  Mike: So, you advocate for games as you said, and for an activity-based approach. I think that what I want to try to do is get really specific so that if I'm a classroom teacher, and I can't see a picture of that yet, can you help paint a picture? Like what might that look like?  Jenny: I love that question because [there are] lots of good games and lots of places. But again, like I said earlier, this thinking really deeply about what game I'm choosing and for what. What do my students need to practice? And then being very intentional about game choice is really important. So, for example, if students are working on their 10-and-some-more facts, then you want to play a game where all the facts are 10-and-some-more facts. That's what they're working on. And then maybe you mix in some that aren't. Or you play a game with that and then they sort cards and find all the solve the 10 and more, or [there are] lots of things they can do. They can play concentration, where the fact is hidden and the answer is hidden and things like that. So, you can be very focused. And then when you get to the strategies, you want to have a game that allows for students to say, allow their strategies.  Jenny: So, I'm a big fan of, like, sentence frames, for example. So, [there are] games that we have in our ‘Math Fact Fluency' book that are in other places that specifically work on a strategy. So, for example, if I'm working on the pretend-to-10 strategy, I like to play the game fixed-addend war, which is the classic game of war, except, there's an addend in the middle, and it's a nine, to start. And then each of the two players turns up a card. So, Mike, if you turn up a seven, then you're going to explain how you're going to use the pretend-to-10 strategy to add it. And I turned up a six, so I'm going to, I'm going to do this then I'll, you can do it. So, I turned up a six. So, I'm going to say, ‘Well, 10 and six is 16, so nine and six is one less, 15.' I've just explained the pretend-to-10 strategy. And then you get your turn.  Mike: And I'd say, ‘Well seven and 10, I know seven and 10 is 17, so seven and nine has to be one less, and that's 16. Jenny: Yeah. So, your total's higher than mine, you win those two cards, you put them in your deck, and we move on. So, that's a way to just practice thinking through that strategy. Notice there's no time factor in that. You have a different card than I have. You have as much time, and we're doing think-aloud. These are all high-leverage practices. Then we get to the games where it's like, you might turn up a six and a five where you're not going to use the pretend-to-10 strategy for that. You've got to think, ‘Oh that doesn't really fit that strategy because neither one of those numbers is really close to 10. Oh hey, it's near a double, I'm going to use my double.' So, you sequence these games to, if you start with one of those open-ended games, it might be too big of a jump because students aren't ready to choose between their strategies. They have to first, be adept at using their strategies. And once they're adept at using them, then they're ready to play games where they get to choose among the strategies.  Mike: So, you're making me think a couple things, Jenny. One is, it's not just that we're shifting to using games as a venue to practice to get to automaticity. You're actually saying that when we think about the games, we really need to think about, ‘What are the strategies that we're after for kids?' And then make sure that the way that the game is structured, like, when you're talking about the pretend-to-10, with the fixed addend. That's designed to elicit that strategy and have kids work on developing their language and their thinking around that particularly. So, there's a level of intent around the game choice and the connection to the strategies that kids are thinking about. Am I understanding that right?  Jenny: That's it. That's exactly right. That's exactly right. And a huge, a lot of intentionality so that they have that opportunity and a no-pressure, a low-stress, think through the strategy. If they make a mistake, they're peer or themselves usually correct it in the moment, and they get so much practice in. I mean, imagine going through half a deck of cards playing that game.  Mike: Yeah.  Jenny: That's 26 facts. And then picture those 26 facts on a page of paper. And then, and again, in the game that you've got the added benefit of think-aloud, and then you're hearing what your peer has said.  Mike: You know, one of the things that strikes me is, if I'm a teacher, I might be thinking like, ‘This is awesome, I'm super excited about it. Holy mackerel, do I have to figure these games out myself?' And I think the good news is, there's a lot of work that's been done on this. I know you've done some. Do you have any recommendations for folks? There's of course curriculum. But do you have recommendations for resources that you think, help a teacher think about this or help a teacher see some of the games that we're talking about?  Jenny: Well, I'm going to start with my ‘Math Fact Fluency' book because that is where we go through each of these strategies, each of the foundational facts sets and the strategies, and for each one supply a game. And then from those games they're easily adaptable to other settings. And some of the games are classic games. So, there's a game, for example, called ‘Square Deal.' And the idea is that you're covering a game board, and you're trying to make a square. So, you get a two-by-two grid taken, and you score a point or five points or whatever you want to score. Well, we have that game housed under the 10-and-some-more facts. So, all the answers are like 19, 16, 15, and the students turn over a 10 card and another card, and if it's a 10 and a five, they get to claim a 15 spot on the game board.  Jenny: Well, that game board can be easily adapted to any multiplication fact sets, any other addition. I like to do a Square Deal with 10 and some more, and then I like to do Square Deal with nine and some more. There's my effort, again, to come back to either pretend-to-10 or making 10. Where they're like, ‘Oh, I just played 10 and some more. Now we're doing the same game, but it's nine and some more.' So, I feel like there's a lot of games there. And there is a free companion website that has about half of the games ready to download in English and in Spanish.  Mike: Any chance you'd be willing to share it?  Jenny: Yeah, absolutely. So, you can just Google it. The Kentucky Center for Mathematics created it during Covid, actually, as a gift to the math community. And so, if you type in ‘Kentucky Center for Math' or ‘KCM math fact fluency companion website,' it will pop up.  Mike: That's awesome. I want to ask you about one more thing before we close because we've really talked about the replacement for worksheets, the replacements for timed tests. But there is a piece of this where people think about ‘How do I know?' right? ‘How can I tell that kids have started to build this automaticity?' And you make a pretty strong case for interviewing students to understand their thinking. I'm wondering if you could just talk again about the ‘why' behind it and a little bit about what it might look like.  Jenny: So, first of all, timed tests are definitely a mistake for many reasons. And one of the reasons— beyond the anxiety they cause—they're just very poor assessment tools. So, you can't see if the student is skip-counting or not, for example, for multiplication facts. You can't see if they're counting by ones for the addition facts. You can't see that when they're doing the test, and you can't assume that they're working at a constant rate; that they're just solving one every, you know, couple of seconds, which is the way those tests are designed. Because I can spend a lot of time on one and less time on the other. So, they're just not, they're just not effective as an assessment tool. So, if you flip that. Let's say they're playing the game we were talking about earlier, and you just want to know can they use the pretend-to- 10 strategy?  Jenny: That's your assessment question of the day. Well, you just wander around with a little checklist ( chuckles ), you know? Yes, they can. No, they can't. And so, a checklist can get at the strategies, and a checklist can also get at the facts like how well are they doing with their facts? So, once they do some of those games that are more open-ended, you can just observe and listen to them and get a feel for that. If they're playing Square Deal with whatever fact, you know. So, what happens is you're, like, ‘I wonder how they're doing with their fours. We've really been working with their fours a lot.' Well, you can play Square Deal or a number of other games where that day you're working on fours. The fixed-addend war can become fixed-factor war, and you put a four in the middle. So adaptable games and then you're just listening and watching.  Jenny: And if you're not comfortable with that approach, then they can be playing those games, and you can have students channeling through where you do a little mini-interview. It only takes a few questions to get a feel for whether a student knows their facts. And you can really see who's automatic and who's still thinking. So, for example, a student who's working on their fours, if you give them four times seven, they might say, ‘Twenty-eight.' I call that automatic. Or they might, they might do four times seven, and they pause, and they're like, ‘Twenty-eight.' Then I'm like, ‘How did you think about that?' And they're like, ‘Well, I doubled and doubled again.' ‘Great.' So, I can mark off that they are using a strategy, but they're not automatic yet. So that to me is a check, not a star. And if I ask, ‘How did you do it?' And they say, ‘Well, I skip-counted.' Well then, I'm marking down the skip-counted. Because that means they need a strategy to help them move toward automaticity. Mike: I think what strikes me about that, too, is, when you understand where they're at on their journey to automaticity, you can actually do something about it as opposed to just looking at the quantity that you might see on a timed test. What's actionable about that? I'm not sure, but I think what you're suggesting really makes the case that I can do something with data that I observe or data that I hear in an interview or see in an interview.  Jenny: Absolutely. I mean this whole different positioning of the teacher as coaching the student toward their growth; helping them grow in their math proficiency, their math fluency. You see where they're at and then you're monitoring that in order to move them forward instead of just marking them right or wrong on a timed test. I think that's a great way to synthesize that.  Mike: Well, I have to say, it has been a pleasure talking with you. Thank you so much for joining us today.  Jenny: Thank you so much. I am again thrilled to be invited and always happy to talk about this topic.  Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.  © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 1 | Episode 14 – Enhanced Tasks for Multilingual Learners Guest: Dr. Zandra de Araujo Mike Wallace: How can educators take concrete steps to enhance tasks for multilingual learners? That's the subject of today's podcast. Today we'll talk with Dr. Zandra de Araujo, the chief equity officer at the University of Florida's Lastinger Center for Learning, about three ways to enhance tasks for multilingual learners and how to implement them in an elementary mathematics setting. We'll also discuss practical strategies and resources for supporting multilingual learners regardless of their age or grade.  Mike: Hey, Zandra. Welcome to the podcast. Zandra de Araujo: Thanks for having me. I'm excited to be here. Mike: I'm super excited to be talking to you. So, I'd love to just start with a quote you and your co-authors wrote. You say, ‘Rather than focus on language before mathematics, research shows that multilingual learners both can and should develop mathematical knowledge and language proficiency simultaneously.' Can you talk a little bit about that statement and share some of the research that informs it? Zandra: Sure. So, basically, if you think about learning a new language, you need to use it to get better at it. And so, in the past, people were more likely to put language first and to hold off on academics until students learned English. And what we've learned since then from brilliant scholars like Judy Moskowitz and others, is that we should simultaneously grow math alongside language development. And there's a couple of reasons for that. One, it helps improve your math learning, your language learning at the same time, which is great. It doesn't put you below grade level for your math learning because you're waiting to catch up with English first. And we know that that proficiency in your first language also will lead to better proficiency in your second language in math and other areas. So, there's only benefits really.  Zandra: And also, if you think about kids who are native English speakers, they're also learning how to talk about mathematics in school and how to use math language. And so, you might as well do it with the whole class and practice discourse and use good multimodal representations and communication skills to enhance everybody's language learning and math learning because you learn through and with language. And so, you can't really put language—or mathematics—on hold completely for kids. It's just not the right thing to do. Mike: I loved where you said you learn through and with language.  Zandra: Um-hm. Mike: Could you just expand upon that? Because it really feels like there's a lot of wisdom in that statement. Zandra: Yeah, I mean, the way that we learn is we listen, we participate, we talk, we discuss. We have to communicate ideas from one person to another. And it's this communication—and communication is not just in one language in one way. And language is more expansive than that. And we need to think about that. And the way you communicate what you've learned is through language. Or you show it visually. But usually as you're showing, you're gesturing and communicating in maybe non-verbal language communication. So, I think we forget that math is inherently language based as we communicate it in schools and as we typically experience it in schools. Mike: Thank you. I want to shift a little bit and talk about the three types of enhancements that you and your co-authors are talking about. So, using and connecting multiple representations, thinking through language obstacles, and contextualizing concepts and problem-solving activities. And what I'd like to do is take time to discuss each one of these. So, to begin, can you talk a little bit about what you mean by using and connecting multiple representations? Zandra: Sure. I tend to put things in my own frame of learning a second language. So, if you think about when you travel to a country that you don't speak the language in fluently, you probably do a lot of gesturing. You look for signs that don't have words in that language, necessarily, if you can't read it. You might draw something, you might do a lot of things. So, visuals and representations are very helpful when we're learning something new or trying to understand something that we already understand, we just can't communicate it. So, in mathematics, a lot of our representations are serving that purpose. They allow us to learn things in a more deep way.  Zandra: So, if you think about, I can show you something like the number five written out. I can show you five unifix cubes, I could show you five tally marks. Those are all different representations that very young children experience. And we're trying to communicate the same concept typically, of five; like the total set of five, the cardinality of five things, typically. And so, kids, when they experience all these different things in different ways, and we connect explicitly across them, it really helps them to understand something in a new or different way. But also, for students who are acquiring English, it allows them to connect the visual with their home language that they're thinking in their brain. And they probably have the words for it in their home language. They may just not understand just the spoken word. But when you see a representation, you have more ideas to anchor on. Mike: Yeah. As you described that, you can see how critically important that would be for multilingual learners and how much that would both support them and allow them to make the connections.  Zandra: Um-hm. And it's not just for multilingual students. I can't imagine the number of times I've been in a classroom and a teacher might model something with base ten blocks and maybe draw on representation of base ten blocks on the board and then never take the extra step to explicitly link it to the numerals that it … Mike: Um-hm. Zandra: … they're representing or the bundles and things like that. But those connections are what we're hoping kids will make. And so, explicitly linking those things and talking across them. And ‘How do you see five here? And how do you see five here?' is really important for all students. But it's especially beneficial if you're still acquiring the language of instruction. Mike: Absolutely. So, let's shift gears and talk a little bit about language obstacles. So, as a monolingual English speaker, this is an enhancement that I'd really like to understand in more depth. Zandra: ( laughs ) As a monolingual also, uh, English speaker that grew up in a Portuguese-speaking household and someone who is trained in mathematics teaching and learning and not in language teaching and learning specifically, this was very interesting to me, too. Essentially, it seems intuitive that you would take away language if that's an obstacle. And that is the main obstacle that students who are acquiring English in school are facing. It's not necessarily that they're below grade level in math. Sometimes they are. But many times they're not. They might be above grade level. But there are specific potential needs for support around English-language proficiency or acquisition. And so, when we think about language obstacles, it's those things that get in the way of learning the mathematics. And there's kind of two ways that you could address them: One is you remove them all, and then two is you scaffold up so that they can access it. Zandra: I'm more in favor of that approach where we scaffold and try to help further their language alongside their mathematics. Because that goes through the very first thing we talked about, is that you're enhancing and developing English alongside mathematics. But there are some times where there's just unnecessary obstacles that are really getting in the way of understanding what you're trying to do in mathematics. And that's kind of what we provided in the article is the list of some of these things. So, for example, a low-frequency term, and we give an example in the article, if you say ‘perusing a menu,' a lot of children do not use that in their day-to-day language, English language learners or otherwise. And so, we might just say, ‘looking at the menu.' It's conveying the same meaning, but it's a more common, frequently used term.  Zandra: So, more students will understand what that means. They're not getting hung up on this word. They're able to actually pursue the math task. Again, you could also say like, “perusing,” oh that's a new word. It means like ‘looking' or ‘reading,' you know, ‘looking over.' And that is certainly an option, but sometimes you just need the kids to understand the task that you're providing, and you don't want to do so much language development on things that are not really going to impact their math. So, as teachers, we make these decisions every day, and I think sometimes we can make these decisions just to eliminate some potential obstacles. There's a lot of other words. A lot of my colleagues and my co-authors have written about words with multiple meanings. Like ‘table.' If you're new to English and you hear table, you're probably going to think of the most commonly encountered table in your life, which is probably like a kitchen table or a table … Mike: Um-hm.  Zandra: … at school and not a mathematical table, which is different. And so, uncovering these things, thinking about them as somebody who's a monolingual English speaker is really important because it just passes by us because it's normal to us. But we need to put ourselves in the shoes of these children as well. Mike: Yeah. I think what it really made me think about is structurally there's lots of challenges if you're trying to make meaning of them for the first time. Like words that have multiple meanings jump out. I found that part of the article really helpful. It helped me see issues with the language structure that having just kind of learned it naturally, they're invisible, right? Zandra: Yeah. I had a colleague at Missouri that taught ESOL classes, and that was her area. And she said, “You say a big, red ball, but you don't say a red, big ball in English.' And I was like, ‘Oh yeah, it's like they're both adjectives,' but we do have patterns that I've never really thought about. But they are common, and you hear them in people that are acquiring the language that like, ‘Oh, it's not how I would say it.' ‘Why not?' And you don't know these rules if you weren't trained in this area. I also had a former graduate student who said—he was Korean—and when he came, he said it was confusing because ‘no, yeah' means ‘yeah.' But' yeah, no' means ‘no.' And it's similar type things that we say, and we don't understand. And ever since he told me that, I'm like, ‘Oh yeah, I totally get that.' And I say it all the time, and I just never noticed how confusing that definitely is. Mike: So, I'm really excited about this last bit, too. I really want to talk about the importance of context and talk a little bit about how context impacts learning, particularly for kids at the elementary level. If I'm an elementary educator using a curriculum, what's your sense of what I might do to build context into my students' mathematical experience? Zandra: So, context helps us make sense of things because we can relate it to our actual uses or things we're familiar with and use that as a sense making tool. So, it's kind of similar to representations in some ways. In elementary school, we're very fortunate that there's so many things that the kids come in contact with because we tend to teach all subject areas in our classrooms. In elementary, we do a lot of counting, for example. And there's so many things that we can count. Or we've been counting every day that we can tie into. That's why a lot of teachers like to use calendar math and things like that because it's interesting, it's something the kids are familiar with. And so that context allows them to think through how they do it in the real world and connect that thinking with the mathematical reasoning, which is really powerful. Zandra: It also is just more interesting to the kids. I think they like it when it's something … I mean, if you want to see a kid get [really] excited, figure out what their pet's name is and make a problem about their, their pet doing something. They get [really] excited because it's, like, personalized to them. And it's not a real deep, meaningful connection. It's not super culturally relevant necessarily, just putting a cat or dog's name in a task. But it's the idea that you're connecting to something that is interesting and matters to the kids, and that they can use that for reasoning and sense making. And that's what we ultimately want. Mike: I'm going to mine what you said for another nugget of wisdom. You said at the beginning, context is a reasoning tool. Did I capture that correctly?  Zandra: Um-hm. Yeah, absolutely. I think I can reason far better with something that I can actually play out and think through the process that I do. And I can connect it to the real world, and then I can think about, like, ‘Oh, what did I actually just do?' Because some things are pretty automatic that we do in every day, and we don't know that they're connected to math or could be. And when we reason through it, it really helps us to reason a little deeper. There's been a lot of math studies—most of them are older now—but about kids who did math in the real world as jobs. Maybe they were, like, working after school when we had real money, ( laughs ) physical money, more frequently, and they could do all these calculations very easily. But they struggled with school mathematics that was decontextualized. So then, as teachers learn how to bring in the context they're familiar with, they know how that works and then they can connect it, the representation to the symbol. So, it's all kind of connected, all three of these enhancements at the end of the day. Mike: Yeah, that makes a lot of sense. So, before we finish, Zandra, I'm wondering if you can point listeners to any kind of additional resources that you think would help them take the conversation that we're having and maybe add some depth to their understanding. Zandra: Sure. So, any three of my co-authors' work is great to find online. Fortunately, I think all three of them consult with the EL Success Forum. It's elsuccessforum.org, I believe. That is a group that has put together amazing banks of resources for teachers and people that work in schools around English language learners, in particular. So, that's a great one that I point a lot of people to. I have a Grassroots Workshop that I made that's on teaching mathematics with English learners that you could find online. And I think beyond that, there's a number of great resources through TODOS: Mathematics for ALL, which is a professional organization. They're an affiliate of NCTM, and they have some amazing resources as well. Mike: That is fabulous. Thank you so much for joining us, Sandra. It has really been a pleasure talking. Zandra: Yeah, likewise. Thanks for having me. I appreciate the opportunity to share about this. Something I'm super passionate about, and I'm always happy to talk about. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 1 | Episode 13 – Keep Calm Guest: Nancy Anderson, EdD   Mike Wallace: We often ask students to share their strategies. But, what does it look like to uncover and highlight the reasoning that informs that strategy? Today on the podcast, we'll talk with Nancy Anderson, a classroom teacher and professional learning developer, about strategies to elicit the reasoning at the heart of the student's thinking.  