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

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

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

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

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

Are you interested in mastering communication in your relationship?   Join us for part two of our conversation with Dr. Mike Frazier MD, host of the "Strong Men, Strong Marriages" podcast. As a renowned psychiatrist specializing in neuroscience, Dr. Mike offers invaluable insights into fostering intimacy and effective communication within relationships.    Delve into practical strategies for improving your relationship dynamics, including the optimal timing for making difficult requests and navigating common conflict areas. Dr. Mike shares inspirational success stories and explores the nuances of forgiveness, faith, and trust, providing clarity on how to strengthen the bond with your partner and foster a thriving, resilient marriage.    “The answer isn't do more, it's actually commit to less and follow through on it” - Dr. Mike   You'll leave this episode with…   The best time for you to make hard requests in your relationship   Inspirational success stories of couples overcoming relationship challenges   The 4 types of people that come in to see Dr. Mike   Why Dr. Mike's program is called “Strong Men, Strong Marriages”   What an intimate conversation entails   How you can be more effective in your communication with your partner   The main framework of communication   How you can deal with the feeling that you're getting run over in your relationship   The main conflict areas in most marriages   Two options of how you can put yourself into your partners shoes   What is the difference between forgiveness, faith, and trust?   The Manhood Experiment of the week that will help you become more clear on what you're asking for of your partner   -----   Leave a Review: If you enjoyed the show, please leave us an encouraging review and tell us why you loved the show. Remember to click ‘subscribe' so you get all of our latest episodes. 
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   What is the Manhood Experiment? 
 It's a weekly podcast where we give you one experiment to level up your mind, career, business, health, relationships and more!   For more tips and behind the scenes, follow us on:
   Instagram @ManhoodExperiment Tiktok @ManhoodExperiment Threads @ManhoodExperiment Submit your questions @ www.manhoodexperiment.com   Resources Mentioned:   Dr. Mike Frazier MD: https://mikefraziermd.com/   Leadership and Self-Deception: Getting Out of the Box - The Arbinger Institute   Influence, New and Expanded: The Psychology of Persuasion - Robert B. Cialdini   Strong Men, Strong Marriages: strongmen.io   The 7 Habits of Highly Effective People: https://www.franklincovey.com/the-7-habits/

What if there were simple yet powerful ways to strengthen and restore intimacy in your relationship?   In this episode, we welcome Dr. Mike Frazier MD, host of the "Strong Men, Strong Marriages" podcast, a renowned psychiatrist with expertise in neuroscience and relationship dynamics. Dr. Mike tackles marriage challenges, from expectations around sex to shared responsibilities, along with insights into therapy, offering guidance on finding the right support and dispelling common misconceptions.    Explore Dr. Mike's passion for his work as he vulnerably shares his own personal struggles within his marriage, along with strategies for overcoming relationship challenges. Delve into Dr. Mike's take on sayings like “Happy Wife, Happy Life” and his advice for happier marriages and learn how to nurture intimacy, set expectations, and communicate effectively in your relationships.    “If you can manage your mind better and your emotions better, that affects everything” - Dr. Mike   You'll leave this episode with…   Tips on how you can strengthen and return intimacy back into your relationship   What are different types of therapy   Dr. Mike's personal journey through marriage struggles, including a year without sex, and how he helps men facing similar struggles today   Dr. Mike's break down of his thoughts on the saying “Happy Wife, Happy Life”   What the mosquito cycle is and how you can change it for a healthier marriage   What it looks like when you treat your marriage transactionally   How do you deal with the expectations you have when you are the main provider for the family   The difference of being a partner or a contractor in your marriage   What you can do if you feel like you are doing too much or more than your partner   An understanding of why seemingly ‘unrelated things' are often connected in woman's minds   How you can make a good request to your partner   The three questions you should ask your partner when having hard discussions   -----   Leave a Review: 
 If you enjoyed the show, please leave us an encouraging review and tell us why you loved the show. Remember to click ‘subscribe' so you get all of our latest episodes. 
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   What is the Manhood Experiment? 
 It's a weekly podcast where we give you one experiment to level up your mind, career, business, health, relationships and more!   For more tips and behind the scenes, follow us on:
 Instagram @ManhoodExperiment Tiktok @ManhoodExperiment Threads @ManhoodExperiment Submit your questions @ www.manhoodexperiment.com   Resources Mentioned:   Dr. Mike Frazier MD: https://mikefraziermd.com/   Leadership and Self-Deception: Getting Out of the Box - The Arbinger Institute   Influence, New and Expanded: The Psychology of Persuasion - Robert B. Cialdini   

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

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

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

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

United recently launched Kinective Media, the airline industry's first media network. Its first-party data could change the future of people's travel experiences.  Episode TranscriptPlease note, this transcript  may contain minor inconsistencies compared to the episode audio. [00:00:00] Damian: I'm Damian Fowler and welcome to this edition of the current podcast this week we're delighted to talk with Mike Petrella, the managing director of partnerships at United airlines. In June, the airline launched a new initiative named connective media, which is the airline industry's first media network. [00:00:17] The network will use data from its customer profiles to create a personalized and immersive travel journey. This launch comes at a time when retail media networks have become one of the hottest topics in ad tech, allowing brand marketers to target consumers using retailers first party data. [00:00:35] We start by asking Mike about why United wanted to move in this direction.  [00:00:40] Damian: United's the first travel focused company to develop its own media network called Connective Media by United Airlines, and how is this a boon for the company and its flyers? [00:00:50] Mike: of course, so we consider ourselves a commerce media network, and we distinguish ourselves as a commerce media network. Given retail media, networks are typically point of [00:01:00] purchase, transaction based. The Commerce Media Network embraces the emotion, the journey, the feelings of all parts of the funnel. [00:01:08] So when you think about our users at time of planning, time of travel, [00:01:13] Damian: and signals [00:01:14] Mike: time of destination, even when they're not thinking about travel, we have 108 million profiles. And the beauty of our data is it's incredibly accurate. You have to be Damien to get on a plane. Your name has to be what it is, your address has to be correct, your phone number, and all the other information. [00:01:33] So the breadth of that information, coupled with the accuracy of it, gives us insights and signals that allow us to create these robust profiles of the user. And it's the user at all points. And the commerce nature of this isn't just a point of sale. We are not microtransactions on a consistent, on a constant basis. [00:01:54] Rather, we think about The interaction of the user at the time of [00:02:00] planning, top funnel. At the time of trip, or even time of purchase in an ancillary mindset. Purchase path typically generates a significant amount of revenue. Be it airline tickets, upgrades, any purchase path. [00:02:11] Regardless of whether it is airline ticket or if you're purchasing a ticket to an event, parking, whatever the case is. So for us, it's the ability to take that journey. To be able to identify when the right time to send the right message to the right user is. And that message could be an advertisement, it could be content, it could be nothing because it's not the right time. [00:02:35] But in each of these cases, you can make a use case for any and every brand based on the scale and depth of our data. [00:02:42] Damian: of our data. Fascinating. And you mentioned that long purchase journey, which is, sometimes it can be a long purchase journey, right? For air travel, or it could be short, but you do have a lot of scope within that context. [00:02:54] Mike: It is. I mean, very few people spontaneously book tickets to destinations, right? [00:02:59] And when [00:03:00] you're in that mindset, you're in a planning mindset, not only of the journey, but think about the insights and intelligence we can extract from the signals we receive to say, Well, this person happens to frequent a specific hotel chain, a car rental chain, a ride share company, when they land, they frequent a food delivery service. [00:03:21] Endemic, but then you think the non endemic piece. And this is the beauty of what we do. The lines of endemic and non endemic are completely blurred. To me at least. Because I think about, when you get on a plane, you may be traveling home to return to normality. Which takes you to food shopping, which takes you to the pharmacy, which takes you to the laundromat. [00:03:39] But my point is, I think the idea of always coupling a travel endemic brand or journey with the traveler is only a piece of it. be it on the road or at home. [00:03:50] I may go see a music event. I may go purchase music. I might play music. I may eat pizza. I will eat pizza just to be clear. But my point is, my behaviors [00:04:00] There are some that are going to be unique based on my journey, and others that are going to overlap with when I'm traveling for leisure, when I'm home. [00:04:08] And so, I love the fact that just, we can essentially meet the interests of the customer, which is the centric piece of this, and provide value to our partners as well. [00:04:21] Damian: It's a very clear example of how non endemic works in a retail media network, I think, because, you know, when you think about other retail media networks, often think about the retailer and what it sells, but, you know, with United, it's a different story. [00:04:34] Mike: Yeah, it's the breath of commerce, and that's what I enjoyed. That was like when I came here, it was eye opening. I had an idea, right? But just to see what we can do and really the validation of just how strong our data is and how valuable it is from a customer standpoint. When I say valuable to the customer, it is to spoon feed customers based on their interests. [00:04:57] Cafeteria style doesn't work. There are too many choices. [00:05:00] So if you're in a planning phase and we can bring about certain things that are of, normality to you, booking a restaurant, booking a golf reservation, simply as getting my ride share, it makes the journey easier. It makes it feel like it's Damien Fowler's journey, not just a customer who purchased a seat in one of our planes. [00:05:20] Damian: Yeah, I love that. And I just want to take that point a little bit further. Can you give some more examples of how, you work with brands, whether endemic or, when I say endemic, that would be travel related, right? Or not. and where that media might appear. [00:05:34] Mike: Sure. So today we are, our media network extends from our dot com, our in app, we have digital signage within the airports, be it in our clubs and lounges, gate information displays, on our planes we have in flight entertainment, or we call IFE, or personal device entertainment on your phone, and so as part of United Next, we made an investment to purchase north of [00:06:00] 800 planes. [00:06:01] And within each of those planes, they will be outfitted with the new IFE system. It's meant to be more of an OTT experience versus the current experience, which quite frankly is, it's legacy, it's the 1950s. It's a small screen with limited choices and it's not what we're used to. we envision this opportunity to have a very personalized experience in which you will have your interests displayed on that screen and every person's screen will be different. [00:06:28] Based on that individual. And so, for us, we will be retrofitting our current fleet, with the exception of a couple planes that will be retired over time. And so, over time, we will have screens in all planes on a, personalized basis. And so, for us too, it's, you extend past that, you have email and such. [00:06:47] It's a true omni channel offering, but most importantly, it's the engagement. We have an average of three and a half hour flight time. And so, when you're at home You can get up, use the restroom, go to the kitchen, whatever, if [00:07:00] a commercial comes on. You cannot do the same in a plane. At the same frequency. I mean, yes, you can get up, but the idea of having the ability to engage in an intimate and targeted manner with our users and to be able to show them things of their interest is huge. [00:07:16] Right? And then you think more, in lounges and clubs, It's not going to be personalized. If Damien walks in, if you walk into the club, you don't want to see. Hello, Damon. How are you? Do you need a new green shirt? That's creepy, right? Yes. So again, there's you can think about. the business traveler travels from Monday at 5 a. [00:07:35] m. to 11 a. m. and Thursdays from 4 to 7. So perhaps we put advertisers endemic to that audience. Families travel on weekends and these are generalities. But through research and through signals, we can begin to capture that. And again, the right message at the right time. [00:07:50] Damian: What customer insights will help connect brands with United Flyers? [00:07:54] Mike: So we capture over 120 targetable segments, or signals, I should say. And that [00:08:00] is, a mix of attitudinal, behavioral, lifestyle, and transactional. And today, our audience indexed to the highly affluent individual. Married, college educated, homeowner, household income of 250, 000 plus. And so you'll see in some of our launch partners, Bottega Veneta, which is a luxury brand, McAllen's, a higher end Scotch. [00:08:21] Very good for that audience, but at the same time, we are very diverse in terms of who is on our plane. We, our launch partner was Televisa Univision. 25 percent of the Chicago population is Hispanic. Is it 63 million, Spanish speaking, Americans in the U. S., right? So the idea of just focusing on one demographic doesn't do anyone justice. [00:08:45] very much. Right? Again, speaks to that scale of data. And so, we, there's a use case for every single brand, every single opportunity. We [00:08:56] Damian: that nuance that you can bring to it, to [00:09:00] advertising, is obviously key to this. what strategies is Connective employ to personalize ads and offer that to these different segments? [00:09:08] We are a very privacy centric, privacy [00:09:10] Mike: privacy safe, conservative approach to what we're doing. We sit atop GAM. we work with, a number of clean rooms. any and everything we do is meant to uphold the integrity of that customer's data. we will never sell the data as a stand alone. It'll always be wrapped with media on a managed basis. [00:09:31] And I say that because the sale of data opens up opportunity for bad actors. Then there are bad actors out there. So when it comes down to it You know, we want to ensure that we are keeping our customers, information, and privacy at the forefront. And then, any and everything we do is in a compliant way. [00:09:51] Data collaborations through clean rooms, proper encryption at all specific times, proper measurement and verification. it's a textbook [00:10:00] approach, knowing full well that,  [00:10:04] Mike: party data is currency, you have to protect it, and you have to use it in the right manner. [00:10:09] Damian: And it feels great, right? The work that we did is meaningful. [00:10:20] Mike: It's been overwhelming, honestly. I used to work, I helped startup advertising. com a long time ago, and all its brand names up through Yahoo. And I was always the one vying for a brand's business. To work on a brand site now has been an eye opening experience because you have the problem of choice. And the reception to what we've been doing has been incredibly positive. [00:10:44] and it feels great, right? The work that we did is meaningful. The work that we did is interesting. but we have to be smart in terms of who we work with. I would say the outreach from partners, we always want to maintain a very premium nature for any owned [00:11:00] and operated supply. I think it's important. [00:11:02] Again, the brand integrity for United is paramount. but at the same time, as I said earlier, there's a use case for all brands. And we're always open to exploration and conversations. And then making the right choice based on United brand, based on the value for our customers and for the overall business. [00:11:21] Damian: Now travel has skyrocketed since pandemic times, and that's been well reported. Can you describe the change United has seen more generally in people coming back to the skies? [00:11:32] Mike: the largest airline in the U. S. right now. and it's, it's a great position to be in because people fly United for the experience. [00:11:39] We do not compete with low cost carriers. That's not our model. People fly for the convenience, for the experience, for the opportunity to increase their loyalty status, for the journey in itself. Our app is the number one rated app in the, in, of all airlines, and if you, you know, I'm not sure if you're a flyer or not, Thank you. [00:11:58] If you are [00:12:00] so you see that app is very intuitive in terms of my baggage goes here. My gate is here. And so against personalization, right? It may not be specific. Damien. This is your journey. Rather, you are flying at this airport. Here is where your luggage is. Here's where your gate is. And it's just it's taking those steps to just again lessen the hassle of travel. [00:12:19] And then, as you get on the plane, our flight attendants, our ground crew, our pilots are just top caliber. it's the friendliness that you see. again, the experience extends beyond [00:12:29] Damian: a traveler's standpoint. [00:12:30] Mike: Connected media provides an opportunity for us to gather what we have from our three core pillars. Travel, loyalty, and media. [00:12:39] And it's that flywheel. we are able to ingest signals based on the profiles that we have. And in doing so, you begin to see the traveler profile as it begins to matriculate to an actual loyalty partner. [00:12:52] 39 million mileage plus loyalty partners. We have a co brand card through Chase. Right. We have our mileage plus [00:13:00] partnerships team, and we think about that from the Avis's, the Marriott's, from a travel endemic standpoint, non endemic, even like the away, I guess away luggage is not therabody, things to that effect. [00:13:10] And so, the ability to accrue and redeem miles as transaction. And then, with the credit card, the ability to redeem miles, or accrue miles, I should say, through transactions. As you go through the flywheel, you come to the media piece, which is the connective tissue. To understanding the middle and lower funnel of that transaction, purchase point, brand affinity, options for our users. [00:13:33] And then back to the first part, the emotion, and the journey, and the actual travel. And as we do this flywheel, we have more travelers, which means more signals, which means more opportunities for media, which means more, and it's a self fulfilling flywheel that essentially, again, with the customer in the middle, or the customer is the focus, it's Creates that opportunity to your point of why people are flying more with United.[00:14:00]  [00:14:00] Damian: What kind of feedback have you had from those customers? what are people's experience, what are people experiencing and how are they setting that back to you? [00:14:08] Mike: think the best part is, they've come up and said I'm so excited you're doing this. Never would have thought of this. like you, you're hearing it from the horse's mouth, right? So there's, in an unbiased manner, what I'm most proud of is the fact that we've come out with a legitimate business with a very, very focused North Star, that is focused solely around the customer. [00:14:31] that's unique. And to bring it to market at the speed that we did. With the help that we had from partners and the support that we've had from the industry has been just, has been amazing. Now the [00:14:43] Damian: the idea now seems like a very good one. And you're describing, you're telling me, Mike, how quickly you brought it to market. What, in under a year, really? I mean, it's a good idea. Do you expect that other airlines are going to want to emulate, what you've done here with your media network? [00:14:59] Mike: is [00:15:00] a very savvy airline. They're a great airline. they're doing certain things [00:15:07] with the connect, that we're connecting. streaming from a device to their, seatback screens. They've done partnerships with Walmart Plus and such. Whether they come out with a full scale media network, I'm not sure. but, United and Delta are the top two airlines in the U. S. [00:15:22] And they are a very savvy brand. So, if they come out, I would not be, surprised. I don't know about the others. You know, for me, it's not one's better than the other. It's just where I see the next. In [00:15:35] Damian: In general, while we're on the topic of predictions, when you look ahead to the rest of this year and to next, as you build this offering out, what are the kind of trends you're looking for in terms of that merging of travel and media that you just talked about? The year into next, what trends are you all looking for? [00:16:09] Mike: It's really, when you and I grew up, you had to pay for HBO, you had to pay for ESPN. it's a similar model, and you're seeing consolidation and M& A start in that sector. There's too many choices for consumers. Today, there's 273 retail media networks. That is not scalable, right? Marketers and agencies already have too many choices to make. [00:16:30] and at the same time, the uniqueness of that data, depending on the sectors. It may not be all that unique. I do think there's going to be consolidation. There has to be. And for me, I would expect that. I think we're in a very good position just given the unique position that we're in. And quite frankly, like the three pillars, right? [00:16:53] Scale, accuracy, and omni channel. And we can say we have that with confidence. I would say like, [00:17:00] to your point of expectations, there has to be consolidation. I think the introduction of AI, it wouldn't be a podcast without saying AI. I've already said flywheel, if there's another one I need to say. [00:17:11] But I do think, the introduction of AI into not only the purchase path, but more importantly, the analytics. Right? Humans know which questions to ask. AI will figure out what other questions to ask. And as we constantly feed these models, you're going to have, just from an analytics standpoint, the ability to extract new data, new intelligence, new insights, and we want to be on the forefront there to ensure that, we modernize our offering at a pace that is quicker, than what the industry is seeing. [00:17:44] Damian: Do you anticipate that your media network and what you're offering might have some kind of partnerships with some of those streaming platforms? I'm just thinking. Yeah, it's my job. So, [00:17:54] Mike: So, like, I do. I think there's opportunity for partnership. Yeah. it's the many versus the [00:18:00] power of one. [00:18:00] Damian: Yeah.  [00:18:01] Mike: You have to be selective, right? If you partner with everyone, you partner with no one. So, I think there's opportunities in the travel space. I do think there's opportunities in the non endemic space, too. We're at really early stages, so Honestly, platform side, I'm not used to this much attention. [00:18:21] and I love it. And we brought friends in to build this business. I'm working with my friends. I absolutely love it. And so together we're kind of sitting down and putting our heads together and say, okay, like we got to the starting line. We bust out our asses for nine months and we got to the starting line. How do we run this race and always be the leader? Because there's going to be people coming up after us. And that challenge with one another is great because we're pushing one another to be better. And it's not intense in the sense that like, any conversations with emotion are meant for constructive and collaboration. [00:18:57] And I think we're all being better because we're constantly pushing [00:19:00] one another. But more importantly, we're supporting one another. [00:19:02] Damian: Yeah. you do see some relationships with broadcasters, with in flight entertainment, but I imagine this is going to go. To a different level. [00:19:11] Mike: this is the early stage of the business. This is the exciting part. we're the bright, shiny object right now, and I think it's good to revel in that just to pat yourself on the back and say, Hey, we did it. [00:19:22] But realistically, like complacency doesn't get you anywhere, right? So everyone else has got has gotten to the starting line. There's been 273 other companies that got to the starting line, and some are running the race faster than others, and some are not even on the same course anymore. so for us, I think it's about heads down, and just constantly push. [00:19:42] And to be the best, [00:19:49] Damian: is highly competitive. Do you feel the pressure? [00:19:53] Mike: I don't feel the pressure from the industry. I feel the pressure to deliver. Like, me personally, I hold the bar very [00:20:00] high for myself, and I'm my worst critic. I know what it's like to be successful. I helped launch advertising. com and I can tell you those first five years were by far like the highlight of my life from a professional standpoint. [00:20:11] these last nine months are on par with that. And if I can make the next four years and three months the same or better, I'm going to do everything I can to do it. And if there's 23 years, 18 more years to follow that, great. I hope to retire at some point in my life. But, um, I'm just excited because. [00:20:30] This is real. And it's good. And, will be responsible for our success. So, yeah, I'm really excited about it.  [00:20:37] Damian: thank you so much for these insights. It's been great. [00:20:40] Mike: to speak with you, Damian. Thank you.  [00:20:42] Damian: And that's it for this edition of The Current Podcast. [00:20:44] We'll be back next week, so stay tuned. [00:20:47] Ilyse: The Current Podcast's theme is by Love Caliber. The current team includes Kat Vesce and Sydney Cairns. [00:20:53] Damian: . And remember, I'm Damian. [00:20:55] Ilyse: I'm Ilyse. [00:20:56] Damian: And we'll see you next time. And if you like what you hear, please [00:21:00] subscribe and leave us a review. Also, tune in to our other podcast, The Current Report.

