Rounding Up

Welcome to “Rounding Up” with the Math Learning Center. These conversations focus on topics that are important to Bridges teachers, administrators and anyone interested in Bridges in Mathematics. Hosted by Mike Wallus, VP of Educator Support at MLC.

Season 4 | Episode 17 – Jana Dean & Heather Byington, Supporting Multilingual Learners During Number Talks 05/07/2026

Season 4 | Episode 17 – Jana Dean & Heather Byington, Supporting Multilingual Learners During Number Talks Jana Dean & Heather Byington, Supporting Multilingual Learners During Number Talks ROUNDING UP: SEASON 4 | EPISODE 17 What might it be like to engage in a number talk as a multilingual learner? How would you communicate your ideas, and what scaffolds might support your participation? Today, we’re talking with Jana Dean and Heather Byington about ways educators can support multilingual learners’ engagement and participation during number talks. BIOGRAPHIES Heather Byington has taught all grade levels over the span of her 27-year career as a bilingual public educator. She currently teaches middle school mathematics and English language support classes in Lacey, Washington. She is also a student at Washington State University pursuing a PhD in Mathematics Education. Jana Dean currently serves as CEO of the Mathematics Education Collaborative and supports a fantastic team of middle school math teachers in North Thurston Public Schools. Her research focuses on the intersection of content learning and language learning. RESOURCES research blog article website Jana Dean email TRANSCRIPT Mike Wallus: Welcome to the podcast, Jana and Heather. I am so excited to be talking with you both today. Jana Dean: Good morning. Yeah, thanks for having us. Heather Byington: Thanks so much for having us. Mike: Absolutely. Jana, before we begin talking about the ways that teachers can support multilingual learners during number talks, I wonder if you can offer a working definition that would help educators visualize what a number talk actually looks like. Jana: Yeah, I'd be happy to do that. A number talk in terms of how we worked with the routine in this project consisted of the teacher providing some sort of visual prompt, starting either with a visual pattern of dots or a computation problem. And then the students get wait time, time to think about how they might solve that problem. And then as they share their strategies, the teacher records and asks them questions about their reasoning for why they approached the problem in the way that they approached it. The teacher creates what I like to think of as a visual mediator of student ideas. So the students’ ideas become visible as they share them. So children who are listening can listen to the dialog or conversation between the person sharing and the teacher, but the ideas actually become visible as they're being shared. And the teacher always verifies with the student whether or not they've been understood. And the goal is not for the student to be right, but for the teacher and student to understand each other. Mike: That's really helpful. Heather, is there anything else you'd add to that? Heather: In terms of the way that we worked with it with multilingual learners and increasing their opportunities for engagement in the routine, we always gave them an option of talking to a partner and rehearsing their answer before they volunteered to share with the whole group. We prioritized calling on multilingual learners if they volunteered. And we also did a final reflection at the end. So those were some enhancements that we added onto the routine. Mike: I think that's really helpful and I'm excited to talk a little bit more about the details of those, Heather. One of the things that really struck me as we were preparing for this conversation was reading about the ways that some of the multilingual learners you worked with, how they described their experience during number talks. And it helped me to see the experience from their perspective and rethink some of the ways that I'd facilitated number talks in the past. And I'm wondering if you could share a bit about some of the feelings students told you that they were experiencing. Jana: Yeah. One of the things we suspected before we started was that as a language learner myself, talking about ideas that you're just forming in a language you're in the process of learning can be really intimidating. It's very challenging. So they were nervous. And when I interviewed fourth graders about their experience in number talks, even facilitated with language acquisition in mind, they talked about how much courage it took them to share their ideas. They also talked about and could very keenly remember moments when they had made a contribution that their teacher made use of or a time when they made a contribution that another student made use of later. So there was a lot of pride they felt in having shared their ideas once they found ways to do that. They also talked about how much easier it was to share our ideas than it was to share my idea. And so if, for instance, we had given them the opportunity—and like Heather said, we almost always gave them the opportunity to talk with a partner—they would often share using the pronoun “we.” “This is how we thought of it.” And we picked up on that and began to ask them if it was OK to attribute a group of students with a unique idea rather than an individual. And that was also consistent with many of their home cultures. It's not every culture in which individual contributions are elevated, but rather when you dare to speak, you're definitely speaking for the group, for a collective. So that collective understanding was really important. There was one child, and I'm really curious about how representative he was of many. He always talked to the same friend, and every time he shared, he, I'm going to say, nailed it. He really had it figured out what it was that he was going to say. And there was one particular day when he did a beautiful job sharing, and I asked him about that day and he said, "To be honest, that day I really didn't want to share, but I knew my teacher wanted to hear my idea, so I did anyway." And so there's that element of love and respect for their teacher that I think was also really motivating for them. Heather: Yeah. Can I add something quickly to that? So one aspect of that, I think that idea of a student sharing because it meant a lot to the teacher, we also tried to utilize individual conferring with students as much as possible and gave them opportunities to confer with us, whether it was just checking in briefly before the number talk started, encouraging them or maybe telling them, "Hey, you can share the idea with me after the number talk if that feels more comfortable to you." So it's giving them multiple opportunities to do that and encouraging them to share their thoughts. Mike: What I appreciate about what you all are doing is even in this initial part of the conversation, really getting specific about the practices and the way that those practices played out for kids. And I think as an educator, one of the things that I've come to over all my years teaching is the need to have humility and also continue to be a learner. And that sometimes really leads me to questions about intent versus impact. Heather, I wonder if you could talk about the parts of the number talk routine or facilitation practices that may have unintentionally provoked some of the anxiety that kids were experiencing. Heather: So for multilingual learners, when I think about what they will need, the supports that they may need to be able to engage in a routine like a number talk, I think about first the processing time that they might need to understand and think about different ways of solving that prompt. And then I think about their understanding of the prompt. And then the other thing I think about is their ability to communicate their thoughts and ideas with others. So naturally, if it seems like there's a lot of pressure because of time, if they don't have much time, if they feel that pressure to do that processing and think of those ideas and share them quickly, that may provoke anxiety because this, of course, is still a language that they're still developing. So that ability to share with a partner and rehearse those ideas and process that with a partner, that really becomes, as Jana mentioned, more of a team effort. And then being able to rehearse the words that they're going to use and the way they're going to convey that message and communicate it to others, that again reduces the anxiety because it's a lot less pressure to share my thoughts and ideas with one person than with a whole group. And if I share those thoughts with one person and they seem to understand what I mean, then now I might feel confident enough to share with more people. So I just think that naturally when it's a time constrained activity, that that naturally can provoke anxiety. Mike: Yeah. I mean, that absolutely makes sense. I will say as a child who was not quick, even in my first language, the impact of that was profound, let alone trying to both process in a language that I was learning and feel like I was under pressure to produce an idea and describe it. That absolutely makes sense. Jana: I want to back up a bit and quote something that you said, Heather, partway through our working together, which was that Heather had some familiarity with number talks before we started working together, but had a healthy skepticism as well. And at one point she said that she wondered if we might not actually be hurting students when we are facilitating a routine that they cannot find entry into. And so it became really like a guiding light or principle of our work together to work hard to help them find entry into the routine. And something that I didn't realize until a year after we began working together and I was really closely tracking the experiences of the multilingual learners themselves—and this is kind of back to your question about intent and impact—when we listen to children's mathematical ideas with the intent of not correcting them, trying to figure out what's right and what makes sense to them, we have to ask them questions about what their ideas are. And for many of the multilingual learners, engaging in that process itself was a huge lift language-wise. So I'm not just going to be able to say the answer or tell my teacher my strategy; I'm going to have to stick with my teacher until my teacher actually gets it. And a few of the multilingual learners that I followed over the course of a year actually said to me, "I don't like it when my teacher doesn't understand me." So while we absolutely, 100%, our intention is golden. It is about understanding them. But putting them in that position of that negotiating meaning with us until we do understand takes a great deal of trust on the part of the student. And so it's on us to develop that trust so that they're willing to do that with us. Mike: I think that's a good segue because Jana, going into this, you mentioned three big ideas as starting points for supporting multilingual learners. One was negotiated meaning, one was the notion of voluntary sharing, and the last was the idea of using ambiguity as a resource. And I wonder if we can start this next part of the podcast with having you describe each of these for the listeners. Jana: Yeah, absolutely. Voluntary sharing means I've made a commitment to not ever put you on the spot as a student. And so any one of us who has learned a second language—which I've learned a couple, none of them to a super high level—but most people can relate to, say, standing in line in a grocery store and rehearsing what you're going to say so that you ask for the bag you want rather than the receipt that you don't want. There's a process in coming to speak, and I think there's a process in coming to speak publicly for just about every learner, especially about ideas that you're in the process of forming, but that pressure—and I've had many, many students over the year thank me for being the kind of teacher in a kind of classroom where they knew that I wasn't going to call on them unless they had volunteered to share. So the level of distraction, I think that that, again, well-intentioned pressure causes, is absolutely not worth it, and especially not for our multilingual learners. Negotiated meaning really is the process of coming to understand each other, and we do it all the time. Unfortunately, often in classrooms, we end up in discourse routines that are actually not about teachers understanding students. They're about teachers asking questions for which students are supposed to have answers, which then the teacher evaluates. So what I would argue that the number talk routine turns that discourse pattern, which is often called I.R.E.—initiate, respond, evaluate—absolutely on its head. The child volunteers their idea, the teacher responds by trying to understand it as best they can, and then the student is the evaluator of whether or not the teacher actually understood them. Mike: Heather, I was hoping we could go granular on a couple pieces that I heard you talk about too. You talk a lot about something very practical, the value of predictability, and I wonder if you can talk about how predictability impacted students and what does that mean for the teacher? Heather: Absolutely. When facilitating these number talks with this goal of engaging multilingual learners or helping them find those entry points, I found it helpful as a facilitator to utilize similar types of approaches to statements I would make during the routine, and then similar ways of asking students if I was seeing things the way that they were seeing them. It seemed to help the students that we were really hoping to engage to feel more comfortable with what was happening in the routine and to lean in more to that engagement. So I think that that is one thing as a facilitator to be aware of. Jana, can you think of anything else that we haven't talked about yet? Jana: There's the whole knowing the rules of the game aspect of really any classroom routine or instructional routine. So if the student knows how this thing goes, whatever “this thing” is, then that lifts off some of the cognitive load in terms of participation because they don't have to be figuring out how to participate. writes about this a lot in her research, and I think she calls it the “sociocultural aspect of learning mathematics,” and she uses the word “ecological”. So the environment itself really matters. And in community, our social environment is made up of all kinds of routines. So I think that part of it is important. My favorite metaphor for it is learning a new card game. The first time you play the game, it is no fun because all you're doing is trying to figure out how the cards move, how the turns go, what the rules are, and how you can play. You can't do any strategy at all. But then as you learn the game, then you can really engage in it in a thoughtful way and have fun with it. So I really think that classroom routines are like that and not only for multilingual learners, but I have the privilege of being an instructional coach now in a middle school and have seen teachers engage in routines that I can tell are 100% soothing of trauma that students have as they come into the classroom, just because they know what to expect. So not only are those kinds of regular routines really helpful for multilingual learners, but they're also trauma-informed teaching. And when I say “routine,” it can be easy to misunderstand and think it's boring. It has to be an open-ended routine so that something inside it that is engaging and fun can happen. Heather: There are a couple of other things that occurred to me in terms of the students participating in the routine. I know that they started to see that we were elevating the status of gestures in terms of the communication to be another way to visualize the thinking in terms of the processing for themselves, but also a way to help others see what they were seeing and to understand their ideas. So that was one aspect of the routine that they could count on, that they could utilize gestures if needed, and that we would reinforce that. If they didn't have a mathematics label for the terminology that would typically be used in that conversation about those mathematics ideas, they could rely on describing what they understood, and then either I, the teacher, the facilitator, or another student, providing those words and the opportunity to practice that specific mathematics language within that routine. So those were some other things that were predictable and happened across all of the different number talks that happened, no matter what the prompt was. Mike: You're making me think that part of what a teacher might do in response to this conversation is really to think about some of the things that they want to make normal, right? Like this notion of using gestures is both normal and accepted and valued. The idea that you are going to use rough draft, informal language, and that's OK, and that's a way that we get to more technical language of mathematics, and that's normal. And so thinking about what are the things that I want to become normal and predictable for kids, maybe homework recommendation number one for an educator that might be listening in. Heather: So another thing that was predictable was the utilization of color-coding. And this is something that many teachers probably do already. But we did, when we were recording the students' ideas, we used different colors for each student, and that made it more accessible. Again, it was a support for our students to be able to distinguish between different chunks of information on the board as they were looking at each other's responses and reflecting on those responses. So really reading that. Mike: Can I ask for a clarification on that, Heather? Heather: Absolutely. Mike: I think what you mean is that you use different [colors] to represent different students' contributions. So if a student shared something, you might write it in red, and if it was a different student, it might be in green. And then you can distinguish what contribution each student made. Heather: Yes. Yes, that was a predictable aspect of the routine, as well as Jana had mentioned earlier, attributing the ideas to students using their initials. And if multiple students contributed to that idea and the original person who was sharing said that, yes, they would like to attribute more people, then we included all the people's initials who contributed to that idea that was shared in that number talk for that idea, that communication. Mike: Speaking of contribution, I want to name something that we talked about in our preparation for this that seems incredibly simple but felt like it was really significant. You all talked about the importance of the teacher consistently—not just once, not just a handful of times—but consistently, on the regular stating to kids that they wanted to hear from all students. And I wonder if you can just talk about what did this sound like to make that happen and what was the impact on kids? Jana, I think this is one I'd love for you to start with. Jana: Yeah, absolutely. It is simple. All you say is, "I'm so glad to be with you today. And let's remember that while we may not hear from everyone today, it's our goal to hear from almost everyone over the course of the week." And if you as a teacher have made a commitment to voluntary sharing, it's essential to say that, to really tell them that you do want to hear their voices. You need to tell them that. Otherwise, they're not going to know that you want to hear their voice. And like I shared a little while ago, there was one student who actually said to me, "I didn't want to share that day, and I knew my teacher wanted to hear from me, and so I did." And then in reflecting back on that share, to get at students' perspectives on what number talks have... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/41183905

