Podcasts about nctm

In this episode of Room to Grow, Joanie and Curtis continue the season 5 series on the Mathematics Teaching Practices from NCTM's Principles to Actions, celebrating its 10th anniversary. This month's practice is “Establish Mathematics Goals to Focus Learning.” This is defined as follows:Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions. In today's conversation, our hosts unpack the key components of this principle. First, they discuss how learning goals, focused on important mathematical understandings, differ from procedural, process goals, which may include skills and procedures that are not directly connected to the underlying mathematical concepts. Next, the discussion turns to situating goals within a learning progression, which helps teachers stay focused on what is relevant to their grade level or course, and provides a venue for students to be active in their progress toward learning. Finally, effective mathematics goals guide instructional decisions, helping educators know which tangents to explore and which are distractions from the intended learning. We hope you enjoy the conversation, and that it extends your thinking on mathematics goals for learning. Additional referenced content includes:·       NCTM's Principles to Actions·       NCTM's Taking Action series for grades K-5, grades 6-8, and grades 9-12·       NCTM's myNCTM forums (membership required).·       How learning goals serve as a guide – NCTM Teaching Children Mathematics blog post·       Rachel Harrington's appearance on the Math Learning Center podcast/blog discussing mathematical goals Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy. 

Welcome to Season 5 of the More Math for More People podcast!!!What if high school mathematics could be truly transformed into something students find engaging, meaningful, and applicable to their lives? In this thought-provoking conversation with mathematics education experts Kris Cunningham and Kristi Martin, we dive deep into the possibilities presented in the NCTM publication "High School Mathematics Reimagined, Revitalized, and Relevant."The discussion opens doors to a reimagined approach to high school mathematics that moves beyond the traditional curriculum model. Both guests highlight the importance of mathematical modeling as a pathway to authentic learning experiences where students tackle complex, multi-variable problems without predetermined solutions. This creates space for creative thinking and problem-solving skills that transcend the classroom.However, transformation doesn't come without challenges. Kristi candidly shares her classroom experiences of struggling to surface the mathematical concepts within engaging modeling activities and the difficulty of finding relevant problems that align with required standards. Kris emphasizes the need for systemic support, including administrative understanding and professional development opportunities that give teachers space to experiment and grow.Perhaps most powerfully, both educators stress the importance of persistence through inevitable frustrations. "If you believe in it and know it's great for students, just keep trying," Kristi encourages, reminding listeners that transformation is a journey rather than a destination. The conversation serves as both inspiration and practical roadmap for anyone interested in creating mathematics education that truly prepares students for their futures rather than simply repeating practices of the past.Send Joel and Misty a message!The More Math for More People Podcast is produced by CPM Educational Program. Learn more at CPM.orgX: @cpmmathFacebook: CPMEducationalProgramEmail: cpmpodcast@cpm.org

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  

In this episode of Room to Grow, Joanie and Curtis continue the season 5 series on the Mathematics Teaching Practices from NCTM's Principles to Actions, celebrating it's 10th anniversary. This month's practice is “Elicit and Use Evidence of Student Thinking.” In Principles to Actions, NCTM describes this teaching practice in this way:Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.This meaty description provides the fodder for today's conversation. Our hosts consider what is meant by “effective teaching,” “assessing progress,” and “adjusting instruction continually,” and tie these ideas back to the important work of classroom educators.Additional referenced content includes:·       NCTM's Principles to Actions·       NCTM's Taking Action series for grades K-5, grades 6-8, and grades 9-12·       Want more ideas for eliciting student thinking in your classroom? Check these out:o   Descriptors of teacher and student behaviors for this practiceo   Thoughts and linked resources from the Colorado Department of Educationo   A classroom observation tool focused on this practice from the Minnesota Department of Education Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy.  

In this episode, Sam is joined by two separate guests. Sarah Powell (University of Texas) describes the origins and aims of the Science of Math efforts, and Rachel Lambert (University of California, Santa Barbara) shares some critiques and concerns about the Science of Math as it is currently instantiated.Introduction (0:00:00)Origins with Sarah (0:03:12)Critiques with Rachel (0:22:26)Responses with Sarah (1:10:42) The Science of Math website https://www.thescienceofmath.com/ Sarah's professional webpage Rachel's professional webpage Rachel's NCTM webinar https://www.nctm.org/online-learning/Webinars/Details/686  List of episodes

In this episode of The Good Life EDU Podcast, Tara Gossman, a Teaching and Learning Specialist at ESU 4, returns to share insights from the continued work with Pawnee City Elementary around high-quality instructional materials (HQIM) in math. This follow-up conversation highlights the critical instructional strategies of purposeful questioning, student discourse, and productive struggle that are the bridge between effective materials and deep student learning. Tara also offers a unique behind-the-scenes look at her recent experience publishing an article on this work with the National Council of Teachers of Mathematics (NCTM). Listen in to learn how ESU collaborations are fostering meaningful instructional growth and inspiring educators across Nebraska. To access Tara's published article, visit the NCTM website at nctm.org or email the ESUCC to request a copy.

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

In this episode of Room to Grow, Joanie and Curtis continue the season 5 series on the Mathematics Teaching Practices from NCTM's Principles to Actions, celebrating it's 10th anniversary. This month's practice is “Use and connect mathematical representations.” Our hosts describe the five representations outlined in Principles to Actions, which include visual, symbolic, verbal, contextual, and physical descriptions of mathematics, but emphasize that the representations are not meant to be a check list to be covered during instruction. Rather, the different representations provide a framework for explore important mathematical concepts through different lenses, allowing students to build and deepen their understanding as they consider these ways of engaging. In addition to deep understanding, teachers' attending to different representations will allow different students in the class to be elevated, as their unique strengths and preferences will have the opportunity to come out and be showcased.  Additional referenced content includes:·       NCTM's Principles to Actions·       NCTM's Taking Action series for grades K-5, grades 6-8, and grades 9-12·       Making Connections Explicit​ (NCTM requires subscription)·       Supporting Understanding Using Representations​ (NCTM requires subscription)·       Three Ways to Enhance Tasks for Multilingual Learners​ (NCTM requires subscription)·       Interpreting Distance-Time Graphs – lesson referred to in this episode  Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy.    