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

Welcome to another exciting episode of Passive Income Pilots! We're diving into a fascinating and often overlooked passive income stream: vending machines. Our guest, Mike Hoffmann, also known as "Mr. Passive," shares his journey of creating a vending machine empire and how this low-barrier investment can yield significant returns. Whether you're a seasoned investor or looking for a side business to try with your kids, this episode is packed with insights and practical advice on turning everyday conveniences into lucrative income streams. Tune in to discover the secrets of the vending machine business and learn how you can cash in on this profitable niche.Timestamped Show Notes(00:00) Introduction by Tait and Ryan(01:24) Mike Hoffman's introduction and passive income journey(03:25) Overview of vending machine types and profitability(06:14) Mike's research and first vending machine purchase(08:34) Cost, EBITDA breakdown, and tax benefits of vending machines(11:41) Vending machine market opportunities and lead generation(13:58) Importance of location and securing vending spots(19:59) Choosing the right vending machine and products(22:57) Operational logistics: stocking process and maintenance(28:34) Expanding operations: remote management and larger routes(34:05) Comparison to other investments: ATMs and self-storage(35:00) Challenges and downsides in the vending business(42:30) Future trends: ad revenue, and demographic data(46:18) Specialty vending machines and innovative solutions(49:00) Closing remarks and how to connect with Mike---You've found the number one resource for financial education for aviators! Please consider leaving a rating and sharing this podcast with your colleagues in the aviation community.Remember to subscribe for more insights at PassiveIncomePilots.com!Join our growing community on FacebookHave questions or want to discuss this episode? Contact us at ask@passiveincomepilots.com or record your question to be featured on the show HERE!Take 10% Off Your Next Uniform - Use code "PASSIVE" on checkout!Legal DisclaimerThe content of this podcast is provided solely for educational and informational purposes. The views and opinions expressed are those of the hosts, Tait Duryea and Ryan Gibson, and do not reflect those of any organization they are associated with, including Turbine Capital or Spartan Investment Group. The opinions of our guests are their own and should not be construed as financial advice. This podcast does not offer tax, legal, or investment advice. Listeners are advised to consult with their own legal or financial counsel and to conduct their own due diligence before making any financial decisions.

Episode Summary:In this gripping episode of "The Nightly Rant," hosts Mike and Toria dive deep into the astonishing reality of adults who lack basic adulting skills. With humor and a hint of disbelief, the two discuss the events that led Mike to express frustration over childish behavior in professional settings. The episode is particularly focused on the interactions with mailing lists and the proper ways to handle unwanted emails.The conversation starts with a recount of what could be considered a trivial issue: someone repeatedly booking appointments in protest of receiving regular emails, despite the availability of a simple 'unsubscribe' link. Instead of discussing the content of these emails, Mike and Toria explore the larger concern of adults failing to navigate basic responsibilities. They elaborate on more effective, mature ways to manage one's inbox and the etiquette surrounding email communications. Their critique of the situation is interspersed with personal anecdotes and their experiences with email list management, making the episode both educational and relatable.Key Takeaways:Adults often fail to perform basic tasks like managing their email inbox properly, leading to unnecessary confrontations.The presence of an 'unsubscribe' link in emails is a tool to manage the receipt of unwanted emails, ensuring recipients have control over their inboxes.Creating unnecessary drama is often the response of individuals who avoid addressing issues in a straightforward, mature manner.Testing the functions of mailing lists, including unsubscribe features, is essential to maintaining good communication and respecting recipients' preferences.The hosts encourage listening to responses and feedback from the email recipients to foster positive and professional interaction.Notable Quotes:"I never knew that there were so many people in this world that don't know how to be an adult." - Mike"You need to learn what that unsubscribe link is for." - Mike"So being the brave soul that I am, I responded by writing an email to my entire list explaining that in order to be an adult in 2024, you need to learn how to handle your inbox." - Mike"I test the unsubscribe after seeing this, thinking, shit, maybe it's broken. No, it's not broken. It unsubscribed me like a champ." - Mike"Stop acting like a damn baby about every little thing." -- MIke  TimestampSummary0:15Adulting Challenges in Today's Society1:10Email Tactics and Unwanted Booking Spam3:29Managing Email Subscriptions and User Responsibilities4:30Estimating Time for Bulk Bookings4:57Unsubscribe Options and Email List Preferences5:30Managing Email Preferences and Frequency Tolerance6:33Circumventing Email Unsubscription Issues7:26Handling Drama and Email Retaliation Tactics10:44Challenging Traditional DEI Hiring Practices11:09Adulting Challenges and T-Shirt Ideas12:11Spreading Slang and the Joy of Podcasting

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

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

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

Rounding Up Season 2 | Episode 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

Unlock the secrets to brand amplification and mortgage mastery with Mike Fitzpatrick, founder of Legendary Mortgage, in this captivating episode. From our chance meeting at GrowthCon3, we explore the transformative power of speaking and networking in brand development, revealing insights into navigating the competitive lending landscape with a distinct value proposition and personal branding strategy.As we journey through the realms of selling, public speaking, and community building, discover the hybrid future of post-pandemic events and the crucial role of self-review in honing communication skills. Dive into the essence of effective leadership, as we discuss the magic of grace, trust-building, and cultivating a collaborative culture that propels businesses to new heights. Whether you're a novice or a seasoned pro, join us for an episode that promises to revolutionize your approach to leadership, speaking, and brand development, leaving you inspired to craft your path to success.NOTABLE QUOTES"Watch the way that people operate, watch their process because you'll learn more from that usually than from what they say on the stage." – Mike"By going around and shaking hands and having conversations with people, that helps build up your credibility and trust with them, also for the event host too." – Philip"It's really good as a speaker if you're willing to stay that extra time because that helps the event host and gives that much more credibility for them and makes the event that much more special and unique too."  – Philip"Continue to find ways to find that spark, and that could be podcasting, that could be speaking, that could be driving value." – Mike"Whenever you build a business, build a vision, people need to fit within that vision." – Philip“Whenever you're speaking, you want to give some people some access to you so they feel important." – Mike“If you truly care about what you're doing, you can learn a lot from listening back to yourself.” – Mike“The more vocabulary you have in your tool belt, the better that you're going to be able to articulate things to somebody.” – Mike“Your body language exudes what you want to try to communicate to somebody, whether it's face-to-face or if it's even voice-to-voice.” – Mike“I don't ask anybody to do anything that I've not done myself or actively doing myself.” – Mike“From a leadership position, make sure that at some point during the year, go roll up your sleeves with the section of the business that you're not super comfortable with.” – Mike“You got to be the lighthouse, the North Star to the people that look to you for leadership.” – Mike“You need to be humbling yourself constantly to try to learn from [your team] at the same time, and not just being a dictator or totalitarian.” – Mike“See the forest outside of the trees, not only seeing the trees themselves.”  – Philip“Make sure that you're operating in grace before you operate in anger.”  – MikeRESOURCESMikeWebsite: https://www.legendarymortgage.com/ Instagram: https://www.instagram.com/realmikefitz/LinkedIn: https://www.linkedin.com/in/mike-f-372b2478/ Facebook: https://www.facebook.com/fitzteam PhilipDigital Course: https://www.speakingsessions.com/digital-courseInstagram: https://www.instagram.com/iamphilipsessions/?hl=enTikTok: https://www.tiktok.com/@philipsessionsLinkedin: https://www.linkedin.com/in/philip-sessions-b2986563/Facebook: https://www.facebook.com/therealphilipsessions Support the Show.

What if there were simple yet powerful ways to strengthen and restore intimacy in your relationship?   In this episode, we welcome Dr. Mike Frazier MD, host of the "Strong Men, Strong Marriages" podcast, a renowned psychiatrist with expertise in neuroscience and relationship dynamics. Dr. Mike tackles marriage challenges, from expectations around sex to shared responsibilities, along with insights into therapy, offering guidance on finding the right support and dispelling common misconceptions.    Explore Dr. Mike's passion for his work as he vulnerably shares his own personal struggles within his marriage, along with strategies for overcoming relationship challenges. Delve into Dr. Mike's take on sayings like “Happy Wife, Happy Life” and his advice for happier marriages and learn how to nurture intimacy, set expectations, and communicate effectively in your relationships.    “If you can manage your mind better and your emotions better, that affects everything” - Dr. Mike   You'll leave this episode with…   Tips on how you can strengthen and return intimacy back into your relationship   What are different types of therapy   Dr. Mike's personal journey through marriage struggles, including a year without sex, and how he helps men facing similar struggles today   Dr. Mike's break down of his thoughts on the saying “Happy Wife, Happy Life”   What the mosquito cycle is and how you can change it for a healthier marriage   What it looks like when you treat your marriage transactionally   How do you deal with the expectations you have when you are the main provider for the family   The difference of being a partner or a contractor in your marriage   What you can do if you feel like you are doing too much or more than your partner   An understanding of why seemingly ‘unrelated things' are often connected in woman's minds   How you can make a good request to your partner   The three questions you should ask your partner when having hard discussions   -----   Leave a Review: If you enjoyed the show, please leave us an encouraging review and tell us why you loved the show. Remember to click ‘subscribe' so you get all of our latest episodes. https://ratethispodcast.com/man   What is the Manhood Experiment? It's a weekly podcast where we give you one experiment to level up your mind, career, business, health, relationships and more!   For more tips and behind the scenes, follow us on:
   Instagram @ManhoodExperiment   Tiktok @ManhoodExperiment   Threads @ManhoodExperiment   Submit your questions @ www.manhoodexperiment.com   Resources Mentioned:   Dr. Mike Frazier MD: https://mikefraziermd.com/   Leadership and Self-Deception: Getting Out of the Box - The Arbinger Institute   Influence, New and Expanded: The Psychology of Persuasion - Robert B. Cialdini 