Season 4 | Episode 16 – Kristin Frang, Understanding the Roots of Fluency with Addition & Subtraction 04/23/2026

Season 4 | Episode 16 – Kristin Frang, Understanding the Roots of Fluency with Addition & Subtraction Kristin Frang, Understanding the Roots of Fluency with Addition & Subtraction ROUNDING UP: SEASON 4 | EPISODE 16 Research suggests that supporting students’ fluency with addition and subtraction hinges on understanding how children’s mathematical thinking develops. So what are the concepts and ideas that play a part in fluency with combinations to 10, 20, and beyond? Today, we’ll explore this question with Kristin Frang, director of instructional programs at Integrow Numeracy Solutions. BIOGRAPHY Kristin Frang is the director of instructional programs for Integrow Numeracy Solutions. She designs resources and services that support states, districts, schools, and individuals in transforming numeracy education. RESOURCES Season 4, Episode 11 of the Rounding Up podcast Integrow Numeracy Solutions book by Lucinda “Petey” MacCarty, Kurt Kinsey, David Ellemor-Collins, and Robert J. Wright TRANSCRIPT Mike Wallus: Welcome to the podcast, Kristin. It is so great to be talking with you today. Kristin Frang: It’s great to be here. I feel so honored to be on this podcast. Mike: Before we dive into a conversation about addition and subtraction, I'd like to do a bit of grounding. So you're currently the director of instructional programs for Integrow Numeracy Solutions. I wonder if briefly you could tell the listeners: What is Integrow Numeracy Solutions, and what's its mission? Kristin: Yeah. Integrow Numeracy Solutions’ mission is to transform numeracy education by connecting research with practice and empowering educators to advance student mathematical thinking and success. But I really want to bring that mission to life through a story, just a quick story, if I can. Prior to my role with Integrow, I was a K–12 mathematics consultant. And one of the things that I did was, when the Common Core [State Standards] were released, I worked with teachers to transition to the then-new standards. We studied many documents together, including progression documents that were included in the standards, and teachers were honestly fascinated by this idea of a progression and that they were embedded into the standard. But I remember an instance where we had been studying these progressions and a teacher came up and said to me, "I know where my students are at; I can see them in these progressions. But how do I get them to the next stage?" And I didn't have an answer (laughs) at that point. I was a former middle school and high school teacher. I was working with elementary teachers. I was studying, just like them, these progression documents, and I could only categorize the reasoning that was in front of us. And so that next step to say, "Oh, this is what I would do and bring into action in the classroom,” I didn't have an answer for. And so that's really where I was introduced to Integrow—formerly [the] US Math Recovery Council, but now Integrow Numeracy Solutions. And at the heart of our mission to empower educators is to bring research to the classroom in accessible and practical ways that advance student reasoning. We do this in professional learning, we do it in supplemental resources, and we also hire and train educators to deliver high-dosage tutoring for students to accelerate their learning. Mike: I want to just linger on something you said, which was—and I really appreciate both the truth of the statement you made and also the vulnerability, which is to say—I think for many teachers, there's this experience of, “I can see my students in these progressions, but I'm not sure what to do when it comes to making moves to shift where they're at or help them move.” And I think that's a profound truth for so many teachers. And I think it's really important that folks like you, who are doing this work, acknowledge that that's a place you were in once as well because that's so true for so many of us. Kristin: Yeah. There's always a new thing where we're watching students, we're thinking about the next steps. And so often it boils down to categorizing the things that students are doing now, but not often figuring out: What are the true actions that we take with real children who are in front of us to get them to progress in their own reasoning? We can tell them the next step, but my belief system that is aligned with Integrow Numeracy Solutions is that the most powerful thing is to help students have those experiences and create that understanding themselves. And to do that, it's more complex than just knowing what the next benchmark is for them. Mike: I think that's a helpful introduction. And I also find it to be a good segue for all the questions that I wanted to explore today. So let me start here: It feels important to acknowledge that supporting students' addition and subtraction fluency actually hinges on understanding how children's mathematical thinking develops. So I wonder if you can talk about some of the concepts and the ideas that play a part in fluency when it comes to combinations of 10, combinations to 20, and even beyond. Kristin: Yeah. The words that we hear associated with fluency right now are “flexibility,” “efficiency,” “accuracy.” So we've moved on from just speed, which I think is a really positive place for us to be in education. But at the heart of flexibility, efficiency and accuracy is a quantitative understanding of arithmetic. I'm really glad that you had Amy Hackenberg on [the podcast] recently who discussed this concept of units coordination because throughout what we'll talk about, you'll see units coordination come out, but she's definitely the expert to explain it. Just a nod. Just [Season 4, Episode 11]. It was amazing. Thinking, though, specifically about fluency—fluency isn't just knowing all of these combinations. In the early stages of counting, students view a number simply as a count or result of a count of single items, and there's this critical shift in developing a unit as a fundamental tool of measurement. And that's the act of unitizing where a student conceives of a collection of items as one unit that's simultaneously made of smaller units. It is a common progression that once a student counts on, that then we would shift to building strategies to solve addition and subtraction within 20, and then of course with 100, and beyond, and then in other domains. But this is all happening in first and second grade for that addition and subtraction to 20 fluency. So attending to this numerical composite—understanding that when a child says “7” and sees that that represents counting from 1 to 7 without having to count—is a really big cognitive shift in their mathematical understanding and can be undermined with, “Oh, now that they're counting on, we're going to tell them these strategies.” And so we really do need to have some intentional instructional strategies to make sure that we're developing that first, that numerical composite, before we try to develop all these strategies for addition and subtraction to 20. Because that is the basis for children to move from a counting-based strategy to compose units. So when they can use a quantity like, "Oh, 8 plus 5, I can break apart this 5 into smaller parts and I can give some of those parts to the 8." So children at that point have to simultaneously hold 5 as a single unit while recognizing the 2 and the 3 make up the 5, but they can be moved to the 8 as well. That's really sophisticated. Mike: So I want to mark that because I think the notion that this is really sophisticated is important for folks to understand because I'll be vulnerable and honest: I didn't recognize the complexity of what children were grappling with when I started teaching, particularly as a person who was teaching kindergarten and first grade. I really saw my job as helping to build a set of rote procedures like counting and number sequence and memorizing combinations and the outcome of being able to count and the outcome of being able to quickly recall those. I think that's not in question, but understanding the mechanics and the evolution of kids' thinking that's going on, that's a big deal. This whole notion that you have a unit and the unit is composed of smaller units. And one of the things that you said that feels like a really big deal that could be lost is the idea that shifting from a counting-based strategy to a strategy that depends on this notion of units that have smaller units inside and that are also still a unit—that's such a big deal. In order to go from counting everything to counting on to being able to look at a number like 8 and say that it has a 5 and a 3 inside of it—all of that is connected to this notion of units inside of units. And I'm so glad you mentioned that. Kristin: Yeah. The mental actions that students are doing, making those visible, when we see children do it developmentally, we just assume it's easy. But the shifts that they're making in their understanding of units to move from that pre-numerical stage of “Everything is a 1 and I have to repeat it” to “Now this word can stand in for the count” to “Now I can embed units inside of other units.” There's so much happening, and they're so young at that age; we have to remember that too. Mike: So let's talk about some other important components of developing fluency. What else is an important primer for how people are thinking about this? Kristin: Yeah. Another important component is supporting students in developing the cognitive structures that allow students to anchor their understanding and quantitative meaning and develop that sophisticated reasoning. Many researchers, many authors have written in different ways and different names about these structures. So like a “mental structure,” “mental residue,” “mental tools,” “patterns of thought.” To name a few people, , , are some people I've read and appreciate their thinking around that. So it's more than just allowing students to use manipulatives to solve problems. There's an intentionality in how we use tools and an explicit process used by educators to bring their mathematical world to life. So first, identifying key settings that emphasize mathematical structures. So the tool in front of them has a big role to play in the “math”—I put that in quotations—in the “math” that they see. 10-frames that highlight a quantity of 10, but also can show other quantities within 10, such as, like, a five or a double. It has an added layer of boxes that contain a number. Some contain a number or a counter and others are empty. So there's ways that kids are coming to understand quantity with the structure. Similarly, a bead rack can show a five structure, a double structure, depending on your representation. They can help kids think about exchanges and really kind of that movement of quantity in a real physical way. Using linking cubes, do you use them all in one color? Are you strategic about the color that you use to bring out mathematical structures for them? So once we think about the key setting and the structure that we're trying to help kids reason about, we want to pose intentional questions that orient students to those structures. So how do they see that 5 inside? How are we going to bring that out? It's obvious to us, but are they seeing that or are they seeing something different in the tool? Are they reasoning about something different? And so the intentionality behind how we question students during those activities also aids to building their cognitive structures. So it's not the tool itself that is the 8. It's that the child is seeing the 8 and they're seeing the 5 and the 3 in some empty boxes. And finally, I think the step that we miss a lot, especially in problem-based instruction or any kind of inquiry-based instruction, is this explicit time where we connect the symbols in formal mathematics directly to represent the child's thinking and the tool that they've been playing around with. So it's not just about knowing I can get an answer on the 10-frame, but it's [that] I'm abstracting that series of actions, and I'm then connecting it to this quantity that I've written in a symbol. And are there connections between those things? And if those things aren't happening—kids are doing all those parts and pieces, but really developing the cognitive structure that they can then themselves use and take with them, I think that's what's so powerful when we talk about fluency is they can take a cognitive structure with them and fill in the mathematics in the future [when] maybe they don't have an educator in front of them asking those questions. But if they've been through those processes, then they have that structure to fill in. Mike: There's a lot that you just said that I think is important and we could probably linger on a lot of it. But on the front end of this conversation, you said it's one thing to be able to see students in a progression, and it's another thing to think about, “What's my role or what are the tools that I have to help them shift?” What I heard in that last part, particularly is this notion of almost like a translation between the physical materials kids are engaging with and the meaning that they're making of that, and then helping them to abstract that in a way where we have symbols that are representing either actions or quantities and the relationships that are happening. That part of the teacher's job and part of the moves that teachers have in their toolbox is this notion of translation—taking what I'm seeing kids doing and how what I'm hearing them say or do to make meaning of it, and then helping them make that abstraction is kind of one of the tools that's really important in a teacher's toolbox when they're thinking about helping kids make moves. In preparation for our interview, one of the things that stayed with me was you described how your own understanding of the meaning and the importance of fluency had shifted over time. And I'm wondering if you can talk about what you used to think and what is it that you think now about fluency. Could you talk about your own personal journey? Kristin: For sure. I used to think that knowing facts, just knowing them in a very static way—like I know the answer to 5 plus 3, I keep coming back to that fact—reduces the cognitive load when they were getting into higher grade levels. Well, they don't need to think about that problem, and they can think about what we're doing in seventh grade math or in algebra. But what I've come to understand is that the ways that students know their facts—more specifically how they're able to work with the units and the way they conceptualize the units that they are given, how they break them apart, how they put them back together—that's what matters as they go. So not just knowing the answer, but that these things can be taken apart and put back together. is a researcher that I really love to listen to. And I listened to him at an Integrow conference once. And he talked about developing mathematics through repeatable mental actions. So this kind of relates back to those cognitive structures. One example of a group of mental actions is this idea of composable, reversible, and associative. So when I think about 8 plus 5, 5 is composed of a 2 and a 3, and I can reverse that to focus on the unit of 2, and then I can associate that quantity with the 8 to make a new unit while keeping intact the unit of 5. That's really complex, but that idea transcends the domains of mathematics. Now, I'm not an expert in units coordination research, so I hope I represented that correctly, but I've certainly experienced students struggling to keep track of different units as they work. So thinking about exponent rules, and they break apart these powers and they're writing them and they're learning all these patterns, but they're struggling to keep track of the units that they're working with. Factoring functions in algebra. We're asking them to break apart something and put it back together in these different forms, and they're losing track of these units. So these actions of composable, reversible, and associative have implications in many domains of mathematics. So the bottom line is we want to develop not the fact itself, but the mental action behind that fact. Anderson Norton, I hope I did that justice. Mike: I want to name something that I think is really important, particularly given the fact that your background is actually in secondary [education]. So what I take from this is this idea of working with units and the mental actions, that transcends arithmetic. It transcends whole numbers and even rational numbers. And it pays dividends and it keeps paying dividends in middle school and high school as kids are working in an algebra context. And I think that's worth saying out loud because it means that doing this work with elementary students to develop fluency is a bit of a twofer in the sense that you do get kids who end up with a bank of facts that they know, but they also have this underlying understanding of units and actions that pays dividends for them in the long run. Mathematics education, students' learning experience, is not a sprint or a series of handoffs. It's really a marathon. And those early experiences, they pay dividends and they keep paying dividends. I think that's really important because it reminds us, particularly as elementary educators, that we're part of a larger project. Kristin: Not only part of a project, but part of building a lifelong interest in mathematics as an actual body of research that's dynamic and not a set of things to memorize and learn so that mathematics does become applicable in these different fields because the way that I approach a problem as an expert mathematician is that I take things apart, I put them back together. That transcends many careers. It's not just about being a math teacher or a math professor. It's about coming to understand that I have autonomy and how I see relationships of things, whether they're numbers or shapes or maybe parts that I'm working on in some sort of creative field that I'm in, but that I can do all of these things and that I can be curious and repeat those actions and see how they play out in that particular study. Mike: That's well said. Well, let's talk about the what, the why, the how of combinations to 10 and 20. To begin, I want to note that we use the term “combinations,” and I'm wondering if you can say more about what you mean when you refer to combinations and why they matter. Kristin: Yeah. I mean combinations not to literally mean “addition,” but that combination is the idea of this relationship between parts and wholes. So that 2, 3, and 5 have this kind of additive relationship. I can put these parts together to make the whole; I can take a part out of the whole and be left with a part. I can have a part and wonder what part I need to make the whole. And so we sometimes talk about these in curriculums as “fact families,” but the emphasis should be on the relationship of the parts to the whole and not filling out that kind of mimicking of like, “I know the four sentences because I know this thing.” So, “If I know this, I also know this.” It feels really nuanced, but in action really quite specific. Mike: So I think that's really helpful and it really does lead me to my next question about how we help kids build their fluency with combinations to 10 and 20 and beyond. So given the why that you just articulated, it seems like the how is going to be substantially different from the ways that many, if not most, adults learn to build fluency. Can you talk about that, Kristin? Kristin: We start from key... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/40979965