Welcome to the podcast, Nancy. I am so excited to talk to you today. Nancy Anderson: Thank you. Likewise, Mike. Mike: I'd like to begin with a quote from your article, “Keep Calm and Press for Reasoning.” In it, you state: “Mathematical reasoning describes the process and tools that we use to determine which ideas are true and which are false.” And then you go on to say that “in the context of a class discussion, reasoning includes addressing the strategy's most important ideas and highlighting how those ideas are related.” So, what I'm wondering is, can you talk a little bit about how eliciting a strategy and eliciting reasoning may or may not be different from one another? Nancy: So, when we elicit a strategy, we're largely focused on what the student did to solve the problem. For example, what operations and equations they might have used, what were the steps, and even what tools they might have used. For example, might they have used concrete tools or a number line? Whereas eliciting reasoning focuses on the why behind what they did. Why did they choose a particular strategy or equation? What was it in the problem that signaled that particular equation or that particular operation made sense? And if the strategy included several steps, what told them to go from one step to the next? How did they know that? And then similarly for the tools, what is it in the problem that suggested to them a number line might be an effective strategy to use? And lastly, listening reasoning sort of focuses on putting all those different pieces together so that you talk about those different elements and the rationale behind them in such a way that the people listening are convinced that the strategy is sound. Mike: That's actually really helpful. I found myself thinking about two scenarios that used to play out when I was teaching first grade. One was I had a group of children who were really engaging with the number line to help them think about difference unknown problems. And what it's making me think is, the focus of the conversation wasn't necessarily that they used the number line. And it's like, ‘Why did this particular jump that you're articulating via number line? What is it about the number line that helped you model this big idea or can help make this idea clearer for the other students in the class?' Nancy: Exactly, yes. So, when I think about reasoning, I think about different pieces coming together to form a cohesive explanation that also serves as a bridge to using a particular strategy for one particular problem, [and] as a tool for solving something similar in the future. Mike: So, I have a follow-up question. When teachers are pressing students for their reasoning, what counts as reasoning? What should teachers be listening for? Nancy: Broadly, mathematical reasoning describes the processes and tools that we use to determine which ideas are true and which are false. Because mathematics is based upon logic and reasoning—not a matter of who says it or how loudly they say it or how convincingly they say it, but rather, what are the mathematical truths that undergird what they're saying? That's sort of a broad definition of mathematical reasoning, which I think certainly has its merits. But then I think about the work of teaching, particularly at the elementary level. I think it's helpful to get much more specific. So, when we think about elementary arithmetic, reasoning really focuses on connecting computational strategies to the operations and the principles that lie underneath. So, in the context of a class discussion, when we have a student explain their reasoning, we're really trying to highlight a particular strategy's most important ideas and how those ideas are related, but in such a way that others can listen and say, ‘Oh, I get it. If I were to try the problem again, I do believe that's going to lead to the correct answer.' Or if it was this problem, which is similar, ‘I think I can see how it might make sense for me to use this approach here with these slight adjustments.' So, do you want to take an example? Mike: Yeah, I'd love to. Nancy: So, for example, in a first-grade class, there might be a class discussion about different strategies for adding seven plus eight. And I think in a lot of classes at one point, the teacher would likely want to highlight the fact that you can find that sum using doubles plus one. So, in this particular instance, if a student were to talk about their reasoning, we'd want to encourage that student and certainly help that student talk about the following ideas: the connection between seven plus eight and seven plus seven, and the connection between their answers, namely because the second addend has changed from seven to eight, and noting the connections between the second addend and the answers, namely, if the second addend increases by one, so, too does the sum. And finally, we'd want to emphasize what it is we're doing here. Namely, we are using sums that we know to find sums we don't know. Nancy: So, that's an effective example of what reasoning sounds like in the elementary grades. It's very specific. So even though reasoning is the thing that allows us to move from specific examples to generalizations in elementary mathematics, it's oftentimes by really focusing on what's going on with specific examples   Mike: Uh-hm. Nancy: … that students can begin to make those leaps forward. Some of my thinking lately about what I do in the classroom comes from the book ‘Make It Stick,' which talks a lot about learning processes and principles in general. And one of the points that the authors make in the book is that effective learners see important connections, for whatever reasons, sometimes more readily or more quickly than others. So, what I try to do with my teaching then is to say, ‘OK, well how can I help all learners see those relevant and important connections as well?' Mike: Absolutely. So, it really does strike me that there are planning practices that educators could use that might make a press for reasoning more effective. I'm wondering if you could talk about how might an educator plan for pressing for reasoning? Nancy: One thing that I think teachers can do is anticipate, in a very literal sense, what is it that they want students to say as a result of participating in the lesson? So, I think oftentimes we, as classroom teachers, focus on what we want students to learn, i.e., the lesson objective or the essential aim. But that can be a big jump from thinking about that to thinking about the words we literally want to hear come out of student's mouths. So, I think that that's one shift teachers can make to thinking not just about the lesson objective as you'd write on the board, but literally what you want students to say, such that when you walk around and you sort of listen in on small groups, those moments where you say like, ‘Oh yeah, they're on the right track.' And then I think another key shift is thinking more towards specific examples rather than generalizations. Nancy: So, as an example, suppose that in a third- or fourth- or fifth-grade classroom, students were talking about fraction comparison strategies, and the teacher had planned for a lesson where the objective was to determine if a fraction was more or less than a half by using the generalization about all fractions equal to a half. Namely, that the numerator is always half of the denominator. So, that certainly could be something that we might see in, you know, teacher's guide or perhaps in a teacher's planning book. But that's different than what we'd want to hear from students as the lesson progressed. For example, I think the first thing that we'd want to hear as the students we're talking, is a lot of examples, right? The kinds of examples that are going to lead to that key generalization. Like if a student was talking about nine sixteenths, I think we'd want to hear that student reason that nine sixteenths is more than half because half of 16 is eight and nine sixteenths is a little bit more than eight sixteenths. Nancy: And so, what's effective about that kind of planning is that it alerts you to those ideas when you hear them in the room. And it can then help you think about ‘What are the pieces of the explanation that you want to press on.' So, in this case, the key ideas are finding half of the denominator, connecting that value to the fraction that is equivalent to one half, and then comparing that fraction to the actual fraction we're looking at so that we can bring those key ideas to the fore, and the ideas become a strategy for students to use moving forward. Mike: You're making me think about two things kind of simultaneously. The first is, I'm reflecting back on my own practice as a teacher. And at that time, my grade-level team and I, we tried to really enact the whole idea of anticipating student strategies that comes out in ‘The Five Practices' book. But what you're making me wonder about is, we went through, and we said, ‘Here are some of the ways that children might solve this. This is some of the strategies.' The step we didn't take is to say, ‘We know that there are multiple ways that children could attack this or could think about this, but what's the nugget of reasoning? What would we want them to say in conjunction with the strategy that they had so that we were really clear on if a student is counting on to solve this problem, what's the nugget of reasoning that we want to either press on or encourage. If their direct modeling, again, what's the nugget of reasoning that we want to press on. If they're decomposing numbers? Same thing. So, really it makes me think that it's helpful to anticipate what kids might do. But the place that really, like, supercharges that is that thing that you're talking about is, what's the thing that we want them to say that will let us know that they're onto the reasoning behind it? Nancy: Exactly. And I think the conversations you're having or have had with your colleagues reflects where we are with the field generally. I think that the field of mathematics education is at a place where, for the most part, we're on board with the use of discussion as a pedagogy. I don't think that it's a tough sell to convince a lot of folks that students should be spending some amount of time talking. But I don't think that we as a field are nearly as clear on what to do next. And again, as you alluded to with ‘The Five Practices' book, and while I would certainly agree that all of these are important aspects of classroom talk, I think that they skip over this essential idea of pressing for reasoning. Namely, staying with the student beyond just their initial explanation so that their ideas become clear, not just to others, but also clear to them. Mike: I love that. I want to go in a direction that you started to allude to, but you really got to in, in your article. This idea that there's a certain number of questions for follow-up that can really have a tremendous impact on kids. I'm wondering if you could talk a little bit about that. Nancy: My article and more broadly, my interest in press for reasoning, is motivated in large parts, uh, by my professional interest in figuring out, you know, what it is about discussion that makes it such a powerful tool for learning. So, although we have enough empirical evidence to support discussion as an effective pedagogy in math class, we as a field are much less clear in knowing which of the aspects of discussion are most efficacious for learning. What are the mechanisms of student talk that help students learn math more deeply? I had the good fortune many years ago to find some compelling research by Megan Franke and Noreen Webb and their colleagues at UCLA who did some digging into press for reasoning. And through their studies, they have shown that follow-up questions, questions that press students to clarify and strengthen their initial explanation, are associated with students giving more robust and more accurate explanations. Nancy: What their research revealed is that it takes two to three specific follow-up questions in order to either have the student say, more math and more accurate mathematics. So, I think about that so often in my work in the classroom because so often I'll ask a student to explain their reasoning and because they're learning, the explanation comes out either partially correct or partially complete, and I need them to say more. And I might ask them the first follow-up question and either they or I suddenly start to worry. The student might think, ‘Am I saying something wrong? Am I totally off track here? Uh, I'm not really sure why I did what I did.' And then I, of course, as the teacher, I'm so worried about, ‘Am I putting the student on the spot? Am I losing the rest of the class?' And in those moments, I hear myself say, ‘Two to three follow-up questions, two to three follow-up questions,' as a way to remind myself to stay with the student. That if we really do believe that students learn by talking, then it only makes sense that we should expect them to need more than just one turn to get their ideas out in such a way that are clear and accurate to them as well as to the listeners. Mike: So, that's fascinating, Nancy. I think there's two things that stood out from what you said. One is, as a classroom teacher, I appreciate the fact that you acknowledge that feeling of, ‘Am I losing the class?' [It] is something that always exists when you're trying to question and support. But I think the thing that really jumps out is, we have research that says that this actually does have a tremendous impact on kiddos. So even though it might feel counterintuitive, staying with the press for those two to three questions really does have a tremendous impact. I'm wondering what it might sound like to take a student's initial response and then follow up in a way that presses for reasoning. Nancy: So, suppose a fourth-grade class is working on strategies for multi-digit multiplication, and one particular strategy that the teacher would like to emphasize, or showcase, is compensation. Namely, how we can change one or both factors in a multiplication to create an easier computation and then make an adjustment accordingly. For example, we can multiply 19 times 40 by thinking about 20 times 40, and then subtracting 40. Let's suppose that students are working in groups and—on this computation—and the teacher overhears a student talking to their partner about how they use this exact strategy, and briefly checks in with the student and asks, you know, if they'd be willing to share their strategy with the whole class. And the student agrees. So, the teacher calls on the student to tell us, ‘How did you compute 19 times 40?' And the student says, ‘Well, I did 20 times 40 minus 40, and I did that because 20 times 40 is easier.' Nancy: Great. So, we've got some ideas on the table, and so now let's unpack. So, maybe the first question to ask the student is for them to interpret 19 times 40. What does that mean? Literally, it says 19 times 40, but can they give a context? Can they provide an interpretation of that expression with the hope of getting the idea out that we can think of 19 times 40 as 19 groups of 40. And similarly, 20 times 40 as 20 groups of 40. So, once we have the idea of groups of a number out there, can the student tell again why it made sense for them to think of 20 times 40? Why is that easier? Then another follow-up question to ask is, ‘Well, what's the connection between changing that first factor to 20 and subtracting 40?' Because if you think about it, if you're a listener who's unfamiliar with compensation, that's a pretty big leap to go from changing the first factor by one to a second step of subtracting 40. Huh? Mike: It sure is. Nancy: ( laughs ) Right? Like, how does changing it by one mean you subtract 40. And so, here the students can talk about the fact that we found 20 groups of 40, which is one too many groups. So, we compensate by subtracting 40. So, those are some follow-up questions that I think we'd want to ask. Mike: This example just makes so many connections. I'm struck by the fact that, simultaneously, that press for reasoning is helping the child who came up with the idea really build a stronger vocabulary and a justification, and at the same time, it's actually providing access to that strategy for kids who didn't come up with it, who maybe kind of wondering, ‘What? Where did that come from?' So, really it's beneficial for the child who brought the reasoning to the table and to everybody else. The other thing that jumped out is, even in that question where you said, ‘Can you offer this in context?' That's kind of connecting representations, right? Like the child was articulating something that might show up in equation form and asking them to articulate that in a contextual form. [That] is actually a way of challenging their thinking as well. Nancy: Exactly, yes. For many students—and, unfortunately, many more adults—symbols are just that, their symbols. Yet, we who engage in mathematics know that many times symbols are linked to not just one representation, but several, that there's certainly a literal interpretation of any kind of symbol string or numeric expression. But then we can interpret what those expressions mean by connecting back to the different meanings of the operation. So yeah, like you said, Mike, there's two things going on here at least: Helping the other students learn about this particular approach and trusting that it works, but also to helping the original speakers see what it takes to convince others. And in this case, part of that includes the fact that, ‘Oh, when I talk about multiplication, it's helpful to remind people that multiplication refers to putting groups together. Or that it's helpful to think about multiplication in terms of putting equal groups together.' Mike: Well, before we close the podcast, Nancy, I typically ask a question about resources because I suspect for some folks this conversation is one that they've been thinking about for a while. And for other folks, this idea of thinking past strategies toward a reasoning might be a new idea. So, I'm wondering if you'd be willing to share resources that you think would help support people maybe taking this conversation we've had and deepening it. Nancy: Sure. So, my work in this field rests upon the shoulders of many brilliant mathematics educators and some of whom, uh, are people I admire from afar, like Megan Franke and Noreen Webb and their team at UCLA. And still others who I've had the honor to work directly with and learn from, uh, over the past 20 years. And two educators, in particular, are Suzanne Chapin and Cathy O'Connor of Boston University, who are a mathematics educator and applied linguist, respectively. Mike: I adore their work. I'm just going to cut in and say, I'm excited for the resource you're going to share because I've read some of their stuff and it's phenomenal. Nancy: They were kind enough and generous enough when I was very new in the field to invite me to collaborate with them on a book called ‘Talk Moves,' which is essentially a teacher's guide to facilitating productive math talk. Many years ago, Cathy, Suzanne and I worked together on a research project where we were using discussion in elementary math classes in the city of Chelsea, Massachusetts, and we realized that there really wasn't a how-to guide out there for doing this kind of thing. So, from our work together came the book ‘Talk Moves,' which is now in its third edition and includes written vignettes in the book showing composite examples of teachers and students using ‘Talk Moves' to learn more mathematics, but also includes a set of video clips that were filmed in actual math classes with real-life teachers and real-life students using productive talk moves, including press for reasoning, to help students talk about their reasoning and respond to the reasoning of others. It's a very user-friendly guide for people who want to dig more deeply and see what this thing called productive math talk looks like in action.  Mike: So, I'll add to your plug. I read that back when I was teaching kindergarten and first grade, and it actually had a huge impact on my practice and just understanding at a granular level what this could look like. Nancy, thank you so much for joining us. It really has been a pleasure talking with you today.  Nancy: Oh, it's been a real pleasure for me too, Mike. Thank you so much for having me. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.  © 2023 The Math Learning Center | www.mathlearningcenter.org 

Rounding Up Season 1 | Episode 12 – Open Tasks Guest: Dr. Kim Markworth Mike Wallus: Lately, terms like ‘rich tasks,' ‘multiple entry points,' and ‘low floor,' ‘high ceiling' are being used so often in the world of mathematics education that many educators are confused about their meaning. Today we talk with Kim Markworth, director of content development at The Math Learning Center, about what these terms look like in practice and how they support student learning. Welcome to the podcast, Kim. It's great to have you. Kim Markworth: Thank you, Mike. I'm really honored and excited to be here. Mike: I would love to start this conversation by talking about what it means for a task to be open-ended and have a low floor and a high ceiling. So, is there a way that you think about these terms that might help educators clarify their meaning? Kim: That's a great question, Mike. In truth, when we think about these terms, they're really all interconnected. And I don't know that anyone has really settled on meanings. And lately there's been a bit of a transition from thinking about low floor and high ceiling to low floor and no ceiling at all. And so, when we think about the variations across continua, this really contributes to how we look at tasks and what we appreciate about tasks. And so, when I think about open-ended, this might correspond with high ceiling or no ceiling. We could keep going with the task. There's not really a defined ending point, uh, but instead many pathways where we could take this task further. And when I think about multiple entry points, this might correspond with low floor. And there [are] so many different ways to approach or enter a task. And rich task, probably my favorite term of them all, corresponds to what we've maybe called a problem historically, where we don't have an anticipated solution path or maybe we haven't solved something like this before. We might have multiple solutions to the task. We might have multiple solution paths or viable strategies. We could have opportunities for extension or generalization. And I love tasks that have an unexpected twist or a need to think about something in a unique way, like where your first inclination, your intuition, might be wrong. And all of a sudden you're like, ‘Ooh, this is really something that I wasn't expecting.' Mike: Can you share an example or a few examples with folks who are listening? There's so much that you said just there about the nature of these things. Are you OK telling us a little bit about one or two of them? Kim: Yeah. One task I'm really excited about is a new third grade task that we've been working into our new curriculum. And this task is positioned at the beginning of a unit on multiplication. And so, for third graders, this is really an introduction to multiplication task. And I'm going to apologize to the audience here because this really does require some mental visualization, but I'm going to describe this image that third graders look at, and it involves a pet store. And so, if you can imagine a pet store and going into a pet store, then we have a dog bone display, and these dog bones are hung on three hooks and Annie took, there are two packages—so that the back package isn't visible because it's behind the front package. So, three hooks, two packages per hook, and in each package there are eight dog bones. And those eight dog bones are arranged in a four-by-two array. Kim: And finally, one additional detail here. Each package is labeled $12. And this is a problem-posing situation for kids where we put this image in front of kids and ask them to think about the different mathematical questions that they might ask about this particular image. So ultimately, the question that we ask them to explore is ‘How many dog bones are on display?' Once you have that image in your head, I want you to be thinking about that final question. And the numbers right now are kind of irrelevant for this discussion, but it's highly unlikely that it's something that kids will already know.  Mike: Hmm.  Kim: So, this is a problem-posing situation for the third graders. It's asking them what mathematical questions could you ask about this particular display? And ultimately, we're going to direct them to how many dog bones are on display, but they could explore other questions with additional time. So, how much for all of these dog bones? How much per dog bone? If you had a given number of dogs, how many dog bones would each get? And so, this to me is a task that is open-ended. We could go multiple ways with this, although we are going to focus in the classroom on a particular question. But it's also rich in that kids are going to connect with it, especially if they've been into shops like this. It has multiple entry points. And so, in a lot of ways it really connects to these different terms that you've brought up earlier. Mike: It's interesting because as you describe it, particularly the fact that there's a visual component to this, it really comes clear how there are multiple ways that a child could think about the question or attack the question that you asked. Is there any role for number choice in thinking about how to design a rich task? Kim: Yeah, the numbers are really important. And it's fun to play with different numbers and see how they pan out. And so, in thinking about the dog bone task, we want to keep the numbers accessible. So, when I think about a single peg or a single hook for the dog bone packages, I can think about eight plus eight. And from where kids are coming from, eight plus eight should be accessible. When I think about what I'm seeing with the dog bones, I'm seeing two groups of four in each of the packages. And so, that's a nice way looking at doubles that students might find useful, but I'm also looking at three packages of eight. And so, I could add eight plus eight plus eight, which could bring the teacher very easily to a three times eight multiplication expression. One that is manageable and a great way to introduce multiplication, the times symbol where we're going with all this, but one that is also still accessible for kids to be thinking about as repeated addition. Kim: Ultimately, the numbers get to a final answer of 48 dog bones, and you could think about this in terms of six times eight. But kids aren't going to know six times eight, at least not very commonly in this point of an early introduction to multiplication. But there's various ways that they could get to the eight. We ultimately landed on these numbers: 2, 4, 8, 16, but also this additional number 3, which as a separate prime number really throws some additional mathematical thinking into the mix that elevates the task itself. There's things that are critical to be thinking about as you imagine this task and how it might play out instructionally. It's really important to think about how this stands in the curriculum sequence. So, it is introductory, it's using numbers that the kids are probably not going to know off the top of their heads—related multiplication fact and an answer—but the numbers themselves might elicit different strategies, and the visual might elicit different strategies. And all of this connects to the commutative and associated properties for multiplication and how they might play out with student thinking.  