Not sure what matters most when designing math improvement plans? Take this assessment and get a free customized report: https://makemathmoments.com/grow/ Math coordinators and leaders – Ready to design your math improvement plan with guidance, support and using structure? Learn how to follow our 4 stage process. https://growyourmathprogram.com Looking to supplement your curriculum with problem based lessons and units? Make Math Moments Problem Based Lessons & UnitsDiscover practical instructional routines from Shannon McCaw's 2025 NCTM Spring Conference session to elevate meaningful mathematical discourse. Learn how structured discourse routines and sentence frames can increase student engagement—especially during direct instruction.With many districts focusing on mathematical discourse, Shannon shared practical instructional protocols to help students engage in meaningful conversations. Learn how structured routines, sentence frames, and intentional strategies can boost participation—especially during direct instruction. If keeping students engaged in discourse is one of your biggest pebbles, this episode is packed with actionable takeaways you can implement right away!What Listeners Will Get From This Episode:Students engage more in discourse when provided with familiar structures and sentence frames.Practical instructional protocols to make direct instruction more interactive and engaging.Implementing small, structured routines can lead to big improvements in student participation.Show NotesLove the show? Text us your big takeaway! Get a Customized Math Improvement Plan For Your District.Are you district leader for mathematics? Take the 12 minute assessment and you'll get a free, customized improvement plan to shape and grow the 6 parts of any strong mathematics program.Take the assessmentAre you wondering how to create K-12 math lesson plans that leave students so engaged they don't want to stop exploring your math curriculum when the bell rings? In their podcast, Kyle Pearce and Jon Orr—founders of MakeMathMoments.com—share over 19 years of experience inspiring K-12 math students, teachers, and district leaders with effective math activities, engaging resources, and innovative math leadership strategies. Through a 6-step framework, they guide K-12 classroom teachers and district math coordinators on building a strong, balanced math program that grows student and teacher impact. Each week, gain fresh ideas, feedback, and practical strategies to feel more confident and motivate students to see the beauty in math. Start making math moments today by listening to Episode #139: "Making Math Moments From Day 1 to 180.

In this episode of Room to Grow, Joanie and Curtis continue the season 5 series on the Mathematics Teaching Practices from NCTM's Principles to Actions, celebrating it's 10th anniversary. This month's practice is “Implement Tasks that Promote Reasoning and Problem Solving.” Our hosts being by expounding on the difference between selecting a task and implementing it, and that selecting a good task does not guarantee good implementation. They bust the idea that the only way to engage students in reasoning and problem solving is with a rich task, by considering how educators can weave together procedural learning with conceptual understanding. Next, they connect reasoning and problem solving to the Standards of Mathematical Practice, particularly practices 7: Attend to and make use of structure and 8: Look for and express regularity in repeated reasoning. Capitalizing on students' natural noticing and shining the light on the underlying mathematics leads to stronger connections and increasing students' ability to generalize their understanding. By building a foundation of reasoning and sense-making, and helping students understand that this is a resource for them tap into, that allows for the learning and engagement beyond rote classroom experiences.Additional referenced content includes:·       NCTM's Principles to Actions·       NCTM's Taking Action series for grades K-5, grades 6-8, and grades 9-12·       Selecting and Creating Mathematical Tasks article from Smith and Stein·       A Teacher's Guide to Reasoning and Sense-Making from NCTM Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy.      

In this episode of Room to Grow, Joanie and Curtis begin a season 5 series on the Mathematics Teaching Practices from NCTM's Principles to Actions, celebrating it's 10th anniversary. This month's practice is “Facilitating Meaningful Mathematics Discourse.” Our hosts first identify what they mean by discourse and why it is important: that students are able to communicate their mathematical thinking in ways that others can clearly understand for the purpose of furthering their own mathematics learning.  Next, Curtis and Joanie unpack how to get students talking in math class, a necessary condition for meaningful math discourse. Classroom culture is a key element to ensure that students feel safe and comfortable enough to share their mathematical thinking. Implied in this is that the teacher must hold themselves to precision of language as well, and should understand when to require precision from students and when to be more flexible with informal language. Finally, their conversation suggests that effective math discourse is not improvisational, but rather something teachers can and should plan for, and use as a strategy for an equitable classroom.Additional referenced content includes:·       NCTM's Principles to Actions·       NCTM's Taking Action series for grades K-5, grades 6-8, and grades 9-12·       Strategies for facilitating math discourse in the classroom·       Latrenda Knighten, NCTM President's message on Let's Give Students the Gift of Time Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy.    

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

In this episode of Room to Grow, Joanie and Curtis reflect on their personal and professional experiences of 2024 and what they learned. Reflect – conferences, books, podcast guests. Thinking differently about teaching and learning math. Hope you'll take the time to reflect and capture your own learning.Curtis and Joanie reference these episodes of Room to Grow which aired in 2024:·       Teaching and Learning Math: Students' Perspectives Part 1 (aired August 28, 2024) and Part 2 (aired September 17, 2024)·       Routines for Supporting Student Thinking with Grace Kelemanik and Amy Lucenta (aired October 16, 2023)·       Unleashing the Mathematical Brilliance of All Students with Rachel Lambert (aired April 10, 2023)·       Balancing Instructional Modalities (aired March 12, 2024)·       Asset-Based Teaching to Transform Math Class with Mike Steele and Joleigh Honey (aired October 15, 2024)·       A Conversation with the National Teacher of the Year with Rebecka Peterson (aired February 13, 2024)·       High School Mathematics Reimagined Revitalized and Relevant with Latrenda Knighten and Kevin Dykema (aired November 12, 2024) Additional referenced content includes:·       The book Transform Your Math Class Using Asset-Based Teaching for Grades 6-12·       The work of Liping Ma, including her book Knowing and Teaching Elementary Mathematics ·       Rachel Lambert's research and resources at mathematizing4all.com ·       Kevin Dykema's President's Message on Balancing Instructional Strategies in the Math Classroom·       NCTM's Reimagining High School Mathematics resources on the NCTM webpage Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomt

How can we reimagine high school math to truly prepare students for the real world? Hear from Kevin Dykema and Latrenda Knighten, two NCTM presidents, one past and one current, who are leading the charge to make math relevant, revitalized, and engaging.High school math often feels disconnected from students' futures, leaving many wondering, "Why do we need this?" This episode dives into the practical steps educators and leaders can take to transform math classrooms into spaces where every student feels the relevance and power of mathematics.Discover actionable strategies from Kevin Dykema, past NCTM president, and Latrenda Knighten, current NCTM president, on making math meaningful for all students.Learn how the High School Mathematics Reimagined framework connects courses and builds coherence across the curriculum.Get inspired by leadership advice on rolling out changes that elevate math instruction without overwhelming teachers or leaders.Press play now to learn how you can make high school math a transformative experience for every student—starting today!Not sure what matters most when designing math improvement plans? Take this assessment and get a free customized report: https://makemathmoments.com/grow/ Ready to design your math improvement plan with guidance, support and using structure? Learn how to follow our 4 stage process. https://growyourmathprogram.com Looking to supplement your curriculum with problem based lessons and units? Make Math Moments Problem Based Lessons & UnitsShow Notes PageLove the show? Text us your big takeaway! Empower Your Students (and Teachers) Using A Professional Learning Plan That Sparks Engagement, Fuels Deep Learning, and Ignites Action!https://makemathmoments.com/make-math-moments-district-mentorship-program/ Are you wondering how to create K-12 math lesson plans that leave students so engaged they don't want to stop exploring your math curriculum when the bell rings? In their podcast, Kyle Pearce and Jon Orr—founders of MakeMathMoments.com—share over 19 years of experience inspiring K-12 math students, teachers, and district leaders with effective math activities, engaging resources, and innovative math leadership strategies. Through a 6-step framework, they guide K-12 classroom teachers and district math coordinators on building a strong, balanced math program that grows student and teacher impact. Each week, gain fresh ideas, feedback, and practical strategies to feel more confident and motivate students to see the beauty in math. Start making math moments today by listening to Episode #139: "Making Math Moments From Day 1 to 180.

Celeste Watson (MM, MS, NCTM) helps musicians achieve artistry through well-coordinated use of the body. She holds credentials in performing arts medicine, piano performance, and music education and coordinates return-to-play mentorships for keyboard musicians who have experienced injury. She owns a conservatory-model studio for pre-collegiate students in Winston-Salem, North Carolina (www.watson-music.com).For more information about the Lister-Sink Institute and teachings, visit https://www.lister-sink.org/For more information about Feldenkrais movement, visit https://feldenkrais.com/Top 5 Songs of Encouragement1) Waltzing Matilda sung by Slim Dustyhttps://www.youtube.com/watch?v=FqtttbbYfSM&list=PLSIRqKhHhcL4nqBuFTgNJZmPqK_7YX-SH&index=22) "Kommt, ihr Tochter, helft mir klagen" from St. Matthew Passion (BWV 244) by Bachhttps://www.youtube.com/watch?v=-9Be9xQrWVU&list=PLSIRqKhHhcL4nqBuFTgNJZmPqK_7YX-SH&index=33) French Suites, by Bach, performed by Andras Schiffhttps://www.youtube.com/watch?v=0sDleZkIK-w&list=PLSIRqKhHhcL4nqBuFTgNJZmPqK_7YX-SH&index=5&t=127s4) Goldberg Variations by Bach, performed by Andras Schiffhttps://www.youtube.com/watch?v=MHFuaaGpGKQ&list=PLSIRqKhHhcL4nqBuFTgNJZmPqK_7YX-SH&index=4&t=55s5) Wayfaring Stranger by Johnny Cashhttps://www.youtube.com/watch?v=Ti-FtTz8t5U&list=PLSIRqKhHhcL4nqBuFTgNJZmPqK_7YX-SH&index=1Support the show

In this episode of Room to Grow, Joanie and Curtis speak with leaders of the National Council of Teachers of Mathematics (NCTM) about their recent publication, High School Mathematics Reimagined, Revitalized and Relevant.  Latrenda Knighten, NCTM President and Kevin Dykema, NCTM Past-President share a great overview of how rethinking how and what we teach in high school math can be improved so that more students leave high school prepared. This preparation involves not only knowing more mathematics, but believing in their capability as math learners and in their preparation for whatever path they have chosen for themselves after graduating.The new “three Rs” of high school math build on NCTM's previous high school publication, Catalyzing Change in High School Mathematics: Initiating Critical Conversations from 2018, and give practical examples and suggestions to engage students in mathematical and statistical modeling, make connections across major concepts, and using mathematical and statistical processes as a frame for student thinking.  We encourage you to explore the resources below, referenced in this episode:o   You can find NCTM's webpage dedicated to supporting the book HEREo   NCTM's webinar about the book was recorded and is available to all HEREo   More information about the Launch Years Pathways work out of the Charles A. Dana Center at the University of Texas at Austin can be found HEREo   Guidelines for Assessment and Instruction in Statistics Education (GAISE) reports can be found HERE Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com . Be sure to connect with your hosts on Twitter and Instagram: @JoanieFun and @cbmathguy.    

How do we create classroom communities that includes and values each member's expertise? In this episode, I'm sharing my NCTM session with you on empowering learners through math discourse. We're chatting about: How to partner with our students in math discussions to leverage each student's insightsStrategies for teachers to use to build a strong math culture that supports mathematical discourse3 student stories and examples of strategies I used to empower them to engage in math discussionsLinks mentioned in this episode: Word Problem WorkshopCultivating Mathematical Hearts by Maria del Rosario Zavala, Julia Maria AguirreA quote from an article titled “Student Representations at the Center: Promoting Classroom Equity” by Imm, Stylianou, and Chae 2008Choosing to See By Pamela Seda & Kyndall BrownMy example video with NairobiGuide to Engaging Math DiscussionsWhile take a break from the podcast to write my book, enjoy these episodes: Episode 85 with John SanGiovanni Episodes 93 & 96 with Peter Liljedahl Episode 73 Best First Week Activities (you can do these anytime) Episode 107 Math Discussions: 3 Ways to Get Students Engaged Episode 102: Talk less so students will talk more Episode 97: 6 Ways to Love Teaching Math Episode 82: Say This to Parents to Explain “New Math” Episode 2: What is Student Centered Math (A look inside my classroom) Have Questions?