Season 4 | Episode 15 – Dr. DeAnn Huinker & Dr. Melissa Hedges, Math Trajectories for Young Learners, Part 2 04/09/2026

Season 4 | Episode 15 – Dr. DeAnn Huinker & Dr. Melissa Hedges, Math Trajectories for Young Learners, Part 2 DeAnn Huinker & Melissa Hedges, Math Trajectories for Young Learners, Part 2 ROUNDING UP: SEASON 4 | EPISODE 15 Research confirms that early mathematics experiences play a more significant role than we once imagined. Studies suggest that specific number competencies in 4-year-olds are strong predictors of fifth grade mathematics success. So what does it look like to provide meaningful mathematical experiences for our youngest learners? Today, we'll explore this question with DeAnn Huinker from UW-Milwaukee and Melissa Hedges from the Milwaukee Public Schools. BIOGRAPHY Dr. DeAnn Huinker is a professor of mathematics education in the Department of Teaching and Learning and directs the University of Wisconsin-Milwaukee Center for Mathematics and Science Education Research. Dr. Huinker teaches courses in mathematics education at the early childhood, elementary, and middle school levels. Dr. Melissa Hedges is a curriculum specialist who supports K–5 and K–8 schools for the Milwaukee Public Schools. RESOURCES website, featuring the work of Doug Clements and Julie Sarama book by DeAnn Huinker and Melissa Hedges TRANSCRIPT Mike Wallus: A note to our listeners: This episode contains the second half of my conversation with DeAnn Huinker and Melissa Hedges about math trajectories for young learners. If you've not already listened to the first half of the conversation, I encourage you to go back and give it a listen. The second half of the conversation begins with DeAnn and Melissa discussing practices that educators can use to provide students a more meaningful experience with skip-counting. Melissa Hedges: One of the things, Mike, that I would add on that actually I just thought about is when you were talking about the importance of us letting the children figure out how they want to approach that task of organizing their count is it's coming from the child. And Clements and Sarama talk about the beautiful work about the trajectory, [which] is that we see that the mathematics comes from the child and we can nurture that along in developmentally appropriate ways. The other idea that popped into my mind is it's kind of a parallel to when our children get older and we want to teach them a way to add and a way to subtract, and I'm going to show you how to do it and you follow my procedure. I'm going to show it. You follow my procedure. We know that that's not best practice either. And so we're really looking at, how do we grab onto that idea of number sense and move forward with it in a way that's meaningful with children from as young as 1 and 2 all the way up? Mike: DeAnn, I was going to ask a question to follow up on something that you said just now when you said even something like skip-counting should be done with quantities. And you, I think, anticipated the question I was going to ask, which is: What are the implications of this idea of connecting number and quantity for processes that we have used in the past, like rote counting or skip-counting? And I think what you're saying is we need to attend to those things that, like the counting sequence, we should not create an artificial barrier between speaking the words in sequence and quantity. Am I reading you right or is there more nuance than I'm describing? DeAnn Huinker: I think you're right on target, Mike. (laughs) Connecting those things to quantity. And I mean, the one that's always salient for me is skip-counting. Skip-counting is such a rote skill for so many children that they don't realize when they go, “5, 10, 15” that they actually have seen, “Oh, there's five [items], there's five more items, there's five more items.” So it's making that connection to quantity for something like skip-counting, but also on the counting trajectory, then we start thinking about, “What's a ten? And what makes a ten?” And, “What is 30? And how many tens are composing or embedded in that number 30?” And again, it's not just to rotely say, “3 tens.” No. “Show me those objects. Can you make those tens?” Because sometimes we find disconnects. Kids will tell us things and then we say, “Can you show me?” And it doesn't match. (laughs) So we continually start thinking about quantities and putting [objects] with quantities. Let me add one more thing. In the counting trajectory—and this was very intentional for Melissa—is when we have kids count, we'd like to give them like 31 or 32 counters to see whether [...] they can actually bridge that decade and to go beyond. The other thing that we did, so getting like beyond a ten, also we find when kids get to the number 100, they stop. They just think that's the end. I got to 100, I'm going to stop. And then we say, "Oh, what would be the next number?" And some will say 110, some will say 200, some will give us something else that we find bridging 100 is on the trajectory. And that's actually a really critical point. And again, we want it with quantities with objects. Mike: I really appreciate this part of the conversation because I think for a teacher who's listening, it helps really get to the specific types of details that would allow them to create the kind of experiences that we think matter for children. I do want to take a step back though and talk about what's going on for students under the hood, so to speak. So as they're engaging in meaningful counting, what are the cognitive processes that they're learning to coordinate? Melissa: This is Melissa. So I'll start that one and then invite DeAnn to jump in as I work my way through my thinking. One of the pieces that, in addition to everything we talked about with all of the skills and ideas and understanding that comes to bear when little ones count, one of the big pieces that we're starting to talk and learn about a whole lot more is this idea of executive functioning. And so executive functioning are those skills that help us manage our attention, help us manage our behavior. They help us stay focused. They help us complete tasks, keep track of things. So hopefully as I'm saying this, what you have in your mind is a little one counting and you're thinking, “Oh my gosh, how do they know where to start?” “How do they know when to stop?” “How do they know when this has been counted with that hasn't been counted?” “What am I going to say next?” All of that tends to be couched very strongly in this idea of executive functions. So when we watch kids count, we know that they're really drawing on those executive functions. And it's actually a really beautiful marriage. So again, we're looking for kids to—are they able to stay on task? Can they keep track? Do they monitor themselves as they go? If someone—this happens a lot—if someone bumps into their collection and their collection gets a little shaky because their desk got moved or someone kicked a counter across the floor, do they remember where that goes and what that stood for in quantity? And for us, that really kind of comes down to some of those higher order skills and in particular, those ideas of the executive functions. So part of what we notice is that in particular with counting, though all of mathematics, much of what we do and ask kids to do, it takes planning, it takes self-monitoring, and it takes kind of a sense of control and agency over their work. We've talked a little bit about some of that other stuff in the way that it's the work of the child, and that's why we will always ask teachers to step back and just watch, just watch what they do, just watch what they do, because it gives us insight into so many skills, understandings, and kind of where they're at. DeAnn: Yeah. This is DeAnn. I was thinking of that same thing, Melissa, about this is the work of the child, right? As adults, we're kind of prone sometimes to say, “Let me show you how to do it.” But if we want to develop these executive function skills, these ideas and cognitive abilities under the hood, we have to give children opportunities. They need the time to think about how to organize that collection. That's always a great one to kind of think about. As adults, we're like, "Well, just line them up." And it's like, oh no, that's actually huge for a child to realize lining them up or organizing them in some way is a strategy, just like we do with larger numbers. It's a strategy for little kids. So again, that work needs to come from the child and they need to do some trial and error and adjustments in order to develop those things under the hood. And as adults, we can't take that opportunity away from children. We need to create the opportunities so they can explore more of their world and the quantitative world that we live in. Mike: Everything that we're talking about has some pretty major implications for instructional practice, but what I find myself thinking about is my own time teaching kindergarten. And when I reflect on that, I sometimes found myself falling into something that I would call a readiness trap. And what I mean by that is I had this notion that kids had to have a certain set of skills in place before they were ready to do something like counting a collection. And I think what you're going to tell me is that perhaps I had it backwards. Am I right? DeAnn: So this is DeAnn and I'm thinking, well, maybe it's not so much backwards, but it's a different perspective. So Melissa and I really struggle with this concept of readiness, and that's because we really frame our work from a developmental perspective. And as we think about learning trajectories, that's what they are. Learning trajectories is a developmental view of children's learning. So what really changes the question for us. We don't ask the question, “Are children ready?” What we ask is, “Oh, where are children currently in their learning?” And then we can start at that spot and then think about the experiences that would help support the next step in their learning. So from a learning trajectory perspective, we really view differences in children's understanding and abilities as just different starting points, OK? They're not deficits, nothing that needs to be remediated. Kids are ready to learn every single day. It's really us as adults. We have to reframe our preconceptions and train ourselves to really look at what children can do, not what they can't do. And that's where learning trajectories are so powerful because they help us identify those starting points and then they help us as educators know more clearly what is the next developmental milestone that we should be working on with that child. So it's our responsibility to be ready for the children that come to us, not the other way around. Mike: I really appreciate this idea of the progression as a series of starting points. I think that's a really helpful framing device, and it certainly puts the work that we do in the kind of light that you're advocating for. One of the other things I wanted to talk about is in the book [], you all make reference to how important it is to develop a playful pedagogy. And I wonder if we could just try to talk about, “Well, what does that mean? What might that look like in a classroom?” Melissa: So this is Melissa. I think in any district or agency that's supporting young children, this is a very hot topic, the idea of play or playful pedagogy. And what I like to do is to think that we can use play as a teaching platform and not just as a break from learning. Play actually can kind of lay the foundation for a lot of those learning experiences. I think it's powerful because playful learning, it nurtures important habits of mind that we can develop in some ways, but for our little ones, they develop very naturally through the idea of play. So we think about curiosity, creativity, problem solving, flexibility, persistence, all of that comes up as kids are playing. And so I think that both DeAnn and I would agree that the idea around playful pedagogy and mathematics learning trajectories really partner well because the trajectories help us see that mathematics develops over time based on experience and opportunity. So the trajectories don't replace play so much as [...] strengthen educators in recognizing during times when kids are playing or during those playful moments that an educator can have a stronger perspective or a more keen eye, I guess, on what they're noticing with their children. And when we think about playful pedagogy, where we're headed is not free play, but this idea of guided play. So in guided play, the teacher's going to set up the environment, they'll have a learning goal in mind. So for example, if I'm working and deepening my understanding as a classroom teacher around the counting trajectory, I'm going to have an idea of where my children are on the trajectory and what questions might I pose during play or ponderings might I provide to the children during play. So it's not me taking over that time or the teacher taking over that time, but it's really supporting or pushing the learning through some subtle prompts or some shared discoveries or maybe some purposeful questions. So, for example, if the kids are in the block area and they're building a tower or they just have blocks all over the floor, they're making a road, I might ask them, “How long is your road?” or “How tall is your tower?” and let them kind of ponder with that. And then, this is always a fun one, “What would happen if I put two more on?” or “What would happen if your tower grew by two more blocks? or “What would happen if three of them fell off?” And really just engaging in some of those playful conversations—not to take over the play, but to capitalize on the playful moment. Mike: I love that, particularly the notion of, “What if three fell off?” or “What if I had four more blocks and I wanted to make it bigger or longer?” It's a lovely way of organically injecting or assessing kids' thinking within the context as opposed to imposing a task in a way that it just has an entirely different feel to it. And yet at the same time, it's really informed by the trajectory in a way that helps it be like, “This is the right point for me to ask that particular question.” Melissa: Exactly. So I can kind of give an example if, I'm thinking of maybe a 5-year-old and so, one of the levels of our counting trajectory is being able to do 1 more or 1 less, and it really sits around that idea of hierarchical inclusion. So if the kids are playing and I know that that's where this child might need or this group of children are ready to take that next step, those are questions I can pose in a very—you're right—in a very low-stress, not-high-stakes setting, and it's still very valuable information. Mike: Actually, that's a great segue because I wanted to ask the two of you about some of the ways that teachers are using the learning trajectories and the assessment protocols that are found in the book to monitor their students' growth. So I wonder if you could say a little bit more about that. DeAnn: This is DeAnn. I'll start and then I'll pass it back to Melissa. So, you ask us about the assessment protocols. So maybe we should explain what an assessment protocol is. One thing that we've done with the trajectories that were developed by Doug Clements and Julie Sarama, we've taken those trajectories, but as we're thinking about making them useful for teachers, we actually have developed some structured assessment protocols that are aligned to the trajectory with [tasks] and prompts that we can use with children to help find those starting points. As I mention in the book, we have five assessment protocols in there, like one for counting, one for subitizing, one for adding and subtracting and so on. And then teachers can take these and use them to [say], “Let me ask this question. Oh, they did great there. Let me jump up a couple levels. Let me ask a question there.” Or maybe I want to back up to a previous level and ask so that we can kind of get a sense of those starting points for then building instruction. All right, and then Melissa, you can share how else teachers are using them in and out in the district. Melissa: I think one of the important aspects that I firmly believe in when a teacher approaches their teaching of mathematics through the lens of a learning trajectory, a mathematics learning trajectory, is that it really does lay the foundation for equitable teaching and learning opportunities. So not only does it lay the path for a developmental approach, it's also incredibly equitable in that we've looked at trajectories as identifying children's strengths. And in that way, it's not what they don't know, it's, “Where are they, and what are those [experiences] that they need?” So it's not that somebody is never going to learn it. Again, they need more experience and opportunity. And that's, I think, probably been one of the biggest takeaways as we've looked at how we are using trajectories here in the Milwaukee Public Schools, in particular the counting trajectory. So to get a really nice handle on where children are developmentally, if we have, for example, in a first grade classroom where they're moving into composing that unit of 10, and we know that we've got kids that are struggling with cardinality, even counting collections of one, two, three, four, five [objects], we know that that's going to be a struggle. So what is it that we can do to accelerate some of those learning opportunities and give more learning opportunities for children so when they get to those big key milestones, we have an idea of why they may be struggling? And it's not that they can't; it's not that they won't; it's not that they don't understand. They just need more experience and more opportunity and more guidance with that work. So that's one of the ways I think that has really allowed us to support our teachers and have our teachers feel a great sense of autonomy in making instructional decisions for their students. That it's not, “The book is telling me to do this or this is telling me to do that.” It’s, “Here is something that's really honored a developmental approach to what kids know, and how can I take that then and apply that in my classroom with my students?” The other thing that it really has helped us do on a big broad level is think about, “Where do we want children to work towards by the end of 3-year-old kindergarten or 4-year-old kindergarten or 5-year-old kindergarten or first grade or second grade in a way that, again, matches the developmental nature of children's mathematical growth?” Mike: What I really appreciate about what you shared is there's certainly the systems level way of thinking about using this as a tool, but I appreciate the fact that as an educator who might be reading the book, I can also see directly into my own classroom practice and think about moves that I can make to support students and also to understand where they are and what comes next for them. That's super helpful. Melissa: Yeah. It's those small little moments. It's really as, just staying keyed in and tuned to those small moments. Mike: I'm going to ask a question at this point in the interview that I suspect is difficult to narrow down an answer, but I want to give it a try just because there's so much from my reading of the book that was powerful. And at the same time, I'm hoping that we can give people a chance to think about how they might start to take action. So here's the question: If you were to, say, recommend one or two small-scale practices for listeners who want to take the ideas we're talking about and put them into action in their classrooms, what might you recommend? DeAnn: This is DeAnn. I'll get us... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/40762650