Kim: And so, it's all connected. All these pieces really fit together into what I think is a really interesting and engaging task for students. It's challenging enough, but it's also very accessible simply by counting, kids could count what they see. And the context is engaging for kids as well, because they might be thinking about displays that they've seen and how this corresponds to trips to the store that they've had. Mike: Part of what you've got me thinking, Kim, is there's the design of the task and then there's the element of how a teacher might go about facilitating it. And I think, I want to come back to that, too, because I heard you not only describe the task—the way that it was designed—but you also described some of the ways that a teacher might introduce it, some of the things that they might pose to kids. I wonder if you'd be willing to talk a little bit about facilitation and some of the things about facilitation that can bring a task to life? Kim: So, one of the things that I like to think about in implementing tasks, Mike, is the very intentional letting go, that kids need time to think. They need time to explore. I think all too often as teachers, we have this desire to go in and help and direct. And sometimes we just need to back off and let them think about the different ways that they might approach that, those different entry points, and let them explore and let them take the time to do that. And so, one of the things that I always used to describe to pre-service teachers was the walk away. That I would go up and talk to a group or talk to some partners working together and listen to what they were doing and maybe pose an additional question and then I'd walk away. I didn't want to hear their answer right away. I wanted them to talk about that amongst themselves. But if I stood there, they would start talking to me. And so, I would walk away, move on to a different group, come back later and hear what their thinking was. But it creates that space for letting kids explore, think about, and also not feel the pressure to be getting to a particular answer in a particular timeframe. And I think that's really important for kids to have that freedom. Mike: Yeah. It also strikes me that you're reframing your role for kids, too, in the sense that by walking away, you're sending the signal that ‘I actually have confidence that you and your partners can think about this and reason about this.' Kim: Absolutely. It is putting some power, some agency with the students themselves. ‘You are capable of doing this, you're capable of thinking about this. You do not need me here to be your sounding board or the mathematical authority.' Mike: Kim, can you talk a little bit about the idea of entry points? I'm wondering for teachers in the field, how would you actually define an entry point? What does that look like? Kim: I'm not sure I have a good definition for it, but I do have an analogy, and I would compare it to on-ramps for highways. And when I think about on-ramps, we can all get on the same highway, but we might do it at different places, and we might make choices for where we get on based on our current location or what we know about the on-ramp. But as long as I have a workable vehicle, I can do it. And maybe that's our prior knowledge. But unfortunately, often kids, they don't think that they have a workable vehicle or teachers might even underestimate the child's vehicle that they have. And so, I've probably gone far enough with this analogy, but kids come onto that mathematical highway at different places with particular problems. And I think making sure that we as teachers, as educators, as curriculum designers, that we're thinking about all those different possibilities for getting into a problem and knowing that we can all go to the same place regardless of where we've gotten on. Mike: That's really helpful. So, if I'm an educator and I'm designing a task, or even if I'm facilitating a task that comes with my curriculum, what guidance would you offer to folks to ensure that there are entry points for kids? Kim: I think it really depends on the mathematics and what you're trying to accomplish. And so, with this one in particular—the dog bones visual—I might be asking myself, ‘Can I do it without multiplication since this is an entry point for multiplication. Can I do it without that or could I do it with basic counting skills?' And so, are those viable entry paths open for kids if they don't have where we're going with the task already in their toolkit? When I think about ensuring that there's entry points, I like to think about stripping away the expectations for where you want to go with the task, really allowing kids to have that freedom for exploration. And it's the variety of entry points that leads to the multiple strategies. And when you have multiple strategies, you can make connections between and among those representations. And then you've got something really robust. Or I might go back to that term rich task. And so, it's about can they do it without where you're going, that mathematical goal, and however they encounter that or engage with it, does it still connect to other strategies that will bring them to your mathematical goal? Mike: That is really helpful. What that has me thinking is we have heard in the field about the idea of the five practices and anticipating. But this is a little bit of a twist on that in the sense that you're evaluating the task and saying, ‘What's possible for a kid to get into this task?' I love the example of multiplication. So, for example, if I only have partial or emergent understanding of multiplication, can I still work my way toward an answer to that? And if the answer is no, then what? Kim: Right? But you mentioned partial or emergent understanding, whereas I think this task, actually, you can get in with no understanding of multiplication. I can look at three sets of eight dog bones, add eight plus eight plus eight to get to 24, and then the teacher has that to latch onto to say, ‘We have another way of writing this. I can write three times eight to represent eight being added three times.' And so even that visual structure leads us to something that we can hook on to, to bring forth the connection to multiplication. Mike: I think that's helpful because it means that there's a value and there's a utility to having ways of doing this that you can ultimately connect to the place where you want to go, right? Kim: Yeah. And I think there's opportunities for that kind of reasoning or openness throughout math education; imagining what we can do with tasks to really not just ensure that there's entry points, but value those entry points as really important connections to all students' prior knowledge and where we're going mathematically. Mike: That totally makes sense. Kim: If we can't imagine that our kids are capable of problem-solving and engaging in challenging tasks, then they won't be able to imagine that themselves. And so, in a way, we have to pass along the agency by sometimes just believing ourselves that, ‘Yeah, maybe they can do this,' and giving them that time and seeing what happens. Mike: Before we close the conversation, I'm wondering if you have any resources that you think would help someone listening to this conversation deepen their understanding of designing or implementing rich tasks? Kim: I think the best way to really think about task design and build your own facility is to do some rich tasks and just engage with them as a learner. So, one of the resources that I frequently turn to is the NCTM journals, both old and new. They have really good problems in there. And you can sometimes take one of those and change it in a way that is making it more challenging or making it a more generalizable situation. There's various problem-solving publications. I could certainly plug my own books, ‘Problem Solving in All Seasons' (K–2, 3–5). Mike: I have read it, Kim. Kim: ( laughs ) Mike: I would absolutely recommend it. Kim: Another resource that I love is ‘NRICH.' It's a website NRICH, which is nrich.maths.org. Those are just incredible tasks that really get you thinking in various ways about, [some] good problem-solving experiences. So, what I would recommend for teachers or other people who are interested in this, is to really do that mathematics and then reflect on what made it interesting for you. What surprised you? Were there twists? Were there things, stuck points where you had to get past? And then also to extend the thinking by asking yourself something like, ‘So does this always work? Or when does this work?' And how could you apply the mathematics to more broad situations? And finally, I think it's really important for teachers to put themselves in the minds of their students and sense what might excite them or challenge them. Mike: Thank you so much for joining us, Kim. It's really been a pleasure talking to you. Kim: Thank you, Mike. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 1 | Episode 11 – Successful Curriculum Adoption Guest: Dana Nathanson Mike Wallus: Adopting a new curriculum is not for the faint of heart. What makes this challenging? Well, beyond the materials themselves, a curriculum adoption may represent many things: changes to longstanding practices, beliefs, and classroom culture. On today's podcast, we'll talk with Dana Nathanson, the elementary math coordinator in Leander, Texas, about how leaders can effectively design, manage and sustain a successful curriculum adoption. Welcome to the podcast, Dana. I'm thrilled to have you and be able to talk with you a little bit about the work that goes into adopting and supporting the implementation of a new curriculum.  Dana Nathanson: I'm excited to be here. Thank you for the opportunity.  Mike: Absolutely. So, in your case, we're talking about the work that you did in Leander, Texas, when you supported the adoption of Bridges in Mathematics. I'd love to start by talking about something that feels really critical when a school or a district adopts a new curriculum: the idea of buy-in. How did you think about building buy-in for teachers when you adopted Bridges in Mathematics in your district?  Dana: I think that's an interesting question, because we do hear a lot about, ‘How do you get people to buy in?' And in our district when we think about buy-in, I think about, ‘That's my idea. And so how am I going to get people on board with my idea?' And so, really, we want to kind of flip the script on that and think about ownership. And so, when we think about, ‘How do I get people to kind of own this idea with me?' Then that is really where we see true empowerment. And so, we really approach this with that kind of lens to be thinking about, ‘How do I get people to own this, um, process and own what good math instruction looks like with me?' So that when we do adopt that we are adopting something that aligns with our vision for mathematics and what we want to see students participating in and being a part of in the classroom.  Mike: That really feels different even just to hear you talk about it. Ownership kind of conveys this idea that there's a shared responsibility as opposed to buy-in, which is, can I convince you to do a thing?  Dana: Right, right. And so, to get that ownership, we were at a time in the state of Texas where we were adopting new standards. And so, it was kind of, like, the perfect timing to think about, ‘How are we going to really get a clear picture of what we want math instruction to look like?' So, we did a lot of work with our teachers up front prior to adoption on what are those standards going to look like and how are we … or what do we feel like is the best way to teach math, really, in the younger grades? And so, we did a lot of learning together, a lot of reading. We really grounded ourselves with some of the work of Cathy Seeley, who is a former NCTM (National Council of Teachers of Mathematics) president. She wrote a book, called ‘Faster Isn't Smarter.' And so, we kind of looked at that as a good starting point for, ‘We want all students to have opportunities to make sense of math, do the math and use the math.'  And that kind of became our foundation. It's not just about procedural fluency, but conceptual understanding and then ultimately, transfer. And so, we grounded our work in that and tried to bring people along as far as owning that vision. And then from there, we really looked at what teachers wanted from a resource. And thinking about the use of continuous improvement tools, we used feedback loops, consent-o-grams to—all along the way—so that we could really feel like everybody was owning. They wanted a parent component. They wanted more technology. They wanted practice opportunities through, through games. And so, when we established a rubric together with teachers and administrators, then that really helped us when we came to adoption because we were looking for something that checked all of these boxes.  Mike: Yeah. The story that I make up as I hear you talk about that, is that you had a level of consensus around what you were looking for, which made it a lot easier to make a decision that you felt good about, that you felt like people could own.  Dana: Right. Exactly.  Mike: So, I think anyone listening to this podcast knows that schools and districts have limited resources. So, the thing that I'm wondering about is, what were some of the supports that you prioritized during the first year of your implementation of Bridges?  Dana: So, I'm fortunate to work in a team of, there's three of us at the district level, to support all of our campuses. We have over, well, we have 28 elementaries and we're about to open 29; and so over a thousand elementary teachers that we support. But we knew that the three of us could not do it alone. And so, we are also fortunate that we have an instructional coach at each campus. Now this instructional coach is not specific to math. They support all content areas, but we had to bring them along. We had to get them to own it, and we had to have them feeling comfortable. And then we also created a teacher-leader system where we had a lead teacher from each campus. And we really focused on the instructional coach and the lead teacher as our early adopters or our campus champions to really help us rally—rally everyone around, um, owning this vision for mathematics and also the implementation of a new resource. And what a great opportunity along with the implementation of our new standards. And so, we did pay our teacher leads a stipend for that year. And having the instructional coaches in place was critical because it's those two groups that we would be able to lead and then they would take back to their campuses. Another thing that was also critical in that first year was administrator support. And I know that we're going to talk a little bit more about that, but I just want to highlight the fact that our campus principals were really great about giving teachers time in that first year of implementation to work as a professional learning community together, to have half days to plan and support the new adoption that we had.  Mike: There's a lot that you shared there … Dana: ( chuckles )  Mike: ( chuckles ) that I'd love to dig into a little bit. I think what strikes me about what you said though, particularly at the last part first, is the way that you worked with and supported administrators in really designing a year one where teachers had space and time to actually really devote mental space to thinking about a new curriculum: how it's designed, giving space to plan. That feels like it was an intentional priority that you worked with your administrative team to create.  Dana: Yes, that was very intentional. And it was evident when we began our first Getting Started trainings that summer. And we also trained our ICS (In-Class Support) and our lead teachers first, so that they could kind of get the buzz going for summer professional learning. And I thought it was also great that we were able to have the resources available. If you attended the training, you left with your resources. And teachers were so excited to get all of the great resources that are provided with Bridges. So, that was kind of a draw for them. But then once they had their resources and you start to dig through everything, there's another level of support that is needed. And so, we actually had what we called open houses prior to school starting so that teachers could go around to different teachers' classrooms in the district to see, ‘How did you set up your Number Corner? How did you provide space or how are you structuring space in your classroom for Work Places?' And so, we had a lot of teachers [who] would go around to other teachers' classrooms at other campuses and kind of explore to see and get ideas from each other, which was really powerful. And we created the space up front for that prior to the school year so that they would have that opportunity. And I also want to say at this time, seven years ago, we had a pretty good Twitter presence during this, so that we could also have people online. And I know Twitter's kind of blown up since then, but we were on Twitter a lot, and just being able to share that way, as well.  Mike: So, I love this idea of giving teachers space and time to get their materials and get set up. And the open house idea feels really supportive. One of the things that I sometimes think about is an adoption and an implementation might be a pedagogical shift. There might be a different understanding of the mathematics. But the truth is for a lot of people, the very first thing is, ‘How am I going to find a home for all of these things? What will my classroom look like?' You're kind of attending to that really important need that people have to have met even before they're trying to grapple with the curriculum itself.  Dana: Right. And so, to give that time for them up front to kind of get settled in—with what's this going to look like and how do I make it work—I think was key. And I talked a little bit earlier about the principals being able to provide some half-day plannings for teams throughout the year. But we also offered what we would term ‘power hours' after school. And we would host these in teachers' classrooms. And so, this month we're going to talk about the Work Places because we thought it was so critical that the teachers played all the Work Places so that they would know. And that's how you kind of get their ownership of that, too, as well. And so, we would have these power hours after school, where they would come and play the Work Places, or maybe the next month we're going to do a Math Forum together. That's coming up. And then the next month we're going to go through all of the Number Corner. Now you guys have all these great videos, but this was before you had those for Number Corner. And so, we were just really trying to get teachers in each other's classrooms sharing and making it easier. And we would all make the charts together so that they would have them ready for the next month. And we would see a lot of people on Twitter posting: ‘Here I am at my son's baseball game with my binder, learning.' ( laughs ) But I mean, that's just part of the process, too, right?  Mike: Well, you've really started to address the next thing that I wanted to bring up, which is, when I think about having been an elementary teacher for 17 years, what strikes me is that in education, we sometimes give ourselves really short windows of time to do a complete ‘implementation' quote- unquote. I can't tell you how many times I've heard this year is literacy. Next year is math. Dana: Right. Mike: I think what you're starting to address, but that I wanted to ask you directly is, as an instructional leader, how have you really tried to maintain the integrity of your implementation over time? Maybe just talk a little bit about how you've thought about that process of maintaining and sustaining.  Dana: So again, we leaned heavily, and we still continue to lean heavily, on our instructional coaches at campuses. So, each nine weeks, especially in the first three years of implementation—but even now— we'll dive into what does that curriculum look like for the upcoming nine weeks? And we'll give them ideas and point out specific things that are coming up so that they know how to share or how to kind of pull these things out when they're planning with the different grade levels. And so, we would continue to meet with them, but we always start with that unit introduction.  Mike: Hmm.  Dana: And if teachers can just take the time to read this, and this was another big sell from our department for Bridges, was the built in PD (professional development). If you read those introductions, just, like, how much learning that the teachers can have. So, those first years we really wrapped ourselves around those introductions and the learning together as teams. But we also took, at the time you guys had an Implementation Guide …  Mike: We still do.  Dana: Then I will plug the Implementation Guide. Now it's expanded a lot more. But we took that and we had teachers really pick what's a strength for you on here so that other teachers could come see that modeled for them. And then, what's your area of growth for this nine weeks or for this year? Are you going to focus just on Number Corner, but what parts of Number Corner? Or you want to work on the Work Places, but you're not really implementing the sentence frames correctly. So, whatever that goal is for you, and then the instructional coach and the campus administrator would know what that is, and they're able to support you or come give you feedback on that. And that has really helped us because that gave also administrators, kind of the look-fors that they should see when they walk into classrooms. And our department is fortunate to be able to walk with administrators and our instructional coaches so that we could all kind of participate in this coaching together around what we want it to look like, and then where it's going well. And we bring teachers across campuses and classrooms to see where it's going well, and really having them focus on some goals that they want to set to improve.  Mike: So, I suspect unless Leander is a magical school district that's different from everywhere else, you don't have exactly the same staff that you did … Dana: ( chuckles ) Mike: … seven years ago when you started your process. So, you probably know where I'm going, which is … Dana: Yes. Mike: … how do you account for the fact that teachers, like everyone else, have lives? And sometimes they move on from the grade level that they're teaching or their families move somewhere else. You have new administrators and educators coming in. How do you account for, kind of, that turnover that's just natural in education?  Dana: Right. So, we have the natural turnover. But also we are one of the fastest-growing school districts in Texas. And we continue to open about one school at least, sometimes two a year. So, we know that training and learning together is so important. And so, we have sent our curriculum specialists have participated in many of the Bridges trainers of trainers, trainers of leaders, and for Getting Started. And so, we still offer a two-day for that every summer and also in the fall. And we offer that special session for our new administrators, and we even have turnover in our cabinet. So, we offer that training, and I sit down with superintendents and our area superintendents, because we all have to own, own this. And so that is just a yearly thing that we do. But then also continuing to use our campus champions. We have continued that teacher-leader program. They support our new-to-district teachers as well, and then our instructional coaches. So, it is an ongoing cycle. And I will tell you, at first we kind of say, like, ‘If you can get Number Corner, your Problems & Investigations, and your Work Places down,' then we kind of introduced then the assessment piece the next year and then the intervention piece. So, we have layered it in that way so that it's not so overwhelming for our teachers. And then it just becomes part of your practice.  Mike: Thank you so much for that, Dana. The next piece that I wanted to go to, and you've alluded to it throughout this, is the role that instructional leaders—be they administrators or instructional coaches— play … I was reading a bit from The Wallace Foundation about how critically important principals are. Anthony Mohammad talks about how administrators are the ceiling on where a building can go. Can you talk in a little bit more detail about the kind of work that you did to bring your instructional leaders, particularly your principals, into the process of owning the adoption and the implementation?  Dana: This is still a journey. And so, I want to make sure that I plug that, that even though we are seven years into this adoption, we're still on a journey. Everybody's on a journey. We're not at the end of the race when we think about best practices and instruction in mathematics. But to bring our administrators along, we are fortunate to have instructional leadership meetings every month. And so, we really focus on curriculum with them. We focus on best practices and really, we bring learning to them. And we use a lot of the resources that The Math Learning Center provides. We will learn through some of the blog posts together, reading those together. But really what we wanted upfront before adoption and through the adoption process was for our principals to really own the fact that all students, each and every student, can learn math; and making that accessible to all of our learners. And so that is a mindset. We did a lot of work around the mindset work with Jo Boaler and Carol Dweck. And so, thinking about how then, we wanted—we're not a district that just throws out the direct instruction piece either. We still value that direct instruction. But we want to see that blended with investigating and exploration for our students. And then also having that small group time where they're able to reinforce through Work Places. And so, we really wanted our principals to be firm in the components so that they would know what to see in the classroom, but also firm in the fact that we want to see visual models. What do our standards say? What are the best practices for mathematics say? And the use of manipulatives. And that our Number Corner is meant to be a routine and why we value that for practice for pre-teaching and reinforcing. And what's the value of playing the games in Work Places? So that they would understand these components and really own that they want to see these in the classroom because that's what we know is best practices in mathematics.  