In this episode of Room to Grow, Joanie and Curtis speak with Mike Steele and Joleigh Honey, authors of the recently released book transform your math class using asset-based teaching for grades 6-12. The book and the conversation explore what is meant by “asset-based,” and why shifting to more asset-based approaches supports a broader range of learners.  Mike and Joleigh unpack ideas around asset-based language, including, the language of mathematics, the language students use to talk about math, and the language educators use to talk about students. They also explore classroom and instructional routines, many of which are already in common use in classrooms, and how to ensure these routines fall more on the asset side of the continuum than on the deficit side. Finally, the conversation shifts to the larger educational structures that could benefit from a more asset-focused lens. We encourage you to explore the resources below, referenced in this episode:·       Mike and Joleigh's book, Tranform your math class using asset-based teaching for grades 6-12 can be found here·       Learn more about Mike Steele here or here and about Joleigh Honey here or here·       Mike and Joleigh both serve on the NCTM Board of Directors  Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com . Be sure to connect with your hosts on Twitter and Instagram: @JoanieFun and @cbmathguy.   

We love the math teaching community! In this episode Pam and Kim are live from NCTM and discuss the many things they are learning from other math teaching educators at NCSM and NCTM in Chicago.Talking Points:Too many great sessions to list! But here are a few.Announcing: Developing Mathematical Reasoning: Avoiding the Trap of AlgorithmsAnnounced: Foundations for Strategies Small Group LessonsPre order Developing Mathematical Reasoning: Avoiding the Trap of Algorithms here: https://us.corwin.com/books/dmr-289132?srsltid=AfmBOor-iPP6FJaP0vQS7MWXGMzv6U6psI-ZbqBwxYbE_y4rGgPHU4fU  Pre order Foundations for Strategies Small Group Lessons here: https://www.hand2mind.com/supplemental-curriculum/math/math-fluency/daily-math-fluency-foundations-for-strategies-small-group-lessons Check out our social mediaTwitter: @PWHarrisInstagram: Pam Harris_mathFacebook: Pam Harris, author, mathematics educationLinkedin: Pam Harris Consulting LLC 

In this episode Ryan Flessner and I are reviewing the end of NCSM 2024 and the start of NCTM in Chicago. Two amazing days of learning, collaborating, growing as leaders and doing LOTS of math! @mathedleaders you put on a heck of a conference and @NCTM is off to a really incredible start! Hats off to the organizers of both conferences and all the hard work that goes into making that happen. And special thanks to all the speakers for doing such an incredible job to provide opportunities for us to learn!  #NCSM24 @NCTMCHI24 @joyfulmaths @teedjvt @iteachthewhy @lynseymathed @pwharris @DrEugeniaCheng @gracekelemanik @amylucenta @tkanold @mathematize4all @jnovakowski_  @ZanerBloser    @GreatMindsEd  @LizRowoldt @IndianaCTM

The National Council of Teachers of Mathematics, or NCTM, is the world's largest mathematics education organization. While many math teachers are members of this organization and may attend its national conference, others do not. Maybe you are someone who has wondered whether or not to join? We wanted to explore this idea more in our latest debate: Is an NCTM membership worthwhile?Follow Desiree Harrison on all platforms: @KidsmathtalkFollow Bobson Wong on Twitter/BlueSky/Instagram: @bobsonwong Check out Bobson's Website:  www.bobsonwong.comFollow Steve Phelps on all platforms: @MathTechCoach  Follow Sunil Singh on Twitter: @MathgardenCheck out Sunil's Website: www.mathsings.comListened to the episode? Now, it's your turn to share! Go to our Twitter: @DebateMathPod to share your thoughts. Don't forget to check out the video version of this podcast on our YouTube channel!Keep up with all the latest info by following @DebateMathPod or going to debatemath.com. Follow us @Rob_Baier & @cluzniak. And don't forget to rate and review us on Apple Podcasts!

Staying active, busy, and productive as a professional musician is not easy. Pianists Elizabeth Yao and Kyunghoon Kim share some of the practical tips and mindset shifts that have helped them with time management, mental health, and making space for personal priorities. Pianist Kyunghoon Kim, D.M., NCTM, is Adjunct Lecturer in Music (Piano) at the Indiana University Jacobs School of Music. He is also a Signature Artist on Musicnotes.com, composing under the name Piano Sandbox. Elizabeth Yao, D.M., NCTM, is Lecturer in Music (Piano) at the Indiana University Jacobs School of Music, where she coordinates the secondary piano program, teaches piano pedagogy, and directs the Young Pianists Program. Subscribe Join Amy's email list Support the Podcast https://pianopantry.com/patreon Transcript Find the full transcript and show notes (including any links mentioned) here: https://pianopantry.com/podcast/episode137 Send Amy a Voice Message