Season 4 | Episode 14 – Dr. DeAnn Huinker & Dr. Melissa Hedges, Math Trajectories for Young Learners, Part 1 03/19/2026

Season 4 | Episode 14 – Dr. DeAnn Huinker & Dr. Melissa Hedges, Math Trajectories for Young Learners, Part 1 DeAnn Huinker & Melissa Hedges, Math Trajectories for Young Learners, Part 1 ROUNDING UP: SEASON 4 | EPISODE 14 Research confirms that early mathematics experiences play a more significant role than we once imagined. Studies suggest that specific number competencies in 4-year-olds are strong predictors of fifth grade mathematics success. So what does it look like to provide meaningful mathematical experiences for our youngest learners? Today, we'll explore this question with DeAnn Huinker from UW-Milwaukee and Melissa Hedges from the Milwaukee Public Schools. BIOGRAPHY Dr. DeAnn Huinker is a professor of mathematics education in the Department of Teaching and Learning and directs the University of Wisconsin-Milwaukee Center for Mathematics and Science Education Research. Dr. Huinker teaches courses in mathematics education at the early childhood, elementary, and middle school levels. Dr. Melissa Hedges is a curriculum specialist who supports K–5 and K–8 schools for the Milwaukee Public Schools. RESOURCES book by DeAnn Huinker and Melissa Hedges website, featuring the work of Doug Clements and Julie Sarama journal article by Greg Duncan and colleagues journal article by Bethany Rittle-Johnson and colleagues TRANSCRIPT Mike Wallus: Welcome back to the podcast, DeAnn and Melissa. You have both been guests previously. It is a pleasure to have both of you back with us again to discuss your new book, . Melissa Hedges: Thank you for having us. We're both very excited to be here. DeAnn Huinker: Yes, I concur. Good to see you and be here again. Mike: So DeAnn, I think what I'd like to do is just start with an important grounding question. What's a trajectory? DeAnn: That's exactly where we need to start, right? So as I think about, “What are learning trajectories?,” I always envision them as these road maps of children's mathematical development. And what makes them so compelling is that these learning pathways are highly predictable. We can see where children are in their learning, and then we can be more intentional in our teaching when we know where they are currently at. But if I kind of think about the development of learning trajectories, they really are based on weaving together insights from research and practice to give us this clear picture of the typical development of children's learning. And as we always think about these learning trajectories, there are three main components. The first component is a mathematical goal. This is the big ideas of math that children are learning. For example, counting, subitizing, decomposing shapes. The second component of a learning trajectory are developmental progressions. This is really the heart of a trajectory. And the progression lays out a sequence of distinct levels of thinking and reasoning that grow in mathematical sophistication. And then the third component are activities and tasks that align to and support children's movement along that particular trajectory. Now, it's really important that we point out the learning trajectories that we use in our work with teachers and children were developed by Doug Clements and Julie Sarama. So we have taken their trajectories and worked to make them more usable and applicable for teachers in our area. So what Doug and Julie did is they mapped out children's learning starting at birth—when children are just-borns, 1-year-olds, 2-year-olds—and they mapped it out up till about age 8. And right now, last count, they have about 20 learning trajectories. And they're in different topics like number, operations, geometry, and measurement. And we have to put in a plug. They have a wonderful website. It's . We go there often to learn more about the trajectories and to get ideas for activities and tasks. Now, we're talking about this new book we have on math trajectories for young children. And in the book, we actually take a deep dive into just four of the trajectories. We look at counting, subitizing, composing numbers, and adding and subtracting. So back to your original question: What are they? Learning trajectories are highly predictable roadmaps of children's math learning that we can use to inform and support developmentally appropriate instruction. Mike: That's an incredibly helpful starting point. And I want to ask a follow-up just to get your thinking on the record. I wonder if you have thoughts about how you imagine educators could or should make use of the trajectories. Melissa: This is Melissa. I'll pick up with that question. So I'll piggyback on DeAnn's response and thinking around this highly predictable nature of a trajectory as a way to ground my first comment and that we want to always look at a trajectory as a tool. So it's really meant as an important tool to help us understand where a child is and their thinking right now, and then what those next steps might be to push for some deeper mathematical understanding. So the first thing that when we work with teachers that we like to keep in mind, and one of the things that actually draw teachers to the trajectories is that they're strength-based. So it's not what a child can't do. It's what a child can do right now based off of experience and opportunity that they've had. We also really caution against using our trajectories as a way to kind of pigeonhole kids or rank kids or label kids because what we know is that as children have more experience and opportunity, they grow and they learn and they advance along that trajectory. So really it's a tool that's incredibly powerful when in the hands of a teacher that understands how they work to be able to think about where are the children right now in their classroom and what can they do to advance them. And I think the other point that I would emphasize other than what moves children along is experience and opportunity. Children are going to be all over on the trajectory—that's been our experience—and they're in the same classroom. And it's not that some can't and some won't and some can; it's just some need more experience and some need more opportunity. So it's really opened up the door many ways to view a more equitable approach to mathematics instruction. The other thing that I would say is, and DeAnn and I had big conversations about this when we were first using the trajectories, is: Do we look at the ages? So the trajectories that Clements and Sarama develop do have age markers on them. And we were a bit back and forth on, “Do we use them?,” “Do we not?,” knowing that mathematical growth is meant to be viewed through a developmental lens. So we had them on and then we had them off and then we shared them with teachers and many of our projects and the teachers were like, "No, no, no, put the ages back on. Trust us. We'll use them well." (laughs) And so the ages are back onto the trajectories. And what we've noticed is that they really do help us understand how to take either intentional steps forward or intentional steps back, depending on what kids are showing us on that trajectory. The other spot that I would maybe put a plugin for on where we could use a trajectory and what would be an appropriate use for it would be for our special educators out there and to really start to use them to support clear, measurable IEP goals grounded in a developmental progress. So that's kind of what our rule of thumb would be around a “should” and “shouldn't” with the trajectories. Mike: That's really helpful. You mentioned the notion of experiences and opportunities being critical. So I wanted to take perhaps a bit of a detour and talk about what research tells us about the impact of early mathematics experiences, what impact that has on children. I wonder if you could share some of the research that you cite in the book with our listeners. DeAnn: Sure. This is DeAnn, and in the book we cite research throughout all of the chapters and aligned to all of the different trajectories. But as we think about our work, there really are a few studies that we anchor in, always, as we think about children's learning. And the research evidence is really clear that early mathematics matters. The math that children learn in these early years in prekindergarten, kindergarten, first grade—I mean, we're talking 4-, 5-, 6-year-olds, 7-year-olds—that their math learning is really more important than a lot of people think it is. OK? So as we think about these kind of anchor studies that we look at, one of the major studies in this area is from , and there was a study published in 2007. And what they did is they examined data from thousands of children drawing information from six large-scale studies, and they found that the math knowledge and abilities of 4- and 5-year-olds was the strongest predictor of later achievement. I mean, 4- and 5-year-olds, that's just as they're starting school. Mike: Wow. DeAnn: Yeah. One of the surprising findings was that they found early math knowledge and abilities was a stronger predictor than social emotional skills, stronger than family background, and stronger than family income. That it was the math knowledge that was predictive. Mike: That's incredible. DeAnn: Yes. A couple other surprising things from this study was that early math was a stronger predictor than early reading. Now, we know reading is really important, and we know reading gets a lot of emphasis in the early grades, but math is a stronger predictor than reading. And then one last thing I'll say about this study is that early math not only predicts later math achievement, it also predicts later reading achievement. So that is always a surprise as we share that information with teachers, that early math seems to matter as much and perhaps more than early reading abilities. There's a couple other studies I'll share with you as well. So there's this body of research that talks about [how] early math is very predictive of later learning, but we're teachers, we're educators. We like to know, “Well, what math seems to be most important?” So there was that looked at children's math learning in prekindergarten, 4-year-olds, and then looked at their learning again back in fifth grade. And what was unique about this study is they looked closely at what specific math topics seemed to matter the most. And what they found was that advanced number competencies were the strongest predictors of later achievement. Now, what are advanced number competencies? So these are the three that really stood out as being important. One was being able to count a set of objects with cardinality. So in other words, counting things, not just being able to recite a count sequence, no. So not verbal rote counting, but actually counting things, putting those numbers to objects. Another thing that they found [that] was really important was being able to count forward from any number. So if I said, “Start at 7 and keep counting,” “Start at 23 and keep counting,” that that was predictive of later learning. And the reason for that is when kids can count forward from a number, it helps them understand the structure of the number system, something we're always working on. And then the third thing that they found as part of advanced number competencies was conceptual subitizing. Now, what that is, is being able to see a number such as 5 as composed of subgroups, like 5 being composed of 4 and 1 or 3 and 2. So subitizing is being able to see the parts of a number, and that was really important for these 4-year-olds to begin working on for later learning. All right. One more, Mike, that I can share? Mike: Fire away! Yes. DeAnn: OK. So this last area of research that I want to share is actually really important as we think about the work of teachers in kindergarten and first grade in particular. So what these researchers did is they looked at children's learning at the beginning of kindergarten and then at the end of first grade. So, wow, think of the math kids learn from 5, 6 years old. And they found that these gains in what children can do was more predictive of later achievement than just what knowledge they had coming in. So learning gains, what children do and learn in math in kindergarten and first grade, is predictive of their mathematical success up through third grade. And then another study took it even further and said: Wait a minute, what they learn in kindergarten and first grade even predicts children's math achievement into high school. So there's just a growing body of research and evidence that early math is really important. The math learning of 4-year-olds, 5-year-olds, 6-year-olds, and 7-year-olds really builds this foundation that determines children's mathematical success many years later. Mike: This feels like a really great segue to a conversation about what it means to provide students opportunities for meaningful counting. That feels particularly significant when I heard all of the ideas that you were sharing in the research. I'm wondering if you could talk about the features of a meaningful counting experience. If we were to try to break that down and think about: What does that mean? What does that look like? What types of experiences count as meaningful when it comes to counting? Could you all talk about that a little bit? Melissa: Yeah, that's a great question, Mike. This is Melissa. So I think what's interesting about the idea of meaningful counting is, the more DeAnn and I studied the trajectory and spent time working with teachers and students, we came to the conclusion that the counting trajectory in particular is anchored, or a cornerstone of that counting trajectory is really meaningful counting. That once a skill is acquired—and we'll talk a little bit more about meaningful counting—but once that skill is acquired, it just builds and develops as kids grow and have more experience with number and quantity. So when we think about meaningful counting, the phrase that we like to use is that “Numbers represent quantity.” And it's just not that kids are saying numbers out loud, it’s that when they say “5,” they know what 5 means. They know how many that is. They can connect it to a context that they can go grab five of something. They might know that 5 is bigger than 2 or that 10 is bigger than 5. So they start to really play with this idea of quantity. And specifically when we're talking about kids engaging in meaningful counting, there's really key skills and understandings that we're looking and watching for as children count. The first one DeAnn already alluded to, is this idea of cardinality. So when I count how many I have—1, 2, 3, 4, 5—if that's the size of my set, when someone asks me, “How many is it?,” I can say “5” without needing to go back and count. So I can hold that quantity. Another one is stable count sequence. So we used to call it rote count sequence. And again, DeAnn referenced the idea that, really, when we're asking kids to count, we're asking more than just saying numbers. So we think about the stability and the confidence in their counting. One of the pieces that we've started to really watch very carefully and think carefully about with our children as we're watching many of them count is their ability to organize. So it's not the job of the teacher to organize the counter, to tell the child how to lay out the counters. It really is the work of the child because it brings to bear counting, saying the numbers, maintaining cardinality, as well as sets them up and sets us up to see where they at with that one-to-one correspondence. So can they organize a set of counters in such a way that allows them to say one number, one touch, one object? And then as they continue to coordinate those skills, are they able to say back and hold onto the idea of quantity? So the other ideas that we like to consider, mostly because they're embedded in the trajectory and we've seen them become incredibly important as we work with children, is the idea of producing a set. So when I ask a child, "Can you give me five?," they give me five, or are they able to stop when they get to five? Do they keep counting? Do they pick up a handful of counters and dump it in my hand? So all of those things are what we're looking for as we're thinking about the idea of producing a set. And then finally, even for our youngest ones, we really place a fair importance on the idea of representing a count. So can they demonstrate, can they show on paper what they did or how many they have? So we leave with a very rudimentary math sketch. So if they've counted a collection of five, how would they represent five on that paper? What that allows then the teacher to do is to continue to leverage where the trajectory goes as well as what they know about young children to bring in meaningful experiences tied to writing numbers, tied to having conversations about numbers. So the kids aren't doing worksheets, they're actually documenting something very important to them, which is this collection of whatever it is that they just counted in a way that makes sense to them. And so I think the other part that I like to talk about when we think about meaningful counting is this idea of hierarchical inclusion. It's that idea that children understand that numbers are nested one within each other and that each number in the count sequence is exactly 1 higher than what they said before. So, many times our reference with that is with our teachers are those little nesting dolls. So we think about 1 and then we wrap 2 around it and then we wrap 3 around it. So when we think about the number 3, we're thinking, “Well, it's actually the quantity of 2 and 1 more.” And we see that as a really powerful understanding in particular as our children get older and we ask them not just what is 1 more or 1 less, but what is 10 more or 10 less, that they take that and they extend that in meaningful ways. So again, the idea of meaningful counting, regardless of where we are on the trajectory, it's the idea that numbers represent quantities. And the neat thing about the trajectory—the counting trajectory in particular—is that they give us really beautiful markers as to when to watch for these. So we tend to talk about the trajectories as levels. So we'll say at level 6 on our counting trajectory is where we see cardinality first start to kind of show up, where we're starting to look for it. And then we watch that idea of cardinality grow as children get older, as they have more experience and opportunity, and as they work with larger numbers. Mike: That's incredibly helpful. So I think one of the things that really jumped out, and I want to mark this and give you all an opportunity to be a little bit more explicit than you already were—this importance of linking numbers and quantities. And I wonder if you could say a bit more about what you mean, just to make sure that our listeners have a full understanding of why that is so significant. DeAnn: All right, this is DeAnn. I'll jump in and get started, and Melissa can add on. As we first started to study the learning trajectory, the one thing we noticed was the importance of connecting things to quantity. Even some of the original levels didn't necessarily say “quantity,” but we anchor our work to developing meaning for our work. And we always think about, even when we're skip-counting, it should be done with objects that we should be able to see skip-counting as quantities, not just as words that I'm reciting. So across the trajectory, we put this huge emphasis on always connecting them to items, to things, or to actions and to movements so that it's not just a word, but that word has some meaning and significance for the child. Mike: I think that takes me to the other bit of language, Melissa, that you said that I want to come back to. You said... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/40515235