Mike: When you think about Bridges, in particular, as a curriculum that you've adopted, were there features of the way Bridges is structured or organized that you really felt like it was important to help people understand going into it? And what I mean by that is, in some ways, Bridges is a departure from a traditional curriculum. And I'm wondering what were the things that you identified that's like, gosh, I've just got to make sure people understand this about how it's designed to work?  Dana: Again, it's kind of the three components that I already alluded to, but really that Number Corner piece. Really thinking about Number Corner as an opportunity for the whole class. And we even kind of connect it to a read-aloud. This is an opportunity for the whole class to come together and to, either it's going to pre-teach some things or it's going to reteach some things. And so how are you making sure that those routines are in place and making sure that we have secured small group time for the Work Places to happen? And that's what we call our small group time, is Work Place time. Because we're talking about how the teacher is floating about the Work Places and observing how they're communicating and playing the game and how they are talking about the math with each other. So, I would say, the Work Places and the Number Corner are really, kind of, the areas that were a little bit harder to bring people along.  Mike: What strikes me about what you said is that you describe the function of those two pieces of the curriculum, Number Corner as a tool to have consistent, long-term opportunities to either reengage with big ideas or pre-engage with big ideas that are coming up. And then the idea that Work Places are an opportunity to practice. But they're so much richer of an opportunity to practice than the worksheets that I remember as a kid, where there were 25 naked number problems and two story problems at the bottom ( chuckles ). They function in the same way in the sense that they're the opportunity for longterm practice. Dana: Right. Mike: And the added bonuses, as you said, when the teacher's moving about the classroom, they can formatively assess and listen to what kids are saying. But they can also jump in and do some miniconferring with children in the moment. Dana: Right. Mike: To help guide them or move them or advance their thinking.  Dana: Exactly. And just thinking about that Work Place time and when teachers are thinking about, ‘Oh, I have to plan something different for this small group.' Well, bring that group together to engage in the Work Place with them so that you are right there observing and having, like you said, that conferring time or that mini-lesson over the Work Place.  Mike: Well, before we close, one of the default questions that I ask anyone who's a guest is, if someone was listening to this podcast and they were charged with leading an adoption or an implementation of a curriculum, what are some of the resources you would recommend for someone who is looking for guidance on how to do this work?  Dana: Well, now I would definitely use the blue ( laughs ) ‘Principles to Actions' NCTM book, because I think this sets the great stage for, what are those teaching practices that we want? But also it talks about the elements. One of the essential elements is specific to curriculum. I didn't mention this earlier, but we also had parents give us feedback along the way. And I think that that is also critical, as well as students. Let your students have some hands-on experiences with the resources so that they're able to even advocate and say, ‘This is how we want to learn math.' There's no denying when you see that students are feeling successful, but also when they are loving what they're doing in the math classroom.  Mike: Well, I was just going to say, everything that you talked about today, I think that the word that comes to mind in addition to ownership is investment. As I've listened to you, I keep thinking, you invested time and energy to make the things that you were looking for come to fruition …  Dana: Uh-hm.  Mike: … to continue the journey, as you said. And without investing in those really important things, the outcome might look really different at this point in time. Dana: Right.  Mike: Well, thank you so much for joining us, Dana. I've learned a lot from the conversation. It's been a pleasure talking to you.  Dana: Thanks, Mike. I appreciate it.  Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 1 | Episode 10: Asset-Based Learning Environments Guest: Dr. Jessica Hunt Mike Wallus: Take a moment to think about the students in your most recent class. What assets do each of them bring to your classroom and how might those assets provide a foundation for their learning? Today we're talking with Dr. Jessica Hunt about asset-based learning environments. We'll talk about how educators can build an asset-based learning environment in their classrooms, schools, and school districts. Welcome to the podcast, Jessica. Thanks for joining us. Jessica Hunt: Thank you. I'm so excited to be here today. Mike: Well, I would love to start our conversation asking you to help define some language that we're going to use throughout the course of the podcast.  Jessica: Sure.  Mike: I'm wondering if you can just describe the difference between an asset-based and a deficitfocused learning environment. Jessica: I think historically what we see a lot of is deficit-based thinking. And deficit-based thinking focuses on perceived weaknesses of students—or even a group of students. And it focuses on students as the problem. And as a result, we tend to use instruction in an attempt to fix students or to fix their thinking. So, an asset-based learning environment means focusing on and beginning with strengths as opposed to what we think kids need or how to fix them. So, this means viewing kids as able and recognizing that the diversity of their thoughts, their culture, their experiences—all of these things are valuable and can actually strengthen and add meaning to classrooms and to instruction. I think assetbased learning environments involve a shift in our own mindset as teachers. And, of course, what we hope results from that is a shift in our practice. We talk a lot about growth mindsets for kids. I think I am referring to growth mindsets that teachers have about kids. We can ask, ‘What do students know and how can I use that? Or how can I build upon that through my teaching?' I've never met a kid that didn't bring something to instruction. Every student that I've met [has] had strengths that they bring to mathematics classrooms and to communities to expand their thinking and also that of their peers. Mike: It's fascinating listening to your description. I find myself thinking about how deficit-based many of the systems and structures … Jessica: Yeah. Mike: … and practices are, even though we do these things with positive intent.  Jessica: Yeah.  Mike: Can you just say more about that? How do you see deficit thinking filtering into some of the systems and then impacting the learning environments in our kids? Jessica: Sure. I think two ways that I see deficit thinking filtering into driving—and driving systems in classrooms—involve things like time and priorities. Time and how it's used in classrooms and schools is one area that deficit thinking can impact in a big way. How are systems recommending that teachers actually spend their time with students in the context of a particular day or a week or even a unit of instruction? And I ask that question because I think that it's one thing to state that we have asset-based approach. Yet it's quite another to consider the need to develop meaningful habits within classroom spaces that can really promote student strengths. Mike: So, one of the things that you just said really struck me, which is this idea of habits in the classroom. I'm excited to hear what you're going to say about that. Jessica: I think one of the key habits that we have in asset-based learning environments is this idea of listening to kids. I've never met a student that didn't have viable and valuable ideas about mathematics. The key for me is having the time and space to uncover and understand what those are. So, we've got to have a way to listen to students' thinking. When we do that, when we understand the reasoning and the strengths that they're bringing, that supports us in selecting instructional tools and strategies that leverage both their individual strengths and those that they bring to the group in order to promote learning. Mike: Let's pick up on that a little bit. This idea of listening to kids and understanding their thinking and understanding of what it means about the assets that they bring. For a person who might be listening, help them form an image of what that might look like in an elementary classroom. Talk to me a little bit about on a day-to-day basis, how might this idea of listening to kids or attending to kids' thinking—and really considering the assets—how might that show up? Jessica: One way it shows up is this focus on learning. And before I go on with that, I want to talk a little bit about how learning and a focus on it is a little different than focusing on performance. So, focusing on performance as opposed to learning, risks looking at change as something that's fast and quick as opposed to something that grows and endures. So, part of focusing on learning means that we're looking more at the process as opposed to only examining quick outcomes or products of what students are experiencing in classrooms. It's actually interesting to think about that in terms of educational equity because there's some research that actually suggests that performance gains don't necessarily equate to learning gains. Mike: I think that's fascinating. You're making me think of two things. One, and I'm going to reference this for people who are listening, is ‘Taking Action,' which is NCTM's work. Really trying to say what do some of the really critical principles of high-quality education look like in grades pre-K through 5? And they have a really specific focus on attending to what do we want kids to learn versus simply what's the performance. Jessica: Yes, absolutely. Mike: I also just wanted to key in on something you said, which is that performance can be short-lived, but learning endures. Jessica: It sure does. If we want to focus on learning, it means that we have to be intentional in our classroom practices. And I also think that links to a lot of things. Like you brought up NCTM, and a lot of the things that they advocate for. I think there are some natural linkages there as well. So, for me, being intentional, one key part of that is ensuring that students are doing the thinking so that teachers can listen to and promote that thinking. So, we want the placement of the learning and the thinking on the students for a good percentage of the instructional time. We want to ensure that we're immersing students in content rather than simply presenting it all the time. And I think another part of that listening involves positioning students and the ideas that they're bringing forward as competent. So, I think, together, what all of this means is that we're supporting students to make meaning for themselves, yet definitely not by themselves. Jessica: Teachers have an intentional, key role. And part of that intentionality involves things like slowing down and thinking carefully about how to structure learning experiences. And taking more time and planning and ensuring that students have access to multiple ways to engage in and represent and express their thinking with respect to those tasks and activities that they're using and drawing upon to learn. And I think that asset-based learning environments allow for that intentionality. It allows for that time and space and planning. And in teaching, it allows for that immersion and thinking and listening and positioning of students as the sense-makers, as the doers and thinkers of mathematics. Mike: I think the connection that I'm making is this idea that there are some shifts that have to happen in order to enable asset-based listening and intentionality. One of the things that comes to mind is it really starts with even how you structure or imagine the task itself. If you're posing a problem, that problem isn't accompanied by a ‘Let me show you how to find the answer.' That actually allows kids to think about it. And there might be some divergent thinking, and that's actually a good thing. We want to understand how kids are thinking so we can respond to their thinking. Jessica Absolutely. Mike: That's a big contrast to saying, ‘Let me show you a task, let me show you how to do the task.' It's pretty difficult to imagine listening in that kind of context because really what you're asking them to do isn't thinking about how to solve it. Does that make sense? Jessica: It sure does. And I think for me, or a hunch that I would have, is that that also goes back to this whole idea of teaching and listening and maybe even assessing, if you will, for what we think kids need versus what they're bringing us versus their strengths. I see some connections there in what you're seeing. Mike: Let's talk about that a little bit.  Jessica: Sure.  Mike: Particularly assessment, I think when I was getting ready for this episode, that was the first thing that came to mind. I found myself thinking about previous PLC meetings or data meetings that I've had where even if we were looking at student work, I have to confess that I found myself thinking about the fact that we were looking at what kids didn't understand versus what they did understand. And I tried to kind of imagine how those conversations would've looked from an asset perspective. What would it look like to look at student work and to compare student work and think about assets versus thinking about what do I need to remediate in the type of thinking that I'm seeing?  Jessica: Uh-hm. I hear you there. I think it speaks to something that if we really want to build assetbased learning environments, we need to make some shifts. And I think one of those shifts is how we look at and use data and assessment. Primarily, I think we need to assess strengths and not needs. I heard that a lot as you were talking. How can we focus on assessing strengths and not needs? I say that to a lot of people and they're like, ‘What's the difference?' ( laughs ) Or, ‘That seems so small.' (laughs) But I think it winds up being a really big deal. If you think about it, trying to uncover needs perpetuates this idea that we should focus on what we see as the problem, which as I mentioned earlier, usually becomes the students or particular group of students. And I think it's very problematic because it sets us up as teachers to keep viewing students and their ideas as something that needs to be fixed as opposed to assets that we can build from or learn from in the classroom. Mike: Yeah. One of the other ideas that we've talked about on this podcast in different episodes is the idea of relevancy and engagement. And it strikes me that these ideas about listening to kids for assets are pretty connected to those ideas about relevancy and engagement. Jessica: Yeah, most definitely. I think, again, figuring out, we sometimes call this prior knowledge, but I look at it as when kids come to school, they bring with them their entire experience. So, what are those experiences and what from their eyes are things that are relevant and engaging and things in which they are passionate about themselves? And what do they know about those things? And how might they connect to what others in the classroom know about those things? And how can we, to borrow a term, how can we ‘mathematize' those things ( laughs ) in ways that are beneficial for individual kids and for the community of learners in our classroom? Like, how can we make those connections? I don't think we can answer those types of questions when we use assessment from this place of, ‘What don't students know?' Or ‘How can I get them to this particular place?' If that makes sense.  Mike: It does.  Jessica: I think we can ask those questions from a strengths-based lens that is curious about and passionate about really getting at, again, this whole experience that kids are bringing with them to school. And how we can use that to not only better students learning, but better the classroom community and maybe even better the mathematics that kids are learning in that community. Mike: Absolutely. Jessica: That's, that's interesting to think about. Mike: So, you started to address one of the questions that I was going to ask, which is, I'm imagining that there are folks who are listening to the podcast and they're just starting to think about what are some of the small steps or the small moves that I might make? What small steps would you advise folks to think about if they're trying to cultivate an asset-focused learning environment? Jessica: It's an interesting question, and I would suggest putting into practice some of the bigger ideas that we're getting at in asset-based learning environments themselves. And the first is, look at your own strengths. And when I say who I'm referencing there, it can be a teacher, it can be a school, it can be a district. If you look at your own strengths first, look at how your practices, your structures, your priorities are uncovering and using strengths. And if they're not, why not? Kind of looking at what's there, what capacities do we currently have that we can build on toward asset-based learning environments? And I think I would pair that with just a commitment to, to action, if you will. You know, start small, but start now. If you're a classroom teacher for instance—I tend to go to that ( laughs ), that grade size a lot ‘cause I still very much, uh, identify as a teacher—start with one task or one day, or part of a day, where you can slow down and use your instructional time to listen for kids' strength. Jessica: What brilliance and valuable ways of reasoning are they sharing with you? And what kinds of activity or task or environment did you need to put in place to uncover that? What did you learn about it? What did you learn about yourself in this process? So, we learn about kids and then we learn about ourselves. It becomes sort of this beautiful back and forth between students and teachers where we're all learning about ourselves and about each other. And I think that learning piece is the third thing that I would suggest. Again, going back to let's focus on learning. Let's celebrate our own learning as teachers and schools and districts and et cetera. Reframing your practices and structures will take time. That's OK. But learn to celebrate the steps that you and your communities are taking toward this asset-based model of instruction. And know that, again, you know, when we work to do that, we enable kids as mathematical thinkers and doers. So, we take that problem off kids, and we place it as a challenge in our instructional design, in our experiences and our interactions between teachers and students. So, I think for me, I would really invite folks to take those small steps, uncover your own strengths, learn to listen, and celebrate your own learning. Mike: Before we conclude the episode, I'm wondering if you can recommend any resources for someone who wants to continue learning about an asset-based approach to elementary mathematics? Jessica: Yeah. There [are] so many good examples of this. I think about my own learning as a teacher and a teacher of teachers, ( laughs ) and a researcher. And I think about things like cognitively guided instruction or the work of the The Dream Project in early childhood or even TODOS, where I know they provide a lot of wonderful examples of asset-oriented resources. I'll also do a shameless plug ( laughs ) for my, for my own book, you know, myself … Mike: Plug away!  Jessica: … ( laughs ) and Jenny Ainslie put together, called, ‘Designing Effective Math Interventions: An Educator's Guide to Learner-Driven Instruction.' And that book came off of a project that I did with, uh, National Science Foundation support, where we looked at kids' thinking over time and designed some tasks and activities to support conceptual understanding of fractions. But there are those. Alnd, and so, so many more. But those are the ones that come to mind immediately. Mike: That's fantastic. And we'll share links to those things with the podcast.  Jessica: Great. Mike: I want to thank you so much for joining us, Jessica, it's really been a pleasure talking to you. Jessica: Oh, thank you. It's been an immense pleasure talking with you as well. And thank you for inviting me. I really appreciate it. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation. dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up  Season 1 | Episode 9: Multilingual Learners  Guest: Jean Harvey, Shannon Lindstedt and Christa Beebe of TNTP (The New Teacher Project) Mike: As a young educator, I was often unsure how to support the multilingual learners in my classroom. And my well-intended attempts didn't always have the impact that I hoped they would. Today we're returning to a topic we've discussed before on the podcast: support for multilingual learners in the mathematics classroom. We'll talk about some of the myths surrounding multilingual learners and dig into specific strategies educators can use to leverage their assets and support meaningful understanding of mathematics. Today we're joined by Shannon Lindstedt, Jean Harvey and Christa Beebe from TNTP (The New Teacher Project). We're going to talk with them about a set of tools and practices they've developed to support educators who serve multilingual learners.  Mike: Welcome, Shannon, Jean and Christa. Great to have you with us today.  Jean: Thanks for having us.  Shannon: Yeah, happy to be here.  Mike: So, Jean, I'd like to start with a question for you. I'm wondering if you could talk a bit about the misconceptions that we have in the education community involving multilingual learners. What is it that we've misunderstood about multilingual learners and how to support them in a mathematics classroom?  Jean: So, one of the most prominent misconceptions is that multilingual learners—MLLs as we call them—cannot engage in grade-level math because they do not yet have the language to understand the task. In MLL Good to Great, we take teachers through a planning protocol that has them assess English-language demands in a task. They consider what mathematical academic language a student needs to know to answer a problem. We ask teachers to also analyze what language in a problem may be new to students, and then they think through what visuals and additional supports could help students to understand the language and the problem. We also think through what language students will need to use to express their understanding. This step is so important because it empowers MLLs to be part of the conversation, and they can grow their language at the same time. When teachers first implement the supports, they're always so delighted how well their MLLs were able to participate in class that day. When the language is supported and MLLs can fully engage in the task, teachers see how capable they are and how eager they are to dig into the rigorous learning.  Jean: The supports also help to dispel another common myth, which is that MLLs might lack the confidence or the ability to engage in class discussions. Sometimes teachers avoid calling on MLLs because they fear embarrassing students. However, when our teachers provide the language supports that help students to understand the task and to produce the language needed to express their understanding, they become part of the conversation. MLLs need that access to critical language, and they'll need some independent think time to craft a response. But they're fully capable of engaging in grade-level math and expressing their understanding. By offering both receptive and productive language supports, MLLs are able to unlock content and demonstrate their incredible learning. We know that actively engaging in class discussions is important for all students, but it's absolutely essential for MLLs. Mike: There was a particular piece that you mentioned. You talked about the need for individual think time. I'm wondering if you can just say a little bit more about that, particularly with respect to MLL students?  Jean: Absolutely. So, one thing that we learned early on was that it's not always instinctive to give kids the think time that they need to gather their thoughts because they're not just processing the math in a given problem, they're also assembling the language that they need to use. In many cases, they're translating from their native language into English and trying to create … figure out how they're going to express their understanding in English. So, giving them that independent think time is incredibly important for MLLs.  Mike: Well, I will say that is most certainly something that is a shift in practice for folks. That level of comfort with what feels like silence—but for the learner is actually think time. That makes a ton of sense to me. Jean, I'm wondering if you could talk in a little bit more depth about the work that you did around vocabulary. And particularly, like, I taught kindergarten and first grade for quite a long time, so this actually feels really relevant to some of the things that I remember thinking about when I had children who may not have been familiar with language, let alone not having the language we were working in be their first language. Can you just talk a little bit about what that process was like for educators as you took them through it?  Jean: We would ask teachers to first think about what's the mathematical academic language that students need to know to access this problem? And so, if it was a problem on ratios, we'd think of ‘What are the terms they might need to use to discuss this problem?' They might not be terms that are specifically listed in the problem, but it's the mathematical academic language that might come up. Then we look at the problem itself, and we wouldn't just focus on vocabulary. There might be phrases in there that are really unfamiliar. We were working with one problem that was about students running a ticket booth and what they were charging for different blocks of tickets. And just the phrase ‘running the ticket booth' was really different because running has multiple meanings. And students know what it means to run, um, you know, using their feet. But running the ticket booth was very different. And so, we supported that with some illustrations and put a sentence by it so that students could make that connection. Sometimes teachers will make some connection to native language supports as well. So, using Spanish or whatever the student's native languages is a bridge to accessing some of the new language and making sure they have that connection as well. And then finally, we'd think about what language the students are going to produce. So, what do they need to say to express their understanding and how can we support them informing the language to express that understanding?  