In today's episode of Building Texas Business, Mike Vellano joins us to share the things he has learned from building his company, Vortex. As the CEO who has steered Vortex's innovative growth, Mike offers a look inside deal-making - including acquiring their significant European branch. Beyond mechanics, it's a celebration of relationships that drive success. We explore personal connections too - from Mike's early job to his passion for Tex-Mex. Plans for an Italian sabbatical link work ambitions with heritage. Mike's gratitude for support systems and understanding of sacrifice offers a holistic view of leading an expanding company. Join us for stories of commitment, strategy and groundedness through change. Mike's experience navigating Vortex's eventful voyage provides actionable insights for any enterprise. SHOW HIGHLIGHTS Mike shares his journey of strategic acquisitions and company growth, emphasizing the significance of people, process, and technology in building a successful enterprise. We explore the legacy of Mike's family in the water infrastructure industry and how this history has influenced his professional path and the founding of Vortex Companies. Chris discusses the challenges and motivations behind starting Vortex in 2015, drawing parallels with the entrepreneurial spirit exemplified by figures like Kobe Bryant. Mike reflects on the transformation from a hands-on CEO to a leader who empowers his team and how he leverages team strengths through delegation. We touch on the importance of maintaining a company's core values, with a focus on the "win as a team" philosophy and the role it plays in Vortex's culture. Mike provides insights on the integration process of new companies and people into Vortex's culture, emphasizing the value of internal sourcing for successful expansion. We discuss the recent acquisition of a foundational manufacturing company and the strategic considerations behind taking calculated risks in business. Mike expresses his personal love for Tex-Mex cuisine and his anticipation for a sabbatical in Italy, highlighting the balance between professional aspirations and personal heritage. Chris and Mike explore the intuitive and emotional aspects of business negotiations and the importance of emotional intelligence in steering deals to success. Mike details the complex nature of legal matters in international deals and the reliance on expert legal counsel to navigate antitrust issues and other legal challenges. LINKSShow Notes Previous Episodes About BoyarMiller About Vortex Companies GUESTS Mike VellanoAbout Mike TRANSCRIPT (AI transcript provided as supporting material and may contain errors) Chris: In this episode, you will meet Mike Vellano, founder and CEO of Vortex companies. Mike and his team have built Vortex through a series of strategic acquisitions since 2015, 17 in total by focusing on people, process and technology. Mike, I want to welcome you to building Texas business. Thanks for taking the time to join me. Mike: You got it, man. Great to be here. Chris: Yeah, let's kind of just dive right in. You're the CEO of Vortex companies. Let's start by just telling the audience you know what that company is and what it's known for. Mike: Vortex is a comprehensive and full portfolio of products and services that serve to repair wastewater and storm water assets, pipes, structures, anything that you, anything that lives in a municipal or a commercial industrial application, and we do it all non-intrusively and through what's called trenchless technology Okay yeah. So find a pipe without excavating. Chris: Okay. Mike: So we get to save the environment a little bit every day. Chris: That's good, I like that. So tell us like this how did you find your way into this industry? Mike: So I'm a, my background, I got a. I have a picture of my great grandfather, paul Villano, laying a water main. In 1925, Italian immigrant came to this country installing a water main and I mean just connected to New York. My family actually fixed that same water main years later. I'm not a finance guy or I don't have an MBA. I'm a. I founded Vortex and I've spent my entire career in this industry, other than you know, cooking in a restaurant and college or bagging a few groceries. I've been married to this trade and and moving that forward, my family was in. My family had a pipe supply business and a and it was in trench shoring and open excavation. My father was one of the pioneers in this industry. In 1994, brought a technology here with my family. The business was actually called trenchless technology and and I left the family business to go to go at a quick stop at a trenchless services business. And then you know the vision to get this thing going. Chris: Wow, so so in. I guess there's a lot of ways. It's kind of a continuation of the family business. Mike: Yeah, I mean, if you look at, my grandfather was using an old steam cable machine to lay a water main to. You know my family delivering and supplying, you know, frames and covers and fire hydrants One of the first water work suppliers in the country to my, my, my father and his brothers and my own goals, who are all entrepreneurs, watching them expand it and bring new technology to, and all of them you know, sort of systematically advancing with the industry, with water infrastructure and buried assets and now vortex is. You know we're an innovative, we're an, we do a lot of R&D and technology development and we and we get to do some really great things with in some pretty, some pretty tough areas. Chris: So it's interesting you mentioned innovation and and R&D. What are some of the things that that you do to kind of instill the innovative side or innovative spirit within the company? Mike: You know, I think, if I think about you know our core values, we have this, our most. Our core values are centered around this, this statement or mantra, called. You know we win big together and then, as we drive down into our culture, we have that. You know we actually have that trademark. It's a big part of what we do. We sign off emails on it and we rally around it if you go into an office, but winning big is thinking big. You know that's a big part of our marketing strategy and you know what we have, the way we think about products that work in infrastructure. You know they don't always come from the lab to the field. They come from the field and they're perfected in a lab and and we have a lot of great people that are that have dedicated their career to this industry, that we've worked with for a long time, and I think everybody at Port Texas is an innovator and I think our culture drives that. Chris: That's great, yeah, so it sounds like you kind of encourage it at all levels, not just some one department that's responsible for things. But are there any things that you do to kind of allow a you know a process for the ideas from the field to bubble up? How do you create that I guess in engagement or pride for people to speak up? Mike: You know, I think having, I think having what what you know Tretches technology is innovative in itself. I mean we're taking liners and pregnant with resin and going through a manhole and we're essentially performing angioplasty in a sewer application. You know we're. When we're coding a large, you know 16 foot diameter culvert with a geopolymer, I mean there's, those solutions don't aren't solved with a material only, it's a system. So it's people, process and technology. I think the way that we innovate and encourage innovation is really, is really through. You know, it can be reactive, it can be proactive. And we started our geopolymer business. We really focused on how the field you know the material led to really, you know, try a ton of innovation and perfection in the field. That was driven by most of our superintendents and most of our group people. So, you know, I think it's, I think it's a combination and it's a, it's chemistry. It's, it's, you know, some of it's organic chemistry, but there is engineering and there is some real science behind it. But there is an art form to what we do. You know there is. You cannot perfect this technology in a closed environment. You need to be able to deal with a lot. You know pipes are pipes of all kinds of construction and diameters, and and condition. Chris: Gotcha, Let me take you back down to the building. You said the beginning. You mentioned that you kind of, you know, broke off and started this. What year was that? Mike: So that would have been 2015. Chris: 15. So you know, that's, you know, the one of the questions I like to ask. You know, guys like you is okay, you had to take, get, I guess, the intestinal fortitude to be ready to kind of take that step, say, okay, I'm actually going to do this on my own. What? How did you know you were ready? What were some of the things looking back that you know, you thought, you know you go, man, that was really tough and had I known it would have been this tough, I may not have done it. Mike: Yeah, I think you know I read it when you're in, when you're in YPO or you're in some of the organizations that are involved. You get some really great case study. And you know there was an article about Kobe Bryant and somebody asked Kobe Bryant how you know how he thought, if he thought he would ever play in the NBA. He's like well, my dad played in the NBA. You know my dad was an entrepreneur, was a CEO. You know I saw that, I saw that spirit and all with all my uncles and my family and my grandfather was in great uncle brought products to this country and innovated waterworks supply and we're one of the first in the country to do something like that. So my risk profile was kind of set out at birth. I mean, I don't, seeing it being around it, seeing your name and the pride of having that on the side of a truck or the side of a building was always something that I wanted to either stay in the business or grow it or, you know, have something that I could, you know, not only make my family proud of but also build on my own and I think, the real working for my family and you know, working for my family, you know we always say we're. You know I would never change that experience. But working around entrepreneurs, and then the stop that I made before I started Vortex, you know, allowed me to get to, you know, the MBA I never wanted you know working around private equity and doing some things that allowed me to recognize, you know, the principles of business that some family businesses don't always fully capture, like HR or fleet management or things that are, or how you know how you manage a P and L and things that are typically, you know, you know, managed at a page in my Italian family, a patriarchal level or you know, however that comes to bear. So I think I always, you know well, my wife, I remember coming home and she had 12, you know Ford envelopes and was like, what are these? And you know I was like, well, we finance some trucks, it's gonna be okay. You know, we had no money in our bank account. So I think some of that stuff is easier for me just because I, you know, back to the like, I had somebody that modeled that risk profile and that ability to say, hey, just go take some chances and you create your own lock and you work. Chris: That's a great story. I mean, it's unique. You're right, cause it sounds like you almost didn't have a choice in it. Right, you were just brought up through from birth to understanding how entrepreneurs work, the risk about it. But yeah, as you were talking, and that has to put a whole another level of pressure on you to say, yeah, this is what I, this is what we do in the Vellano family, I better make sure it works, cause everyone else looking back uncles, grandfathers, and they've all made it work. I better make this work. Mike: That has to be a whole another level of pressure 100%, and I think that's, I think that's, I think that is a driving principle, right, you know people, you can harness pressure and use it, use it effectively. I mean, sometimes I probably run myself a little too hard, but I still am afraid to be late to work. Chris: You know it's just. I love that, yeah, Especially in today's world right that the CEO is afraid to be late and there's that level of kind of accountability that, as a leader, you want to instill in everybody in the organization. Mike: What are some of the? Chris: things you do to kind of to do that to demonstrate that. I mean, obviously it starts with you know your actions, but how do you try to show up as a leader to make sure that those values that you grew up with get infiltrated throughout the organization? Mike: Yeah, I remember going to the first management meeting with my family and and you know we it was you know if you've been around Italians, you, there's always a big dinner. There's always, you know, more appetizers than entrees on the table, good wine, or you know definitely a few cocktails, and you know you sit there and you go and you run and you come in town for a meeting and I remember one of our branch managers. I remember one of our branch managers showing up late and my dad locking the door and just saying you know it's like that, it's two minutes late, it's like, well, we, this meeting started at seven o'clock. Chris: And. Mike: I've done that a couple of times. I'm mellowing out a little bit, but when I first started this business, I mean as a 30 year old guy, you know you're trying to prove a point and you realize that there's better way. You know that's not always the way to do it, but that left a left an impression on me. I think you know there are definitely times where I get overextended and I got to move calls around and I'm not. I want to be, I want to be more, but this is. It's a big job with a lot of responsibility and you got to prioritize differently. But I think you know our core values are you know we are a driven business and I think them seeing drive and our organization, not only by me but our team I mean our team is our team is a very close group, you know. I think, like when we went through our core values, we talked about how teams win. You know families fight, you know, so I don't know he's, I don't. You know we think like a family, but we work like a team, we are a team and I think them knowing that we are going to win as a team, I think that, look, they know that we're all going to get up, that nobody's, that nobody's fallen behind. And I think in a dynamic organization where that's a driven organization and you know they see every time they walk into a vortex office, they see that core value, that we are driven and core values are for all. You know some people. We stand behind ours and I think some of them have become a little cliche, but I think that's how we keep people. I think that's how we, I think that's a model that comes from our entire executive team down to the organization. Chris: Yeah, I would say you know we talk about core values. In my experience, if you identify them right, then their behavioral characteristics for the behavior that you, as the organization, want to see, expect to see and, almost you know, demand to see for those that are going to be successful in your organization. So you talk a lot about I love the win as a team and I thought it was. I love that family's fight, tim's win, that's a good one. I'm going to use that again. But so culture is definitely important to any organization. It sounds like it is. You know, obviously, the years with that. The win is a team mantra. To me, that, then, means that the hiring and onboarding process is critical to making sure you're getting the right people during the interview process as you're bringing them on and building the team. So what are some of the things that you do at Vortex to try to make sure you've got the right processes in place in the interview and integration process to add successful members to the team? Mike: I think that onboarding is we have a great HR person that our onboarding process when we first started was where are we going to grab a steak and or where are we going to where's a fun place to do an interview? And we've created some. We Brooke has really helped us to kind of formalize some things, and our head of shared services that to make that first day you know that first day at Vortex their most exciting day. I mean that's it's got to be. You know whether it's the type of swag they get or their waybook or whatever those things are, and what their introduction is to an organ, to a high performing organization. I mean they got to feel like they're. They got to feel that fire and that encouragement day one. You know when you I think that if you build your team right and you build your people, you know we have we've acquired seven companies now and our executive team has leadership across our. The leadership across our organization has come from those, has come from those transactions and those folks have moved into pretty dynamic roles in the industry and high level leadership roles. So we've brought some of their core values along. But we've also elevated several members of you know an acquisition we did in Maine. We have three people running different parts of the business. Our COO has mentored several you know of our younger project managers and sales people into roles that they're going to be leaders in our business and we have the same role, you know, for me as sort of a sales support role that you know young men that started around 22,. One of those guys is is our is our senior VP of services now and he's been with us for 12 or 14 years through through other places. So I think we focus on you know it is and it's a sink or swim Sometimes. We're, sometimes, you know we do, we do sort of police, our crew and we in our drive gets in the way of really understanding, you know, and wanting everybody to be more than maybe they should be in some cases. But yeah, I think that was a bit of a ramble but hopefully I answered your question. Chris: Oh yeah, you did. I think it's great. So I think in that answer that you hit on a couple of things. I want to follow up on 17 acquisitions in that sense 25. Yeah, that's small to large. Mike: I know that we've got. Chris: Yeah, I know that you recently just closed a new transaction, so it sounds like there's been a lot of growth through acquisition. You know entrepreneurs or people that started a company. I mean, they're faced with right. You know, how do I grow? Do I grow organically? Do I grow by acquisition? Do I do both When's it right? So what are some of the things I guess that you could share from your kind of strategic thinking about? When you felt it was right to make those acquisitions, how did you vet that to go Okay, this is a good fit, knowing, I guess, there's no sure bet. Mike: Yeah, we're acquiring businesses from $1 million to $15 million, $20 million in revenue. So a few of these businesses have been smaller tuck-ins or technologies or somebody would call them an asset deal. So we have, and in all of those we've never hired a banker. We source them internally because we either have a customer or a vendor relationship and we have some, you know, we have some. We have a little bit of a matrix that we use, you know, in the sense of do they have personnel, do they have technologies? Do they support the things that we do? Are there people innovative enough to expand? And, if we can, we add value. We're not going to buy a business that we can't grow organically or turn into something that is truly going to make that business better and make our business better. Sometimes it's technology and sometimes it's, but there is always an organic element to everything that we do. We start, we add crews every day, we implement our technology developments into our own service businesses or into others, and a true differentiator in how we go to market, like how we go to market up until, you know, up until this recent transaction, which is a products company, a subsidiary that I can get into that in a few minutes, but this is the biggest, the largest acquisition we've made and most of our product product's business was developed and grown organically through some smaller, what I would call partnering opportunities with. You know our we're so proud of our business out in Utah. You know we grew we've grown that business, by you know, 20 times from when we acquired it in 2019 with because we had a partner that understood the chemistry, we understood the operation and commercial side and together, you know, great products and a great strategy work. So that's been our that's been a big part of our strategy. And the product side and the services side. We've bought, we've acquired some mature businesses. We just bought a business in the UK that we're really excited about. It's got an excellent market opportunity. It's a service business but they fit our DNA. They're not afraid to do, they're not afraid to go out and work with difficult customers or on difficult projects or take on emergency jobs. You know we live in a municipal world and we really do focus on selling. We go after negotiated work. We don't just go and low bid work like a lot of, like a lot of municipal contractors do or have to do because they don't have the resources or you know, or some of the, I think, some of the talent that we have to go utilize procurement networks or emergency contracts. So it's a steady diet of acquiring to build on or just doing it. Crash roots organic. Chris: Gotcha. So then the next question comes. You do all these acquisitions. Acquisitions sound great and sexy and you go close a deal, but it will only be successful if you are successful in the integration process. You've done 17 of different sizes. It sounds like you've gotten pretty good at the integration process, so I want to talk to you about something about that. It's clearly not happening by accident if you're good at it. So what are some of the tips that you could share about what you all have done there? Processes you've you've developed I'm sure they haven't all been successful. You've learned from some failures, so talk to us about that, tell us kind of you know how that's evolved and vortex for you and your team to make sure you get the integration piece right. Mike: I think we start integrating a deal before it's close and I think that's important and I don't think that's a strategy that can be. I don't know if you can hear that thing. No. I don't know, and I don't know if that's a. I don't think that's a strategy that everyone has a luxury to support. But part of our story is we you know we've never hired as never hiring a banker. We are very familiar with the acquisitions that we're going in to make, both from a personnel perspective, the technologies that they support and how they think about the world. You know we want, we want these acquisitions to be as excited to join vortex as we are to acquire them. The other thing that we don't do is buy 100% of anything. We're typically partnering with a seller that is going to come into the business and continue on and you know we and we do a really good job of I think of, of, you know creating the right level of support. I'm understanding what their skill sets are. At a seller come to me after we bought his company and say you know what? I always wanted to own a business but I never wanted to run it and I'm like, ok, I can see that, but let's put you in a row Like you're. We're here because you were doing something right. Let's figure out what you're, what you're good at, and I think you know our team collectively took a step back and was like this is what he's going to be good at. And he's been and he's one of our best in that role now and still a shareholder and had enough, you know, hasn't had enough. You know really believed in what we're doing and we believed in him as a in this role. And you know, some of a big part of integration is understanding what everyone's thinking and being transparent and saying, hey, you know, you really think of this. We need to get you a better finance person, like, yeah, your finance person can go do this. We were not here to. I think our real focus is finding people and the other thing that we don't do and we buy cash flowing businesses that have good roots and have good people and we don't buy distress businesses. We're not. That's just not who we are and that's understanding what your capabilities are. We don't we want to manage. We want to manage a business, to grow it and build it and make it better. Like I said before, and I think when you really look at, hey, this is we. As you grow up, you know you you'll learn to deal with those, with those situations. We just had a business in Colorado. We kind of took a step back and said, hey, we're breaking even here. This is a distraction. Maybe we can move these assets somewhere else and focus somewhere else, and this is how we can do it in a way that will be more productive for everybody involved. So I think a big part of integration is really understanding our deals and having that luxury by being part of the diligence and really, I think, starting integration almost before we close an acquisition. Chris: Yeah Well, and what I hear you saying is look, we're very thoughtful about and transparent about, the process, and those are two key elements, I think, to anything being successful, so that you're going to have clear communication. It also sounds like you take a little page out to Jim Collins. Good to grade and get the right people on the bus, but in the right seat. Yeah, so your example of I want to always want to be an owner but didn't want to run it. I mean, that's someone that probably should be on the bus, obviously, but he just sure on the right seat for him. Mike: Yeah, I think that you know it's the my father would call it the Holy Trinity. It's sales, operations and finance. But you know, good to grade is that's one of the one of the big takeaways of our best businesses have just really a really great balance across all three of those, all three of those areas. I mean anything R&D, anything project level, any good strategy you have to have. You have to hit all three of those areas. Chris: Yeah for sure. So you talked about a lot of these acquisitions being either vendors or partners that you have interacted with over time, so that, in first to me, you do a really good job at Vortex of creating some really strong relationships with your vendors and your business partners, you know. So let's talk about that a little bit. What are some of the things you do to create in your people, I guess, to foster those strong relationships that sometimes lead to these add-on acquisitions and then become part of the team? Any tricks or things there you really encourage? Mike: You know, like you made the comment about innovation before, I mean, I think everybody, you know we have 800 employees now I think we have, you know, 700 of them want to be investment bankers. I get calls all the time hey, you know someone's supposed to say, hey, you know this, hey, I bet you that guy would be really interesting in this. You know, in this role, or we should look at this company, you know, I think, and some of those have come to fruition and look, you know, when you do that many deals in that time frame which seems like a lot more than it is, but you know, we did four deals in COVID which was crazy, but we all. But you know, I think a big part of that is, yeah, I think a big part of that is like we look at hundreds of deals. I mean we still we can turn. You know we're not going to get into early diligence, like there's some things I can look at with our team and our team is in the higher, you know, the more involved, the more involved we get with a company. You know we might say you know, man, these guys don't even know how to order material, like they're unorganized. You know, I know they want to put like, this is going to be a lot to fix. You know, maybe we can work together and there's been times where we've worked with companies for years three, three, four years talking to them about a deal why we work parallel as a vendor or them as a sub, and then they become an acquisition and I think we both got better together. You know, I think that some of that stuff is, but there's that's also a luxury in our strategy, but that makes our strategy sound. You know, we don't have, we don't have. You know you look at these deals and, hey, if half of them went well with our organic growth which is in the, you know just, you know, high teens to 20s and archaegers in the mid 20s, you know it's, it's, it's a great story. I think that was like as you should be. Chris: So let's talk a little bit. You know more about you and your evolution as a leader. You talked about it a minute ago, just referencing how you're probably a little bit harder, maybe from the lessons you saw from your dad, and you've evolved. Let's dig in a little bit there. I mean, how would you describe your leadership style and how do you think that's changed and evolved? You know, over the year since you started, you know this company and and grown into where it is today. Mike: You know, I think I know what. I know what I'm not, and I know that I have to surround myself with people, that that you know you backfill by weakness, I think. But at the same time, you know our CFO has been my partner for 15 years. You know we've had, we have, people in our team that we've worked with for all of that and even before and even going back 20 years in some cases. You know, I think leadership, I think leadership to me has evolved as I've really understood what I'm truly. You know what I'm really good at, and having enough and sort of having, I think, being intentional and having some humility and being able to say, hey, I'm not. You know, I believe in the one minute manager. I don't need a ton of detail, I just need to know what the issue is, and I've gotten better at really understanding why that detail is important. But, you know, as a leader, I think I think gaining perspective and, you know, and listening, I think YPO has helped me in a lot of ways. You know, being around, I think peer development is something that I never, you know that was not. That's not something. When you grow up in an Italian family in upstate New York. You know, therapy isn't something I don't know. That's something I don't know if I knew the meaning of that word until I was married, but those aren't things you don't. You don't share family business, you don't share problems, you fix problems, and I think that I think in some ways, that's good. Like you know, you, I know I want to run into I think I wanted to run into every birding building until about five years ago, and I've started to realize that there's people on my team that are better at things, that I am and can execute on that. So you know, I think it's a combination of a lot of things, but I do enjoy leading, I do love my team and I, like I love seeing them be successful. Chris: That's good. Now I think it like I said, I think it starts with drive right. You have to have the drive and the want to and a little bit of that risk profile to take the risk and then I think over time you learn maybe humility and empathy and you can let others do some things, backfill where you're not as strong or you want to provide the opportunity. Mike: But yeah, don't get me wrong, the New York Italian does come out. Chris: It's a, it's a, it's real, so Well let's, we'll test that a little bit then on, say, because I like to ask people you know I'm a big believer in I think we learned from failure right and so you know, is there a situation or decision or circumstance you can think of? You know there was a failure right when you got it wrong but you were able to recover, or what you learned from that moving forward that made you better, stronger, you know, leader person, whatever that might be. Mike: I think I, you know, I probably learned from the. You know we always say when we're hiring somebody from a competitor, you know, try to be a good leaver. I think the way I left my family business was probably not. You know it was. It's never perfect. If I could go back, I'd probably. I would probably do that differently. You know there's been technologies that I have died on a hill for and you take a step back and you go. God, why did I think that would work? You know, and I think now I'm having conversations that people had with me 15 years ago where they're going, hey, what do we should buy this company? I'm like, yeah, you know, I'm like that's not the right company for us. And I think there's times where I know I need to explain myself better. But yeah, I think there's been some. We've had. I've had some bad partners that we had to buy out and that I separated with. And you know, when you're going through that, you realize like, hey, this is this guy you know you want to. It's all there fall in your point, in the finger and then, as you get past it, you go you know, I didn't. I perspective in time or time changes perspective, right? So, yeah, I think there's certainly no shortage of things that I failed at. Chris: Gotcha. Well, the other thing I kind of like to ask people as we start to wrap things up is if you think about one or two things you would impart to an aspiring entrepreneur that if you're going to, if you're going to go chase that dream, you know here, here are a thing or two that I think you should keep in mind. Or you know, maybe it's a do this and don't do that type of thing. What would be that from you to kind of that generation, next generation of entrepreneurs, man? Mike: I love move fast and break stuff. Like and you know what, like when I did at first, I felt like we're breaking more. I feel like we've got a point where we move, really we move. It seems like we're moving at the speed of light. But, you know, something like this deal we just closed, which is, you know, sort of a dream deal for us. It's a company that is a founding, founding manufacturer business. It's applied, they're one of the largest producers of liners in our space and have go back to Eric Wood, who's really a, you know, a founding father of our business, who started a company called in situ forum. And you know, we've been working on this deal for two and a half years. I mean, this isn't like 90 days. We got the book and we hired a big New York law firm and our banker, you know, handled all the conversation and we walked in the door. You know, and and here are the changes we're going to make Like we have a foundational, innovative, pioneering you know industry giant that we want to, that we want to take to what we want to take into into its next evolution. And, you know, I think about moving fast. I think move fast breaks, but maybe we're not moving as fast as you think, but at the same time, you know, I do think that's a big you know. Don't be afraid to risk. Focus on what you're good at, because in the minute don't get distracted from that. And and don't be afraid to partner with. You know you're going to have some bad partners, but you'll. You'll if you can find the right ones. That means all the difference. Chris: That's really good. You know the point to that. Similar with employees. Right, you're going to make some bad decisions, whether it's a partner or personnel. But once you realize that move fast, right to cut, because a bad employee can be kind of road culture or a bad partner obviously can run a business down. But once you know that definitely want to move fast. Mike: Yeah, I mean, if it seems like it's, if it seems like you know, bring a raincoat. If it's raining every day, you know we're sitting here having these calls, like this business every month is having the same challenges. Okay, you know, you know, and we've gotten the point of my partners and the key leadership on our team where we look each other in the eye and we go all right, it's, yeah, we need to make a change, and then you know that that leads to some quick and real thoughtful action. So, yeah, I totally agree. Chris: Well, let's appreciate all that. I mean, I think, your success at more taxing your teams I know it takes more than just you in seven acquisitions since 2015 and the growth up to 800 employees is is anything. It's very impressive. So congrats on all that and the new acquisition. I want to ask a few personal questions Just before as we wrap up. What was your first? Mike: job. That was actually my icebreaker at my integration meeting today. My first job was was that Vellano brothers? Chris: Okay, doing what? Mike: I was I think it was counting T bolts because I wasn't old enough to drive a forklift or picking up paper. I remember my father told me hey, I want you to go pick up paper on the. I'll give you a dollar for every piece of paper you pick up on the warehouse floor. And I went out and I had like a grocery bag. I'm like gotta be $1,000. I think you gave me here's five bucks. Go buy a soda. That's right. That was my first job. Chris: Okay, I know you're from New York. You've been in Texas a while, so I ask all my guests do you prefer Tex max or barbecue? Mike: I prefer Tex max. Okay, I like a margarita. I like some good, proper cheats, texas cheese enchiladas and a good margarita. Chris: That's right. So last question is if you could take a 30 day sabbatical, where would you go? What would you do? Italy, no hesitation there. Mike: Anywhere in Italy would be okay. Chris: Very good, very good. Well, that's a popular answer, by the way, but I guess you have family ties that would make it your answer to you. Mike: So some family ties, and I like one that makes both of us. Chris: So, mike, thanks again for taking the time to come on the podcast. Really enjoyed getting to know you and hear your story and wish you nothing but the best of success in 2024. Mike: Awesome. Thank you for your time. I appreciate it. Chris: All right, we're going to end the recording there. Special Guest: Mike Vellano.