On episode 238, Emily Kircher-Morris talks with Lindsay Kapek and Katie Tabari about the challenges neurodivergent students face in learning math. The conversation explores strategies for creating inclusive math classrooms that foster confidence and support for all students, regardless of their neurodivergence, and they talk about the importance of understanding individual learning styles. They also discuss using low floor, high ceiling tasks and three-act tasks to engage students and promote problem-solving skills. If you're a teacher, or if you have kids in school, this is an episode you can't miss. Takeaways Neurodivergent students face barriers in math education, but their unique traits can be leveraged as assets. Creating a supportive and inclusive math classroom starts with building a connection with students. Implementing strategies like low floor, high ceiling tasks and three-act tasks can engage students and promote problem-solving skills. Language and mindset play a crucial role in fostering confidence and growth in math. Understanding students' learning styles and providing individualized support is essential for their success. Register here for our free annual fall event we've created specifically for educators who are passionate about creating neurodiversity-affirming learning environments for students. The event will be held on Monday, September 23, and you can sign up to join Emily and a live panel of experts, who will be discussing ways to best support students of all neurotypes. Lindsay Kapek and Katie Tabari are experienced K-8 math educators with a passion for making math accessible to all students. They are accomplished leaders in education, leading school-wide professional development efforts, consulting with schools and school leaders regarding the implementation of skill-based instruction, and speaking nationwide at conferences including NWAIS as well as NCTM. Katie has extensive experience working with K-5 students, nearly all of whom were neurodivergent learners. She is also a mom to three kids, one of whom is neurodivergent. She is passionate about ensuring every student feels seen, heard, and included in the math classroom. Lindsay has extensive experience working with K-8 students who have been diagnosed with ADHD. Lindsay herself has ADHD and is a huge advocate for celebrating the incredible gifts that students with ADHD bring to the math classroom. BACKGROUND READING Prep Set Grow Tools referenced during the interview Facebook Instagram Blog Pinterest

The late Sir Ken Robinson once quipped that many of us feel like, "Math is a party to which we have not been invited." Today's guest, Peter Coe, wants to make sure those invitations get delivered. As the ​​Founder and Lead K-12 Mathematics Consultant at Coe Learning, LLC, Peter works to make sure students not only get invited to the party, but that once they arrive it's party worth staying for. Peter works with schools, districts, and organizations on the equitable mindsets, technical skills, resources, and infrastructure required to provide rich, engaging mathematical learning experiences for K-12 students.  A mathematician by training, he has taught in both district and charter schools, and served as a mentor teacher, department chair, and instructional coach. He is a recipient of the Math for America's Master Teacher and School Leader fellowships.  ​Peter helped lead the development of the EngageNY mathematics curriculum, as well as realignment of the state assessment program in mathematics. He also helped found and served as Chief Academic Officer of UnboundEd, leading the development of the Standards Institute mathematics pathway and advising numerous organizations and school districts on K-12 mathematics strategy. Peter has been a speaker at the NCTM and NCSM National Conferences, SXSWEdu, Learning Forward, and other national conventions. Today he shares advice to districts in how to shift both their practices and resources to raise engagement and achievement in math classrooms.    Resources:  Episode sponsor, Mathseeds is an award winning early math program designed to help build students' confidence and enthusiasm for math in the early years. The engaging program combines highly structured lessons with fun motivational elements, ensuring key concepts are learned in depth. Mathseeds is trusted by teachers for its curriculum alignment to state standards and proven effectiveness, earning ESSA Level II evidence certification. Inquire HERE about a pilot opportunity   Want to partner with Peter Coe to build a world where all students are included in the math party? Go to coelearning.org Learn more about Peter Coe Work with Peter Coe Connect with Peter Coe on LinkedIn Explore all of Peter's resources Read Peter's Blog     More great stuff: Explore our Micro Professional Learning ExPLorations fun and free, 1-hour digital, on-demand Professional Learning for teachers from all content areas and grades levels EdCuration's Blog: Learning in Action

Today's guest is Christina Whitlock. We're going to be chatting about some fun hidden truths about the teaching life. Christina Whitlock, MM, NCTM, fancies herself a friend and philosopher to the piano teaching profession. Connecting primarily with audiences through the Beyond Measure Podcast, she enjoys supporting the piano teaching community in many ways. Check out ChristinaWhitlock.com for more details on her varied offerings.Find out more about membership at vibrantmusicteaching.com.

Send us a Text Message.Today's episode is an interview with Elizabeth Grace. Her's is a story of incredible strength and resolve in discovering how to play the piano with ease and without pain through the Taubman Approach. Her harrowing journey in the Taubman work, through the support and encouragement from her family and husband, was accompanied with superior focus and determination to learn all that she could. The end result is stunning. She is a master teacher and is performing at the highest level. Don't miss a second of this interview. www.bethgrace.com Elizabeth Mueller Grace, NCTM, enjoys a multi-faceted career as performer, teacher, clinician and adjudicator. An award-winning pianist, her performances have been praised for their “depth of sound, intelligent interpretation and fluid technique.” (Omaha World Herald)A frequent collaborator, Ms. Grace is a member of the Capriole Duo with Barbara Leibundguth, former co-principal flutist in the Minnesota Orchestra. The acclaimed ensemble has performed extensively together throughout the United States since 1984. The Duo was chosen to perform at the National Flute Conference in Chicago, toured under the auspices of the Midwest Arts Council and was featured on the Ruel Joyce Series in Kansas City. Ms. Grace has appeared with the Ives Quartet on the Chamber Music Tulsa Series, and has performed in ensembles nationwide, including at the University of Texas, University of Denver, Drake University and the Lawrence Conservatory of Music. Ms. Ms. Grace regularly concertizes with Dr. Janet Fetterman in duo-piano and four-hand collaborations. The duo was featured as the Conference Artists for the Missouri Federation State Convention.Ms. Grace was selected as a 2021 Steinway and Sons Top Teacher and was the winner of the 2014 Kansas Outstanding Teacher of the Year. A dedicated and enthusiastic teacher, she has taught at Rice University, Houston, Texas and Creighton University in Omaha, Nebraska. Her students frequently win top prizes in competitions and festivals.Ms. Grace is certified at the Master Level of the Taubman Approach through the Golandsky Institute, New York City, and has been involved with the Taubman Approach since 1986.  She regularly coaches with Edna Golandsky and John Bloomfield, New York City. She has extensive experience teaching the principles of injury prevention and recovery. Her work with injured musicians is described in Preventing and Resolving Piano Injury and is featured onMajoringinMusic.com and MTNA.org.The recipient of numerous prizes and awards, Ms. Grace served as Keyboardist in the Houston Symphony, Topeka Symphony and as PrincipaThis Summer, Edna Golandsky, renowned pedagogue and leading expert on the Taubman Approach will release her first book with Amplify Publishing Group. Entitled ‘The Taubman Approach To Piano Technique: A Comprehensive Guide To Overcome Physical Limitations and Unlock Your Full Pianistic Potential.' Visit: www.ednagolandsky.com to learn more.The Golandsky Institute's mission is to provide cutting-edge instruction to pianists based on the groundbreaking work of Dorothy Taubman. This knowledge can help them overcome technical and musical challenges, cure and prevent playing-related injuries, and lead them to achieve their highest level of artistic excellence.Please visit our website at: www.golandskyinstitute.org.