Mike: That's fascinating. What strikes me is how often the work that you're describing stops with the mathematical vocabulary and doesn't actually do that next piece, which feels really important. Like this idea: What is it about the vocabulary that we're using that we assume people understand, but that, like, ‘running the booth,' that's ( chuckles ) as you say it, and actually think and contemplate it. That's confusing.  Jean: Yeah, it's very confusing. And once teachers realize that that's what it takes to support language, you don't have to have an advanced degree in linguistics. It doesn't have to be deeply complicated. You're just really planning for what students might need to know to understand the mathematics in that task.  Mike: What are some of the moves that educators can make when they discover this language that we take for granted as everyone understanding? Would you be willing to talk a little bit about, what are the adaptations or the steps that folks take to help unpack that for children?  Jean: Yeah, absolutely. I think once you've identified different terms within just that day's lesson versus your academic language, you're going to want to have some consistent supports in your classroom. So, a lot of teachers will create a word wall. But a word wall isn't really effective unless students are using it. So, terms, definitions, and I'd also say having an illustrated word wall can be a game changer for some of the common vocabulary you're going to see within a unit—having that up so students can continually reference it and understand what it means. When we looked at the vocabulary and the phrases within the problem, we also connected it to visuals so we can explain what it means. We can provide students a written definition, but when you're still learning a language, the visuals are so essential to actually understanding what the term means or understanding it in context.  Mike: So, one of the things I'm curious about is, what are some of the understandings, the ahas, and the practices that you saw emerging as teachers engaged in this cycle of PL (professional learning)?  Shannon: I can respond to this one. We work with teachers to implement specific instructional strategies during their math classes, such as those mathematical language routines or the five practices. So, by using the variety of language supports incorporated in the program, we have definitely seen teachers develop a more nuanced understanding of what makes an appropriate scaffold and how to differentiate support for students based on their levels of English proficiency. It's not uncommon for teachers and the program to voice concerns that the tasks that we're using or how we're asking students to participate is too hard. And we know that this is coming from a good place. Teachers want their students to feel supported and be successful. So, we talk a lot about productive struggle and the role that it plays in students' meaning making and development in math class, and how critical it is that multilingual learners also get those opportunities to grapple with deep math concepts.  Mike: I think you're hinting at my next question, too, which is: Can you talk a little bit about the impacts that you observed on student identity and their learning as a result of this work?  Christa: Yeah, I'll take this one. This is my most favorite thing to talk about, cause I think this is where we saw the biggest impact, um, in the work that we were doing. And when we think about student identities, we almost had to take a step back and think about teacher identities. Especially when we think about mathematics and the role that that plays. We know that there's been a big emphasis on mindset and, and how important it is when we're learning mathematics to have this growth mindset and recognize that mistakes are OK and good, and that's how we learn. But we also know that math classes historically haven't been set up that way, right? We focus on a right or a wrong answer. So, there's not a lot of opportunity for kids in a traditional math class setting to experience the joy of making a mistake and working through it.  Christa: The hard thing about that is, we want teachers to create that type of math class for kids, but they may not have experienced that type of math class as a learner. So, in Good to Great, we give teachers the opportunity to reflect on who they are as math learners, who they were as math learners, and what their experiences were. And it's not surprising that many of our stories were the same, right? Like, we didn't see ourselves as math people, math is not our favorite subject, you know, on and on. And when we started to reflect on, ‘Well, how does that come through in our teaching?' Some things kind of bubble to the surface. Some teachers would look at that and say, ‘Math is hard for me, so I want to make it easier for my kids.' They want to make this a more positive experience, trying to make it easier for them to, to solve the problem.  Christa: So inadvertently, they're kind of taking away that power, making that mistake, and learning through it. And so, teachers had the opportunity to pause and think about, ‘Who did I position as mathematically capable today?' Really what that means is, ‘Who did we give the opportunity to be seen as a mathematical thinker, who got to answer the questions, who got to share their thinking?' And when teachers were reflecting on that, some of them started to realize that ‘No, I may not be giving my multilingual learners the same opportunities as my native English speakers.' And once we had those discussions, we pulled in those tools that support that productive and receptive language, and we challenged teachers to call on their multilingual learners the next day. And let's see what happens. They did the supports in class, called on those kids, and what we noticed in those debriefs that came after that: The teachers were starting to share, ‘Once I gave them those tools, they ran with it.' We heard things like, ‘My kids enjoy math class, they, they want to participate. They're raising their hands.' All of this from providing the right supports, digging in deeper to some of these mindset issues that we may have ourselves as math learners. And then how do we shift that experience for students so that they can develop their mathematical identities in this?  Mike: The psychology of all of this is fascinating because you're making me think about the idea of intent versus impact, right? So, the intentions of an educator who might be making some of the choices that you're talking about are positive, right? Like they're genuinely in a spot where it's like, ‘I don't want to make a child feel embarrassed.' On the other hand, the child doesn't know that. They just know that they're not getting called on, and they're making up their own story about why that's true. And that's also true for all the other kids in the class who are noticing that as well. And I think the thing that I'm coming around to is, it really does come back to the practices. You all gave them a set of tools to allow them to feel comfortable calling on those kids because they felt they could support them in the moment, and that produced a massive shift.  Christa: Yeah, absolutely. Once, they had the tools, they were able to see what their kids had in them all along.  Mike: You know, one of the things that jumps out for me is, there are a lot of demands on teachers' time. But what you described, I can imagine this happening in a grade-level team. I can imagine it happening at a PLC, and really investing in the types of practices that you all just described feels like the payoff is pretty solid. So, I wanted to ask you all for educators or instructional leaders who are interested in learning more about the Good to Great professional learning that you all have built, designed, and implemented, where can they go to actually learn more?  Jean: Sure. Thanks for asking that. So, we recently published a free toolkit that contains many of our MLL Good to Great resources, including the planning and reflection tools that we've been talking about today, as well as videos and exemplars. So, if someone just wants to learn a little bit more, they can go to the toolkit and see what some of the tools look like. The toolkit is called ‘More Than Right Answers: Math Instruction for Multilingual Learners,' and it's available on tntp.org. So, the toolkit also includes links to contact us at TNTP with any additional questions. And anyone interested in learning more could also email me directly. It's jeanine.harvey@tntp.org.  Mike: Thank you all so much for this conversation. I've learned a lot, and it was a pleasure talking to y'all. Jean: Thank you so much for having us.  Shannon: Thanks, Mike. It was great to be here.  Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.  © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up Season 1 | Episode 8 – Unpacking ICUCARE Guests: Dr. Pamela Seda & Dr. Kyndall Brown Mike Wallus: What does it mean to offer our students a culturally relevant experience in mathematics? This is a question on the minds of many, particularly elementary mathematics educators. Today we're talking with Pamela Seda and Kyndall Brown, authors of “Choosing to See: A Framework for Equity in the Math Classroom.” We'll talk with our guests about what culturally relevant mathematics instruction looks like and identify practical steps educators can take to start this important work in their classrooms. Mike: So, hello, Pam and Kyndall. Welcome to the podcast. We're so glad to have you with us. I'm wondering if both of you would be willing to take a turn and just talk a little bit about what brought you to writing the book. Pamela Seda: OK, well I'll start. This book really started with my dissertation research. And when I started my Ph.D. program, I was very well aware of the achievement gap and the lack of opportunities for so many students, and I just wasn't satisfied that there was a gap. I had to find answers. And so, my Ph.D. program was my quest to find answers. In the process of finding answers, I created this framework that came out of my study, and I had the opportunity to think about how to support teachers. Firstly, implement it in my own classroom and then figure out how to help teachers implement this. And it was just one of those things that I knew that there were a lot of people who wanted to do better for their kids, but they weren't quite sure how to do it. And so, therefore, this book was really kind of a nuts-and- bolts place to start. Mike: And, Kyndall, if you can pick up the story, how did the two of you start collaborating around the book? Kyndall Brown: So, I met Pam at the National Council of Supervisors of Mathematics Conference in Boston in 2015. I was doing a keynote presentation focused on equity and mathematics, and Pam was in the audience. And at the end of the presentation, she approached me and suggested that we start doing presentations together. So ever since then, we were collaborating to do presentations at national conferences. I had been approached by a publisher about writing a book focused on equity in mathematics. So often when those of us who've been doing this equity work over the years, what we hear from math teachers in particular is, ‘What does it look like in the math classroom?' In language arts, you can read the literature that's reflective of your student population. And [in] a social studies class, you can study the cultures of the student populations in your classroom. But math teachers were always wondering, ‘What does equity look like in a math classroom?' And so, one of the first things Pam did when we met was, she introduced me to her ICUCARE framework. It just made perfect sense to use her framework. I asked if she would like to collaborate. She said yes, and this is what we did during the pandemic. Mike: Well, I'm wondering if the two of you could just start and unpack the premise of the book and describe the framework that you all have proposed for people who may not have read it yet. Pamela: Well, ICUCARE is the acronym. The first part is, ‘I include others as experts; C, be critically conscious; U, understand your students well; and then the second C is, use culturally relevant curricula.' Kyndall, you want to take it from there? ( laughs ) Kyndall: ( laughs ) Sure. The next principle is, ‘Assess, activate, and build on prior knowledge; then comes release control; and the final principle is, expect more.' Mike: You know, we could do a podcast episode for every component of the ICUCARE framework, but today we're really focused on using culturally relevant curricula. I suspect there are many educators listening who are kind of in the shoes that Kyndall was describing earlier, this idea that they're interested in the work, but they're not sure how to start, particularly in the math classroom. So, I'm wondering if you all could just spend a little bit of time talking about the guidance you would offer folks when it comes to culturally relevant curricula in a math classroom. Kyndall: Well, first of all, in order to make a task or your curriculum culturally relevant, you have to know who it is that you're teaching, right? You can't make assumptions and assume that you know who they are based upon some physical characteristic or some other information that you might have with your students. The first thing you have to do is get to know who they are, what their interests are, what their concerns are, and then you can begin to start making the curriculum culturally relevant. Mike: Hmm. Pamela: I always say, if we're talking about a task, let's start with something that is cognitively demanding; something that is accessible but also cognitively demanding. And so, oftentimes we describe that as a low-floor, high-ceiling task. And it's real important that students have that opportunity to be able to have cognitively demanding tasks. I say that's a good place to start. We can use textbook problems, we can go to websites—things like Jo Boaler and Achieve the Core and Bridges—those kinds of things. And that's a good place to start. And so, then you might say, ‘OK, well how do I know that's culturally relevant?' Well, that's what we start with, the good task, and then we're going to take that and make it culturally relevant. And one way I say to take a baby step is, take that task and then just change the names and put some names in there that are meaningful to your students. Pamela: And I say, put your students' names in there rather than just trying to come up with some ethnic-sounding names. Put your students' names so that they can see themselves in there. Put your school's names, put the other teacher's names. The key is students need to be able to see, ‘I am a part of mathematics, that mathematics is a part of who I am, a part of who we are.' And so, I think that's a very good baby step to take is just put meaningful names in there. I know that it was very effective. My students really enjoyed it. I could tell, like, even I purposely oftentimes would do that on tests to help reduce the anxiety level of taking a test. And my students, you would see them kind of smile and look around for the persons that they saw whose name was mentioned in the problem. Pamela: So, that's a good first step. And then I would say, the next thing you could do after you've changed the names is then change the context. Change the contexts to things that are meaningful. But as Kyndall said, this is going to require you understanding something about your students. And some things that you can do to understand your students: You can interview your students. And one of the things we talk about in our book is empathy interviews that you can do. You can have listening conversations. Just have conversations with your students in the hall. What are they talking about in the hall? What are they talking about at lunch? What are they talking about at the bus stop? Just pay attention to those conversations, those social conversations, to figure out what's important to them. And then just do community walks. Find out what's in the community. What are popular places that kids hang out, that they go? What's meaningful to them and their families? And incorporate those contexts into problems. And then after that, if you've gotten used to changing the context, then I suggest what I call go to a Stage Four Task. And then you try to engage their agency and help them understand that math can be a tool to use. Mike: I would love for you to—either of you—to talk a little bit more about that last bit that you mentioned, Pam, when you talked about ways to build up kids' sense of agency. Would you be willing to indulge and just go a little bit further down into that conversation? Pamela: Absolutely. So oftentimes, even if we have these wonderful contexts that students will solve problems and become engaged problem-solvers, there's always the question is, like, ‘So what now? What do I do with this? Why is this important to even get this answer?' And it has to be more than, ‘Well, it's going to be on the test,' right? ( laughs ) And so, helping students understand and solve problems that help them see that they can be a part of solutions [to] things that are important to them. So, for example, I remember taking a problem. And it was something about increase in numbers. There was something about what percent did this increase? And I changed the context to the housing market because we had just actually had some storms that had come through our state and had created a lot of damage to houses and homes. And so, then the very next step was I started having them think about, ‘Well, how much might it cost to rebuild these homes? Were some houses damaged more than others?' Pamela: And ‘What could you possibly do to help?' Those are just some kinds of things to help kids understand that, ‘Oh, well, I'm not just trying to find percent increase or decrease, but there's some contexts here that matter, and it may cause me to do some more research.' And even thinking about, ‘Well, if there are neighborhoods that were impacted, what are some things that I can do? Could there be some money that we raise? If I'm going to rebuild the house, how much might I need to spend? How much might I need to invest so that this maybe doesn't happen again?' Those are just all different types of questions to help students understand that you can use math as a part of your community. I also talk about an example of how I was teaching a unit on regression equations, and I know this is an elementary audience, but it was just an example of the fact that we give tests all the time. Pamela: We give those state standardized tests, and I decided to use our district's data for the schools in our district, and things like that, to actually do the mathematics. And students care about that. They got to see their state scores, and they got to see the scores of their neighborhood, of friends who maybe go to a school down the street. And then not only did they get to do the math with that, then they got to have some input. I gave them that opportunity to basically talk to fellow students, talk to fellow teachers, talk to fellow administrators about, ‘What do you think should be different now that you've analyzed and looked at this data?' Kyndall: And I would just add that Lisa Delpit, an education scholar, wrote this book in the early 2000s called ‘Multiplication is for White People.' And that's an extremely provocative title, but it was actually a quote from an African American student of ours. And it kind of spoke to that student's math identity. The actual quote was, ‘Multiplication is for white people, addition and subtraction is for Black people,' right? And so that speaks to what that student's identity was about. The ability of certain people to do math based upon their racial or ethnic background. So, it is very easy to go through the U.S. educational system and come to the conclusion that mathematics is pretty much the domain of mostly white, European men, right? Mike: Certainly. Kyndall: When nothing could be further from the truth. There's an excellent book called ‘The Crest of the Peacock: Non-European Roots of Mathematics' that shows very clearly that mathematics is a cultural endeavor. It's a humanistic endeavor that all humans all over the planet have engaged in. And that other cultures have made significant contributions to the field of mathematics. And so, we need to do a lot better job of exposing students to that so that we can make sure that they see mathematics is as much a part of their culture as any other racial or ethnic group. And they need to see examples of people that look like them in the math textbooks, on the walls of their classrooms, as another way to help build that mathematics identity. Mike: You know, and I think that is actually one of the things that I really appreciated about the way that you all structured the book. I know that I've heard other people who have read it say how much they appreciated being able to hear the stories from your own classrooms, the experiences that you had with students, and really being able to put those out there in a way that help people see where there might be pitfalls and where there might be opportunities. I'm curious if either of you would be willing to share a story about culturally relevant curricula and the impact that you saw on a particular student. Kyndall: Well, Pam has a couple of really good stories in that chapter, so I'm going to let her ... Pamela: ( laughs ) Yeah. So, one of the things I talk about is Jasmine. Jasmine was one of my students who, we'll just say we didn't see eye to eye on most things ( laughs ). Jasmine was very openly hostile towards me, and I was expending a lot of my energy just trying to get her to do anything. And she just made it very clear to me she wasn't interested in doing anything I asked her to do. And so I gave her that project that I talked about, where we decided to look at our test scores, our standardized test scores throughout the district, and applied the math content of the standard that we were using to this, to where she got to make an analysis and be able to see if there was a relationship between the percentage of Black students in our school and then our college and career readiness index, and those kinds of things. Pamela: And I was just really amazed about the transformation that happened with her. Because previously, not only was she not willing to work with me, she didn't want to work with her classmates either ( chuckles ). Mike: Mm. Pamela: And she, as a result of working on this project, asked to be a part of a group. When she found out that she had made some mistakes on some of the data, she willingly stayed after school to fix her mistakes. And I even remember the day that the project was due. She stayed late to put her finishing touches on it. And so, I just was amazed. She was just ... became pleasant. And as a result, I wanted to talk with her about the impact that this project had on her. And she said she really wanted to do it. It wasn't like it was just for a grade. She really wanted to learn the information. And the other thing that was kind of interesting is she didn't really see it as math. She didn't really think that what she was doing was really math, even though she was using Excel spreadsheets and she was using formulas. What that told me was how her perception was that school math wasn't what real math was, and that what we were doing that was connected to her community didn't feel like math. And I felt like that's something that we really need to change. Mike: Yeah. Kendall, I saw you nodding on the other ... (this podcast was recorded via Zoom with video) Kyndall: Well, I think the general public has come to believe that the only thing that counts as math is what you do in school, in a math classroom, right? Mike: Uh-hm. Kyndall: That all of these ways that people are engaging in mathematical thinking and reasoning all day, every day, they don't see as math. And so, they don't see themselves as math people, right? Because they were not successful at school math. Right? Mike: Right. Kyndall: And so how do we undo that perception and get people to recognize the myriad of ways that they're engaging in mathematical thinking and reasoning all the time? Mike: Absolutely. Yeah. I was just going to ask you if there's anything in particular you think might be important for an elementary math educator to be thinking about when they're trying to apply the ideas, some of the suggestions that you all have when it comes to ‘Choosing to See.' Is there anything in particular that folks who are operating at the elementary level might consider or might think about that has come to y'all as you've brought the book out into the world and had people interact with it? Pamela: Well, one thing that I've come to understand is that, while we do need to have good tasks—and the work that we ask students to do needs to be meaningful and needs to be accessible—tasks don't teach kids. And we need to think about how do we structure how kids experience the mathematics in our classrooms? And that to me is what the framework does. It's a lens to help teachers think about, ‘How do I engage my students? How do I structure the instruction so that kids have a positive experience around the mathematics?' So, it should not be thought of as, ‘Oh, this is just once I get the math, then I'm going to go and think about this as a add-on.' Mike: Hmm. Pamela: There are myriads of strategies out there. It's not saying that you should throw out everything that you've ever done before. It's just look at the strategies and the things, the rituals and routines that you've been using in your classroom. And think about them in terms of this lens. If you're getting ready to do an activity, you might say, ‘OK, here's a routine that I normally have. How can I adapt it so I can include others as experts, so I'm not the only one that's doing all the talking? How can I engage my students so that I expect more out of them?' Right? So that they're doing more of the work? So, it's really a lens of how to think about the work that you do and the work that they do. Mike: That totally makes sense. Kyndall: Right. And the research shows that tracking begins very early in elementary school, right? And so elementary teachers need to be conscious of all of these different issues so that they can be on guard at the very early stages to not allow that tracking to begin. Mike: For educators or instructional leaders who are new to the conversation, in addition to reading ‘Choosing to See,' are there other resources that you think would be helpful in supporting people in learning more about equity in the mathematics classroom? Pamela: Well, yes, I know that I've just started reading recently, it's a new book this out called ‘Engaging in Culturally Relevant Math Tasks: Fostering Hope in the Elementary Classroom.' And it's by our good friends Lou Edward Matthews, Shelly M. Jones, and Yolanda Parker. It's at Corwin books, and I definitely recommend that that is a great resource. Kyndall: There's a new book that just came out. It's called ‘Middle School Mathematics Lessons to Explore, Investigate, and Respond to Issues of Social Injustice,' by Robert Berry and his colleagues. In 2020, they released a high school version of the book. And in the fall of 2022, they're planning on releasing an upper- and lower elementary version of these books. And the first section of the book is really talking about the kind of pedagogy needed to implement social justice tasks. And then the second part of the book has lessons aligned to the different content strands that are social justice focused, a lot of digital resources. And so, I think that is an excellent resource for teachers. Mike: That's fantastic. Pam and Kyndall, I want to thank you both so much for being here with us today, for sharing the book with us. It's really been a pleasure talking with both of you. Kyndall: Thank you. Pamela: Well, thank you. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Rounding Up  Season 1 | Episode 7 – Cognitively Guided Instruction: Turning Big Ideas into Practice  Guest: Dr. Kendra Lomax  Mike Wallus: Have you ever had an experience during your teaching career that fundamentally changed how you thought about your students and the role that you play as an educator? For me, that shift occurred during a sweltering week in July of 2007, when I attended a course on cognitively guided instruction. Cognitively guided instruction, or CGI, is a body of research that has had a massive impact on elementary mathematics over the past 20 years. Today on the podcast, we're talking with Kendra Lomax, from the University of Washington, about CGI and the promise it holds for elementary educators and students. Well, Kendra, welcome to the podcast. It's so great to have you on.  Kendra Lomax: Well, thanks for having me.  Mike: Absolutely. I'm wondering if we can start today with a little bit of background; part history lesson, part primer to help listeners understand what CGI is. So, can you just offer a brief summary of what CGI is and the questions that it's attempted to shed some light on?  Kendra: Sure, I'll give it my best try. So, CGI is short for cognitively guided instruction, and it's a body of research that began some 30 years ago with Tom Carpenter and Elizabeth Fennema. And there's lots of other scholars that since then have kind of built upon that body of research. They really tried to think about and understand how children develop mathematical ideas over time. So, they interviewed and studied and watched really carefully what young children did as they solve whole-number problems. So, you may have heard about the book ‘Children's Mathematics,' and that's where you can read a lot about cognitively guided instruction and [it] summarizes some of that research. And they really started with whole-number computation and then have kind of expanded into areas like fractions and decimals, learning about how kids develop ideas about algebraic thinking, as well as early ideas around counting and quantity.  Mike: Uh-hm.  Kendra: So, there's a couple of books that are kind of in the CGI family. ‘Young Children's Mathematics' includes those original authors, as well as Nick Johnson and Megan Franke, Angela Turrou, and Anita Wager. That fractions and decimals work was really led by Susan Empson and Linda Levi. And then, like I mentioned, ‘Thinking Mathematically' is the text by the original authors that kind of talks about algebra. So, in all of those texts that summarize this research, basically, we're trying to understand how do children develop ideas over time? And Tom and Liz really set an example for all of us to follow in how they thought about sharing this research. They had a deep respect for the wisdom of teachers and the work that they do with young children. So, you won't find any sort of prescription in the CGI research about how to teach, exactly, or a curriculum. Because their approach was to share with teachers the research that they had done when they interviewed and listened to all of these many children solving problems, and then learn from the teachers themselves. What is it that makes sense to do in response to what we now know about how children develop mathematical ideas?  Mike: I mean, it's kind of a foundational shift in some ways, right? It reframes how to even think about instruction, at least compared to the traditional paradigm, right? Kendra: Yeah, it's less a study of how best to teach children and really a study and a curiosity about how children bring the ideas that they already have to their work in the math classroom, and how they build on those ideas over time.  Mike: Definitely. It's funny, because when I think about my first exposure I think that was the big aha, is that my job was to listen rather than to impose or tell or perfectly describe how to do something. And it's just such a sea change when you rethink the work of education.  Kendra: Definitely. And it feels really joyful, too, right? You get to be a student of your students and learn about their own thinking and be really responsive to them in the moment, which certainly provides lots of challenges for teachers. But also, I think, just a sense of genuine relationship with children and curiosity and a little bit of joy.  Mike: Definitely. So, I'm wondering if we could dig into a little bit of the whole-number work, because I think there's a bit that we were talking about with CGI, which is really the way in which you approach students, right? And the way that you listen to students for cues on what they're thinking is. But the research did reveal some ways to construct a framework for some of the things you see when children are thinking.  Kendra: So, if you read the book ‘Children's Mathematics,' you might notice or recognize some of those ideas, because CGI is one of the research bases for the Common Core state math standards. So, when you're looking through your grade-level standards and you see that they're suggesting particular problem types, number sizes, or strategies that children might use, much of that is based on the work of cognitively guided instruction, as well as other bodies of research. So, it might sound familiar when you read through the book yourself. And what CGI helps reveal is that there's a somewhat predictable sequence: That young children develop strategies for whole-number operation for working with whole-number computational problems.  Mike: Yeah. Can you talk about that, Kendra?  Kendra: Yeah. So, young children are going to start out with what we call direct modeling, where they are going to directly model the context of the problem. So, if we give them a story problem, they'll act out or model or show or gesture, to show the action of the problem. So, if it describes eating something ( makes eating sounds ), you can imagine, right, the action that goes along with eating? And we're all very familiar with it. So, they're going to show maybe, the cookies, and then cross out the ones that get eaten …  Mike: Uh-hm.  Kendra: … right? So, they're really going to directly model the action or relationship described in the problem. And they're going to also represent all the quantities in the problem, which is different. What they learn over time is to count on or count back. So, some of the counting strategies where they learn, ‘Gosh, I don't want to make all the quantities in this problem.' It becomes too difficult, too cumbersome. And they learn that they could count on from one of the quantities or count back. So, in that cookie example, maybe there are seven cookies on a plate, and I have two of them for dessert, right? ( makes eating sounds) They go away. So, in direct modeling, they're going to show the seven cookies. They're going to remove those two cookies that get eaten, and then count how many are left. Where in counting on—so they have had lots of experiences of direct modeling—they can say, ‘Gosh, I don't really want to draw that seven. I'm going to imagine the seven … ‘  Mike: Uh-hm.  Kendra: ‘ … And I can maybe count backwards from there.'  Mike: So, like, 7, 6, 5.  Kendra: Yeah. Right. So, I don't have to make the seven. I can just imagine it. And I keep track of those two that I'm counting back.  Mike: That totally makes sense. And as a former kindergarten and first-grade teacher, it's an amazing thing to actually see that shift happen.  Kendra: Right? And it's really specialized knowledge that teachers develop to pay attention to that shift. It's easy for everybody else to kind of miss it. But for teachers, it's a really important shift to pay attention to.  Mike: I used to say to parents, when I would try to describe this, it's something that we almost aren't conscious of being able to do. But it's a gigantic step to go from imagining a quantity as a set of ones to imagining a quantity that is a number that you can count back from or count forward from. It's a gigantic leap. Even though to us, we've forgotten what big of a leap that was because it's been so long since we took it.  Kendra: Yeah. That's one thing I love about studying children's mathematics, is, like, you get to experience that wonderment all over again …  Mike: Uh-hm.  Kendra: … in the things that we kind of, as adults, take for granted in how we think about the world.  Mike: Yeah. I think you really clearly articulated the shift that kids make when they move from direct modeling, the action and the quantities, to that kind of shift in their thinking and also their efficiency of being able to count on or count back. Is there more to, kind of, the trajectory that kids are on from there?  Kendra: There is, yeah. So, after children have had lots of experiences to direct model, and then learn to become more efficient with that, and counting on or counting back, then they might start inventing. We call them invented algorithms, which is a fancy way to say that they think about the relationship between quantities and start putting them together and taking them apart in more efficient ways. So, they might use their understanding of groups of 10, right? So, in that example, with the cookies—seven cookies and eating two of them—I might know something about the relationship with fives …  Mike: Uh-hm.  Kendra: … Five and two make a seven. So, they start to develop some sense of how numbers go together, and how the operations really behave. So, in addition, I can kind of add them in any order that I want to, right? So, we see these called in the Common Core standards, Strategies Based on Place Value, Properties of Operation, and the relationship between addition, subtraction, or multiplication, division.  Mike: That's super helpful to actually connect that language in Common Core to what you might see, and how that translates into, kind of, what one might read about in some of the CGI research.  Kendra: Right. It'd be lovely if we all had the exact same ( laughs ) names, wouldn't it?  Mike: Definitely. One of the questions that I suspect people who might be new to this conversation are asking is, what are the conditions that I can put in place? Or what are the things that I might, as a teacher, be able to influence that would help kids move and make some of these shifts. Knowing that the answer isn't direct instruction. I could get a kid to mimic counting on, but if they're still really thinking about numbers in the sense of a direct modeler, they haven't really shifted, right? So, my wondering is, how would you describe some of the ways that teachers can help nudge children, or kind of set up situations that are there to help kids make the shift without telling, or …  Kendra: ( chuckles)  Mike: … like, giving away the game?  Kendra: Totally. Yeah. That's one takeaway that I'm always on the lookout for when people hear about CGI and this trajectory that's somewhat predictable.  Mike: Uh-hm.  Kendra: Let's just teach them the next strategy then, right?  Mike: Right.  Kendra: And what's important to remember is that these are called invented algorithms for a reason.  Mike: Uh-hm.  Kendra: Because children are actually inventing mathematics. It's amazing. Kindergartners are inventing mathematics. And so, our role is really to create the right opportunities for them to do that important work. And like you're saying, when they're ready for the next ideas that they're building on their existing knowledge, rather than us kind of coming in and trying to create that artificially.  Mike: Uh-hm.  Kendra: So, again, like, Liz and Tom really kind of taught us to be students of our students as well as students of teachers.  Mike: Uh-hm.  Kendra: So, what we've learned over time … some of the things that teachers have found really productive for supporting students to kind of move through this trajectory, to create increasingly efficient strategies, is really about thinking about carefully choosing the problems that we've put in front of students.  Mike: Uh-hm.  Kendra: So, paying attention to the context. Is it familiar to them? Is it reasonable for the real world? Are we helping kids see that mathematics is all around them.  Mike: Uh-hm.  Kendra: Paying attention to the quantities that we select. So, if we want them to start thinking about those relationships with five and 10, or as they get older with hundreds and thousands, that we're intentional about the quantities that we choose for those problems.  Mike: Right.  Kendra: And then, of course we know that students learn a lot from not just us, but their relationships and their discussions with their classmates. So, really orchestrating classroom discussions, thinking about choosing students to work together so that they can both learn from one another, and really just finding ways to help students connect their current thinking with the new ideas that we know are on the horizon for them.  Mike: I would love for you to say a little bit more about number choice. That is such a powerful strategy that I think is underutilized. So, I'm wondering if you could just talk about being strategic around the number choices that you offer to kids. Can you say more about that?  Kendra: Sure! It's going to depend on grade level, of course, right?  Mike: Uh-hm.  Kendra: Because they're going to be working with very different quantities early in elementary and then later on … One thing I would say, across all of the grade levels, is to not limit students whenever possible. So, sometimes we want to give problems that kids are really comfortable with, and we know they're going to be successful. But if I'm thinking of how they develop more efficient strategies, sometimes the growth comes in making it a little tricky. So, giving quantities that are just a little bit beyond where they're counting as young children, so they develop the need to learn that counting sequence. Or, as we're working with older students, if we know that particular multiplication facts are less familiar to students. Giving them that nudge by creating story context, where they can really make sense of the action of the relationship that's happening in it, but maybe choosing that times seven that we know has been tricky for kids, right?  Mike: Yep.  Kendra: So, I would just encourage people to not shy away from problems that we know pose some challenge to students. That's actually where a lot of the meat and the rigor happens. And, but then we also want to provide support inside of those, right? So, working with a partner.  Mike: Definitely. Kendra: Or making sure they have access to those counting charts. That's one thing I would say across grade levels.  Mike: Yeah. So, you made me think of something else. It's fascinating to have this conversation, Kendra, 'cause it reminds me of all the things that I had to learn over time. And I think one of the things that I'm wondering if you could talk a little bit more about is, the types of problems and how the problem that you choose for a given group of students might influence whether they're direct modeling or they're counting on or whether they're using invented algorithms. Because I think, for me, one of the things that it took a while to make sense, is that the progression isn't necessarily linear, right? Like, if I'm counting on in a certain context, that doesn't mean I'm counting on in all contexts or direct modeling or what have you. So, I'm curious if you could talk a little bit about problem types and now how those influence what things students sometimes show us.  Kendra: Yeah. I'm glad you brought that up. When we describe, kind of, that trajectory of strategies, it sounds really nice and tidy and organized and like it is predictable in some ways. But like you're saying, it also depends on the kind of problem and the number size that we're putting in front of children. So that trajectory kind of iterates again and again throughout elementary school. So, as we pose more complex problem types … so, for example, the cookies problem where I have seven cookies, I eat two of them and the result is what's at the end of the story, right? The cookies left over.  Mike: Uh-hm.  Kendra: If I now make that problem, I have some cookies on a plate. I ate two of them, and I have five left over. All the kindergarten teachers, actually all the elementary school teachers …  Mike: ( laughs ) Yes.  Kendra: … can automatically recognize that's going to be a more tricky problem, right?  Mike: Uh-hm.  Kendra: Where do I start!? Especially if I'm direct modeling, right? We know they start and follow the exact action of the story.  Mike: Absolutely ( chuckles ).  Kendra: So, as we pose more complex problem types, you're right. You're going to see that they might use less efficient strategies because they're really making sense. They're like, ‘Wait, what's the relationship that's happening in this story? Where do I begin? Where are the cookies at the beginning, middle, and end of this story?' So, we see that happen throughout elementary school. So, it's not that direct modeling is for kindergartners. And that invented algorithms are for fifth grade. It's that as new ideas get introduced, as we make problems more complex, maybe increasing the number size or now we're working with fractions and decimals …  Mike: Uh-hm. Kendra: We see this happen all over again. Kids begin with direct modeling to make sense of the situation. Then they build on that and get a little bit more efficient with some counting kinds of strategies. And then over time with lots of practice with that new problem type, those new numbers, um, they develop those invented algorithms again.  Mike: So, this makes me think of something else, Kendra. How would you describe the role of representation in this process? That could mean manipulatives that students choose to use. It could mean things that they choose to draw, visual models. How does representation play in the process?  Kendra: Yeah. So, oftentimes I hear people say, ‘This student used cubes. That was their strategy.' Or ‘This student used a drawing. That was their strategy.' And that's really not enough information to know the mathematical work that that child is doing. Did they use cubes as a way to count on?  Mike: Uh-hm.  Kendra: Are they keeping track of only one of the quantities but using cubes to do so? Are they doing a drawing that actually represents groups of 10? And they're using ideas about place value inside of it, which is different than if they're just drawing by ones, right? So, there's lots of detail inside of those representations that's important to pay attention to.  Mike: Yeah. I'm thinking about one of my former kindergartners. I remember that I had some work that she had done in the fall, and then I had another bit of work that she had done in the spring. And the fall was this ( chuckles ) very detailed drawing of, like, a hundred circles. And then in the spring, she was unitizing, right? She had a bunch of circles and then within [them] had labeled that each of those were 10. And it just struck me, like, ‘Wow, that is a really tangible vision of how she was drawing in both cases.' But her representation told a really different story about what she understood about math, about numbers, about the base 10 system.  Kendra: Right. And those might be very different starting points. As you, the teacher, you walk over and you see those two different kinds of drawings …  Mike: Uh-hm.  Kendra: … your conversation or your prompt for them … your next step for them might be pretty different.  Mike: Absolutely.  Kendra: Even though they're both drawing.  Mike: Yeah. Well, let me ask you this, 'cause I think I struggled with this a little bit when I first started really thinking about CGI. I had gone to a training and left incredibly inspired and was excited. And one of the things that I was trying to reconcile at that time is, like, I do have a curriculum resource that I'm using, and I wonder how many teachers sometimes struggle with that? I've learned these ideas about how children think, how to listen … what are some of the teacher moves I can make? And I'm also trying to integrate that with a tool that I'm using as a part of my school or my district. So, what are your thoughts about that?  Kendra: That makes a lot of sense. And I think that happens a lot of time in professional learning, where we learn a new set of ideas and then we're wrestling with how do they connect with the things I'm already doing? How do I use them in my own classroom? So, I really appreciate that challenge. I guess one way I like to think about it is that the trajectory that CGI helps us know about how children develop ideas over time is a little bit like a roadmap that I can use regardless of the curricular materials that I have in front of me. And it helps me understand what is on the horizon for that child. What's next for them and their learning? Depending on the kind of strategies that they're using and the kinds of problems that we're hoping to be giving them access to in that grade level, I can look at my curricular materials in front of me and use that roadmap to help me navigate it. So, we were talking about number selection. So, I might take that lens as I look at the curriculum in front of me and think about, ‘Are these the right numbers to be using? What will my students do with the problem …  Mike: Uh-hm.  Kendra: … that is suggested in my curricular materials?' To anticipate how my discussion is going to go and what kinds of strategies I might want to highlight in my discussion. So, I really like to think of it as the professional knowledge that teachers need in order to make sense of their curriculum materials and make informed decisions about how to use those really purposefully.  Mike: Yeah. The other thing that strikes me, that I'm connecting to what you said earlier, is that I could also look at the problem and think about, ‘Does the context actually connect with what I know about my children? Can I somehow shift the context in a way that makes it more accessible to them while still maintaining the structure, the problem, the mathematics, and such?'  Kendra: Right. Yeah. Are there small revisions I can make? Because, uh, I don't envy curriculum writers ( chuckles ) at all because there's no way you can write the exact right problem for every day, for every child across the country. So, as teachers, we have to make really smart decisions and make those really manageable. Because teachers are very busy people.  Mike: Sure.  Kendra: But those manageable, kind of, tweaks or revisions to make it really connected to our students lives.  Mike: Yeah. I think the other thing that's hitting me is that, when you've started to make sense of the progression that children go through, it's a little bit like putting on a pair of glasses that allow you to see things slightly differently and understand that skill of noticing. That's universal. It doesn't necessarily come and go with a curriculum. It's something that's important. Knowing your students is always going to be something that's important for teachers, regardless of the curriculum materials they've got.  Kendra: Yep. That's right.  Mike: So, here's my, I think my last question. And it's really, it's a resource one. So, if I'm a listener who's interested in learning more about CGI, if this is really my first go at understanding the ideas, what would you recommend for someone who's just getting started thinking about this and maybe is walking away thinking, ‘Gosh, I'd like to learn more.' Kendra: Sure. Well, I mentioned the whole laundry list of great texts that you can dig into more. So, ‘Children's Mathematics' being the one on whole-number operation across grade levels. I find that, like, preschool through first- or second-grade teachers have found ‘Young Children's Mathematics' incredibly impactful. It helps connect ideas about counting in quantity with these ideas about problem-solving and operation. And then kind of connects them and helps us think about how to support students to develop those really important early ideas.  Mike: Uh-hm.  Kendra: Anybody who I have talked to that has read ‘Extending Children's Mathematics: Fractions and Decimals' has found it incredibly impactful.  Mike: I will add myself to that list, Kendra. It blew my mind.  Kendra: Yeah, us, too! Everybody who read it was like, ‘Ohhh, I see now.' It points out a lot of really practical ways for us to pay attention. It offers a trajectory much like whole-number about how children develop ideas and also kind of suggests some problems that will help us support students as they're developing those ideas. So, [I] definitely recommend those. And then, ‘Thinking Mathematically' is another great text that helps us connect arithmetic and algebra, as we're thinking about how to make sure that students are set up for success as they start thinking more algebraically. And [it] digs into a little bit of—I talked about young children inventing mathematics—I think even further describes the ways that they invent important properties of operation that can be really interesting to read about.  Mike: That's fantastic. Kendra, thank you so much for joining us. It's really been a pleasure talking to you today.  Kendra: Thanks for having me.  Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.  © 2022 The Math Learning Center | www.mathlearningcenter.org