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  

Unlock the secrets to sales success with Mike Bauer of Delta Defense, the genius behind a team that turned into a $135 million sales powerhouse. In our latest episode, discover how 'humble confidence' propelled his team to unparalleled heights during the 2020 firearms boom. Mike's insights aren't just about seizing opportunities; they're a goldmine for anyone aiming to master the art of sales with finesse.Join Mike and me as we explore the power of communication in sales. Learn how silence can speak louder than words and gain actionable strategies to connect with clients and captivate audiences. Our conversation is your playbook for refining your pitch, potentially your ticket to sales stardom. From storytelling to adaptability, we provide a battle-tested framework—memorize, internalize, customize—ensuring you're armed to forge genuine connections and achieve professional triumphs. Join us for a dialogue that's enlightening and invigorating, as we prepare you not just to meet the bar but to raise it.NOTABLE QUOTES"When we position ourselves well enough amplified by opportunity, when the wave [comes], we are ready to surf and surf." – Mike"Someone who is humbly confident is, you are confident in the things that you have done and that you're doing, but you're humble enough to understand that you always have the opportunity to learn more." – Mike"Being humble is knowing that there's more ahead, that they aren't the cream of the crop, they're not at the very top of the peak." – Philip“I'm a huge advocate of lifelong learning." – Mike"If someone was calling us and it was the wrong phone number, we trained our team to take care of that call and have the one-call resolution, even if that person is calling about something that has nothing to do with our company. So we were truly customer service based first to get those sales." – Mike"Most often, it is not advantageous to correct someone in their line of thinking right out of the gate. I think that it's much better to go down the line." – Mike“We share what we share and if we happen to forget something, if it's not that important, don't worry about it." – Philip"Focus first on what's the main priority at hand . . . and then we can figure out the rest." – Mike"Just focus on them. This is exactly how it is when we go to speak somewhere, and that's where sales and public speaking go hand in hand." – Philip“Getting the local presence, being in person with people, is such a big deal.” – Philip“If you can find a way to talk about something that interests [potential clients] outside of the business that's at hand, you will often be surprised at how much more friendly the business at hand gets.” – Mike“As salespeople, if we can have a conversation around something that interests them? Know, like, and trust—that's going to skyrocket because they're going to see that this guy isn't just here as a salesperson, he's a true human being.” – Mike“You've got to start with being insanely curious.” – Mike“It's not necessarily that you have to have a bunch of facts that you rely on. It just comes down to you naturally having to be curious, with the understanding that eventually you have to get down to business.” – Mike “Have constant curiosity and strive to continue to get better.” – MikeRESOURCESMikeFacebook: https://www.facebook.com/mike.h.bauer.7/Instagram: https://www.instagram.com/mike.f.bauer/LinkedIn: https://www.linkedin.com/in/mike-f-bauer-293a1b7b/PhilipDigital Course: https://www.speakingsessions.com/digital-courseInstagram: https://www.instagram.com/iamphilipsessions/?hl=enTikTok: https://www.tiktok.com/@philipsessionsLinkedin: https://www.linkedin.com/in/philip-sessions-b2986563/Facebook: https://www.facebook.com/therealphilipsessions Support the Show.