It's National Sprinkle Day! So, of course, Joel and Misty first discuss how they use sprinkles, different names for sprinkles, and some other interesting facts about them. How will you celebrate? Then they have the final part of their conversation about productive struggle in math class with Kevin Dykema, the President of NCTM. If you missed parts one and two, then rewind back to June 25 to start from the beginning. To connect with Kevin:X: @ kdykemaInstagram:  dykemamathLinkedIn: kevin-dykemaThen, Joel and Misty google some higher level math jokes. Maybe you can explain some of them to us?Send Joel and Misty a message!The More Math for More People Podcast is produced by CPM Educational Program. Learn more at CPM.orgX: @cpmmathFacebook: CPMEducationalProgramEmail: cpmpodcast@cpm.org

Join Joel and Misty as they celebrate National Sugar Cookie Day. Learn about when sugar cookies first were made and revel in fun facts like Pillsbury's world record for most cookies iced in one hour. Next, they continue their thought-provoking conversation with Kevin Dykema, president of NCTM, about promoting productive struggle in math classrooms. They talk about how teachers can balance encouraging perseverance with timely intervention to foster critical thinking and problem-solving skills. They'll also discuss moving beyond procedural knowledge to make math more engaging and relevant, and highlight the importance of teacher collaboration and active listening to enrich students' learning experiences.To connect with Kevin:X: @ kdykemaInstagram:  dykemamathLinkedIn: kevin-dykemaFinally, enjoy a new segment on the podcast, Dear CPM. This letter requesting advice from Misunderstood in Minneapolis in about how to connect with their students. Bri Ruiz, one of the Professional Learning Specialists, answers the question. And you'll enjoy the math joke of the podcast from Tom!Send Joel and Misty a message!The More Math for More People Podcast is produced by CPM Educational Program. Learn more at CPM.orgX: @cpmmathFacebook: CPMEducationalProgramEmail: cpmpodcast@cpm.org

It's Global Beatles Day! So first of all, Joel and Misty talk about the impact of the Beatles and their memories of them. Then it's part 1 of a conversation with Kevin Dykema. Kevin is the current President of NCTM and a 25-year middle school teacher from Michigan. He's also the co-author of Productive Math Struggle: A 6-Point Action Plan for Fostering Perseverance. Part 1 includes how to get stakeholders on board with productive struggle in math class.To connect with Kevin:X: @ kdykemaInstagram:  dykemamathLinkedIn: kevin-dykemaAlso, Morgan Normand, one of the copy editors for CPM chimes in about her recent experience as a "student" in Inspiring Connections!And, of course, we have a math joke of the podcast for you!Send Joel and Misty a message!The More Math for More People Podcast is produced by CPM Educational Program. Learn more at CPM.orgX: @cpmmathFacebook: CPMEducationalProgramEmail: cpmpodcast@cpm.org

Note: this is a special episode of The Disagreement. What you're about to hear is a live recording from the New Schools Summit, one of the most important education events of the year. This our first ever live taping and we had a blast. Huge shout out to the NewSchools team for making it happen.And we should add that we're taking our podcast on the road! Would you like The Disagreement to come to your conference, event, off-site, college, synagogue, or mosque? We want to hear from you! Email podcast@thedisagreement.com.--Today's disagreement is on The Math Wars.For some context, “the math wars” is a debate happening in K-12 education about the best way to teach math. Broadly speaking, there are two camps that have conflicting pedagogical approaches:Explicit instruction focuses on procedural fluency, guided practice, and repetition.Inquiry-based instruction focuses on conceptual understanding, open-ended problems, and productive struggle.This is an incredibly high-stakes debate — especially if you have children or loved ones that are currently receiving K-12 math instruction. To explore its contours, we've brought on two math education experts:Kevin Dykema is President of the National Council of Teachers of Mathematics (NCTM), an international organization with more than 30,000 members. Kevin has been a passionate advocate for inquiry-based instruction and NCTM is one of the method's leading proponents. Kevin is also a teacher — currently in southwest Michigan — and he has taught 8th grade mathematics for over 25 years.Holly Korbey is an independent education journalist, whose work has appeared in The New York Times, The Washington Post, The Atlantic, and many more. Holly also writes and produces The Bell Ringer, a Substack newsletter about the science of learning.Today we ask a wide range of important questions about the Math Wars:How do children actually learn math, and what's the best way to teach them?Which approach has a more compelling body of evidence on its side?What is the best way to teach students from low-income and marginalized communities?Show NotesWhy the math wars are consequential [03:20]Inquiry-based instruction overview [05:19]Cognitive science [06:52]Relationship between conceptual understanding and fluency [11:26]Productive struggle [13:15]Research overview [20:05]What does explicit instruction look like? [23:50]Income and race [25:13]Arithmetic automaticity [29:19]What would change your mind? [32:01]Steelmanning [34:24]Episode Previeww/ Alex Grodd and Producer Catherine Cushenberryxoxo,The Disagreement Team

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

Learning to teach math teachers better with Dr. Christa Jackson, Professor of Mathematics, Science, and STEM Education at St. Louis University. She shares her infectious joy about learning and mathematics. Christa is currently serving as the editor of a new book series from NCTM, Powerful Mathematicians Who Changed the World. Four of the planned thirteen books are currently available. Show notes: Jackson, C., & Delaney, A. (2017). Chapter 10: Mindsets and Practices: Shifting to an Equity-centered Paradigm. (https://www.nctm.org/Store/Products/Access-and-Equity--Promoting-High-Quality-Mathematics-in-Grades-6-8/) _Powerful Mathematicians Who Changed the World _series (https://www.nctm.org/Store/Products/Powerful-Mathematicians-Who-Changed-the-World-Series/) Mathematics Teacher Educator Podcast (https://mtepodcast.amte.net) Special Guest: Christa Jackson.