Season 4 | Episode 13 – Dr. Mike Steele, Pacing Discourse-Rich Lessons 03/05/2026

Season 4 | Episode 13 – Dr. Mike Steele, Pacing Discourse-Rich Lessons Mike Steele, Pacing Discourse-Rich Lessons ROUNDING UP: SEASON 4 | EPISODE 13 As a classroom teacher, pacing lessons was often my Achilles’ heel. If my students were sharing their thinking or working on a task, I sometimes struggled to decide when to move on to the next phase of a lesson. Today we're talking with Mike Steele from Ball State University about several high-leverage practices that educators can use to plan and pace their lessons. BIOGRAPHY Mike Steele is a math education researcher focused on teacher knowledge and teacher learning. He is the past president of the Association of Mathematics Teacher Educators, editor in chief of the Mathematics Teacher Educator journal, and member of the NCTM board of directors. RESOURCES Journal Article Books TRANSCRIPT Mike Wallus: Well, hi, Mike. Welcome to the podcast. I'm excited to talk with you about discourse-rich lessons and what it looks like to pace them. Mike Steele: Well, I'm excited to talk with you too about this, Mike. This has been a real focus and interest, and I'm so excited that this article grabbed your attention. Mike Wallus: I suppose the first question I should ask for the audience is: What do you mean when you're talking about a discourse-rich lesson? What does that term mean about the lesson and perhaps also about the role of the teacher? Mike Steele: Yeah, I think that’s a great question to start with. So when we're talking about a discourse-rich lesson, we're talking about one that has some mathematics that's worth talking about in it. So opportunities for thinking, reasoning, problem solving, in-progress thinking that leads to new mathematical understandings. And that kind of implicit in that discourse-rich lesson is student discourse-rich lesson. That we want not just teachers talking about sharing their own thinking about the mathematics, but opportunities for students to share their own thinking, to shape that thinking, to talk with each other, to see each other as intellectual resources in mathematics. And so to have a lesson like that, you've got to have a number of things in place. You've got to have a mathematical task that's worth talking about. So something that's not just a calculation and we end up at an answer and that the discourse isn't just, “Let me relay to you as a student the steps I took to do this.” Because a lot of times when students are just starting to experience discourse-rich lessons, that's kind of mode one that they engage in is, “Let me recite for you the things that I did.” But really opportunities to go beyond that and get into the reasoning and the why of the mathematics. And hopefully to explore some approaches or perspectives or representations that they may not have defaulted to in their first run-through or their first experience digging into a mathematical task. So the task has to have those opportunities and then we have to create learning environments that really foster those opportunities and students as the creators of mathematics and the teacher as the person who's shaping and guiding that discussion in a mathematically productive way. Mike Wallus: One of the things that struck me is there is likely a problem of practice that you're trying to solve in publishing this article, and I wonder if we could pull the curtain back and have you talk a bit about what was the genesis of this article for you? Mike Steele: Absolutely. So let me take us back about 20 or 25 years, and I'll take you back to some early work that went on around these sorts of rich tasks and discourse-rich lessons. So a lot of this legacy comes out of research or a project in the late nineties called the Quasar Project that helped identify: What is a rich task? What is a task, as the researchers described it, of high cognitive demand that has those opportunities for thinking and reasoning? The next question that that line of research brought forward is, “OK, so we know what a task looks like that gives these opportunities. How does this change what teachers do in the classroom? How they plan for lessons, how they make those moment-to-moment decisions as they're engaged in the teaching of that lesson?” Because it's very different than actually when I started teaching middle school in the nineties, where my preparation was: I looked at the content I had for that day, I wrote three example problems I wanted to write on the board that I very carefully got all the steps right and put those up and explained them and answered some questions. “Alright, everybody understand that? OK, great, moving on.” And then the students went and reproduced that. That's fine for some procedural things, but if I really wanted them to engage in thinking and reasoning, I had to start changing my whole practice. So this bubbles up out of the original work of the 5 Practices for Orchestrating Productive Discussions [book] from Peg Smith and Mary Kay Stein. I had the opportunity actually to work with them both in the early two thousands at the University of Pittsburgh. And so as we were working on this five-practices framework that was supposed to help teachers think about, “What does a different conceptualization of planning and teaching look like that really gets us to this discourse-rich classroom environment where students are making sense of and grappling with mathematics and talking to each other in a meaningful way about it?” We worked with teachers around that and the five-practices [framework] is certainly helpful, but then as teachers were working with the five practices and they were anticipating student thinking, they were writing questions that assess and advance student thinking, some of the things that came up were, “OK, what are the moment-to-moment decisions and challenges related to that as we start planning and teaching in this way?” And a number of common challenges came up. A lot of times when we were using a five-practice lesson, we were doing kind of a launch, explore, share, and discuss sort of format where we've got the teacher who's getting us started on a task, but we're not giving the farm away on that task. We're not saying too much and guiding their thinking. And then we let students have some time individually and in small groups to start messing around with the mathematics, working, talking. And then at some point we're going to call everybody together and we're going to share what the different ways of thinking were. We're going to try to draw that together. Peg Smith likes to talk about this as being more than a show-and-tell. So it's not just, “We stand up, we give our answer, we do that. Great.” Next group, doing the same thing, and oftentimes they start to look alike. But there's some really meaningful thinking that goes on in that whole-class discussion. So one of the really pragmatic concerns here is, “How do I know when to move?” So I've got students working individually, and maybe I gave them 3 minutes to get started. Was that enough? What can I see in the work they're doing? What questions am I going to hear to tell me, “OK, now it's a good moment to move to small groups.” And then similarly, when you've got small groups working, they're cranking away on a task. There might be multiple subquestions in that task. What's my cue that we're ready to go on to that whole-class discussion? We were in so many classrooms where teachers were really working hard to do this work, and this happens to me all the time. I have somehow miscalculated what students are going to be able to do—either how quickly they're going to be able to do it, or I expected them to draw on this piece of prior knowledge and it took us a while to get there, or they’ve flown through something that I didn’t expect them to fly through. So I'm having to make some choice in a moment, saying, “This isn't exactly how I imagined it, so what do I do here?” And frequently with teachers that get caught in that dilemma, the first response is to take control back, [to] say, “OK, you’re all struggling with this. Let's come back together and let me show you what you should have figured out here.” And it's done with the best of intentions. We need to get some closure on the mathematical ideas. But then it takes us right away from what we were trying to do, which was have our students grapple with the mathematics. And so we do this lovely polished job of putting that together and maybe students took the important things away from that, that they wanted to, maybe they didn't, but they didn't get all the way they were on their own. So that's really the problem of practice that this helps us to solve is, when we get in those positions of, “OK, I’ve got to make a call. I've got this much time left. I've got this sort of work that I see going on in the classroom. Am I ready? What can I do next?” That really keeps that ownership of the mathematics with our students but still gives me some ability to orchestrate, to shape that discussion in a way that's mathematically meaningful and that gets at the goals I had for the lesson. Mike Wallus: Yeah, I appreciated that part of the article and even just hearing you describe that so much, Mike, because you gave words to I think what sat behind the dilemma that I found myself in so often, which was: I was either trying to gauge whether there was enough—and I think the challenge is we're going to get into, what “enough” actually might mean—but given enough time, whether I was confident that there was understanding, how much understanding was necessary. And what that translates into is a lack of clarity around “How do I use my time? How do I gauge when it's worth expending some of the time that I maybe hadn't thought about and when it's worth recognizing that perhaps I didn't need all of that and I'm ready to do something?” So I think the next question probably should be: Let's talk about “enough.” When you talk about knowing if you have enough, say a little bit more about what you mean and perhaps what a teacher might be looking and listening for. Mike Steele: Absolutely. And I think this is a hidden thread in that five-practices model because we say: “OK, we want that whole-class discussion to still be a site for learning where there are some new ideas that are coming together.” So that then backs me up to thinking about the small-group work. I'm putting myself in that mode where I've got six groups working around the classroom. I'm circulating around; I'm asking questions. I of course don't see every single thing at any given moment that the small groups are doing. So I'm getting these little excerpts, these little 2- to 3-minute excerpts, when you stop into a group. So I think when we think about “enough,” I want to think about, with that task that I'm doing, with what my mathematical goals are and knowing that we're going to have time on the backend of this whole-class discussion to pull some ideas together, to sharpen some things to clarify some of the mathematics. Do I have enough mathematical grist for the mill here in what the small groups are doing to be able to then take that and make progress with students’ thinking at the center—again, not taking over the thinking myself—to be able to do that work. So, for any given mathematical idea, as I've started thinking about this when I plan lessons using the five-practices model, I am really taking that apart. What's the mathematical nugget that I'm listening for here, that I'm looking for in students' work that tells me: “OK, we've gotten to a point where, if I were to call people together right now and get them thinking about it, that there's more to think about, but we're well on our way.” And also when I'm looking for that, knowing that I'm also not looking at those six groups all at exactly the same time. So, I want to look for those mile markers along the way that tell me we're getting close, but we're not all the way there. Because if I pick one that's, we’re pretty much all the way there, that’s the first group I come to and I'm going to circulate around to five more. They're going to have run out of interesting things to do, and they're off talking about, thinking about something else. So, that really becomes the fine line: “What are those little mathematical ideas along the way that are far enough that get us towards our goals, but still we've got a little bit of the journey to go that we're going to go on together?” Mike Wallus: This is so fascinating. The analogy that's coming together in my mind is almost like you're listening for the ingredients for a conversation that you want to have as a group. So it's not necessarily “Has everyone finished?” And that's your threshold. It's actually “Did I hear this idea starting to bubble up? Did I hear elements of this idea or this strategy start to bubble up? Is there an insight that's percolating in different groups?” And it's the combination of those things that the teacher is listening for, and that's kind of the gauge of enoughness. Is that an accurate analogy? Mike Steele: It is, and I love that analogy because it reminds me of a favorite in our household as we're relaxing. We love to watch The Great British Baking Show. So, you're watching people take something from ingredients to a finished product. Now as you're watching that 20-minute segment, they're working on their technical challenge and they're all baking the same thing. I don't have to wait until the end of that, where they've presented their finished product, to have a good idea of what's going to happen. As I'm going through, as I'm watching 'em through that baking process, we're at the middle, my wife and I are talking, like, “Ooh, I’ve got concerns about that one. That one's looking good though.” We get an idea of where it's going. So I think the ingredient analogy really lands with me. We don't have to wait. We're looking for those pieces to be able to pull that together, those ingredients. We're not waiting until there's a final product and saying—because then, what is there to say about it? “Oh, look, that looks great. Oh, that one, maybe not exactly what we'd intended.” So, it's giving us those ingredients for that whole-class discussion. Mike Wallus: The other thing that struck me as I was listening to you is: We're not teaching a task; we're teaching a set of ideas or relationships. The task is the vehicle. So, it's perfectly reasonable, it seems, to say, “We're going to pause at this point in the task, or at a place where students might not be entirely finished with the task. And we might have a conversation at that point because we have enough that we can have part of the conversation.” And that doesn't mean that they don't go back to the task. But you're really helping me recognize that one of the places where I sometimes get stuck, or got stuck, when I was teaching, is task completion was part of my time marking. And I think really what you're challenging me and other educators to do is to say, “The task is just the vehicle. What's going on? What's percolating around that task as it's happening?” How does that strike you? Mike Steele: Yeah, absolutely. And it was the same challenge with me and sometimes still is the same challenge with me. (laughs) Yeah, you give this task, and we think about that task as our unit of analysis as a teacher when we're planning. And so we want our students as we're using it to get to the end of it. It's a very natural thing to do. And let me make this really concrete. If I'm doing a visual pattern task with third graders, we have, I think there's one of the elementary [5 Practices in Practice] book called “Tables & Chairs.” So you've got these square tables that have four seats around them, and you're putting a string of tables together and asking kids to get at the generalization. “If you have any number of tables, how many people can you seat?” And so I think early when I started giving those tasks, I was looking for, “OK, has everybody gotten to the rule? Have they gotten to that generalization? OK, now we can talk about it.” And we can talk about the different ways people made sense of that geometrically and those connections, and that's what I want to get out of the whole-class discussion. But we don't even have to get there if groups have a sense of how that pattern is growing, even if they haven't gotten to the formal description of the rule. Because if they've gotten to that point, they've made some sense of the visual. They've made some of those connections. They've parsed that in different ways. That's plenty for me to have a good conversation, that we can come to that rule as a group and we can even come to it in different ways as a group. But it frees me up from being like, “OK, everybody got the rule? Everybody got the rule? Everybody got the rule?” Because that often resulted in, I'd have a couple of groups that maybe had been a little slower getting started and they're still getting there. And then I'm sitting there and I'm talking to them, I'm giving them these terribly leading questions. “Can we just get to the rule? Come on, let's go. You're almost there. We got it. We got it.” And that then is, again, me taking over that thinking and not giving them the space for those ideas to breathe. Mike Wallus: What else is jumping out for me is the ramifications for how thinking this way actually might shift the way that I would plan for teaching, but also how it might shift the way that I'm looking for evidence to assess students' progress during the task. So I wonder if you have situations or maybe some recommendations for: How might a person plan in ways that help them recognize the ways that the task can be a vehicle but also plan for the kind of evidence that they might be looking for along the way? Could you talk a little bit about that? Mike Steele: Absolutely. So I'll give kind of a multi-layered description of this. When we're using a task that's got multiple solution paths that has these opportunities for diverse thinking, the five-practices framework tells us anticipating student thinking is a critical part of it. So, what are the different solution paths that students can take through it? So, if it's a visual pattern task, they may look at it this way with a visual. They may think about those tables like the tops and the bottoms and then the sides. They may think about the two ends of the tables having different numbers of chairs and the ones in between having a different number of chairs and parsing it that way. And we can develop those. It's actually, for me, quite a lot of fun to develop those fully formed solutions that students can do. And early on when I was enacting lessons like this, I would do that. I'd have those that I was looking for. I'd also think about questions I'd want to ask students who are struggling to get started or maybe were going down a path that may not be mathematically productive and the questions I might ask them to get them on a more mathematically productive path. And I'd go around and I'd look for those solutions, and I'd use that to think about my selecting, my sequencing, my connecting my whole-class discussion. So, great, check. That's layer one. I think responding to the challenge of what's enough requires us to then take those solution paths apart—both the fully formed ones, maybe the incomplete thinking—and say, “OK, within that solution, what are the things that I want to see and hear that gives me some confidence that we're on this path, even if we're not at the end of this path, and that give me enough to think about?” So, if I think about, I'll go back again to this visual pattern task analogy. If I see groups that are talking about increases, so when we add a table, we're adding two... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/40157040