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

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

The Nightly Rant: Examining Society from a Sarcastic Point of ViewWelcome to The Nightly Rant with your hosts, Mike and Torya. In this show, we take a sarcastic look at society and dive into various topics that provoke thought and conversation. Today, we want to discuss the importance of being prepared to answer questions and engage in meaningful dialogue when expressing opinions publicly. We believe that adults should act like adults and handle disagreements in a mature manner. Let's dive into the details.The Importance of Being Prepared to Answer QuestionsMike: "People in general should not ask questions unless they're prepared to hear any answer."Torya: "I'll take what you just said a step further to also include stating your opinion publicly."Mike: "If you state your opinion publicly on something, then you need to be prepared to answer questions. Some people are going to be confused. You put your opinion out there that that is influential. It means something. I don't care who you are. Be prepared to answer questions. That's all."Torya: "Influences at least one other person in the universe."Mike: "Be prepared to answer questions. That's all I ask. Is that bad?"Torya: "Well, why are you asking the questions? What is your goal with asking people questions about their opinion?"Mike: "To understand their viewpoint on that issue."Torya: "Because to tell them how wrong they are, that they're dumb or that their opinion is factually incorrect. No."Mike: "In fact, usually these days especially, I will say your opinion is completely valid, but it isn't one that I completely connect with because I don't understand this aspect."Torya: "Can you explain?"Mike: "And I blah out whatever the question is and then they attempt to answer. And most times they don't even attempt to answer because they don't care. They just growl back at you and it's like, no, I was truly trying to understand where you were coming from."Torya: "Well, then if your intention is to just understand, then no, right, you're not doing it."Mike: "Exactly. There's nothing wrong. Then if your intention is to drag them into a trap and then pounce on them and beat them to death, yeah, that'd be pretty wrong."Torya: "Or at least mean and nasty. Sure."Mike: "Which is wrong sometimes."Torya: "It's fun."Mike: "Well, it can be. I won't lie. It can be."Torya: "Especially when somebody has a really obscure and ridiculous and factually incorrect opinion. But I digress. We don't need to go there."Mike: "Well, yeah, and why can't somebody disagree about that and not make it be such a big deal? I mean, come on, there's a specific incident in my head. And the minuteness of the topic, if you think about it, it was like less than a grain of sand in the grand scheme of everything. Right?"Torya: "We've had disagreements about things way bigger than that and laughing together about something else five minutes later."Mike: "Nothing. That's like nothing. That's what I'm trying to say. It's like fart dust is a bigger deal than that."Torya: "Fart dust pretty bad."Mike: "And yet people who are allegedly adults make it about them and, oh, we need to quit the friendship. And that's literally what people do these days. Grown ass adults. Yeah, grown ass adults just leave and don't talk to you anymore. They don't have even the balls to say I'm going to say it. They don't even have the balls to say, fuck you."Torya: "Wow."Mike: "They don't have the balls to say anything. Yes, I'm the reason never for the E. It's always all me now."Torya: "I feel like all the words are fair game, though."Mike: "But here's the thing. It's like they don't even have the guts to say goodbye. Like, I'm done. That's a woosy move. I mean, those are the kind of people that disappear from their family, too. No BS."Torya: "I had a great idea earlier, and I was thinking that it'd be great if society would just chew up these cocksucking assholes that you're describing. The people who don't function as part of society because they're just too fucking wrapped up in their own self."Mike: "Yeah, they're not adults."Torya: "Chew them up and shit them out into outer space using the Earth's giant rectum. Yeah, the Earth is going to grow a giant rectum and it needs to shit these people out because they're destroying."Mike: "The world in many senses of the word. Yes, they are."Torya: "They're the most hostile people yet. They're the people who will call everybody else hostile."Mike: "Well, and that's the thing. There's also this issue where adults can't be adults, they just can't. Like we're talking about ghosting. That's not an adult move ghosting people. That's a little baby's move ghosting."Torya: "Right?"Mike: "Oh, where's your friend Johnny? Oh, I don't talk to him anymore. It's what the little kid does. It's not what an adult does."Torya: "Right? And then there's poor Johnny crying in the corner because he doesn't know why nobody likes him. And also, Johnny will continue to be an asshole for the rest of Johnny's life because nobody has ever told him why they don't like him."Mike: "Which in my opinion, makes you the asshole for not pointing it out to."Torya: "Right?"Mike: "I mean, if you pointed it out to him and he continued down the pathway, you pointed it out to him, and he gets to continue down the pathway if he wants to."Torya: "You've got to tell people how you feel about things. You know, it's interesting. When I was in Canada, not this most recent time, but the time before, I was hanging out with Alicia, you know, how she has miniature humans. Well, the boy miniature human punched the girl miniature human. And she came screaming and crying to know kid stuff. And Alicia told her that she needed to go tell miniature boy human how it made her feel so that he would apologize to her."Torya: "And she did, and he apologized and then gave her a hug for a five year old. People. Yeah."Mike: "And, you know, the honest truth is there's no reason whatsoever for someone to act like everything's okay when there are obvious signals that everything's not okay. There's no reason for anyone to ever do that to anybody. That's why I think we talk about this all the time. Our relationship works because it takes you longer than me to get there, but we tell each other, well, that bothered me when that happened."Torya: "I know that I'm a little bit irrational. Okay, whatever. Don't even no commenting. Not allowed. I know that I'm a little bit irrational, and sometimes I don't know if I'm actually annoyed with you or if I'm being crazy, so I need to take some time to decipher if I am being crazy."Mike: "But see but that's fair, because that means instead of that even makes you even more reasonable, because instead of putting our relationship through a roller coaster of crap, you take the time to filter it yourself. Here's the thing, though. A lot of people would bitch at you for that, but I commend you for it because you still come forward with the issues to get them fixed. You do."Torya: "Think about it. You're doing something that's annoying the crap out of me. Okay? Not right now. This is a hypothetical you are doing something."Mike: "Well, it's a hypothetical reality. It happens."Torya: "Yeah. Anyway, I could say something right then when you're being annoying in my hyper irrational, super annoyed state sure. Which what would happen bad?"Mike: "Let's just say doom would ensue. Always."Torya: "Nobody needs that. Or I could stew quietly about it for a while and then come back to it the next day when I'm not crazy and decide if I was actually annoyed with you or not. And then if I was actually annoyed with you, I could be like, hey, Mike, you did this thing. Please don't do it again."Mike: "Yeah, you know what, though? I obviously am the same person as you, so I obviously approach that same issue the same way as you. And I, though, have one time only with you, followed the completely reasonable give her the benefit of the doubt approach. I've done it many times, but once and only once did I regret doing it. And you did something like you had a comeback of, like I can't even remember the comeback at this point, but it was really this really sassy."Mike: "You didn't deny that you were doing something and that it would have bothered me. You didn't deny that at all, but instead, you just sort of ignored it. And went like, what about this? And it's like, wow, man. That isn't how we're supposed to deal with each other. We're supposed to deal with each other face to face. We're not supposed to try to duck around one another. And that's how that felt, right?"Torya: "That's why everybody has to calm down."Mike: "Before issues should be correct. Exactly. And that's the thing. There's times when you do have to wait, and there's times when you should take time to think about things. And I think it's more adult like to wait rather than overreact. However, I'm going to say something different. One last thing. When someone does that to you, they overreact. It's best for you to quote overreact back and protect yourself. That's what I think."Torya: "Well, because then you're going to get the whole thing out of the way right there, instead of you stewing that I overreacted and making it a fight the next day and the next day and the next day. I agree with you. If one person has already gone off the deep end, well, you might as well just have the knockdown drag out fight right there. Just get it over with."Mike: "I think we agree with that. And it's healthy. That's the healthy way, and we act like adults about it. And that's the key thing, though. You have to be able to speak your mind without the other person getting offended. And honestly, I think a big thing that most adults just don't have any longer than they used to is the ability to separate things. Just because you're not the most empathetic person in the world doesn't mean you're stupid."Torya: "Yeah, that's what I was going to say before you had something else you had to say. If I didn't take the minute to calm myself or minute or hours or six days, whatever is necessary, all arguments I had with everybody would be like, my fight with the Sam's Club lady where I called them an idiot and."Mike: "They walked wasn't your that wasn't your proudest moment."Torya: "Or my slight disagreement with that soccer mom that one time that I won't repeat."Mike: "Well, what's funny about that? What's funny about that is both of those situations turned out okay in the end, but they had the potential not to be. But here's the thing. By us being reasonable people, 98.5% of the time, you can get away with a slip up like that. And the rest of the people are like, in particular the soccer incident, the rest of the parents, you were like, oh, I'm so sorry that I said that in front of you. And they're like, Are you kidding? I would have said worse to her."Mike: "She deserved what you said. And then everyone that was literally the opinion, they would be like, oh, I would have said worse. Oh, she deserved it. Blah, blah, blah, blah, blah. Not a single person sided with the other person. Not a single person. Now, here's the thing. We still set our apologies to everybody. To everybody."Torya: "Even though the person I exactly."Mike: "Even though they supported us, we still apologized."Torya: "Well, to be fair, I used the worst word Americans can possibly use in front of, like, twelve year old children."Mike: "Well, the twelve year old children were out on the soccer field, though Mitchell."Torya: "Said he heard it."Mike: "Well, it is what it is, man. You did apologize."Torya: "That's what I felt like I needed."Mike: "But you apologized. You did. To everybody. And yet the point is, they were supportive of us because we had always been reasonable people. We didn't yell and scream on the sidelines at our kid, at the referee. We didn't do that stuff. We sat there and we cheered."Torya: "Called the referee a sight."Mike: "We talked to each other."Torya: "Only when you were egged on by other groups of people, though, too."Mike: "We would just talk to each other and ignore everyone. I mean, that's just how we handled things. And so it's sort of annoying that people go down roads that they don't even bother to think about. Well, are they the type of people that would act that way?"Torya: "So can we officially shoot these hyper aggressive snowflake motherfuckers into space?"Mike: "Yeah, with the rectum. You said this already, and I fully agree with you."Torya: "Well, I need to know if other."Mike: "People."Torya: "We're not feeding the rectum Taco Bell."Mike: "Both of those are going to make a great audio club. That's the little shorty. Munch, munch, munch kapow. And we're not feeding in Taco Bell. Just wow. All right. Well, I think we have beaten this topic to death. I didn't even expect us to talk about it for this long. Here's what I want to kind of close up with tomorrow, which is the day after we record this, which will be weeks from the time you hear it."Mike: "We are getting involved in our very first official sporting event together. We are going to play fantasy hockey with the rest of our family. Yes, we are. And we suck at this for my ultimate failure. We're going to learn this quick. So with that, that is all I've got for you people."Torya: "Good night, everyone."Mike: "Hasta La Bye bye."Conclusion and Future OutlookIn this episode of The Nightly Rant, Mike and Torya discuss the importance of being prepared to answer questions and engage in meaningful dialogue when expressing opinions publicly. They emphasize the need for adults to act like adults and handle disagreements in a mature manner. The hosts share personal anecdotes and observations to highlight the negative consequences of ghosting and avoiding confrontation.The conversation delves into the significance of open communication and the ability to separate personal opinions from personal attacks. Mike and Torya stress the importance of understanding different viewpoints and seeking clarification rather than resorting to hostility. They also touch upon the need for self-reflection and taking the time to assess one's own emotions before engaging in discussions.The hosts conclude the episode by announcing their participation in a fantasy hockey league, highlighting the importance of learning new skills and embracing new experiences. They encourage listeners to approach disagreements with maturity and respect, fostering a culture of open dialogue and understanding.Moving forward, it is crucial for individuals to recognize the impact of their words and opinions on others. By being prepared to answer questions and engage in meaningful conversations, adults can foster a more inclusive and understanding society. The Nightly Rant serves as a reminder that communication is key, and it is essential to approach disagreements with empathy and respect.TimestampSummary0:00:15Introduction to the podcast and topic of the day0:01:34Importance of being prepared to answer questions when stating opinions0:03:34Adults making small disagreements a big deal0:05:02Criticism of people who ghost others without explanation0:06:34Society's inability to handle conflicts maturely0:08:25The importance of open communication in a relationship0:09:11The need to address issues face-to-face rather than avoiding them0:10:59Reacting to overreactions to protect oneself0:11:46Having a knockdown drag out fight to resolve conflicts0:12:22Lack of empathy and offense to criticism of empathy0:12:27Torya talks about needing time to calm herself before arguments0:12:55C mentions the soccer incident and how it turned out okay0:13:48They discuss apologizing to everyone involved in the incident0:14:11C talks about how they were always reasonable people0:14:54Torya suggests shooting hyper aggressive people into space0:15:55They mention their upcoming fantasy hockey event0:16:22Closing remarks0:31:54Mike thanks listeners and asks for a rating0:32:06End of transcript

Upcoming Event!How Can Mindfulness Help You Reach Financial Independence?Do you want to reduce money anxiety, but don't know who to trust?Would you like to learn how to set up and manage your own retirement plan?Do you want to know how we create a passive income stream you can't outlive?If yes, join us and learn how to answer the 4 critical financial independence questions:Am I on track for financial independence?What do I need to do to get on track?How do I design a mindful investing portfolio?How do I manage that portfolio and my income over time through changing markets?Learn more: https://courses.mindful.money/financial-independence-bootcampMike Van Pelt is an entrepreneur, author, speaker, and men's life coach leader. He is the founder of True Man Life Coaching and host of the popular men's podcast, True Man Podcast. Mike has served in leadership roles for most of his career, bringing over two decades of engagement and expertise in account management, consulting, and leadership development.Today, Mike joins the show to discuss the work he's doing mentoring and guiding men, what positive masculinity means to him, and why so many men tie their identities to their careers.

Ahead of the release of the J.D. Power 2023 North America Airport Satisfaction Study, listen to Michael Taylor, Practice Lead for J.D. Power Travel & Hospitality Intelligence explain optimal airport designs. What makes a great airport? Some insights from Mike: -- You can design an airport for planes or for people/ -- A little distraction for the eye can be a good thing, especially in high traffic areas, like restrooms. -- Signage and layout are important to the overall experience. -- The approach to your airport should also be considered. Does a traveler have plenty of time to get in the correct lane, find the right parking garage, etc.? Listen to the full podcast for more insights!

“As leaders no matter what business you're in, you're tasked with leading change,” states host Mike Horne. The first half of season four has contained many valuable insights from guests who are leaders of authentic change within their organizations. They have spoken on everything from coaching to organizational and executive development, diversity and inclusion, and the ai revolution. Today, Mike shares highlights from episodes 72 through 85 for this special 100th episode celebration of Authentic Change Podcast.  In Episode 72, Dr. Toby Travis spoke on the high level of trust required for those in leadership roles to be successful. Then, in Episode 73, Dr. Beth Banks shared her tips for architecting change on an organizational level. In Episode 74, John Sanders talked about the importance of building and optimizing teams, explaining that leaders regardless of industry are always responsible for leading change. Next, Gina Riley spoke about the importance of networking and the types of things that can derail the career transition process. In Episode 76, Kevin Palmieri shared his love of podcasting, discovered while falling out of love with his corporate career. In 77, Fatima Mirza discussed AI and the tools that are available to assist people with their job searches. Then, Brenda Pack spoke about designing inclusive workspaces and the unfortunately common experience of feeling like an outsider as a minority woman in the workplace. In Episode 79, Hortense la Gentil spoke on being authentic and connecting with your true self, flowing into an episode with Laurie Smith on utilizing the power of your voice to unleash executive presence. In Episode 82, Douglas Spencer shared personal branding strategies, explaining how the generations differ in their approach to brand loyalty. In 83, Robert White shares that good leadership is based on good relationships. Then, in 84, Dr. Jan Freed discussed leaving a breadcrumb legacy. Lastly, episode 85 featured Beth Ridley who went into a deeper explanation of diversity and the importance of recognizing all the things that make each person unique.  Part one of season four was filled with wisdom from the best leaders in their fields and the second half of the season was no exception. Be sure to subscribe to the Authentic Change Podcast if you have not already to be among the first to listen to Episode 101 which will feature highlights from episodes 86 through 99.  Quotes: “Episode 100 is major milestone for the Authentic Change Podcast.” (2:34 | Mike)  “One of the things that Toby said that absolutely resonated with me was, without a high level of trust in those who are in leadership positions, organizations do not experience the kind of growth or success that they desire.” (3:07 | Mike) “As leaders no matter what business you're in, you're tasked with leading change.” (4:30 | Mike) “You have to be aligned if you want to be authentic, a powerful reminder expressed eloquently by Hortense la Gentil in episode 79 of Authentic Change.” (6:51 | Mike) “In our episode, organizational culture consulting with Beth Ridley, she says, when we think of diversity in the United States, we tend to start and end with the visible things that we can see, race and gender, which are really important. But I think if people really want to get the most out of your talent, you've got to really appreciate everything that makes people unique.” (9:11 | Mike)  Connect with Mike Horne: Schedule a meeting with Mike: https://calendly.com/mikehorne/15min Learn more about Mike Horne on Linkedin https://www.linkedin.com/in/mikehorne1/ Email Mike at mike@mike-horne.com Learn More About Executive and Organization Development with Mike Horne     Podcast production and show notes provided by HiveCast.fm