In this episode of Room to Grow, our hosts look for the balance between instruction that is teacher-driven, traditional lecture-style, and inquiry-based, discovery-style lessons. They recognize the value of both types of teaching, understanding that there is a time in learning for both exploration and for direct and explicit teaching. The conversation offers explanation of what conditions may require different teaching strategies, based on the goals and content of the lesson as well as how students are responding to and progressing (or not) toward intended learning. The common theme between these approaches is student sense-making, and our hosts each share a personal example of taking opportunities to encourage sense-making in students.We encourage you to explore the resources below, referenced in this episode:NCTM President Kevin Dykema's President's message that sparked this episode: https://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Kevin-Dykema/Balancing-Instructional-Strategies-in-the-Math-Classroom/ TI's Building Concepts lesson on structure in solving equations: https://education.ti.com/en/t3-professional-development/for-teachers-and-teams/online-learning/on-demand-webinars/2016/building-concepts-foundations-for-success-in-expressions-and-equations A sample problem-based curriculum for middle school (NOT the one Curtis' son uses!): https://curriculum.illustrativemathematics.org/MS/teachers/what_is_pbc.html Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com. Be sure to connect with your hosts on Twitter and Instagram: @JoanieFun and @cbmathguy. 

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 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  

Christa Jackson from Saint Louis University discusses the book series that she is editing for the National Council of Teachers of Mathematics, Powerful Mathematicians Who Changed the World. Christa's professional webpage https://www.slu.edu/education/faculty/christa-jackson.php  NCTM's bookstore https://www.nctm.org/Search/?query=powerful%20mathematicians List of episodes

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

In this episode, we discuss the NCTM standards and teaching practices and how they compare to the NGSS standards. NCTM Principles to Action (https://www.nctm.org/uploadedFiles/Standards_and_Positions/PtAExecutiveSummary.pdf) Things that bring us joy this week: College Football: Michigan vs. Ohio State (https://www.espn.com/college-football/game/_/gameId/401520434) Tootsie Roll Snowballs (https://shop.tootsie.com/Tootsie-Roll-Snowballs-35-oz-Bag/p/CHR-TR021354&c=TootsieRoll@Christmas) Intro/Outro Music: Notice of Eviction by Legally Blind (https://freemusicarchive.org/music/Legally_Blind)

Learning to teach math teachers better with Dr. Sarah Bush, Professor of K-12 STEM Education and Lockheed Martin Eminent Scholar Chair in the School of Teacher Education at the University of Central Florida. Sarah shares about her roles within NCTM as a Board Member (2019-2022) and Task Force Chair and Lead Writer for Catalyzing Change in Middle School Mathematics: Initiating Critical Conversations. She also speaks about the Master Teacher Fellows and their work in a Noyce Track 3 grant, Empowering STEM Teachers with Earned Doctorates. Links mentioned in this episode: Empowering STEM Teachers with Earned Doctorates (Noyce Track 3 Grant) https://ccie.ucf.edu/noyce-mathematics-education/ Noyce Blog post for AAAS https://aaas-arise.org/2022/05/26/elevating-voices-catalyzing-change-a-partnership-approach-to-supporting-k-8-mathematics-teacher-leaders/ NCTM's Catalyzing Change Series: https://www.nctm.org/change/ Catalyzing Change in Middle School Mathematics: Initiating Critical Conversations https://www.nctm.org/Standards-and-Positions/Catalyzing-Change/Catalyzing-Change-in-Middle-School-Mathematics/ Success Stories from Catalyzing Change https://www.nctm.org/Store/Products/Success-Stories-from-Catalyzing-Change/ Simplifying STEM [6-12]: Four Equitable Practices to Inspire Meaningful Learning By: Christa Jackson, Kristin L. Cook, Sarah B. Bush, Margaret Mohr-Schroeder, Cathrine Maiorca, Thomas Roberts https://us.corwin.com/books/simplifying-stem-285696 Simplifying STEM [PreK-5]: Four Equitable Practices to Inspire Meaningful Learning By: Christa Jackson, Thomas Roberts, Cathrine Maiorca, Kristin L. Cook, Sarah B. Bush, Margaret Mohr-Schroeder https://us.corwin.com/books/simplifying-stem-285694 STEM Education Report https://www.energy.gov/sites/default/files/2019/05/f62/STEM-Education-Strategic-Plan-2018.pdf Amidon Planet E096: The Path to Professor with Dorothy White https://amidonplanet.com/episode96/ Melissa Adams Corral: Teaching as Community Organizing https://www.teachingmathteachingpodcast.com/82 Special Guest: Sarah Bush.

This episode includes information about the 3 plenary presentations at the 2023 NCTM Annual Meeting and 2 summaries from presenters -- Gail Burrill from Michigan State University, Amanda Huffman-Hayes from Purdue University, and Lindsay Gold from the University of Dayton. The NCTM Annual Meeting and Research Conference were held in Washington, DC. NCTM Website https://www.nctm.org/ 2023 Program https://www.nctm.org/uploadedFiles/Conferences_and_Professional_Development/Annual_Meetings/DC2023/DC2023_Annual_Program.pdf List of episodes

Recorded live from NCTM! Pam and Kim discuss some of their experiences and takeaways from the 2023 National Council of Teachers of Mathematics conference.Talking Points:Travel storiesAs kids grow, teachers grow as wellKnow your content know your kids to choose Problem Strings for your studentsTools that focus on answer getting stifle students reasoningUsing a neutral response to keep students and adults continue their thinkingWe all need to understand the content - teachers and learning Check out our social mediaTwitter: @PWHarrisInstagram: Pam Harris_mathFacebook: Pam Harris, author, mathematics educationLinkedin: Pam Harris Consulting LLC