Season 4 | Episode 12 – Kyndall Thomas, Building a Meaningful Understanding of Properties Through Fact Fluency Tasks 02/19/2026

Season 4 | Episode 12 – Kyndall Thomas, Building a Meaningful Understanding of Properties Through Fact Fluency Tasks Kyndall Thomas, Building a Meaningful Understanding of Properties Through Fact Fluency Tasks ROUNDING UP: SEASON 4 | EPISODE 12 Building fluency with multiplication and division is essential for students in the upper elementary grades. This work also presents opportunities to build students' understanding of the algebraic properties that become increasingly important in secondary mathematics. In this episode, we're talking with Kyndall Thomas about practical ways educators can support fluency development and build students’ understanding of algebraic properties. BIOGRAPHY Kyndall Thomas serves as a math interventionist and resource teacher with the Oregon Trail School District, focusing on data-driven support and empowering teachers to spark a love of numbers in their students. TRANSCRIPT Mike Wallus: Hi, Kyndall. Welcome to the podcast. I'm really excited to be talking with you today. Kyndall Thomas: Hi, Mike. Thanks for having me. I'm excited to dive into some math talk with you also. Mike: Kyndall, tell us a little bit about your background. What brought you to this work? Kyndall: Yeah. I started in the classroom. I was in upper elementary. I served fifth grade students, and I taught specifically math and science. And then I moved into a more interventionist role where I was a specialist that worked with teachers and also worked with small groups, intervention students. And through that I was able for the first time to really develop an understanding of that mathematical progression that happens at each grade level and the formative things that are introduced at the lower elementary [grades] and then kind of fade out, but still need to be brought back at the upper elementary. Mike: So I've heard other folks talk about the ways students can learn about the algebraic properties as they're building fluency, but I feel like you've taken this a step further. You have some ideas around how we can use visual models to make those properties visible. And I wonder if you could talk a little bit about what you mean by making properties visible and maybe why you think this is an opportunity that's too good to pass up? Kyndall: My thought is bringing visual models back into the classroom with our higher upper elementary students so that they can use those models to build a natural immersion of some of the algebraic properties so that they can emerge rather than just be rules that we are teaching. By supporting students’ learning through building models with manipulatives, we're able to build a bridge in a student's mind between their experience with those models and then their mental capacity to visualize those models. This is where the opportunity to bring properties to life is too good to pass up. Mike: OK, so let's get specific. Where would you start? Which of the properties do you see as an opportunity to help students understand as they're building an understanding of fluency? Kyndall: So, when I begin laying the foundation for understanding of the operations and multiplication and division, I intentionally layer in two other major algebraic properties for discovery: the commutative property and the distributive property. We're not setting our students up for success when we simply introduce these properties as abstract rules to memorize. Strong visual models allow students to discover the why behind the rules. They're able to see these properties in action before I even spend any time naming them. For example, they get to witness or discover how factors can switch order without changing the product, how grouping affects computation, and how numbers can be broken apart and recombined for efficient counting and solving strategies. By teaching basic facts in this structured and intentional way through the behavior of numbers and the authentic discovery of properties, we're not only building fluency, but we're also developing deep conceptual understanding. Students begin to recognize patterns, understand rules, make connections, and rely on reasoning instead of rote memorization. That approach supports long-term mathematical flexibility, which is exactly what we want our students to be able to do. Mike: I want to ask you about two particular tools: the number rack and the 10-frame. Tell me a little bit about what's powerful about the way the [10-frame] is set up that helps students make sense of multiplication. What is it about the way it's designed that you love? Kyndall: The [10-frame] is so powerful because it's set up in our base ten system already. It introduces the tens in a way that is two rows of 5, which is going to lead into properties being identified. So, let me break that up into each individual thing that I love about it. First, the [10-frame] being broken up into the two rows of 5. That's going to allow students to be able to see that distributive property happening, where we're counting our 5s first and then adding some more into each group. So, when we're seeing a factor like 8 times 2, we're seeing that as two groups of 5 and two groups of 3. Mike: I think what you're making me remember is how it's difficult to help kids visualize that, right? It's a challenge. You can say “‘4 times 4’ is the same as ‘4 times 2 plus 4 times 2,’” but that's still an abstraction of what's happening, right? The visual really brings it to life in a way that—even if you're representing that with an equation and doing a true-false equation where it's 4 times 4 is the same as 4 times 2 plus 4 times 2—that's still at a level of abstraction that's not necessarily accessible for children. Kyndall: And as we're talking through this, if I see students and they're working on four groups of 3 and they're seeing those 3s as a double fact plus one more group, I'm on the board writing out the equation, and I'm using the parentheses as that introduction to what this looks like abstractly. They're building it, and they're building those visuals both with their hands and with their minds, and then I'm bringing it to life in the equation on the board. Mike: So, I think what I see in my mind as I hear you describe that is, you have kids with a set of materials. You're doing, for lack of a better word, a translation into a more abstract version of that, and you're helping kids connect the physical materials that they have in front of them to that abstraction and really kind of drawing the connection between the two. Am I getting that right? Kyndall: Yeah. As the students are doing the physical work of math, I'm translating it into its own language up on the board. Absolutely. Mike: I think what's clear to me from this conversation is the way that the tools can illuminate the property, and I think this also helps me think about what my role is as a teacher in terms of building a bridge to an abstraction. Do you actually feel like there's a point where you do introduce the formal language of it? And if you do, how do you decide when? Kyndall: So, the vocabulary kind of comes after the concept has been discovered. But I don't like to introduce the vocabulary first as a rote memorization tool because that has no meaning to it. Mike: I think if I were to summarize this, you're giving them a physical experience with the properties. You're translating that into an abstraction. And then once they've got an experience that they can hang those ideas on top of, then you might decide to introduce the formal language to them at some point. Kyndall: Yeah, absolutely. Mike: So, just as a refresher, for folks who might teach upper elementary and don't have a lot of lived experiences with the number rack—be it the ten or twenty or the hundred—can you describe a little bit about the structure, and maybe what about the structure in particular is important? Kyndall: The structure of a number rack has rows, and each row has 10 beads in it. And typically those beads are divided into two sets of 5: five red beads and five white beads. Then we typically move into a number rack that has two rows so that we're working within 20. Now, my thought is to take that [to] our third, fourth, and fifth grade, our upper elementary students, and use the hundreds rekenrek [i.e., number rack], where now we have 10 rows and we have 10 beads in each row—still split up into five red [beads] and five white—so that we can use that to teach things. If we're looking at the zero property, students are starting to notice that the rows represent the groups—the rows with the beads on it, that's one group. And so, if we're building zero groups of 3, we don't have a group that we can access to put three beads in. If we're looking at it with the commutative property, students are able to say, “One group of 3. We have one row and we're putting three beads in it.” But what happens when we switch those factors? Now we're utilizing three of our rows, but we're only sliding over one bead. The number rack is also so important when we get to the distributive property because of the way that they have separated those colors. So when we're looking at a factor like 7 times 6—seven groups of 6—then we're gonna be accessing seven rows with six beads in each. That is already set up in the structure of the tool to have five red beads and one white bead showing seven groups of 5 and seven groups of 1 put together. Mike: That is super powerful. One of the things that really jumped out that I want to mark is: If I treat the rows like the groups and then I treat the beads like the number of things in each group, I can model one group with three inside of it, or I can model three groups with one inside of it, and I can really make the difference between those things clear, but also [I can make] the way that the product is still the same clear, right? So, I've got an actual physical model that helps kids understand what was often a rule that was just like 1 times 3 is the same as 3 times 1, because it is. But you're actually saying this is a tool that helps us make meaning of that. The other thing that jumps out from what you said is: If I'm doing 6 times 5 or 6 times 7 and I push over six [beads], and six looks like five red, one white, I'm automatically set up to make sense of the distributive property because the visual helps me see it. Am I getting that right? Kyndall: Yes, except let me correct you on that last one. You said “6 times 5,” and you said, “If I slide over six,” Now, six is our group number. We have to be deliberate; that’s six groups of 5. So, we're grabbing our groups first, but absolutely, yes. That is the key structure there. [laughs] That's the idea. Mike: This is why this would've been very helpful for a young Mike Wallus. Kyndall: [laughs] Mike: Well, before we go, are there any resources that you'd recommend to a listener that have either informed your thinking or that might help someone take what you've been talking about and put these ideas into action? Kyndall: Yeah. I've been putting this practice into play here at my own district and tracking its progress for a while now. After seeing the success in my own halls here in Sandy, [Oregon,] I've started to reach out and work with other educators on purposeful tool use and mathematical progression. If it resonates with you, whether you're in the classroom or in a leadership role, I would genuinely love to connect and learn alongside you. You're always welcome to reach out to me directly at . I anticipate more conversations in collaboration, and I'd love to bring them to life through trainings moving forward. I believe that when teachers are confident in their own understanding, they build that same confidence in students. Mike: I think that's a great place to stop. Kyndall, thank you so much. It has really been a pleasure talking with you and learning from you. Kyndall: Thank you so much for having me. It's been fun. 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. © 2026 The Math Learning Center | /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/40141940

Season 4 | Episode 11 – Dr. Amy Hackenberg, Understanding Units Coordination 02/05/2026