In September 2012, a news story broke about Pastor Saeed Abedini, an ordained minister with duel American-Iranian citizenship who was arrested in Iran. He had been involved with setting up house churches in Iran and was working on setting up an orphanage. In 2013 he was sentenced to eight years in prison where he was beaten and denied medical treatment. Saeed's former wife, Naghmeh, worked tirelessly for Saeed's release, which took place in 2016. Shortly before Saeed was released, Naghmeh revealed that Saeed had been addicted to pornography and physically, emotionally, and verbally abused her throughout their marriage. In this first of two episodes, Naghmeh shares her story. Episode Transcript SPONSOR: This program is sponsored by Blazing Grace Ministries. ANNOUNCER: This radio program is PG13. Parents strongly cautioned - some material may be inappropriate for children under the age of 13. Jesus's mission was to comfort those who mourn, bind up the broken-hearted, proclaim liberty to captives, and open prison doors for those who are bound. For those who want more than status quo Christianity has to offer, Blazing Grace Radio begins now. And here is your host, Mike Genung. MIKE GENUNG, HOST, BLAZING GRACE RADIO: Hey, Mike Genung here, and welcome back to Blazing Grace Radio. I'm coming to you from another 115 degree day here in Phoenix, AZ. I think this is like #21 in a row, with more to come! So in September 2012, a news story broke about Pastor Saeed Abedini. He was an ordained minister with dual American-Iranian citizenship and he was arrested in Iran. He had been involved in the past, with setting up house churches there, and was working on setting up an orphanage. Then in 2013, he was sentenced to 8 years in prison. And the stories coming out of that prison were pretty dark, where he was beaten, and denied medical care, and it was pretty rough treatment. And I remember watching that story back at that time and praying for his release. And his wife Naghmeh was... she seemed to be everywhere, petitioning for his release, and getting involved with the US government, and some big name Christian ministries got involved, and... and then Sabadini - or Abedini - was released from prison on January 16th, 2016. And then shortly afterward there were started... stories were starting to surface where he had been abusing his wife, and he was - he had some porn problems. And as we know in our ministry, those two can often go hand in hand, with porn and abuse. It could be physical, emotional or verbal abuse. So, today I have Saeed's former wife Naghmeh with me to tell her side of the story. She made national news when she publicly advocated for the release of her then-husband, Saeed Abedini. Through Saeed's imprisonment, Naghmeh was able to bring worldwide attention to the plight of persecuted Christians, and able to proclaim the Gospel to millions across the globe by speaking at human rights groups, major news outlets, the United Nations at Geneva, the European Parliament Congress, and she had personal meetings with both President Barack Obama and [President] Donald Trump. When it came to light that she had been abused throughout her marriage by her husband, the Christian community suddenly changed on her. So, Naghmeh, welcome to the program. NAGHMEH ABEDINI PANAHI, GUEST: Thank you for having me. MIKE: So let's get started, and have you share your story. NAGHMEH: Yeah, I, since I can... I became a Christian from Islam when I was nine years old. My passion had always been to missions and preaching Muslims for Christ. It wasn't until my husband, who was very abusive, went to prison in Iran, that God started building me up. Because up to that point I have been so abused, so controlled, that I wouldn't then rely on my own thinking. I was completely controlled by my husband, and his imprisonment is actually what set me free, where I was able to... I guess I drew close to the Lord, reading His word, and praying more. And through that time is when God revealed to me the abuse I was under and set me free. And, as you mentioned, porn was a big part of our marriage, and it was considered as "godly" to watch and to try to please my husband in a way that would... in a way, it was twisted to show that it was not sin, and that it was... you know, a way that I could serve my husband. MIKE: You talked about abuse. What did that look like? NAGHMEH: Well, it was very subtle. I didn't even... when I met Saeed in 2002, I had no idea what narcissism was, or even any clue about emotional or psychological abuse. I knew about physical abuse, but even then I thought it's someone that gets beaten up all the time and.... it started with, just when I was meeting him, it started with a lot of verbal putting me down, my looks, wanting me to change certain things about myself, where I had entered the relationship, I was very confident. I was becoming more and more... not confident and believing lies about myself and that I wasn't desirable. Also, around that time, it was the isolation. I didn't realize that's what it was, but just criticizing all my friends... At that time I was a missionary in Iran when I met Saeed, so I didn't have a whole bunch of friends, but I had made some friends. I had come to Iran about a year early before I met Saeed, so some friends. And then my family would visit, and he was undermining them as not being spiritual, as Saeed was… Saeed was very Pentecostal, casting out demons, and a lot of signs and wonders, and he was basically... really, because I did see a lot of signs and wonders, I guess, he made me really trust in him and not to go to my family members or friends that I used to go for council. So he... Saeed became the only source of truth in my life. And there was... there was a few physical... it was some pushing and shoving and... but it wasn't a full-on beating until about a year into our marriage. About a year and a half into our marriage is when the first physical abuse happened. But before then, there was a lot of pushing me away, isolating me. The silent treatment, which is abusive as well, not speaking to me for weeks or months, and or... days, weeks, or months, depending on how much he wanted to punish me, and me begging to talk to him. And yeah, just some, I guess some physical, but at that time I wouldn't have considered it abuse. Some pushing and shoving and yeah, that's how it was. Until about a year, we were so busy with the house church movement, pretty much when I met my husband, we... focused on, we were leading one of the largest house church movements in Iran, so we were busy building disciples. I was actually really busy with that, and traveling, and starting churches, and so our relationship was not so much the focus, even though I knew something was off. But we were so busy, until... November of 2005, we had to flee Iran. So three years. If I met Saeed in 2002, I didn't... I met Saeed in 2002, I didn't marry him until 2004. So about three years after we met and about a year after marriage, we had to flee Iran. We were getting arrested a lot, and... I personally had guns pointed to me and told to deny my faith, and it was just getting so intense that we believed that it would actually endanger the house churches if we stayed longer. And, so, we went to Dubai, and a missionary family... it was around Thanksgiving, and a missionary family had gone to America for sabbatical for six months. So they told us we could use their apartment in Dubai until Saeed could get a visa to America. And the first night that we landed there I was pregnant with my daughter, and I was so tired, I was throwing out... I was searching through the suitcase for my pajamas and Saeed got upset and said "You're making a mess," and I said "Who cares?" and that's when my first full-on beating happened was... MIKE: Oh. NAGHMEH: He just beat me, kicked me, punched me; head, stomach, everywhere, I was bruised, and I thought I was going to die. I crawled into the bathroom, and I mean I had bumps coming out of my head. I crawled into the bathroom and called his parents. I called my mom. They were, of course, his parents were back in Iran, and my parents were in America. And he never said sorry, I mean... he ended up saying it was just the demonic forces in Dubai that had made him do it, but never true repentance. And at that time I was pregnant with our first... child, our daughter. So, having come from the Middle East culture, and also the American culture, church culture of purity movement, where if a girl kisses a guy, she's lost her purity and you're not supposed to, you know... you're supposed to keep yourself for your husband, I felt like I was damaged goods. So I didn't see a way out, in terms of walking away from that marriage. I thought, you know, "I'm stuck and I'm also pregnant, I don't want to be a single mom." So I... from that moment on, I guess I learned my lesson? I was walking on eggshells and did everything not to upset him again. And we ended up coming to America and we had our daughter in 2006, and our son in 2008. So that's... that's my... that's before his imprisonment. And I ended up, I was working until... he, yeah, I was working and we were raising the kids until he... he was traveling back and forth to Iran starting in 2009, when my son was about a year old. And in 2012, he was arrested. And that's when my whole life changed, I guess, from what I thought was the worst thing that could have happened to me, ended up being... actually, God rescued me. MIKE: You met him in 2002. So in the two years that you courted, was there any clue of the physical abuse, or any type of abuse, or the pornography? NAGHMEH: No, he was so deceptive. No, because I, well now looking back, of course there was emotional abuse. There was spiritual abuse, he would use Bible verses, but not something... It was very subtle. And even with his porn addiction, he was very deceptive. We had satellite so we could watch TV shows that were outside of Iran, which sometimes they had kissing, and so when the scene would come up with people kissing, he would look away and I thought, "Wow, this is a really pure man that can't even watch... like, he doesn't want to watch that." So no, I... the porn stuff I had no idea. I actually thought, "Wow, this is a very pure person." I knew of... of his past that was not very... was... yeah, his past was not good. I didn't know about the... he didn't talk about any porn addiction, but I knew he had relationships, there was even an incident of... gang rape, and his behaviors towards women. But all of that, he would say, was before he became a Christian, and he'd become a different person. And so, yeah. I just thought he's a Paul. I would call him a Paul, because he has such a radical background, where he was trained by Hezbollah terrorist group to attack Israel, and then he was about to murder a pastor, and he got saved. And so... I just thought that's the past. He's... when I met him he was a baby Christian, probably two years in the Lord, and he seemed on fire. And so I didn't... his past didn't bother me. It wasn't til we came to America and I was pregnant with my daughter, and I would look next to my bed and he was gone. And, of course, I'm usually a sound sleeper, but when you're pregnant, you go to the bathroom a lot [laughs] So it was towards the end of my pregnancy, and when I would wake up, and then I'd catch him in our living room, which was, you know, a good walk from the bedroom. It was, it was... I would search for him, where could he be? And I would find him in the living room and... he was watching porn. And he would turn it off and pretend it didn't happen. And so that was my first wake up call of what's going on. And then when I had my daughter, when she was about 10 months old, I was pregnant again with my son, Jacob, and we... he was again, there was some abuse, physical abuse, where he had grabbed me and I called the police. He ended up being charged for domestic abuse. It was in 2007, because, yeah, because my son was born in 2008. So my pastor at that time suggested that we move in with my - or I move in with my parents. He said, "Get away, it's not a safe place," you know, "go somewhere away from him." And so that's when I moved in with my parents. And then just over time he left... you know, I don't want to go into the details. There was a season where he left that then he came back, and ended up actually weaselling himself back into my parents house, coming back. And again, that's when it came up. It was... my parents now would walk in, and he had the TV on and he was watching porn, and he'd get so embarrassed and turn it off. And then it became to a point where he would just, if I walked in... he ended up watching it in the playroom, which was not an area my parents would normally come up at, because it was like the second floor. But then I would walk in, he was watching it, and it became to a point where he was no longer ashamed. He was just like, "Uh huh. I'm watching it. It's not sin." And so I didn't know, I guess, what to say to that. He had good arguments that when he was going to... Bible school, I guess they talked about sexually learning to do things, and it was okay to do things in marriage, and they had, he had taken a class on Song of Solomon. And so he was basically justifying it. And then soon... well, yeah, after my son was born, then he was demanding it. He was saying, "We need to watch it, and you need to do things like I tell you to do." And I was resisting watching it. I watched it a few times, but I couldn't stomach it. I just... the Holy Spirit within me was just so much against it. So I refused to watch it, but then he was watching it without shame. And but... and then he was, because of his porn addiction, sexually abusing me. He's just like wanting to do certain things that I didn't want to do. He wanted, you know, certain positions, and forceful, and aggressive, and... which again, at that time I wouldn't have labeled as sexual abuse. But it was. It was... very forceful, and demanding, and not listening to my... I guess my... I'm not wanting, you know, certain things and... so, but at that time I didn't see any of that as abuse, I just thought I have a hard marriage, and I didn't know what to do with it. So yeah. MIKE: You mentioned something about a rape in his past? NAGHMEH: Yeah, he had mentioned... he had mentioned that as a young boy in Iran, they had... it was rape. That he had a relationship with older women and younger women, but he mentioned a specific situation where there was a street girl. I don't know if she was a prostitute or if she was... she was being.... she was sex slavery or whatever her situation was, but she was pretty young, probably a teenager, and he was 18-20, because he got saved at 20. So he was... she was definitely much younger, and they kind of gang raped her. It was him and a group of friends, and... at that time when he told me I was, we were still dating, and so I couldn't even, I had to have no sexual relationship, I couldn't even fathom what he was saying, and... I... I didn't realize the red flag of... a person, basically raping, what that, another human being, what that would mean. MIKE: Mhmm. NAGHMEH: And then years later, I... other women approached me and talked to me, and I realized he's done that to many, many people, even in our house churches. He would pray for people, he would turn off the light, and he would pray for people and they would fall. Like Benny Hin. He was... Benny Hin was Saeed's hero. And later, so many girls at my church said he was fondling with them, and he was sexually molesting them. And when they were on the ground and everyone else's eyes were shut, Saeed was the only one praying and moving around, and praying for people. And years later, people start coming forward and saying that he was sexually abusing them and had used that opportunity to molest them, and things like that. MIKE: Mhmm [sighs] NAGHMEH: But it wasn't until after I came out with the abuse that all these other people started coming to me about things that they had seen. I'd even had other pastors, house church pastors, come forward to me and said, "Now that you say it, like I'd noticed this and this and this, and I didn't know if I was seeing it correctly, but Saeed was doing all this stuff with these girls," and... so until the abuse and his sexual addiction became known to me, and I even understood how damaging, like you said, a porn addiction is, and how it goes hand in hand with abuse, because a woman is no longer viewed as a human being, that they're objectified so... so usually porn addiction and abuse usually go hand in hand... I didn't know any of that until my eyes were opened up to abuse. And, so yeah, and the way they were opened up was when Saeed was in prison. So in 2012, as you said, Saeed was arrested. And at that time, right before his arrest, I remember crying out to the Lord because I could barely read my Bible. I could barely pray. The Bible was used to manipulate me, to control me - MIKE: Mmm. NAGHMEH: - to call me a bad wife. So I couldn't even read God's word because anytime I opened it, it was condemning. It was oppressive. And so when Saeed was arrested, I start opening up the word of God and I start praying. But before his arrest, I remember crying out and asking God to help with the marriage. I thought, "this is going to be my life for the rest of my life. This is going to be my marriage for the rest of my life." Saeed had been in a hotel room, before his arrest, with another woman. And I had called and she had picked up, and I became hysterical. And instead of apologizing, he said I'm crazy and I need to go see a doctor because I was hysterical that another woman - MIKE: Mmm. NAGHMEH: - answered the phone in his hotel. And so at that time, I remember thinking, "Wow, like he's saying I'm crazy for thinking that he cheated on me? They're just sleeping in the same room?" And I thought, this is my... I didn't even see a way out. I never thought divorce would be an option. So I remember crying out to the Lord, like, what's going on? Lord is, you know, and for the first time, just pouring my heart out to the Lord. And then a few hours later, both this girl and my husband were arrested and put in the prison, or actually put under house arrest first, and then put in prison. And from that moment on, I just start praying and reading my Bible. And in the effort of trying to get my husband out, God was building my confidence in Him and growing my walk with Him. And in 2015, I was speaking at a mega church in North Carolina, and the pastor's name was David Chadwick, Pastor David Chadwick. And he... I just... Saeed had a smartphone inside of the prison, and he had gotten a smartphone about a year before his release and he was messaging me, he was seeing how famous he was getting - MIKE: Mmm. NAGHMEH: - and I noticed he was getting access to my Amazon account, and he was watching things like "50 Shades of Gay", which I clicked on and it was just... it was making me like sick to my stomach, the sexual content. And so I noticed he's definitely watching porn again, so I mean, from Iranian prison, and then and it was seemed like there was a lot of more gay porn which made me wonder about what's happening in prison. And then I finally, I didn't share that part of the sexual stuff, because I was so ashamed. I didn't share it with the pastor, but I shared with him, like, "I don't understand. Saeed has a phone inside of the prison and he's sending me really rude messages saying me I'm a whore, I'm a... I'm a Jezebel," and so this pastor looked at me, and I shared everything of what has happened in our marriage, and I couldn't make sense of why Saeed was putting me down when I was trying so hard to get him out. Later it made sense that Saeed was noticing that I was becoming confident, I wasn't the same girl as I was before, where I would just shrivel up and submit to him. So this pastor, after I mentioned everything, and I showed him some of the text messages, he looked at me. He said "Naghmeh, you know, I'm not just a pastor." And I said, "No, I didn't know that." He said. "I'm a doctor." I - MIKE: Naghmeh, Naghmeh I need you to, I need to interrupt for a moment because we're out of time for this first show. But your story is very powerful, and we're going to continue this interview with Nagmeh next week, so I encourage you to join us. And then we'll talk to you next week. ANNOUNCER: Blazing Grace is a nonprofit international ministry for the sexually broken and the spouse. Please visit us at blazinggrace.org for information on Mike Genung's books, groups, counseling, or to have Mike speak at your organization. You can e-mail us at email@blazinggrace.org or call our office in Chandler, AZ at 719-888-5144. Again, visit us at blazinggrace.org, e-mail us at email@blazinggrace.org, or call the office at 719-888-5144. SPONSOR: This program was sponsored by Blazing Grace Ministries.

Mike Kiss and his wife, Kristin, are friends of Whitney that she's known since high school. Mike and Kristin Kiss like challenges. The two former athletes set out 10 years ago to run a half- marathon in all 50 states, a goal that they accomplished in four years. Since then, their priorities have expanded to include two children, but one thing remains: their willingness to take on big, audacious goals. Today, Mike and Kristin joins the show to talk with Whitney about his unique goals, the hardest race he's ever competed in, and the biggest challenges he's faced along the way.Episode SponsorRunner Click Pro – https://pro.runnerclick.com/Key Takeaways00:54 – Whitney Heins welcomes Mike and Kristin Kiss to the show to talk about the importance of stretching, what they love about running, and why they set such lofty goals 17:43 – Mike and Kristin talk about their unique and audacious goals 21:35 – Racing in Las Vegas and running 50 half marathons in all 50 states25:37 – The hardest race Mike and Kristin ever ran 28:47 – Getting their kids involved and another arduous goal 33:42 – How running has fortified Mike and Kristin's relationship 41:51 – The biggest challenges Mike faced when trying to achieve his goals52:44 – What's next for Mike and one piece of advice for anyone looking to take on a big, audacious goal 58:23 – Whitney thanks Mike for joining the show to share his storyTweetable Quotes“If I'm so prescribed, I don't enjoy it. And, at the end of the day this is all about having fun and enjoying running. I don't want it to become work. If it becomes work, I'm not gonna want to do it. So, I'm just looking at the adventure side of it.” (13:03) (Mike) “You plan as well as you can and then things don't go well. And then you don't plan, or maybe you make the wrong decision, and things can still go well. You never know.” (28:13) (Mike) “The difference between a half and a full, as you're aware, is that you can get by with ‘okay weather' in a half marathon and it's not gonna make a huge impact on the day or the time. ‘Okay weather' for a full marathon can be a lot more challenging.” (44:45) (Mike) “Everything is a competing priority. There's finite resources. You have to make decisions and concessions, but I haven't walked back from a trip that we've done with our kids regretting that we did it or that we spent money to do it.” (48:18) (Mike)“I just encourage people to look at it in that light and put yourself out there. Find resources, find people that are gonna support you. I think you'd be surprised at how many people will be in your corner to help you along the way.” (56:35) (Mike/Kristin)Resources MentionedWhitney's LinkedIn – https://www.linkedin.com/in/whitney-heins-02ba3b5The Mother Runners Club – https://www.themotherrunners.com/Mike's Email - mikeakiss@gmail.comMike's Facebook - https://www.facebook.com/mikeakiss50in100 - https://www.50in100.com/

Whether you end up working as an agency officer, get placed in the office, or work on the field, you'll come to understand that all of these play an equal part in the protection and security of the country.Mike Howard, Former Chief Security Officer of Microsoft and former CIA operative continues the conversation today and shares about his days in the CIA, from starting out in operational support work to going around the world working security details.Mike also talks about his experience over the years across different departments in the CIA. He also shares some of his favorite rules on leadership and gives advice on how others can start to implement them.Learn about all this and more in this episode of The Fearless Mindset Podcast. GOLDEN NUGGETSMike on being assigned to career development"Those experiences taught me things I wouldn't have learned had I just stayed in the operations circuit I eventually went back and finished out my career in operations but those three or four years I wouldn't trade for anything. It taught me new skills and gave me an awareness of what other people do in the agency."The leadership rule on perpetual optimism being a force multiplier - Mike"You as a leader, there are days when you're dragging ass or you're just not feeling it. But for your troops, they can't see that. They need to see you positive and can-do and no matter what obstacles are in front  of you, you and the team can nail this thing together." Get to know more about Mike:LinkedIn: https://www.linkedin.com/in/mike-howard-3423051/Website: https://www.mikehowardauthor.com/To hear more episodes of The Fearless Mindset podcast, you can go to https://the-fearless-mindset.simplecast.com/ or listen to major podcasting platforms such as Apple, Google Podcasts, Spotify, etc. You can also subscribe to the Fearless Mindset YouTube Channel to watch episodes on video.