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

Learning to teach math teachers with Courtney Baker, Associate Professor in the Mathematics Education Leadership program at George Mason University, and Melinda Knapp, Assistant Professor of Education at Oregon State University-Cascades, as they share their experiences and advice on being mathematics teacher educators. Melinda and Courtney also share their book from NCTM, Proactive Mathematics Coaching: Bridging Content, Context, and Practice. Links from the episode Proactive Mathematics Coaching: Bridging Content, Context, and Practice (https://www.nctm.org/Store/Products/Proactive-Mathematics-Coaching--Bridging-Content,-Context,-and-Practice/) NCTM Book Study: Unveiling the Proactive Coaching Framework (12/13/23) (https://www.nctm.org/Store/Products/NCTM-Book-Study--Unveiling-the-Proactive-Coaching-Framework-(12/13/23)/) National Center for Faculty Development and Diversity (https://www.facultydiversity.org/) Mathematics Teacher Educator Podcast (https://mtepodcast.amte.net/) Special Guests: Courtney Baker and Melinda Knapp.

Kelly Spoon a tenured faculty member at San Diego Mesa College. She teaches mathematics and statistics within the Mathematics department. She's had a number of different roles on campus and within my department, including STEM Professional Learning Coordinator and is involved in statistics education and focused on culturally responsive teaching. Amy Hogan is an NYC high school teacher. Currently teaching AP Statistics, Math Analysis (sophomore math team), and Algebra 2. She is involved with stats education at the K-12 level with the ASA and NCTM, and served on the committee for USCOTS 2023. Daniel Kaplan is a nationally recognized college professor in Statistics, Data Science, and Applied Mathematics. Author of multiple university-level textbooks, award winning teacher and curriculum developer. Strong innovator in teaching with professional-level software and author of several R packages for teaching data science, statistics, and calculus.

Are you curious about travel teaching? In this episode, piano teacher and composer Chrissy Ricker shares her experiences as a traveling piano teacher: the advantages, disadvantages, and lessons learned from over a decade of teaching piano in students' homes. Chrissy Ricker, NCTM, is a pianist, composer, and arranger from North Carolina. She enjoys sharing her ideas with other piano teachers and writing motivating music for pianists of all ages. Learn more about Chrissy at her website, chrissyricker.com. For the rest of the show notes, including links mentioned, [CLICK HERE]. --- Send in a voice message: https://podcasters.spotify.com/pod/show/piano-pantry-podcast/message

In this episode of Room to Grow, Joanie and Curtis talk about teaching strategies for remembering in mathematics, such as mnemonic devices, tricks, and gimmicks. They challenge the notion that teaching with tricks is inherently bad, and discuss how to determine when a strategy intended to help students learn might actually work against their understanding of the underlying mathematics. For instance, “FOIL” and “SOH-CAH-TOA” are both frequently taught in high school math classes, yet one is a way to remember mathematical definitions (not a trick!) and the other is a random association for a limited procedure (a trick!). So what about if a student creates their own strategy or trick while learning math?  Join our hosts in trying to make sense of how and when remembering strategies are helpful and when they might be more harmful.We encourage you to explore the resources below, referenced in this episode:Nix the Tricks – available to download for freeThirteen Rules that Expire is an article from Teaching Children Mathematics (NCTM membership required) about commonly taught ideas in elementary school that don't support long-term learning. This blog about the article does not require NCTM membership.Twelve Math Rules that Expire in the Middle Grades is a similar publication from Mathematics Teaching in the Middle School (NCTM membership required) with middle school-specific ideas.This EdWeek article includes a commentary from Dr. Hilary Kreisberg about “nixing tricks.”Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com. Be sure to connect with your hosts on Twitter and Instagram: @JoanieFun and @cbmathguy. 

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 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

In this episode, we are going to chat about one of my favorite business practices for piano teachers: Reflecting on your piano teaching business so you can determine the specific strategic goals you need to set for the coming quarter. This will help you run your piano studio with relevance, action, and efficiency so you can make decisions on how to operate your studio with your strategic goals in mind! Dr. Melody Payne, NCTM, is a piano teacher and educational resource author who believes that all students deserve the best musical experiences possible. She publishes pedagogical teaching materials and articles for piano teachers at melodypayne.com. Melody and her husband, Greg live in Marion, Virginia, where she teaches children and adults of all ages. For the rest of the show notes, [CLICK HERE]. --- Send in a voice message: https://podcasters.spotify.com/pod/show/piano-pantry-podcast/message

In this live taping of the Make Math Moments That Matter Podcast Jon & Kyle visit the 2022 Annual NCTM Conference in Los Angeles. The discuss how conferences can “fill your bucket” as well as hear key takeaways from conference attendees. Stick around and you'll learn how you too can fill your bucket. You'll Learn: Why conferences are so much more than just learning sessions; Why mimicking is a teaching technique we should avoid; How to leverage constraints vs. freedoms; How you can fill your own bucket when you can't attend a conference; Resources: The 2022 Make Math Moments Virtual Summit [REGISTER NOW]District Leader/Mentor Summit Sharing Resources [DOWNLOADABLE GOODIES]District Leader Resources:The Make Math Moments District Planning Workbook [First 3 pages] Are you a district mathematics leader interested in crafting a mathematics professional learning plan that will transform your district mathematics program forever? Book a time to chat with us!Other Useful Resources and Supports: Make Math Moments Framework [Blog Article]Make Math Moments Problem-Based Lessons & UnitsEmpower your educators to elevate their pedagogical practice and deepen their mathematics content knowledge.Your Educators can attend this FREE LIVE Virtual Summit on Friday November 18th, Saturday November 19th and Sunday November 20th, 2022.Learn More: https://makemathmoments.com/summitdistrict/ Join Kyle and Jon in this 5 module course as they teach you how to structure your assessment practices so you can use assessment to empower yourself and your students to promote their learning instead of using it as punishment.Learn here --> https://makemathmoments.com/afg/