Season 4 | Episode 11 – Dr. Amy Hackenberg, Understanding Units Coordination Amy Hackenberg, Understanding Units Coordination ROUNDING UP: SEASON 4 | EPISODE 11 Units coordination describes the ways students understand the organization of units (or a unit structure) when approaching problem-solving situations—and how students' understanding influences their problem-solving strategies. In this episode, we’re talking with Amy Hackenberg from the University of Indiana about how educators can recognize and support students at different stages of units coordination. BIOGRAPHY Dr. Amy Hackenberg taught mathematics to middle and high school students for nine years in Los Angeles and Chicago, and is currently a professor of mathematics education at Indiana University-Bloomington. She conducts research on how students construct fractions knowledge and algebraic reasoning. She is the proud coauthor of the Math Recovery series book, Developing Fractions Knowledge. RESOURCES by Amy J. Hackenberg, Anderson Norton, and Robert J. Wright TRANSCRIPT Mike Wallus: Welcome to the podcast, Amy. I'm excited to be chatting with you today about units coordination. Amy Hackenberg: Well, thank you for having me. I'm very excited to be here, Mike, and to talk with you. Mike: Fantastic. So we've had previous guests come on the podcast and they've talked about the importance of unitizing, but for guests who haven't heard those episodes, I'm wondering if we could start by offering a definition for unitizing, but then follow that up with an explanation of what units coordination is. Amy: Yeah, sure. So unitizing basically means to take a segment of experience as one thing, which we do all the time in order to even just relate to each other and tell stories about our day. I think of my morning as a segment of experience and can tell someone else about it. And we also do it mathematically when we construct number. And it's a very long process, but children began by compounding sensory experiences like sounds and rhythms as well as visual and tactical experiences of objects into experiential units—experiential segments of experience that they can think about, like hearing bells ringing could be an impetus to take a single bong as a unit. And later, people construct units from what they imagine and even later on, abstract units that aren't tied to any particular sensory material. It's again, a long process, but once we start to do that, we construct arithmetical units, which we can think of as discrete 1s. So, it all starts with unitizing segments of experience to create arithmetical items that we might count with whole numbers. Mike: What's really interesting about that is this notion of unitizing grows out of our lived experiences in a way that I think I hadn't thought about—this notion that a unit of experience might be something like a morning or lunchtime. That's a fascinating way to think about even before we get to, say, composing sets of 10 into a unit, that these notions of a unit [exist] in our daily lives. Amy: Yeah, and we make them out of our daily lives. That's how we make units. And what you said about a ten is also important because as we progress onward, we do take more than 1 one as a unit—like thinking of 4 flowers in a row in a garden as a single unit, as both 1 unit and as 4 little flowers—means it has a dual meaning, at least; we call it a composite unit at that point. That's a common term for that. So that's another example of unitizing that is of interest to teachers. Mike: Well, I'm excited to shift and talk about units coordination. How would you describe that? Amy: Yeah, so units coordination is a way for teachers and researchers to understand how children create units and organize units to interpret problem situations and to solve problems. So it originated in understanding how children construct whole number multiplication and division, but it has since expanded from just that to be thinking more broadly about units and structuring units and organizing and creating more units and how people do that in solving problems. Mike: Before we dig into the fine-grain details of students' thinking, I wonder if you can explain the role that units coordination plays in students' journey through elementary mathematics and maybe how that matters in middle school and beyond middle school. Amy: So that's where a lot of the research is right now, especially at the middle school level and starting to move into high school. But units coordination was originally about trying to understand how elementary school children construct whole number multiplication and division, but it's also found to greatly influence elementary school children's understanding of fractions, decimals, measurement and on into middle school students’ understanding of those same ideas and topics: fractions ratios and proportional reasoning, rational numbers, writing and transforming algebraic equations, even combinatorial reasoning. So there's a lot of ways in which units coordination influences different aspects of children's thinking and is relevant in lots of different domains in the curriculum. Mike: Part of what's interesting for me is that I don't think I'm alone in saying that this big idea around units coordination sounds really new to me. It's not language that I learned in my preservice work[, nor] in my practice. So I think what's coming together for me is there's a larger set of ideas that flow through elementary school and into middle school and high school mathematics. And it's helpful to hear you talk about that, from the youngest children who are thinking about the notion of units in their daily lives to the way that this notion of units and units coordination continues to play through elementary school into middle school and high school. Amy: Yeah, it's nice that you're noticing that because I do think that's something that's a strength of units coordination in [that] it can be this unifying idea, although there's lots of variation and lots of variation in what you see with elementary students versus middle school students versus high school students versus even college students. Some of the research is on college students' unit coordination these days, but it is an interesting thread that can be helpful to think about in that way. Mike: OK. With that in mind, let's introduce a context for units coordination and talk a little bit about the stages of student thinking. Amy: Yeah. So, one way to understand some differences in how children up through, say, middle school students might coordinate units and engage in units coordination is to think about a problem and describe how solving it might happen. Here's a garden problem: “Amaya is planting 4 pansies in a row. She plants 15 rows. How many pansies has she planted?” There are three stages of units coordination, broadly speaking—we've begun to understand more about the nuances there. But a stage refers to a set of ways of thinking that tend to fit together in how students understand and solve problems with whole numbers, fractions, quantities, and multiplicative relationships. It's sort of about a nexus of ideas, and—that we tend to see coming together and students don't usually think in a way that's characteristic of a different stage until they've made a significant change in their thinking, like a big reorganization happens for them to move from one stage to the next. So students at stage 1 of units coordination are primarily in a 1s world and their number sequence is not multiplicative. That's going to be hard to imagine. But they can take a group of 1s as one thing. So, they can make a composite unit and that means in the garden problem, they can take a row of pansies as 1 row as well as 4 little ones, and they can continue to do that over and over again. And so they can amass rows of 4 pansies and keep going. And what it usually looks like for them to solve the problem is they'll count by 1s after any known skip-counting patterns. So, in this case they might be like, “Oh, I know 4 and 8; that's two rows. 9, 10, 11, 12; that's three rows.” Often using fingers or something to keep track, or in some way to keep track, and continuing to go up and get all the way, barring counting errors, to 60 pansies. And so for them the result, 60 pansies, is a composite unit. It's a unit of 60 units, but they don't maintain the structure that we see at all of the units of 60 as 15 fours. That's not something—even though they did track it in their thinking—they don't maintain that once they get to the 60, it's really just only a big composite unit of 60. So their view of the result is very different than an adult view might be. So, the students at stage 1 can solve division problems, which means if they give some number of pansies and they're supposed to make rows of 4, they can definitely do it, they can solve that. But they don't think of multiplication and division as inverses. So let me say what I mean by that. If they had this problem next, so: “Amaya's mom gave her 28 pansies. How many rows of 4 can she make?” A student at stage 1 could solve that problem, and they would be able to track 4s over and over again and figure out that they got to 7 fours once they get to 28. But then if immediately afterwards a teacher said, “Well, so, how many pansies are there in 7 rows of 4?,” the student at stage 1 would start over and solve the problem from the beginning. They wouldn't think that they had already solved it. And that's one telling sign of a student operating at stage 1. And the reason is that the mental actions they engage in to do the segmenting or the tracking off of the 4s and the 28 pansies are really different to them than what they use then the ways of thinking they use to create the 7 rows of 4 and make the 28 that way. And so they don't recognize them as similar, so they feel like they have to engage in new problem solving to solve that problem. So, to get back to the garden problem, students at stage 2 have a multiplicative number sequence, so they think of 60 as a one that they could repeat. Iterating is a term we often use. They could imagine it just being repeated over and over again. And this is a contrast to students at stage 1 who think of 60 as like, “Oh, I got to have all 60 pansies there if I'm going to think about a number like 60.” Whereas students at stage 2 do have a multiplicative number sequence and so they think, “Oh, I don't have to have all my 60 pansies. I can just think about one pansy and I just repeat it however many times I need, to have however many pansies I want to imagine in my problem solving.” So they anticipate 60 as 1 sixty times. And that's obviously a great relief for kids who are dealing with big numbers. You can imagine it feels really onerous to think about 1,000 if you feel like you have to have 1,000 items in your mind, “Oh, how could I possibly do that?” But, “Oh, I don't have to have 1,000; I can just have 1 and I can repeat it.” That's a great economy, efficiency in thinking that happens. So in terms of the garden problem, students at stage 2 also have constructed a row as a thing to count, so a composite unit’s one item as well, so 4 little items. And they can amass 4s just like I was talking about with students at stage 1. But what they are also able to do is break apart 4s as they go along. They might say, “Well, I've got 4 and 4 is 8 and one more [4] is 12 and one more is 16 and one more is 20 and one more is 24 and one more is 28.” Maybe at that point they say, “Oh, let's see. I don't know what one more 4 is, but two more [4s] is 30 and then two more is 32.” So they can take the row apart. They don't all do this, but they can; they have the mental capabilities to do that because they're not right in the midst of making the coordination happen. They're sort of a little bit able to stand above the coordination and take their rows apart if they need to. Mike: It sounds like part of what happens at stage 1 is you might have a kid who potentially could count by 4s for lack of a better way of saying it. And they might say, “Well, 4 and 4, so 2 sets of 4s, [is] 8.” And then at some point it kind of breaks down where that memorized list of what happens when you count by 4. And then kids are back to saying, “OK, 12, 13, 14, 15, 16.” And if you were watching this, listeners, you would see that I stuck out four fingers and then I'm like, “OK, so that's 3 fours, and so on.” And so I would see a student who might appear to be thinking about units, but tell me if I'm correct in thinking that it's more a function of that they know a set of numbers in accounting sequence for counting by 4s. Amy: So students at any stage may vary in the skip-counting patterns they know. I call it knowing a skip-counting pattern, to know automatically, like, 4, 8, 12, 16, or whatever it is. So you could have a student at stage 2 who doesn't know their skip-counting patterns very well, and you also could have a student at stage 2 who counts by 1s. So that's the issue there, is you can't always tell just from what you see if you have to do more than the test of what I'm saying. It's just to give a sense of the stages. But the main thing is the outer boundary of what they can do at stage 2 is they don't have to count by 1s. They can do other things because of the fact that their composite units have this special feature where they're multiplicative in nature. I mean the fancy term for it is they have iterable units of 1. But let me say a little bit more about what happens when they get to 60. So, let's say a student at stage 2, they've gotten up to 60, there are 60 pansies and there are 15 rows of 4. They will think of the 60 as 15 fours as they make it. So we call it a three levels of unit structure. 60 is a unit of 15 units, each containing 4 little ones. They'll think about [it] that way as they solve the problem, but as they continue to work further and add more pansies on or do a further extension of the problem, they wouldn't maintain that three levels of units structure that we see. So that's important because it has implications for how they can build from what they've done. Mike: How would you know that they hadn't maintained it? What might they say or do that would give you that cue? Amy: Well, so you see it most if, let's say I say, “Oh, guess what? We got 12 more pansies and you're going to put 'em in rows of 4. Can you put those on?” And then they put 'em on. OK, they find out it's 72 now. “OK, so how many rows are we talking here?” It would be a new problem for them to figure that out. It wouldn't be like they would be able to maintain that, “Oh, I had 15 rows and then I now have the 3 more added on.” Mike: Got you. OK. Amy: So, you see they're having to remake stuff as adult learners. We would think, “Oh, you should already know that that's 15 fours, right?” But they'll have to redo that in solving an extension of the problem like I was talking about there. So students at stage 3, they also can definitely take 4 as a row of 1 and also 4 pansies. They can arrive at 60 and view it as a unit of units, but they also can view it as a unit of 15 units, each containing 4, and they maintain that. So, if they were asked a further problem, like, “Hey, we're going to rearrange this garden; we're going to actually 3 rows together at a time. Can you do that, and how many rows would you have and how many pansies in each row? And what would be the total?” They'd be able to say, “Oh, yeah, I can, let's see, put my 3 rows together, that's going to be 12, and then I'm going to end up with 5 of them.” And now they've created 60 as a unit of 5 rows, each containing 12, and they can still think of 60 as a unit of 15 units, each containing 4, or 15 rows, each containing 4. So they can switch between different unit structures. It doesn't mean they automatically know it without thinking it through, but they can do it and they can go back and forth. And that has great implications for anticipating and for solving division problems and seeing them as inverses of multiplication and a whole lot of stuff: proportional reasoning, fractions, lots of things. [laughs] Mike: I think what's really interesting about this is I really appreciate you walking through the mental processes or maybe even the mental scripts that the kids might engage in to help see behind the curtain, for lack of a better word. Because what strikes me is that there is a point, probably early in my teaching career, where I would've attended and focused mostly on, “Did they get the answer?” And I think what you're helping remind me of is that it's the “how,” but there are particular ideas. And now I think I understand why the notion of units—plural—units coordination matters so much because a lot of what's happening is their ability to coordinate a unit made of units and then to be flexible with the units within that unit of units. Am I making proper sense of that, Amy? Amy: Yeah, for sure. That's great; that's exactly it. So the process and what units get created and how they get thought about and used is actually really, really important in trying to support kids' multiplicative thinking among other kinds of thinking too. Mike: I think this is a great segue because I suspected a lot of teachers are wondering about the kinds of tasks or practices or questions that they might use that could nudge students' thinking regarding units coordination. And I'm wondering: What are some ideas you'd recommend for teachers as they're trying to think about how they assess but also advance their students' thinking when it comes to units coordination? Amy: That's a great question. And, I mean, the big response is: Have students engage in lots of reasoning with units—composite units, breaking apart numbers strategically, thinking about different solution pathways. So not just one solution pathway, but can you come up with multiple solutions for the problem? Really sharing student solutions that involve breaking apart units. So if you're doing something like 5 sevens and finding out that kids are thinking of it as 5 fives and 5 twos, let's share that. How else could we break apart the 5 sevens? 5 fives and 5 twos? Why is that maybe helpful compared to other ways we might think about it? We might know 5 fives and 5 twos more easily than other ways of breaking it apart. And then even how are kids thinking about the 5 twos and the 5 fives and evaluating each of those. So basic things like that are super important. How many rows can we make with 36 flowers with 4 per row? Thinking strategically about that, like: I know that 5 fours is 20 and I need 16 more flowers, so that's 4 fours because it's double 2 fours, so 8, so that means 9 rows total. So I'm just kind of really briefly talking through, but posing these kinds of tasks and then asking for how students can break them up and think about them and presenting and making public that kind of thinking and reasoning. So valuing it in that way and sharing it. Same thing with lots of even more advanced multiplication problems. So for example, my daughter's in fourth grade right now, and so we've been working with her on, like, 30 times 20 and doing something other than knowing 3 times 2 and then putting 0s on because she doesn't remember that. So to do 30 times 20, we asked her about 10 twenties. Oh, she can figure that out; that's 200. And then can I iterate? Oh yeah, another 10 twenties, another 10 twenties. And then we did like 40 thirties, which was definitely harder. And so as part of the process of that, after she figured out 10 thirties, when she was iterating her thirties, that was harder than iterating the twenties. She had to break apart numbers. When she got to 90 plus 30, she had to think about 90 plus 10 plus 20. So doing embedded, breaking apart of units with the prospect of trying to figure out a larger multiplication... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/39899160

Season 4 | Episode 10 – What Counts as Counting? Guest: Dr. Christopher Danielson, Part 2 01/22/2026