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

無情工商時間～ 暑假就要來了，家長們還在思考該怎麼讓孩子們度過一個充實又有趣的暑假嗎？（當肯：把拔，暑假要幹嘛？）喜歡籃球的你們可以參考我們跟PLG合作開設的「OhMyBasketball!我的籃球英文」，這堂課從籃球基本配備到戰術講解，由淺入深帶你用英文認識籃球。同時搭配比賽經典畫面，暑假覺得太熱？沒關係我們就留在家裡看比賽畫面學籃球英文，偶爾再去球場實戰，你說dribble我說運球，邊學英文邊打籃球。暑假…就是這麼chill…親子共學的機會，不要錯過… 結帳前記得輸入podcast聽眾專屬優惠碼 roundball150，還可以再折150元！ 歡迎到節目資訊欄點選連結試看課程！https://lihi2.com/cst50 快速幫你複習一下這集的主題句 ＆ 單字： 你又放我鴿子 / 爽約 You stood me up again! *stand sb up 放鴿子、放鳥 不要放我鴿子、記得要來 Don't stand me up this weekend. 補充學習 stand up for sth/sb 支持某人/事 遵守約定 keep a promise / keep one's word 跟某人有約 -跟特定人士/專業人士(醫生、老師) have an appointment with -跟對象/情侶 have a date with 情境對話 Mike：當肯，不要忘了我們下週有約。 Duncan, don't forget we have a date next week. Duncan：下週？下週我要帶我的狗去參加戶外教學。 Next week? I have to take my dog on a field trip next week. Mike：你又放我鴿子！而且你什麼時候養了一隻狗？ You are standing me up again! And since when did you have a dog? Duncan：很抱歉，但狗勾真的需要人家陪。 Forgive me, but my doggo really needs my company. 小額贊助支持本節目： https://open.firstory.me/user/ckf6dwd77euw20897td87i5wj 留言告訴我你對這一集的想法： https://open.firstory.me/user/ckf6dwd77euw20897td87i5wj/comments Powered by Firstory Hosting

Today, we're really excited to share the 8th installment of our Ask Me Anything (AMA) series.The ever-insightful and entertaining O'Neil Cespedes is back to co-host this month's AMA and – per usual – we have some fun diving into your questions. On this episode, we chat about: What to do if you're feeling burnt out and overworkedGrappling with the need to prove yourself to others How to build culture and hire appropriatelyThe difference between high performance and masteryFeeling like you're “behind” in lifeAnd, one question to chew on… Are you a thermostat? Or a thermometer?Lastly – as always – reply to this email with feedback or any questions you have for future AMAs. Wishing you an epic week.With Fire, Dr. Mike-----You can WATCH this episode on our YouTube channel.Connect with us on our Instagram.For more information and shownotes from every episode, head to findingmastery.com.See Privacy Policy at https://art19.com/privacy and California Privacy Notice at https://art19.com/privacy#do-not-sell-my-info.

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

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

Patrick talks with an ex-Catholic who converted to Judaism for his family asks if his soul is at risk. Indian official drained a reservoir to get his smartphone Trans extremists have threatened to bomb at Target in Utah for removing Satanist's LGBTQ+ products targeted at kids due to backlash Chase – I was born Catholic and converted to Judaism and I love my life. Is my soul at risk? Ryan - Saw an article about the Sisters of St. Joseph announcing on their website that they were celebrating World trans day. Are they not supposed to be Catholic? Mike - You should have been more positive about the Chase's call. In CCC819, The Sanctification is not restricted to the Catholic Church, the fullness is in the Catholic Church. Elena - Works the night shift and sleeps in the parking lot of the Church waiting for daily Mass in the morning. Many times she falls asleep and can't make it to Mass. Is it a sin? I feel bad. Anne - Views on Ecumenism in the Catholic Church and the promotion of learning about other faiths by taking Catholics to other religious places. When typing on my phone 'God' it always comes up something else.

When it comes to history and culture, it doesn't get much better than Athens, Greece. This European destination is an amazing city to tour, and today Ryan and I are joined by a recent traveler to the “glorious city”.Travel to Athens from Airport (taxi vs Train)Where to stay in Athens (the Plaka Neighborhood), Hotel PlakaHighlights of Athens (what to see, how to do it)Food, culture, peopleTips for visiting AthensA “Highlight” story from our time in Athens (no, not the barbershop story Mike…)You'll also enjoy Cruising the Greek Isles!Question: What would you like to know about visiting Athens, Greece? **Let us know on Facebook and Instagram.**~~~~~~~Grab your free download: Top Items to Buy Before Your Theme Park Vacation to Save You Money and Stress!Ready to plan your vacation? Ryan and Shayne are both Travel Advisors with Creating Magic Vacations and will make YOU the travel-planning superhero. Contact Ryan at DisneyTravelDad.Com. Join the travel conversations and the fun in the All Things Travel Show Facebook group! Please share the show with your travel buddies!! Click this link and share the show! Never miss an episode and help us take you to the top with us by following and leaving a 5-Star review on your favorite podcasting app!

一秒切換無情工商時間～ 近期因為魔獸跟林書豪的加盟，讓台灣籃球的話題聲勢也跟著提升，這一次我們跟ＰＬＧ所開設的「Oh my basketball 我的籃球英文」，是全台第一堂將籃球與英文結合的課程，由ＰＬＧ的英文主播Ryan所主講，在課程裡可以學到籃球英文轉播中常聽到的用字，老師會用實際比賽的經典畫面講解，讓你上課就好像在球場看比賽！還有啦啦隊賣力應援的畫面幫你上課提神顧目珠！ 歡迎點進節目資訊欄試看課程，結帳前不要忘記輸入podcast聽眾的專屬優惠碼 roundball150，就可以折150塊！ https://lihi2.com/cst50 快速幫你複習一下這集的主題句 ＆ 單字： 你真掃興== You are (such) a buzzkill/ downer /party pooper. You're no fun. *downer 1.鎮定劑 2.掃興的人（或事） 補充學習 你真難約。 You're always busy. / You never have time. 下次再約。（下次好了，下次啦，緩和語氣） Next time, then. Maybe next time. 情境對話 Mike：我們下禮拜要去台東玩，你要去嗎？ We're gonna take a trip to Taitung. Do you wanna come along? Duncan：下次好了。台東太遠了。 Not this time. Taitung is too far. Mike：你真掃興== 台東這麼好玩欸！ You are such a buzzkill. Taitung is such a good place to visit. Duncan：沒錢沒時間啦，除非你幫我安排私人飛機我就去。 I got no time or money. I will join if you arrange a private plane for me. Powered by Firstory Hosting

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

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

「你忘了關電燈～」 英文怎麼說？ 你常常會忘記關燈或是關冷氣嗎？ “你忘了關冷氣”、“你水龍頭沒關好”、“電費漲價了耶”的英文怎麼說呢？ 這一集文化閒聊，Duncan 要跟我們分享， 美國家庭平均一個月的電費大概是多少？比台灣貴還是便宜呢？ 除了電費之外，Duncan 在台灣是第一次看到付費買專用垃圾袋丟垃圾嗎？ 快來聽這一集內容，聽聽看你忘了關電燈的英文怎麼說。 英文口說救星：賴世雄幫你自信開口說英文 賴世雄老師自學口說三心法 - 教你開口說正確、自然的英文。 課程優惠中，馬上點進試看，結帳前輸入 speak200 再折 200 元喔！ https://lihi1.com/qTXgp 快速幫你複習一下這集的主題句 ＆ 單字： 你忘了關電燈（冷氣)～ You forgot to turn off the light(s). (AC). 補充學習 你忘記關瓦斯爐 You left the gas on. 你水龍頭沒關好 The faucet is still running. 電費漲價了耶 Electricity prices went up / increased. 情境對話 Mike：你忘了關電燈～！ You forgot to turn off the light. Duncan：喔～抱歉，剛剛走出來忘記關了。 Oh, my bad. I forgot to turn it off when I left just now. Mike：下次要記得隨手關燈，現在電費很貴耶～ Next time, remember to switch it off as you leave. Electricity is expensive these days. Duncan：好喔～我知道了。 All right. I get it. 學英文吧網站https://ivybar.com.tw/?c=3 或追蹤 iVY BAR 學英文吧的 IG，上面圖文版 podcast 複習也很棒喔！https://pse.is/39vede 現在我們也有影音版的 Podcast 實境秀喔https://pse.is/3ahupl Powered by Firstory Hosting

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

Since Morgan Glennon wasn't a full-time Supergirl Radio co-host until the end of Supergirl Season 1, she wanted to revisit the episodes she missed out on discussing! In this episode, Rebecca Johnson joins her LIVE [and WIRED] to hear her thoughts about "Truth, Justice and the American Way". Note: Rebecca's memory failed her! Jeff Branson played Ronan Malloy on The Young and The Restless. Watch the Live Stream Episode Links Supergirl Radio's Lexi Alexander Interview Supergirl Radio Season 1 - "Truth, Justice annd the American Way" ‘Sheba' Drama From Chantelle Wells, Azie Tesfai & Ryan Coogler's Proximity Media In Works At Onyx Collective Master Jailer in DC Comics "We Can Be Heroes" Boardroom or Ballroom - Master Jailer Facet Krypton on CW Seed Rachel Finds Out About Chandler and Monica (Friends) Gay Gals Watch: Trashy TV and GBBO with Morgan and Mike You can find Supergirl Radio on: Social Media: Facebook – Twitter – Instagram  Subscribe: Apple Podcasts – Stitcher Radio – DC TV Podcasts - Google Play - Spotify Playlist - iHeartRadio Support: DC TV Podcasts TeePublic Store – Patreon Contact: supergirlradio@gmail.com 

Rob - I've been homeless for 10 years and have been making it work well. Albert - I am a truck driver and technically I am homeless because my wife divorced me and I gave her the house so she could raise the kids. I am now back on my feet but still not wanting to buy a house because of the market and my Job as a trucker. Mike - Because of you, I went from being a homeless drug addict to starting my own business and making a career for myself. Michelle - What should I do if someone is not paying rent? I don't want to kick them out but I need to make money from their rent? Alicia - Jordan Peterson posted a message to churches encouraging men to return to church. Mike – You can't get out of drug addiction alone. You need a sponsor and a spiritual advisor Mike - People who are homeless choose to be there and don't want help. I don't want to enable drug addicts. Terri - I went to confession and had a complete healing from my addiction Brother Sean - I am a religious brother and we work with the homeless on the street.