Rounding Up  Season 1 | Episode 6 – Cultivating a Positive Math Identity  Guests: Nataki McClain and Annelly Rodas  Mike Wallus: Today I'd like to start our episode with a bit of a thought exercise. I'd like you to close your eyes and picture your childhood self, learning math in your elementary school. What are some of the memories and feelings that come to mind? And when you reflect on those memories, what do you think the unspoken messages you may have absorbed about what it means to be good at math were? And then maybe most importantly, how did those early experiences with mathematics shape your belief about yourself as a doer of math? Today on the podcast, we're talking about identity; specifically, math identity. What is it? And how can we as teachers shape our students' math identities. Let's get started.  Mike: Well, hey, everyone. Welcome to Rounding Up. I'm excited to have our friends Nataki and Annelly joining us today. And I think I'll just start by welcoming the two of you. It's great to have you on the podcast.  Nataki McClain: Hi, Mike. Thank you for having us.  Annelly Rodas: Thank you, Mike.  Mike: Absolutely. So the two of you are currently curriculum consultants for the Math Learning Center. And I'm wondering, before we get started with the topic of the day, can you tell us just a little bit about your teaching background and your experience in education? And, Nataki, I'm wondering if you'd be willing to go first?  Nataki: Sure. Well, I have been in education in some capacity for about 25 years. I spent 16 years in the classroom. Fourth grade was my favorite year of all time. And then I've spent eight years as a math specialist. This past year, I am now a curriculum consultant for the Math Learning Center.  Mike: Annelly, how about you?  Annelly: So I started my career as a pre-K teacher at a head start program, and then I moved to the New York City public school system, where I taught second grade and fourth grade. Later, I had the opportunity to work as a math coach at my own school. And I supported pre-K to eight.  Mike: Fabulous. Thanks to both of you. So let's jump into the topic of the podcast: Cultivating a Positive Math Identity. Getting ready for this, what I found myself thinking about is that there is so much conversation in the field right now around math identity. And CTM has position statements about the importance of supporting a positive math identity. There's a ton of research that validates that need. I think I'd like to start by just asking you, from your perspective, how would you describe math identity to a listener who's new to this conversation?  Annelly: I think that it is important to understand that math identity is our own personal view on how we engage with mathematics, right? And it has to do with our disposition and our beliefs on our mathematics ability. I know for me, this topic is really close to my own personal journey in mathematics because I grew up thinking that I was not a math person and that changed with my experiences really late in life. So it has become my mission that kids get to experience math in a different way, and that they feel comfortable engaging with mathematics.  Nataki: And Nelly, um, I have to agree with you. I share a similar experience in that, I guess in my elementary school days, I didn't think of math as something that you got to either enjoy or not. It was just kind of, it's just there and you do it and you learn it. But then in high school I did not have a positive experience. I was made to feel like math was not my thing. And so, Mike, to address that question about what is math identity, it really—to Nelly's point—it really is how you view yourself as a mathematician. And again, my experience in high school was such that I did not feel like I was a mathematician. So to everyone's surprise, when I go off to grad school I'm studying math and now I'm working at the Math Learning Center, right? It's kind of a big deal. And I think it's important that everyone feel like a mathematician.  Mike: Yeah, gosh, you know what you two are saying, I suspect that it resonates with so many people who, whether they're teachers or parents or folks who are just kind of going about living their lives, think this resonates so much. I really resonate with what you said, Nataki, about this idea that math was just there.  Nataki: Uh-hm.  Mike: It was about a series of procedures that you do quickly and that you try to always find the answer as soon as possible. And get it correct the first time. And if you didn't, that meant something about who you were, what your ultimate capacity as a mathematician was.  Nataki: Uh-hm.  Mike: And I think for a lot of folks, that really shapes their belief about what school math is and what math is in general.  Nataki: Absolutely.  Mike: Yeah. So I'm really curious, when you think about the resources that helped you all build your understanding of math identity, what are some of the kind of seminal pieces of work that helped you begin to think about this idea?  Nataki: Well, Anelly and I are reading this book. It's called ‘Choosing to See.' It's written by Pamela Seda and Kyndall Brown. And I have found that this is a relevant resource, especially to our work at the Math Learning Center, because it focuses on equity specifically in the math classroom. And as you're reading it, hopefully you'll find, like we have, that the authors do a really good job in describing those instructional strategies that help teachers to build positive math identities for students. Right away in the introduction, Kyndall Brown outlines a framework for the principles that guide equity, agency, and also identity in the classroom. And he uses an acronym. I see you care. So it's I, the letter C-U-C-A-R-E. And that stands for Including others as experts; being Critically conscious; Understanding your students; Using Culturally relevant curricula; (Assess), activate, and also to build (on) prior knowledge; Releasing control; and Expecting more. And the idea here is to be intentional about what you see, to also be compassionate and purposeful enough to respond. And when we allow this mindset to be prevalent in our classroom, it really does help to support a positive student math identity. But it also serves as a guide to help the teacher understand what, particularly, is at stake. Annelly: And I love that resource. The two of us are, are reading that book and always have conversations about it. But I also think that a starting point for a teacher should be examining their own journey with mathematics, right? Like I talked about how I didn't feel as a mathematician. And I taught, at the beginning of my career, I taught the way that I was taught: very procedural. Expecting quick answers. And the more I started putting my students at the center of my teaching, I started realizing that I was not meeting the needs of all my students. So I would say another research—and I'm going to do a plug in here for our blog—'A Summer Dive into Teacher Math Identity.' That might be something, like a starting point, right? We have to examine our own thinking and our own role before we can create those opportunities for students to develop a positive math identity.  Nataki: I like that, Annelly, that's a good one.  Mike: Hmm. Yeah. I think one thing that jumps out for me is, it would be hard for me to imagine that there's a lot of people who disagree with the aspiration of helping children build an identity about mathematics. That's positive. But I think what's hitting me is you all are kind of highlighting that there are actual practices and things that one does that actually helps build that. And, Annelly, I think I'm really struck by the statement that you made, where you said, ‘I realized that I needed to put kids at the center of my instruction.' And I'm wondering if you can just talk a little bit about, for you, in your journey as a math educator, what did it look like to do that in your classroom?  Annelly: What happened to me was that I started exploring my own math identity at the same time as I was teaching. And one of the things that I noticed is that for me, I need processing time and I needed visuals. So I started playing with that in the classroom to see what my students needed, right? I started bringing in visuals, and we started thinking about—I started thinking about—like, processing time for my kids, giving them time to think, slowing down their thinking. And that made a huge difference for my kids. And it provided a lens where I was pushed to, to think about and really pay attention to, what are the other things that they need? How can I open up space for them to share their thinking? And also, where are the opportunities for them to develop that agency as well? Where they can feel like, ‘I can tackle this,' even though it's hard.  Mike: Hmm. Nataki I, I was going to also offer, like, from your perspective, what did this journey look like for supporting students?  Nataki: Well, kind of similar to Annelly, you know. When I, when I am reflective of my own experiences as a math student, but also reflective in my practices as a teacher, one of the things that I noticed that was missing is the element of fun, right? And also how that fun factor makes room for accessibility. When students start having fun, then the math is accessible to them. And so one of the things that I can say that absolutely was consistent in my classroom, is that we were having fun. Now, of course, fun looks different for different people. And for me, it wasn't just, ‘We're being goofy and being silly.' But fun meant that we are enjoying thinking about the math, doing the math, talking to our friends about the math, looking at math in different ways. In fact, I remember many days when we were at recess and students would come up to me with something that they'd noticed on the playground, right? Being that, ‘Oh, you know, Ms. McClain, that this merry-go-round is a circle. And it's going around and around and around and around. And it spins in the same, in the same distance from the center all the time.' That's something that I didn't teach them. It was something that they noticed because they were having fun on the playground. And they were able to bring in the math concepts from the classroom into their own fun spaces. Mike: You know, one of the things that I find myself thinking about is a really old piece of research. And gosh, I forget the actual researcher. But this idea that teaching is a cultural experience, right? That there are certain cultural narratives around mathematics education that exist just under the surface for lots of people. They're the scripts that they learned when they were in childhood. And that's the picture that shows up in people's heads when they think about math education. So part of the work really is offering kind of a counternarrative to that cultural script. Where I'm going with this is, my cultural script is: Teacher stands in front, shows me what to do, we practice it, and then I go and I sit and do 15 problems, and then two story problems at the end. And that's kind of the cultural script.  Nataki: Right.  Mike: And I suspect that it's fairly difficult to make that kind of cultural script fun. So it makes me wonder, ‘What did your classroom look like to make things fun?'  Nataki: Well, one of the things that was really important to me is that students could see themselves in the math that we are doing. So there wasn't a division problem that wasn't accessible to all students in the beginning, right? So we had to make it accessible. And then I would always find ways to turn everything into a game. To provide, again, that level of fun for kids. So whether it's that I've watched a game show like ‘Jeopardy' … well, ‘How could I use this game show to create a math lesson or a math event or an experience for students?' And so sometimes I could do that in the planning stages. OK, thinking about the content that I wanted students to learn, and then, ‘How can I make it fun? How can I make it engaging?' And then sometimes it just happened in the moment. You know, if you read the room and you discover that, mmm … they're not really having a lot of fun. And again, fun looks different for different people. And for me, I knew that it was fun when all students were engaged and all students had access to the learning.  Mike: So you all are really making me think about the fact that part of building identity is task structure, right? The way that you design tasks, the context that you provide that helps kids connect to it, and also really knowing your kids and knowing the fact that if I'm in second grade, you know, having the agency to actually use some of the materials and have choice around that, that's part of being fun, right? I have a question for you. When you all think about the fact that you also supported a Bridges implementation, what's your lived experience with the places where you see opportunities for building math identity within the structure of the Bridges curriculum. Um, how did that play out for you? How did that connect to the story that you're telling about your own journey?  Nataki: Kids would come barging in the room expecting Number Corner to happen. They were just so excited to discover the next pattern. Or, what are we collecting this month, right? And then, I mean, talk about fun. Work Places was just a natural place for that fun to happen. So I would say Number Corner and Work Places were the places in which I saw kids just really engage. And it was also a great time for teachers to help build that math identity in students, right? To offer supportor just to be there next to students, watching them as they're playing the Work Place games. Those were two components where I saw the most where students really were engaged and having a lot of fun. And not only students. Cause I have to admit that I might have been on a couple of floors, and I might have been caught playing a couple of games, and laughing and chuckling myself ( chuckles ).  Mike: ( chuckles ) Annelly, how about for you? Because I know that you actually, you were not only a Bridges teacher for quite a while, but you also supported the implementation in your building. Annelly: I think that something that we saw when we implemented Bridges was the opportunity to allow kids to show their thinking. And I think that was so big, right? Like in thinking about, ‘There are so many subtle ways.' Like when we ask kids, ‘Can you show me eight on your number rack,' right? We're not dictating how they should think about it. They're jumping in and creating their own strategies and their own learning. And I think that that's an important way to develop that math identity. Because we are telling kids, ‘You can do it. You have all of the skills to do this.' So I see it in that. I see it also in, when we ask kids to write their own math problems—this is something that I've been thinking about a lot—like, when we give kids the opportunity to become authors in the math classroom, we want to hear their ideas and their strategies.  Nataki: Uh-hm.  Mike: How does the role of the teacher shift in a classroom that's really supporting a positive mathematics identity? Part of what's on my mind is that idea of a cultural script, where the teacher is the knower and the place where all of the knowledge lives. And then it's really just kind of beamed out to the kids. What's the shift? If I'm trying to just reconceptualize what teaching looks like in a classroom where I am actively building a positive math identity for my students, how would you describe that?  Annelly: Like, I think that, for that I'm going to connect to my years when I was a coach. I used to love going into classrooms where I wouldn't know where the teacher was.  Nataki: Right.  Annelly: And it's even physical, right? The teacher is not in the front of the room. The teacher might be, like Nataki said, on the floor, playing with the kids. Or at a table, meeting with them. And I think that's a sign that shows you how the teacher is moving away from a teacher-center into a more of a student-center. Also, when we can see kids thinking. Where we can see strategies being named after kids. Again, it seems as something so simple, but it's so powerful for them. It gives them validation that what you are thinking is important. I value your strategies. I used to say, ‘Even if they take you down to a rabbit hole value, their thinking … ‘  Nataki: ( laughs )  Annelly: ( laughs )  Mike: That is really powerful. And, Nataki, how would you answer that question?  Nataki: Everything that Annelly said, I 100 percent agree with. I also think where there are opportunities to ask questions of students, to take those opportunities. Particularly when you have a student who doesn't always get to shine in the class, you know, when that student does something that you think the entire class should hear, find time and find moments to highlight that again. That's giving the student a different feeling about math and a different feeling about where that student finds himself or herself in that math classroom. It makes them feel like they are a mathematician. So I think asking questions and finding moments to allow all students to shine.  Mike: You know, I'm trying to put myself back into the world of a classroom teacher. I wonder if for a lot of folks, part of the hesitation is this fear of, what happens if kids say something that quote unquote is wrong or incorrect? And especially if that happens publicly in front of other children. I think there's this hesitation on the part of people. Because, again, the cultural script is, ‘I'll correct that and show you and tell you exactly what to do.' And I wonder, when you've been faced with that spot where you have used questioning, you've been building discourse, and something just comes out of left field … When you think about a classroom again, where you're supporting identity, what does it look like in that moment for a teacher who's working to support identity, and they have some information that kids are putting out that they're concerned? Like, what do I do?  Nataki: Right.  Mike: Yeah, tell me about your thinking on that.  Nataki: Before we start to build discourse, we need to take some time at the very beginning to build a classroom community where everyone in the room feels free to share their thinking. No matter if it's quote correct or incorrect. And I always find opportunities to kind of press more when those incorrect answers come out, because we can learn a lot from those incorrect answers. We don't just learn from the things that are right. We learn from the things that are incorrect. So can you tell me more about that? Or maybe we could write the ideas on sticky notes and revisit them, right? If there are conjectures, which we talk a lot about in our classroom. Conjectures are always meant to be proven right or wrong, not just in that moment, but for as long as we are in the classroom. We're going to be thinking about the conjecture that Sally made. And the students love—and it's fun for them—when they can prove or disprove Sally's conjecture. That's fun for them. But because we've built the community, it's safe to do that.  Annelly: I love that, Nataki. I think that also creating a culture where it's OK to make a mistake and also modeling from teachers, right? Modeling that, ‘Oh, I made a mistake.' But what I love about math is that I just think, ‘Cross it out and, and kind of like, think about it again.' The one tip that I will give teachers that are just starting with math discourse, and they're afraid to get into gray areas: Do a turn-and-talk and listen to your kids before you ask them to share. And then you can kind of like select which kids are going to share, and you know where they're going. The other thing is that you have to do the math before you do the lesson, right? So that you know where they can go. One of the things that we used to do is, uh, we used to sit down and think about all the different ways that kids can answer a question, like a problem string. What are all the different ways kids can tackle problem strings? And then that gives you kind of like the foundation, right? Granted, you might have some kids that want to be really creative, and they might break it apart into ways that you were not even thinking about. But I think those two are, like maybe two tips, that open up the space for kids to share their ideas.  Nataki: And, Annelly, I think that's an important thing to mention because that anticipation of student responses that comes in the planning. And so it's important for teachers to remember that planning is part of your teaching. That we just don't show up and just start teaching, right? That there has to be some thought that we're giving to the anticipated responses.  Mike: Yeah. I mean, I think when you say that you, gosh, I'm so glad that we talked about this question. I mean, a few things jump out: 1) the idea of positioning student thinking as not being immediately judged right or wrong by the teacher, but as an opportunity to actually build an understanding, to actually have kids justify, to have kids turn to one another and talk about, ‘What is your understanding of this?' And then to build the conversation. So again, it goes back to agency, right? Nataki: Uh-hm.  Mike: You are not the source of right or wrong. You're actually asking them to engage in thinking about that. But I think, Annelly, I'm really keying on what you said earlier about the idea that you have to anticipate where kids might go, because it actually means something. Regardless of whether they've arrived at the correct answer or whether they've arrived at something that shows partial understanding, they're telling you something, and you can use that place to help build an understanding for the whole group. Cause if one kiddo says it, it strikes me that there's probably a fairly good amount of other kiddos who might be thinking the exact same thing.  Annelly: And I think that's another way to build that math identity when we tell them, ‘It's OK if you just have the beginning of an idea' …  Nataki: Uh-hm.  Annelly: … right? ‘Can you share with us? And we can build on that.' Because what Nataki was saying before: We have the power to position kids in a positive light with the rest of the class …  Nataki: Uh-hm.  Annelly: And that it's also so important.  Mike: I just want to thank the two of you for joining us and sharing your thinking. One last question, I think before we have to close things out. You know, if I'm a listener, we've covered a lot of territory in the last bit. If I'm thinking about taking some steps in my classroom, where do you see opportunities for people to get started? Particularly if they're using the Bridges curriculum.  Nataki: I'd say one of the first places—not only a teacher, but any person in, in a school building could start—is taking a look at the blogs that are posted about math identity. One of the blogs, I think Annelly mentioned earlier is, helping teachers to be reflective of their own math journey. And I think that's an important step. So reflection, I would say, is a great place to start. And it starts perhaps by reading the blog.  Annelly: I would say don't be afraid to have conversations with your kids. And letting them lead some of those discussions.  Mike: Hey, thanks so much to both of you for joining us today. It was really a pleasure to hear your thinking and to have you on the podcast.  Annelly: Thank you, Mike, for having us.  Nataki: Yes. Thank you, Mike. This was a lot of fun. But listen, next time … can you bring cookies?  Mike: Hey, you got a deal, my friend. Thanks so much.  Nataki: Thank you. Bye now. Mike: This podcast is brought to you by the Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.  © 2022 The Math Learning Center | www.mathlearningcenter.org  