About MikeBeside his duties as The Duckbill Group's CEO, Mike is the author of O'Reilly's Practical Monitoring, and previously wrote the Monitoring Weekly newsletter and hosted the Real World DevOps podcast. He was previously a DevOps Engineer for companies such as Taos Consulting, Peak Hosting, Oak Ridge National Laboratory, and many more. Mike is originally from Knoxville, TN (Go Vols!) and currently resides in Portland, OR.Links Referenced: @Mike_Julian: https://twitter.com/Mike_Julian mikejulian.com: https://mikejulian.com duckbillgroup.com: https://duckbillgroup.com TranscriptAnnouncer: Hello, and welcome to Screaming in the Cloud with your host, Chief Cloud Economist at The Duckbill Group, Corey Quinn. This weekly show features conversations with people doing interesting work in the world of cloud, thoughtful commentary on the state of the technical world, and ridiculous titles for which Corey refuses to apologize. This is Screaming in the Cloud.Corey: This episode is sponsored in part by our friends at AWS AppConfig. Engineers love to solve, and occasionally create, problems. But not when it's an on-call fire-drill at 4 in the morning. Software problems should drive innovation and collaboration, NOT stress, and sleeplessness, and threats of violence. That's why so many developers are realizing the value of AWS AppConfig Feature Flags. Feature Flags let developers push code to production, but hide that that feature from customers so that the developers can release their feature when it's ready. This practice allows for safe, fast, and convenient software development. You can seamlessly incorporate AppConfig Feature Flags into your AWS or cloud environment and ship your Features with excitement, not trepidation and fear. To get started, go to snark.cloud/appconfig. That's snark.cloud/appconfig.Corey: Forget everything you know about SSH and try Tailscale. Imagine if you didn't need to manage PKI or rotate SSH keys every time someone leaves. That'd be pretty sweet, wouldn't it? With Tailscale SSH, you can do exactly that. Tailscale gives each server and user device a node key to connect to its VPN, and it uses the same node key to authorize and authenticate SSH.Basically you're SSHing the same way you manage access to your app. What's the benefit here? Built in key rotation, permissions is code, connectivity between any two devices, reduce latency and there's a lot more, but there's a time limit here. You can also ask users to reauthenticate for that extra bit of security. Sounds expensive?Nope, I wish it were. Tailscale is completely free for personal use on up to 20 devices. To learn more, visit snark.cloud/tailscale. Again, that's snark.cloud/tailscaleCorey: Welcome to Screaming in the Cloud. I'm Cloud Economist Corey Quinn, and my guest is a returning guest on this show, my business partner and CEO of The Duckbill Group, Mike Julian. Mike, thanks for making the time.Mike: Lucky number three, I believe?Corey: Something like that, but numbers are hard. I have databases for that of varying quality and appropriateness for the task, but it works out. Anything's a database. If you're brave enough.Mike: With you inviting me this many times, I'm starting to think you'd like me or something.Corey: I know, I know. So, let's talk about something that is going to put that rumor to rest.Mike: [laugh].Corey: Clearly, you have made some poor choices in the course of your career, like being my business partner being the obvious one. But what's really in a dead heat for which is the worst decision is you've written a book previously. And now you are starting the process of writing another book because, I don't know, we don't keep you busy enough or something. What are you doing?Mike: Making very bad decisions. When I finished writing Practical Monitoring—O'Reilly, and by the way, you should go buy a copy if interested in monitoring—I finished the book and said, “Wow, that was awful. I'm never doing it again.” And about a month later, I started thinking of new books to write. So, that was 2017, and Corey and I started Duckbill and kind of stopped thinking about writing books because small companies are basically small children. But now I'm going to write a book about consulting.Corey: Oh, thank God. I thought you're going to go down the observability path a second time.Mike: You know, I'm actually dreading the day that O'Reilly asks me to do a second edition because I don't really want to.Corey: Yeah. Effectively turn it into an entire story where the only monitoring tool you really need is the AWS bill. That'll go well.Mike: [laugh]. Yeah. So yeah, like, basically, I've been doing consulting for such a long time, and most of my career is consulting in some form or fashion, and I head up all the consulting at Duckbill. I've learned a lot about consulting. And I've found that people have a lot of questions about consulting, particularly at the higher-end levels. Once you start getting into advisory sort of stuff, there's not a lot of great information out there aimed at engineering.Corey: There's a bunch of different views on what consulting is. You have independent contractors billing by the hour as staff replacement who call what they do consulting; you have the big consultancies, like Bain or BCG; you've got what we do in an advisory sense, and of course, you have a bunch of MBA new grads going to a lot of the big consultancies who are going to see a book on consulting and think that it's potentially for them. I don't know that you necessarily have a lot of advice for the new grad type, so who is this for? What is your target customer for this book?Mike: If you're interested in joining McKinsey out of college, I don't have a lot to add; I don't have a lot to tell you. The reason for that is kind of twofold. One is that shops like McKinsey and Deloitte and Accenture and BCG and Bain, all those, are playing very different games than what most of us think about when we think consulting. Their entire model revolves around running a process. And it's the same process for every client they work with. But, like, you're buying them because of their process.And that process is nothing new or novel. You don't go to those firms because you want the best advice possible. You go to those firms because it's the most defensible advice. It's sort of those things like, “No one gets fired for buying Cisco,” no one got fired for buying IBM, like, that sort of thing, it's a very defensible choice. But you're not going to get great results from it.But because of that, their entire model revolves around throwing dozens, in some cases, hundreds of new grads at a problem and saying, “Run this process. Have fun. Let us know if you need help.” That's not consulting I have any experience with. It's honestly not consulting that most of us want to do.Most of that is staffed by MBAs and accountants. When I think consulting, I think about specialized advice and providing that specialized advice to people. And I wager that most of us think about that in the same way, too. In some cases, it might just be, “I'm going to write code for you as a freelancer,” or I'm just going to tell you like, “Hey, put the nail in here instead of over here because it's going to be better for you.” Like, paying for advice is good.But with that, I also have a… one of the first things I say in the beginning of the book, which [laugh] I've already started writing because I'm a glutton for punishment, is I don't think junior people should be consultants. I actually think it's really bad idea because to be a consultant, you have to have expertise in some area, and junior staff don't. They haven't been in their careers long enough to develop that yet. So, they're just going to flounder. So, my advice is generally aimed at people that have been in their careers for quite some time, generally, people that are 10, 15, 20 years into their career, looking to do something.Corey: One of the problems that we see when whenever we talk about these things on Twitter is that we get an awful lot of people telling us that we're wrong, that it can't be made to work, et cetera, et cetera. But following this model, I've been independent for—well, I was independent and then we became The Duckbill Group; add them together because figuring out exactly where that divide happened is always a mental leap for me, but it's been six years at this point. We've definitely proven our ability to not go out of business every month. It's kind of amazing. Without even an exception case of, “That one time.”Mike: [laugh]. Yeah, we are living proof that it does work, but you don't really have to take just our word for it because there are a lot of other firms that exist entirely on an advisory-only, high-expertise model. And it works out really well. We've worked with several of them, so it does work; it just isn't very common inside of tech and particularly inside of engineering.Corey: So, one of the things that I find is what differentiates an expert from an enthusiastic amateur is, among other things, the number of mistakes that they've made. So, I guess a different way of asking this is what qualifies you to write this book, but instead, I'm going to frame it in a very negative way. What have you screwed up on that puts you in a position of, “Ah, I'm going to write a book so that someone else can make better choices.”Mike: One of my favorite stories to tell—and Corey, I actually think you might not have heard this story before—Corey: That seems unlikely, but give it a shot.Mike: Yeah. So, early in my career, I was working for a consulting firm that did ERP implementations. We worked with mainly large, old-school manufacturing firms. So, my job there was to do the engineering side of the implementation. So, a lot of rack-and-stack, a lot of Windows Server configuration, a lot of pulling cables, that sort of thing. So, I thought I was pretty good at this. I quickly learned that I was actually not nearly as good as I thought I was.Corey: A common affliction among many different people.Mike: A common affliction. But I did not realize that until this one particular incident. So, me and my boss are both on site at this large manufacturing facility, and the CFO pulls my boss aside and I can hear them talking and, like, she's pretty upset. She points at me and says, “I never want this asshole in my office ever again.” So, he and I have a long drive back to our office, like an hour and a half.And we had a long chat about what that meant for me. I was not there for very long after that, as you might imagine, but the thing is, I still have no idea to this day what I did to upset her. I know that she was pissed and he knows that she was pissed. And he never told me exactly what it was, only that's you take care of your client. And the client believes that I screwed up so massively that she wanted me fired.Him not wanting to argue—he didn't; he just kind of went with it—and put me on other clients. But as a result of that, it really got me thinking that I screwed something up so badly to make this person hate me so much and I still have no idea what it was that I did. Which tells me that even at the time, I did not understand what was going on around me. I did not understand how to manage clients well, and to really take care of them. That was probably the first really massive mistake that I've made my career—or, like, the first time I came to the realization that there's a whole lot I don't know and it's really costing me.Corey: From where I sit, there have been a number of things that we have done as we've built our consultancy, and I'm curious—you know, let's get this even more personal—in the past, well, we'll call it four years that we have been The Duckbill Group—which I think is right—what have we gotten right and what have we gotten wrong? You are the expert; you're writing a book on this for God's sake.Mike: So, what I think we've gotten right is one of my core beliefs is never bill hourly. Shout out to Jonathan Stark. He wrote I really good book that is a much better explanation of that than I've ever been able to come up with. But I've always had the belief that billing hourly is just a bad idea, so we've never done that and that's worked out really well for us. We've turned down work because that's the model they wanted and it's like, “Sorry, that's not what we do. You're going to have to go work for someone else—or hire someone else.”Other things that I think we've gotten right is a focus on staying on the advisory side and not doing any implementation. That's allowed us to get really good at what we do very quickly because we don't get mired in long-term implementation detail-level projects. So, that's been great. Where we went a little wrong, I think—or what we have gotten wrong, lessons that we've learned. I had this idea that we could build out a junior and mid-level staff and have them overseen by very senior people.And, as it turns out, that didn't work for us, entirely because it didn't work for me. That was really my failure. I went from being an IC to being the leader of a company in one single step. I've never been a manager before Duckbill. So, that particular mistake was really about my lack of abilities in being a good manager and being a good leader.So, building that out, that did not work for us because it didn't work for me and I didn't know how to do it. So, I made way too many mistakes that were kind of amateur-level stuff in terms of management. So, that didn't work. And the other major mistake that I think we've made is not putting enough effort into marketing. So, we get most of our leads by inbound or referral, as is common with boutique consulting firms, but a lot of the income that we get comes through Last Week in AWS, which is really awesome.But we don't put a whole lot of effort into content or any marketing stuff related to the thing that we do, like cost management. I think a lot of that is just that we don't really know how, aside from just creating content and publishing it. We don't really understand how to market ourselves very well on that side of things. I think that's a mistake we've made.Corey: It's an effective strategy against what's a very complicated problem because unlike most things, if—let's go back to your old life—if we have an observability problem, we will talk about that very publicly on Twitter and people will come over and get—“Hey, hey, have you tried to buy my company's product?” Or they'll offer consulting services, or they'll point us in the right direction, all of which is sometimes appreciated. Whereas when you have a big AWS bill, you generally don't talk about it in public, especially if you're a serious company because that's going to, uh, I think the phrase is, “Shake investor confidence,” when you're actually live tweeting slash shitposting about your own AWS bill. And our initial thesis was therefore, since we can't wind up reaching out to these people when they're having the pain because there's no external indication of it, instead what we have to do is be loud enough and notable in this space, where they find us where it shouldn't take more than them asking one or two of their friends before they get pointed to us. What's always fun as the stories we hear is, “Okay, so I asked some other people because I wanted a second opinion, and they told us to go to you, too.” Word of mouth is where our customers come from. But how do you bootstrap that? I don't know. I'm lucky that I got it right the first time.Mike: Yeah, and as I mentioned a minute ago, that a lot of that really comes through your content, which is not really cost management-related. It's much more AWS broad. We don't put out a lot of cost management specific content. And honestly, I think that's to our detriment. We should and we absolutely can. We just haven't. I think that's one of the really big things that we've missed on doing.Corey: There's an argument that the people who come to us do not spend their entire day thinking about AWS bills. I mean, I can't imagine what that would be like, but they don't for whatever reason; they're trying to do something ridiculous, like you know, run a profitable company. So, getting in front of them when they're not thinking about the bills means, on some level, that they're going to reach out to us when the bill strikes. At least that's been my operating theory.Mike: Yeah, I mean, this really just comes down to content strategy and broader marketing strategy. Because one of the things you have to think about with marketing is how do you meet a customer at the time that they have the problem that you solve? And what most marketing people talk about here is what's called the triggering event. Something causes someone to take an action. What is that something? Who is that someone, and what is that action?And for us, one of the things that we thought early on is that well, the bill comes out the first week of the month, every month, so people are going to opened the bill freak out, and a big influx of leads are going to come our way and that's going to happen every single month. The reality is that never happened. That turns out was not a triggering event for anyone.Corey: And early on, when we didn't have that many leads coming in, it was a statistical aberration that I thought I saw, like, “Oh, out of the three leads this month, two of them showed up in the same day. Clearly, it's an AWS billing day thing.” No. It turns out that every company's internal cadence is radically different.Mike: Right. And I wish I could say that we have found what our triggering events are, but I actually don't think we have. We know who the people are and we know what they reach out for, but we haven't really uncovered that triggering event. And it could also be there, there isn't a one. Or at least, if there is one, it's not one that we could see externally, which is kind of fine.Corey: Well, for the half of our consulting that does contract negotiation for large-scale commitments with AWS, it comes up for renewal or the initial discount contract gets offered, those are very clear triggering events but the challenge is that we don't—Mike: You can't see them externally.Corey: —really see that from the outside. Yeah.Mike: Right. And this is one of those things where there are triggering events for basically everything and it's probably going to be pretty consistent once you get down to specific services. Like we provide cost optimization services and contract negotiation services. I'm willing to bet that I can predict exactly what the trigger events for both of those will be pretty well. The problem is, you can never see those externally, which is kind of fine.Ideally, you would be able to see it externally, but you can't, so we roll with it, which means our entire strategy has revolved around always being top-of-mind because at the time where it happens, we're already there. And that's a much more difficult strategy to employ, but it does work.Corey: All it takes is time and being really lucky and being really prolific, and, and, and. It's one of those things where if I were to set out to replicate it, I don't even know how I'd go about doing it.Mike: People have been asking me. They say, “I want to create The Duckbill Group for X. What do I do?” And I say, “First step, get yourself a Corey Quinn.” And they're like, “Well, I can't do that. There's only one.” I'm like, “Yep. Sucks to be you.” [laugh].Corey: Yeah, we called the Jerk Store. They're running out of him. Yeah, it's a problem. And I don't think the world needs a whole lot more of my type of humor, to be honest, because the failure mode that I have experienced brutally and firsthand is not that people don't find me funny; it's that it really hurts people's feelings. I have put significant effort into correcting those mistakes and not repeating them, but it sucks every time I get it wrong.Mike: Yeah.Corey: Another question I have for you around the book targeting, are you aiming this at individual independent consultants or are you looking to advise people who are building agencies?Mike: Explicitly not the latter. My framing around this is that there are a number of people who are doing consulting right now and they've kind of fell into it. Often, they'll leave one job and do a little consulting while they're waiting on their next thing. And in some cases, that might be a month or two. In some cases, it might go on years, but that whole time, they're just like, “Oh, yeah, I'm doing consulting in between things.”But at some point, some of those think, “You know what? I want this to be my thing. I don't want there to be a next thing. This is my thing. So therefore, how do I get serious about doing consulting? How do I get serious about being a consultant?”And that's where I think I can add a lot of value because casually consulting of, like, taking whatever work just kind of falls your way is interesting for a while, but once you get serious about it, and you have to start thinking, well, how do I actually deliver engagements? How do I do that consistently? How do I do it repeatedly? How to do it profitably? How do I price my stuff? How do I package it? How do I attract the leads that I want? How do I work with the customers I want?And turning that whole thing from a casual, “Yeah, whatever,” into, “This is my business,” is a very different way of thinking. And most people don't think that way because they didn't really set out to build a business. They set out to just pass time and earn a little bit of money before they went off to the next job. So, the framing that I have here is that I'm aiming to help people that are wanting to get serious about doing consulting. But they generally have experience doing it already.Corey: Managing shards. Maintenance windows. Overprovisioning. ElastiCache bills. I know, I know. It's a spooky season and you're already shaking. It's time for caching to be simpler. Momento Serverless Cache lets you forget the backend to focus on good code and great user experiences. With true autoscaling and a pay-per-use pricing model, it makes caching easy. No matter your cloud provider, get going for free at gomemento.co/screaming That's GO M-O-M-E-N-T-O dot co slash screamingCorey: We went from effectively being the two of us on the consulting delivery side, two scaling up to, I believe, at one point we were six of us, and now we have scaled back down to largely the two of us, aided by very specific external folk, when it makes sense.Mike: And don't forget April.Corey: And of course. I'm talking delivery.Mike: [laugh].Corey: There's a reason I—Mike: Delivery. Yes.Corey: —prefaced it that way. There's a lot of support structure here, let's not get ourselves, and they make this entire place work. But why did we scale up? And then why did we scale down? Because I don't believe we've ever really talked about that publicly.Mike: No, not publicly. In fact, most people probably don't even notice that it happened. We got pretty big for—I mean, not big. So, we hit, I think, six full-time people at one point. And that was quite a bit.Corey: On the delivery side. Let's be clear.Mike: Yeah. No, I think actually with support structure, too. Like, if you add in everyone that we had with the sales and marketing as well, we were like 11 people. And that was a pretty sizable company. But then in July this year, it kind of hit a point where I found that I just wasn't enjoying my job anymore.And I looked around and noticed that a lot of other people was kind of feeling the same way, is just things had gotten harder. And the business wasn't suffering at all, it was just everything felt more difficult. And I finally realized that, for me personally at least, I started Duckbill because I love working with clients, I love doing consulting. And what I have found is that as the company grew larger and larger, I spent most of my time keeping the trains running and taking care of the staff. Which is exactly what I should be doing when we're that size, like, that is my job at that size, but I didn't actually enjoy it.I went into management as, like, this job going from having never done it before. So, I didn't have anything to compare it to. I didn't know if I would like it or not. And once I got here, I realized I actually don't. And I spent a lot of efforts to get better at it and I think I did. I've been working with a leadership coach for years now.But it finally came to a point where I just realized that I wasn't actually enjoying it anymore. I wasn't enjoying the job that I had created. And I think that really panned out to you as well. So, we decided, we had kind of an opportune time where one of our team decided that they were also wanting to go back to do independent consulting. I'm like, “Well, this is actually pretty good time. Why don't we just start scaling things back?” And like, maybe we'll scale it up again in the future; maybe we won't. But like, let's just buy ourselves some breathing room.Corey: One of the things that I think we didn't spend quite enough time really asking ourselves was what kind of place do we want to work at. Because we've explicitly stated that you and I both view this as the last job either of us is ever going to have, which means that we're not trying to do the get big quickly to get acquired, or we want to raise a whole bunch of other people's money to scale massively. Those aren't things either of us enjoy. And it turns out that handling the challenges of a business with as many people working here as we had wasn't what either one of us really wanted to do.Mike: Yeah. You know what—[laugh] it's funny because a lot of our advisors kept asking the same thing. Like, “So, what kind of company do you want?” And like, we had some pretty good answers for that, in that we didn't want to build a VC-backed company, we didn't ever want to be hyperscale. But there's a wide gulf of things between two-person company and hyperscale and we didn't really think too much about that.In fact, being a ten-person company is very different than being a three-person company, and we didn't really think about that either. We should have really put a lot more thought into that of what does it mean to be a ten-person company, and is that what we want? Or is three, four, or five-person more our style? But then again, I don't know that we could have predicted that as a concern had we not tried it first.Corey: Yeah, that was very much something that, for better or worse, we pay advisors for their advice—that's kind of definitionally how it works—and then we ignored it, on some level, though we thought we were doing something different at the time because there's some lessons you've just got to learn by making the mistake yourself.Mike: Yeah, we definitely made a few of those. [laugh].Corey: And it's been an interesting ride and I've got zero problem with how things have shaken out. I like what we do quite a bit. And honestly, the biggest fear I've got going forward is that my jackass business partner is about to distract the hell out of himself by writing a book, which is never as easy as even the most pessimistic estimates would be. So, that's going to be awesome and fun.Mike: Yeah, just wait until you see the dedication page.Corey: Yeah, I wasn't mentioned at all in the last book that you wrote, which I found personally offensive. So, if I'm not mentioned this time, you're fired.Mike: Oh, no, you are. It's just I'm also adding an anti-dedication page, which just has a photo of you.Corey: Oh, wonderful, wonderful. This is going to be one of those stories of the good consultant and the bad consultant, and I'm going to be the Goofus to your Gallant, aren't I?Mike: [laugh]. Yes, yes. You are.Corey: “Goofus wants to bill by the hour.”Mike: It's going to have a page of, like, “Here's this [unintelligible 00:25:05] book is dedicated to. Here's my acknowledgments. And [BLEEP] this guy.”Corey: I love it. I absolutely love it. I think that there is definitely a bright future for telling other people how to consult properly. May just suggest as a subtitle for the book is Consulting—subtitle—You Have Problems and Money. We'll Take Both.Mike: [laugh]. Yeah. My working title for this is Practical Consulting, but only because my previous book was Practical Monitoring. Pretty sure O'Reilly would have a fit if I did that. I actually have no idea what I'm going to call the book, still.Corey: Naming things is super hard. I would suggest asking people at AWS who name services and then doing the exact opposite of whatever they suggest. Like, take their list of recommendations and sort by reverse order and that'll get you started.Mike: Yeah. [laugh].Corey: I want to thank you for giving us an update on what you're working on and why you have less hair every time I see you because you're mostly ripping it out due to self-inflicted pain. If people want to follow your adventures, where's the best place to keep updated on this ridiculous, ridiculous nonsense that I cannot talk you out of?Mike: Two places. You can follow me on Twitter, @Mike_Julian, or you can sign up for the newsletter on my site at mikejulian.com where I'll be posting all the updates.Corey: Excellent. And I look forward to skewering the living hell out of them.Mike: I look forward to ignoring them.Corey: Thank you, Mike. It is always a pleasure.Mike: Thank you, Corey.Corey: Mike Julian, CEO at The Duckbill Group, and my unwilling best friend. I'm Cloud Economist Corey Quinn and this is Screaming in the Cloud. If you've enjoyed this podcast, please leave a five-star review on your podcast platform of choice, whereas if you've hated this podcast, please leave a five-star review on your podcast platform of choice along with an angry, annoying comment in which you tell us exactly what our problem is, and then charge us a fixed fee to fix that problem.Corey: If your AWS bill keeps rising and your blood pressure is doing the same, then you need The Duckbill Group. We help companies fix their AWS bill by making it smaller and less horrifying. The Duckbill Group works for you, not AWS. We tailor recommendations to your business and we get to the point. Visit duckbillgroup.com to get started.Announcer: This has been a HumblePod production. Stay humble.

Patrick talks with a man who has been homeless for 10 years, a homeless truck driver, and a homeowner with tenants who stopped paying rent. They talk about drug addiction and the importance of going to confession. Rob - I've been homeless for 10 years and have been making it work well. Albert - I am a truck driver and technically I am homeless because my wife divorced me and I gave her the house so she could raise the kids. I am now back on my feet but still not wanting to buy a house because of the market and my Job as a trucker. Mike - Because of you, I went from being a homeless drug addict to starting my own business and making a career for myself. Michelle - What should I do if someone is not paying rent? I don't want to kick them out but I need to make money from their rent? Alicia - Jordan Peterson posted a message to churches encouraging men to return to church. Mike – You can't get out of drug addiction alone. You need a sponsor and a spiritual advisor Mike - People who are homeless choose to be there and don't want help. I don't want to enable drug addicts. Terri - I went to confession and had a complete healing from my addiction Brother Sean - I am a religious brother and we work with the homeless on the street.