Rounding Up Season 1 | Episode 5 – Learning Targets Guest: Dr. Rachel Harrington Mike Wallus: As a 17-year-veteran classroom teacher, I can't even begin to count the number of learning targets that I've written over the years. Whether it's writing ‘I can' statements or developing success criteria, there's no denying that writing learning targets is an important part of teacher practice. That said, the thinking about what makes a strong learning target continues to evolve and the language that we select for those targets has implications for instructional practice. Today on the podcast, we're talking with Dr. Rachel Harrington from Western Oregon University about creating powerful and productive learning targets. Welcome to the podcast. Rachel Harrington: Thank you for having me. I'm excited to be here. Mike: Sure. So I'd love to just start our conversation by having you talk a little bit about how the ideas around learning targets have evolved, even just in the course of your own teaching career. Rachel: I started out as a pre-service teacher in the late '90s and got a lot of practice in undergrad teacher education, thinking about writing those objectives. And we were always told to start with, ‘The student will be able to … ,' and then we needed to have some skill and then it needed to end with a percentage of performance. So we need percent of accuracy. And so I got a lot of practice writing things that way, and we always were very strategic with our percentages. We might say 80 percent because we planned to give them five questions at the end and we wanted four out of five to be correct. And then we could check the box that the students had done what we wanted. And I felt like it was really critical. We always were kind of drilled into us that it must be measurable. You have to be able to measure that objective. And so that percentage was really important. Rachel: In my experience though, as a teacher, that, that didn't feel as helpful. And it wasn't something that I did as a classroom teacher very often. As I transitioned into working in teacher preparation, now we have shifted the way we talk about things. Instead of saying a learning objective, we talk more about learning targets. And we talk about using active verbs that, when we phrase the learning target or the learning goal, it's using a verb that is more active and not so much ‘Student will be able to … .' And so we might use verbs like compare, explain, classify, analyze, thinking more about that. And then, rather than thinking about an assessment at the end, with five questions where they get four correct, we want to think about multiple times throughout the lesson where the teacher is assessing that learning goal and the progress towards that goal. Sometimes those assessments might be more classroom-based. Other times you might be looking more at an individual student and collecting data on their progress as well. But it's more progress towards a goal rather than something that's met at the end of the lesson with a certain percentage of accuracy. Mike: You named the thing that I think stood out for me, which is you're moving from a process where you're thinking about an outcome versus what's the action, be that cognitive or in the way that students are solving. The focus is really on what's happening and how it's happening as opposed to just an outcome. Rachel: Uh-hm. And I feel like when I started in teacher preparation, the standards were a little more siloed by grade level. It was sort of like, this is what we do in fourth grade and it starts and ends in fourth grade. Whereas with the Common Core State Standards, we see these learning progressions that stretch across the child's whole math experience. And so I think that's shifted a little bit the way we think about targets as well and learning goals and whatever title you've given them. Now, we don't think so much as, ‘What are you accomplishing at the end of today?' but sort of your progress across a learning progression and, and what progress are you making towards a longer-term goal? Mike: I think that's a really profound shift though. There are two things that come to mind: One is really thinking about how that impacts my practice as a teacher. If I'm just thinking about what happens at the end of today, in all of these little discreet iterations, versus what's the pathway that the child is on, right? I'm really interested in, how is their thinking shifting? And that the end of the day is not the end of that shift. It's really something that happens over time. Does that make sense to you? Rachel: Definitely. And I think it's really critical when we're teaching in a mixed-ability classroom, and we're thinking about children making progress at their own pace and not expecting every child to learn the same thing every single day, but we can have individual goals for our kids. We can have ideas about, as long as they are making progress in their math journey, then we're going to be OK with that. And we're helping them in that progress. And I think it's also more evidence as to why curriculum needs to cycle back to previously taught concepts because those concepts may or may not be mastered by all the children or understood by all the children at the end of the lesson. We're going to keep revisiting it. And children get multiple opportunities to think about this idea, and they will make progress on their own at their pace. Mike: Well, that's in stark contrast to my own childhood math experiences. You got through your unit on fractions in fourth grade, and… Rachel: Yep. Mike: ... if you didn't get it, well … Rachel: So sad. Mike: ... good, good luck in fifth grade! Rachel: ( laughs ) Mike: ( laughs ) Um, but it's really an entirely different way of thinking about the child's development of ideas.  Rachel: Yep. I remember teaching multiplication of fractions on a Monday followed by a division of fractions on a Tuesday. It was really just like, you know, when we moved past this idea that multiplication of fractions is a procedure that, that students will master. Then we need to start thinking about it as happening more than just on Monday. Mike: We've already started to address the second question I had, which is: What are some of the pitfalls that schools and teachers might fall into or might encounter when they're thinking about learning targets?  Rachel: I think some folks have put pressure on teachers to take the idea of a learning target and phrase it into an ‘I can' statement or a student-friendly language—which, I am not at all opposed to the idea of making things into student-friendly language. I think that's actually really critical in math class. Mike: Uh-hm. Rachel: But I think it can be problematic. When we start the lesson with an ‘I can' statement, are we giving away the ending of the lesson right at the beginning? Mike: Yeah. Rachel: Are we taking away their joy of that discovery and that excitement of finding out this, understanding this new concept? I don't want to remove that magic out of math class by just saying, ‘Hey, I'm going to tell you the ending right before we get started.' And I also worry a little bit that sometimes those ‘I can' statements and those things that we put up on the board at the beginning of class are done under the guise of ‘holding teachers accountable,' which I think is a phrase that is very ( chuckles ) problematic.  Rachel: I tend to err on the side of trusting teachers; that they can be trusted to know what they're doing in the classroom and that they have a goal in mind. And I assume that they are planning for teaching without telling me exactly and explicitly on the whiteboard that they are doing that. But I also recognize that the presence of that learning target or that ‘I can' statement on the board at the beginning is an easy thing to check off. All of the different things that are happening in math class are really complex and really hard to understand and notice. And it can take years and tons of experience before we're able to notice all the things that are happening. And so as an administrator that maybe has limited experience teaching mathematics, I could see where it would be difficult coming into the classroom and really being able to recognize what is happening. You might look around the room and be like, ‘Is this some kind of birthday party? What's going on? All these kids are cutting things out and gluing things. This doesn't look like math class.' Rachel: But if I can see that statement written up on the board, that's something that's kind of concrete and measurable. I also just think this idea of capturing learning as a daily objective can be problematic, especially when we're thinking about building really complex ideas in mathematics. You know, that's not going to happen in one lesson, in one session of curriculum. It might build over multiple days. It might cycle back into multiple units. And so we need to make sure that students are developing alongside their peers and, but maybe not out at the same pace. And I think that's OK. Mike: Yeah. You made me think about a couple different things, Rachel. One is the idea that the way that learning targets have been kind of introduced into classrooms really feels more like compliance as opposed to something that has value in terms of your instructional practice. And I, I've lived that world, too, as a classroom teacher. I think the other thing that really hits me from what you said is, I started thinking about whole-number multiplication, right? If I'm just thinking about the end product—meaning students being able to perform multiplication—there's so much richness that has been missed ( chuckles ) in that process. Rachel: ( chuckles )  Mike: I mean, we're trying to help children move from thinking additively to thinking multiplicatively. You're going to move along that kind of continuum of understanding over time. Honestly, I would say it shouldn't happen in one day. Rachel: Yeah. What can you really learn in just one lesson? And learn, not, I wouldn't say just perform a skill. Mike: Yeah. Rachel: I think skills, performing a skill and memorizing an algorithm, that is something that can be taught in a really concrete chunk of time, potentially. But the real conceptual understanding of what's happening with multiplication—how it's connected to addition, how it's connected to geometric concepts and things like that—that all comes and builds. And I feel like it also builds in fits and spurts. Some kids are going to make a big leap at one point and then make some smaller steps before they make another big leap. It's not a linear progression that … Mike: Right. Rachel: … they're going through. And so we have to allow that to happen and give room for that to happen. And if we say everyone in the class will do this by the end of the lesson with this amount of accuracy, we don't make room for that to happen. Mike: Yeah. I think what you're highlighting is the difference between what I would call like a learning goal and a performance goal. And I'm wondering if you could help unpack that. Because for me, when I started thinking about learning targets in that framework, it really opened my eyes to some of the places where I'd gotten it right in the classroom and some of the places where, boy, I wish I had a do-over. Rachel: Yeah. I think the language that the National Council of Teachers in Mathematics has brought to us, is this idea of contrasting performance goals with learning goals. And I find myself turning to the ‘Taking Action' series of books. Specifically, K–5 when we're thinking about elementary. There's a chapter of that book I have found to be really powerful. Sadly, I think it's one that we can sometimes gloss over a little bit in our reading. Because for some folks, they look at that and they say, ‘Well, I don't choose the learning goal. My curriculum chooses the learning goal or my school district tells me what the learning goal is.' But when you really look at what a learning goal is, as opposed to a performance goal, that's really not what's dictated by your curriculum or by your school district. And so in the 'Taking Action' book, I think they do a really nice job of contrasting the difference between a learning goal and a performance goal. And I would say a performance goal is sort of what I described earlier when I was talking about ‘The student will be able to … ' Mike: Uh-hm. Yeah. Rachel: … at a certain amount of accuracy. So, an example. If you do have access to the book, it talks about ‘Students will solve a variety of multiplication word problems and write the related multiplication equations.' And (given) that, I could see that as the type of thing I would've written maybe with a certain amount of accuracy ( laughs ) at the end of it. And I would've given them maybe five word problems and then assessed if they could get at least four out of the five correct equations. And so that's a really good example of a performance goal. And, and they talk about this idea of a performance is, what is the student doing? What's something that we can look and observe and measure and count.  Mike: That's so hard though! Because what's missing in that goal is ‘how'!  Rachel: Right.  Mike: You know ( laughs ), like …  Rachel: Or ‘why'! ( laughs )  Mike: ( laughs) Or ‘why'! Right?  Rachel: Yep, yep.  Mike: Like when you actually look at the student's work, what does that tell you about how they arrived there? And then what does that tell you about what that child needs to continue making sense of mathematics? You gave an example of a performance goal around multiplication and word problems. What might that sound like as a learning goal instead?  Rachel: So an example of that same—probably aligned to the exact same standard and the Common Core State Standards—would be that students will understand the structure of multiplication as comprising equal groups, within visual or physical representations, understand numbers and multiplication equations, and connect those representations to equations. So that learning goal really describes what you're hoping the students learn. Not just what they do, but what do they carry forward with them as they move into more and more complex mathematics? I think you'll also recognize the verbs in there are much more complex. In the previous performance goal, we talked about students solving and writing. They're solving, and they're writing. But in the learning goal, we're looking at understanding, connecting, and representing those different ways of thinking about it and bringing them together. Putting those pieces together. And again, that might be something that develops over a long period of time. They might be working on one piece of it, which is looking at an array and connecting that to an equation. But maybe later on, they're connecting the context of the task to the equation. Or they're taking a context and recognizing, ‘Wouldn't an array model be a great way to solve this? And wouldn't an equation model be a great way to solve this?'  Mike: Uh-hm.  Rachel: And that's really developing over time.  Mike: Yeah. I was just going to say, you mentioned ‘Taking Action.' The, the chapter on learning goals is actually my most dogeared, uh, chapter in the book. I want to read you something that I think is really powerful though. Very first chapter on learning goals, the way that they describe it is: ‘Identifying what students will come to understand about mathematics rather than focusing on what students will do.' I've read that, underlined it, highlighted it. And I've got a Post-It note on that page because I think it just fundamentally changes what I think my role is as a teacher in preparing and also in a moment with children.  Rachel: Yep. It's not so much about, they're going to be able to cut this out and do this thing and perform this action. But it's really, what's the purpose? Why are we doing this? Why would they cut that out? Why would they do this action? What is that contributing to their long-term understanding? I do appreciate NCTM's guidance on this. I think they're leading the pack. And this is really cutting-edge … Mike: Yeah. Rachel: … thinking about how we set goals for our classroom. It's not commonly held in the field or applied in the field yet. Mike: Uh-hm. Rachel: But I think folks are really starting to understand its importance. That if, as we change the way we teach mathematics and the outcomes we expect for students, we have to start thinking differently about how we set up learning goals. We can't keep having these performance goals and expecting what's happening in the classroom to change. If we're really going to go towards the type of instruction we want to see in a classroom, we've got to think about learning goals instead of focus so much on just performance. Mike: I actually had a chance to talk to DeAnn Huinker, who's one of the co-writers of ‘Taking Action,' and she used the phrase, ‘What are the mathematical conversations you want children to have?' And I was really struck by, like, that's a really interesting question for me to think about if I'm thinking about my learning goals. But even if I'm just thinking about planning and preparing for a lesson or a unit of study. Rachel: Definitely. I don't think that's something that's thought a lot about. I mean, I might see for my students and their lesson plan: ‘Turn and talk to your neighbor.' But if you don't really think carefully about what kind of conversation you want to happen during that turn and talk … . Or I'll see in their lesson plan that ‘We will have a discussion about students' various solutions.' And what does that mean? You know, what's going to happen in that time? What's the point … Mike: Uh-hm. Rachel: … of that time? I can't remember who, I think it was Elham Kazemi that said something once about, ‘In math class folks will present,' and it's like that old football cheer, you know, ‘stand up, sit down, clap, clap, clap.' That's what we do in math class. Mike: Yeah. Rachel: We have kids stand up, we sit down, we all politely listen, and then we clap. And that's it. We move on. But if you really focus on those conversations that you want kids to have, what are the interesting things that you want them to be thinking about? That's a complete shift in how we've taught math. Mike: Yeah, it really is. It makes me think about, on a practical level, if I'm a person who's listening to this podcast, what I might be starting to think about is, ‘How do I take action'—no pun intended—'on this idea of thinking deeply about learning goals, integrating them into my practice?' And, for me at least, the first place I went when I read this was to think about shifting what I did in my preparation and my planning. Rachel: Uh-hm. But I think when it comes to planning, we need to be thinking, first of all, kind of the three parts that ‘Taking Action' talks about, is setting a goal that's clear. It should be clear in your mind what the children are learning. And so that can take some reading, right? It can take reading through the session, reading through the overviews, thinking about the learning progressions, always keeping your eye on that mathematical horizon, making those learning goals clear. But then also thinking about the fact that I am situating those learning goals into a learning progression. And I'm thinking about what this lesson that I'm doing on Tuesday, where does it fit in the math journey? So that makes me think about two things. First, what is this lesson building on? What foundation do these students come with that I can build on? But then also, what is it leading toward? Rachel: Where are we going from here? And what is the important role that this idea we're looking at today plays in the whole mathematical journey? And then using that as your foundation for your instruction. So if you're finding that the activity that you had planned isn't meeting that learning goal. So it isn't helping you with this clear understanding of what you want them to know. If it isn't helping build toward something that you want them to be able to understand, then what are the changes you need to make? Mike: Uh-hm. Rachel: What are some things you want to adjust? Where do you want to spend more time? How do you add those conversations? Things like that. Mike: Uh-hm. I think you led back to the thing that I wanted to unpack, which is: I worried that at different points in this conversation, people might think, ‘Well, they're just suggesting that learning goals or learning targets don't really have a role.' We're not saying that. We're saying that they really stretch over time. And I think your description was really elegant in thinking about, what does this session contribute to that larger goal of understanding the meaning of multiplication? What is the intent of this session in helping that development proceed? Rachel: Yeah. What is the big idea? What is this leading towards? Because if you don't see it, then that's when you, as a teacher, need to make some decisions. Do I need to do more reading? Do I need to do more understanding about this particular content area? Do I need to adjust the lesson itself? Is there something that I need to change or add or incorporate so that it does play a stronger role? Plus, you know your students. So if we're thinking about this session being a part of a learning progression, and it's building on something they already have, if you feel like maybe they don't have what they need to engage with today's lesson—now I'm going to think about some ways to reengage them with this content. I think especially over the next few years, that's going to be critical. But yeah, I definitely agree with you, Mike. Cause I think NCTM, the authors would say the first thing about a learning target or a learning goal is that it has to be clear, and it has to guide and be the foundation for instruction. And so, they're really important. It's just maybe the way that we've talked about them in the past hasn't been helpful. Mike: Yeah. The other place you bring me to, Rachel, is the idea that if I'm really clear on my learning goal, what is it that children will come to understand? And where is this lesson situated in that journey? That actually has a lot of value because I can think about, ‘What are some of the questions that I want to ask to try to either assess where kids are at or advance their thinking?' Or when I think about what children might do, ‘Which kids do I want to strategically highlight at a closure?' So I think understanding that learning goal really does have value for folks. It's just a different way of constructing them. And then also thinking, what do you do next? Rachel: And I also think, again, I'll take this back to the idea of assessing those learning goals. 'Cause I do think assessment and goals cannot be separated. You're going to always be thinking about that, right? Why set a goal if you don't have any way of knowing whether students are making progress towards that goal? When you establish them in that way and you think about them as less of something that's going to be accomplished by the end of this session, we allow room for students to progress at different ways and learn different things in the class. And then that's when we can have those rich conversations at the end, when we're drawing things together. If every child's going to do everything the exact same way in my classroom, then there's no opportunity for interesting conversations. The interesting conversations happen when kids are doing things differently and making progress in different ways, and heading in different directions towards the same goal. Rachel: Then we start learning from each other. We can see what our partner is doing and try to understand what they're doing. That's when interesting math happens. And I want to encourage teachers to feel confident in thinking about these as the idea of a learning goal. And even starting to incorporate this into student-friendly language. You know, a learning goal doesn't have to be written as an ‘I can' statement for kids to be able to understand it. And I also want teachers to feel confident in their abilities for advocating. Um, when they see learning goals being used in a problematic way, when we see pitfalls and things that we talked about at the beginning happening in their classroom—be confident in your abilities and your knowledge and what you know is best for students. You know your students better than anyone else does. The teacher does. And you know how to think about those individual needs and the individual growth of each child in your classroom. Rachel: So rest assured in that confidence. But go to the resources that are available to you as well. When you're struggling with the idea of where these lessons or these concepts or these ideas you're teaching fit, go to the learning progressions, go to the ‘Taking Action' book, go to the NCTM resources. Um, read your session overviews in your curriculum. Have conversations with your colleagues. Have conversations with the colleagues that teach grades above you and grades below you. That's really critical if we're think about taking away this silo idea of teaching mathematics, we need to start thinking about have these conversations across grade levels. And, and knowing, you know, if you're struggling with where this idea is going, talk to the teacher who comes next. And even just ask them, ‘What reason do you think a child would need to learn this?' Mike: Yeah. Rachel: You know, and then they might be able to help you see where it fits in the progression. Mike: Well, and I was going to say, look at the scope and sequence and notice, where do the ideas come back? How are they coming back? How are they being developed? And then the icing on the cake would be to do what you said. Let's take a look at how this manifests itself in the next grade or perhaps in the grade prior. Rachel: I think that's also a role for math leaders in elementary and in the building instructional coaches, that's a vision that they can help teachers with 'cause they get the opportunity to be in multiple grades in multiple classrooms. And they also have more space to read through the progressions, and they might have more time for those sorts of things. And so I want to push math leaders to be doing that as well. Not just the classroom teachers, help your teachers to see where these ideas carry across into future grades and how they build on previous content and facilitate those conversations. Mike: Yeah. You know, I'm so glad that you brought that up. Because it makes me think about, there are some things about the way that we've organized education that just, are givens, right? We have primarily grade-level classrooms, right? And so, I taught first grade for eight years. I intimately knew my first-grade standards. I did not clearly have a vision of necessarily how that was going to play out in second grade and third grade and fourth grade and so on. And I think that's one of the inadvertent problems that we're stuck with is, if we don't have a vertical understanding of: How are these ideas going to support children over time? It might be easy to say, ‘Well, I just need them to be able to do X by the time they get out of third grade.' Not really understanding that, actually I need to have them understand X, so then they can, in fact, understand all these other concepts that are coming. Rachel: I've just seen this year, so much, what is happening in fifth grade is dictating how you understand algebra. You know, it's like … Mike: Yes! Rachel: … what we see in the fifth-grade standards. If you are not really understanding those concepts, you might be OK for a little while. And then once you're into your algebra classes, you're realizing that all of that foundational knowledge came from what you learned in fifth grade and what you understand about rational numbers. And so, I totally agree. I don't think we've done a good job in education in general of those cross grade-level conversations. But I think we're getting better with this idea of having instructional leaders, instructional coaches that are really there to support the instruction … Mike: Yeah. Rachel: … that's happening. So I know I work with math leaders and that's one of the things I really encourage them, is not only should they know the entire curriculum or continuum, but how are they helping their classroom teachers understand that? 'Cause I think there's a lot of power in having a teacher spend eight years in first grade and really knowing those standards intimately. But there's also some value in, in once you've taught third grade going back to first grade and realizing, ‘Wow, this is where it was all going.' Mike: Absolutely. Yeah. I had a role at one point where I was a K–12 curriculum director for math. Rachel: Oh, yeah. Mike: And it was the most eye-opening experience because, as you said, you recognize how, if kids walk out of elementary school without a deep foundational understanding—and if it's just really a surface set of performance skills ... wow—that catches up with kids when they get into sixth, seventh, and eighth grade. Rachel: Yep. For sure. And those concepts become more abstract when we start this idea of variables and thinking about things algebraically. That if you didn't have that foundation in the concrete, the abstract is too much. It's too much to ask of kids. And so then we find ourselves reteaching and wondering, ‘What happened?' And yeah, I just, I wish more conversations were happening across those grade levels.  Mike: Absolutely. Well, thank you again, Rachel.  Rachel: Yeah!  Mike: It was lovely to have you. I think a lot of folks are going to find this really helpful, and maybe validating in the experience they've had. And also a vision for what they might do in the future. And hopefully we'll have you back at some point.  Rachel: I'm always here for you. ( laughs )  Mike: Thank you so much. All right, bye bye.  Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.  © 2022 The Math Learning Center | www.mathlearningcenter.org