Season 4 | Episode 10 – What Counts as Counting? Guest: Dr. Christopher Danielson, Part 2 What Counts as Counting? with Dr. Christopher Danielson ROUNDING UP: SEASON 4 | EPISODE 10 What counts as counting? The question may sound simple, but take a moment to think about how you would answer. After all, we count all kinds of things: physical quantities, increments of time, lengths, money, as well as fractions and decimals. In this episode, we'll talk with Christopher Danielson about what counts as counting and how our definition might shape the way we engage with our students. BIOGRAPHY Christopher Danielson started teaching in 1994 in the Saint Paul (MN) Public Schools. He earned his PhD in mathematics education from Michigan State University in 2005 and taught at the college level for 10 years after that. Christopher is the author of Which One Doesn’t Belong?, How Many?, and How Did You Count? Christopher also founded , a large-scale family math playspace at the Minnesota State Fair. RESOURCES by Christopher Danielson by Christopher Danielson by Simon Gregg by Jo Boaler and Cathleen Humphreys TRANSCRIPT Mike Wallus: Before we start today's episode, I'd like to offer a bit of context to our listeners. This is the second half of a conversation that we originally had with Christopher Danielson back in the fall of 2025. At that time, we were talking about [the instructional routine] This second half of the conversation focuses deeply on the question “What counts as counting?” I hope you'll enjoy the conversation as much as I did. Well, welcome to the podcast, Christopher. I'm excited to be talking with you today. Christopher Danielson: Thank you for the invitation. Delightful to be invited. Mike: So I'd like to talk a little bit about your recent work, the book In it, you touch on what seems like a really important question, which is: “What is counting?” Would you care to share how your definition of counting has evolved over time? Christopher: Yeah. So the previous book to How Did You Count? was called , and it was about units. So the conversation that the book encourages would come from children and adults all looking at the same picture, but maybe counting different things. So “how many?” was sort of an ill-formed question; you can't answer that until you've decided what to count. So for example, on the first page, the first photograph is a pair of shoes, Doc Marten shoes, sitting in a shoebox on a floor. And children will count the shoes. They'll count the number of pairs of shoes. They'll count the shoelaces. They'll count the number of little silver holes that the shoelaces go through, which are called eyelets. And so the conversation there came from there being lots of different things to count. If you look at it, if I look at it, if we have a sufficiently large group of learners together having a conversation, there's almost always going to be somebody who notices some new thing that they could count, some new way of describing the thing that they're counting. One of the things that I noticed in those conversations with children—I noticed it again and again and again—was a particular kind of interaction. And so we're going to get now to “What does it mean to count?” and how my view of that has changed. The eyelets, there are five eyelets on each side of each shoe. Two little flaps that come over, each has five of those little silver rings. Super compelling for kids to count them. Most of the things on that page, there's not really an interesting answer to “How did you count them?” Shoelaces, they're either two or four; it's obvious how you counted them. But the eyelets, there's often an interesting conversation to be had there. So if a kid would say, "I counted 20 of those little silver holes," I would say, "Fabulous. How do you know there are 20?" And they would say, "I counted." In my mind, that was like an evasion. They felt like what they had been called on to do by this strange man who's just come into our classroom and seems friendly enough, what they had been called on to do was say a number and a unit. And they said they had 20 silver things. We're done now. And so by my asking them, "How do you know? " And they say, "I counted." It felt to me like an evasion because I counted as being 1, 2, 3, 4, 5, all the way up to 20. And they didn't really want to tell me about anything more complicated than that. It was just sort of an obvious “I counted.” So in order to counter what I felt like was an evasion, I would say, "Oh, so you said to yourself, 1, 2, 3, and then blah, blah, blah, 18, 19, 20." And they'd be like, "No, there were 10 on each shoe." Or, "No, there's 5 on each side." Or rarely there would be the kid who would see there were 4 bottom eyelets across the 4 flaps on the 2 shoes and then another row and another row. Some kids would say there's 5 rows of 4 of them, which are all fabulous answers. But I thought, initially, that that didn't count as counting. After hearing it enough times, I started to wonder, “Is it possible that kids think 5 rows of 4, 4 groups of 5, 2 groups of 10, counted by 2s and 1, 2, 3, 4, all the way up to 19 and 20—is it possible that kids conceive of all of those things as ways of counting, that all of those are encapsulated under counting?” And so I began because of the ways children were responding to me to think differently about what it means to count. So when I first started working on this next book, How Did You Count?, I wanted it to be focused on that. The focus was deliberately going to be on the ways that you count. We're all going to agree that we're counting tangerines; we're all going to agree that we're counting eggs, but the conversation is going to come because there are rich ways that these things are arranged, rich relationships that are embedded inside of the photographs. And what I found was, when I would go on Twitter and throw out a picture of some tangerines and ask how people counted, and I would get back the kind of thing that was how I had previously seen counting. So I would get back from some people, "There are 12." I'd ask, "How did you count?" And they'd say, "I didn't. I multiplied 3 times 4.” “I didn't. I multiplied 2 times 6." But then, on reflection through my own mathematical training, I know that there's a whole field of mathematics called combinatorics. Which if you asked a mathematician, "What is combinatorics?," 9 times out of 10, the answer is going to be, “It's the mathematics of counting.” And it's not mathematicians sitting around going “1, 2, 3, 4” or “2, 4, 6, 8.” It's looking for structures and ways to count the number of possibilities there are, the number of—if we're thinking about calculating probabilities of winning the lottery, somebody's got to know what the probabilities are of choosing winning numbers, of choosing five out of six winning numbers. And the field of combinatorics is what does that. It counts possibilities. So I know that mathematicians and kindergartners—this is what I've learned in both my graduate education and in my postgraduate education working with kindergartners—is that they both think about counting in this rich way. It's any work that you do to know how many there are. And that might be one by one; it might be skip-counting; it might be multiplication; it might be using some other kind of structure. Mike: I think that's really interesting because there was a point in time where I saw counting as a fairly rote process, right? Where I didn't understand that there were all of these elements of counting, meaning one-to-one correspondence and quantity versus being able to just say the rote count out loud. And so one way that I think counting and its meaning have expanded for me is to kind of understand some of those pieces. But the thing that occurs to me as I hear you talk is that I think one of the things that I've done at different points, and I wonder if people do, is say, “That's all fine and good, but counting is counting.” And then we've suddenly shifted and we're doing something called addition or multiplication. And this is really interesting because it feels like you're drawing a much clearer connection between those critical, emergent ideas around counting and these other things we do to try to figure out the answer to how many or how did you count. Tell me what you think about that. Christopher: Yeah. So this for me is the project, right? This book is an instantiation of this larger project, a way of viewing the world of mathematics through the lens of what it means to learn it. And I would describe that larger project through some imagery and appealing to teachers' ideas about what it means to have a classroom conversation. For me, learning is characterized by increasing sophistication, increasing expertise with whatever it is that I'm studying. And so when I put several different triangular arrangements of things—in the book, there's a triangular arrangement of bowling pins, which lots of kids know from having bowled in their lives and other kids don't have any experiences with them, but the image is rich and vivid and they're able to do that counting. And then later on, there's a triangular arrangement of what turned out to be very bland, gooey, and nasty, but beautiful to photograph: pink pudding cups. Later on, there are two triangles of eggs. And so what I'm asking of kids—I'm always imagining a child and a parent sitting on a couch reading these books together, but also building them for classrooms. Any of this could be like a thing that happens at home, a thing that happens for a kid individually or a classroom full of children led by a teacher. Thinking about the second picture of the pudding cups, my hope and expectation is that at least some children will say, "OK, there are 6 rows in this triangle and there were 4 rows previously. So I already know these first four are 10. I don't have to do any more work, and then 5 plus 6, right?” And then that demonstrates some learning. They're more expert with this triangle than they would have been previously. I'm also expecting that there's going to be some kid who's counting them 1 by 1, and I'm expecting that there are going to be some kids who are like, "You know what? That 6 up top and the 1 makes 7 and the 5 and the 2 make 7, and the 4 and the 3. So it's 3 sevens. There's 21.” I'm expecting that we're going to have—in a reasonably large population of third, fourth, fifth graders, sort of the target audience for this book—we're going to have some kids who are doing each of these. And for me, getting back to this larger project, that is a rich task, which can be approached in a bunch of different ways, and all of those children are doing the same sort of task. They're all counting at various levels of sophistication representing various opportunities to learn previously, various ways of applying their new learning as they're having conversations, looking at new images, hearing other people's ideas, but that larger project of building something that is rich enough for everybody to be able to find something new in, but simple enough for everybody to have access to—yeah, that's the larger project. Mike: So one of the things that I found myself thinking about when I was thinking about my own experiences with dot talks or some of the subitizing images that I've used and the book that you have, is: There's something about the way that a set of items can be arranged. And I think what's interesting about that is I've heard you say that that arrangement can both reveal structure, in terms of number, but it can also make connections to ideas in geometry. And I wonder if you could talk a little bit about that. Christopher: Yeah. I'll draw a quick distinction that I think will be helpful. If you've ever seen bowling pins, right? It's four, three, two, one. The one [pin] is at the front; the [row of] four is at the back. Arranged so that the three fit into the spaces between the four as you're looking at it from the front. Very iconic arrangement. And you can quickly tell that it's a symmetric triangle and the longest row is four. You might just know that that's 10. But if you take those same bowling pins and just toss them around inside of a classroom or inside of a closet and they're just lying on the floor, so they're all in your field of vision, you don't know that there's 10 right away. You have to do a different kind of work in order to know that there are 10 of them. In that sense, the structure of the triangle with the longest row of four is a thing that you can start to recognize as you learn about triangles and ultimately what mathematicians refer to as triangular numbers. That's a thing you can learn to recognize, but learning to recognize 10 in that arrangement doesn't afford you anything when it's 10 [pins] scattered around on the floor. Unless you do a little abstraction. There's a story in the book about a lovely sixth grader who proceeded to tell me about how the bowling pin arrangement matches a way that she thinks about things. Because if she's ever going about her life, I don't know, making a bracelet or buying groceries, collecting pencils for the first day of school or whatever. If she wants to count them, and it looks like there's probably fewer than 100 but more than 5, she will grab a set of 4, a set of 3, a set of 2, a set of 1, and she'll know that's 10. Unprompted by me, except that we had this bowling pin arrangement. So there are ways to abstract from that. You can use these structures that you've noticed in order to do something that isn't structured that way, but the 4, 3, 2, 1 thing probably came from recognizing that 4, 3, 2, 1 made this nice little geometric arrangement. So our eyes, our brains, are tuned to symmetry and to beauty and elegance, and there is something much more lovely about a nice arrangement of 4, 3, 2, 1 than there is about a bunch of scattered things. And so a lot of those things are things that have been captured by mathematicians. So we have words for square numbers—3 times 3 is 9 because you can make 3 rows of 3 and you make something that looks nice that way. Triangular numbers, there are other figurate numbers like hexagonal numbers, but yet innate in our minds, there is an appeal to symmetry. And so if we start arranging things in symmetric patterned ways that will be appealing to our brains and to our eyes and to our mathematical minds, and my goal is to try to tap into that in order to help kids become more powerful mathematicians. Mike: So I want to go back to something you said earlier, and I think it's an important distinction before I ask this next question. One of the things that's fascinating is that a child could engage with this kind of image, and there doesn't necessarily have to be an adult in the room or a teacher who's guiding them. But what I was thinking about is: If there is a student or a pair of students or a classroom of students, and you're an educator and you're engaging them with one of these images, how do you think about the educator's role in that space? What are they trying to do? How should they think about their purpose? And then I'm going to ask a sub-question: To what extent do you feel like annotation is a part of what an educator might do? Christopher: Yes. One thing that teachers are generally more expert at than young children is being able to state something simply, clearly, concisely in a way that lots of other people can understand. If you listen to children thinking aloud, it is often hesitant and halting and it goes in different directions and units get left off. So they'll say, “3 and then 4 more is 8” and they've left off the fact that the 4 were—I mean, you could just easily get lost. And so one of the roles that a teacher plays can certainly be to help make clear to other students the ideas that a particular student is expressing and at the same time, often helping make it more clear for that student, right? Often a restating or a question or an introduction of a vocabulary word that seems like it's going to be helpful right now will not just be helpful to other people to understand it for the whole class, but will be helpful for the student in clarifying their own ideas and their own thinking, solidifying it in some kind of way. So that's one of the roles. I know that there are also roles that involve—and I think about this a lot whenever I'm working with learners—status, right? Making sure that children that have different perceived status in the classroom are able to be lifted up. That we're not just hearing from the kid who's been identified as “the math kid.” So I think intellectual status, social status, those are going to be balances, right? I also understand that teachers have a role in making sure that children are listening to each other. If I'm working with learners, I can't always be the one to do the restating. I've got to make sure there are times where kids are required to try to understand each other's thinking and not just the teacher's restatement of that thinking. There are just so many balances. But I would say that some top ones for me, if I'm thinking about how to make choices, thinking about raising up the status of all learners as intellectual resources, making good on a promise that I make to children, which is that any way of counting these things is valid and not telling a kid, "Oh no, no, no, we're not counting 1 by 1 today" or, "Oh no, no, no, that's too sophisticated. That's too advanced of a—We can't share that because nobody will understand it." So making good on that promise that I make at the beginning, which is, "I really want to know how you counted." Making sure that learners are able to get better at expressing the ideas that are in their heads using language and gesture and making sure that learners are communicating with each other and not just with me as a teacher. Those seem like four important tensions, and a talented and experienced elementary teacher could probably name like 10 other tensions that they're keeping in mind all at the same time: behavior, classroom management, but also some ideas around multilingual learners. Yeah, a lot of respect for the kind of balances that teachers have to maintain and the kinds of tensions that they have to choose when to use and when to gloss over or not worry about for right now. So you ask about annotation and, absolutely, I think about multiple representations of mathematical ideas. And so far I've only focused on the role of the teacher in a classroom discussion and thinking about gesture, thinking about words and other language forms, but I haven't focused on writing and annotation is absolutely a role that teachers can play. For me, the thing that I want to have happen is I want children to see their ideas represented in multiple ways. So if they've described for the class something in words and gestures, then there are sort of two natural easy annotations for a teacher to do or a teacher to have students do, which is, one, make those gestures and words explicit in the image. And that's where something like a smartboard or projecting onto a whiteboard—lots of technologies that teachers use for this kind of stuff—but where we can write directly on the image. So if you said you put the 1 and the 4 together in the bowling pins and then the 3 and the 2, then I might make a loopy thing that goes around the 4 and the 1, and I might circle the 3 and the 2, right? And so that adds both some clarity for students looking, but also is a model for: Here's how we can start to annotate our images. But then I'm also probably going to want to write 4 plus 1, maybe in parentheses, plus 3 plus 2 in parentheses, so that we can connect the 4 to the four [items] that are circled, the 1 to the one that is circled, the 4 plus 1 in parentheses, identifying that as a group,... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/39806720

Season 4 | Episode 9 - Dr. Todd Hinnenkamp, Enacting Talk Moves with Intention 01/08/2026

Season 4 | Episode 9 - Dr. Todd Hinnenkamp, Enacting Talk Moves with Intention Dr. Todd Hinnenkamp, Enacting Talk Moves with Intention ROUNDING UP: SEASON 4 | EPISODE 9 All students deserve a classroom rich in meaningful mathematical discourse. But what are the talk moves educators can use to bring this goal to life in their classrooms? Today, we're talking about this question with Todd Hinnenkamp from the North Kansas City Schools. Whether talk moves are new to you or already a part of your practice, this episode will deepen your understanding of the ways they impact your classroom community. BIOGRAPHY Dr. Todd Hinnenkamp is the instructional coordinator for mathematics for the North Kansas City Schools. RESOURCES TRANSCRIPT Mike Wallus: Before we begin, I'd like to offer a quick note to listeners. During this episode, we'll be referencing a series of talk moves throughout the conversation. You can find a link to these included in the show notes for this episode. Welcome to the podcast, Todd. I'm really excited to be chatting with you today. Todd Hinnenkamp: I'm excited to be here with you, Mike. Talk through some things. Mike: Great. So I've heard you present on using talk moves with intention, and one of the things that you shared at the start was the idea that talk moves advance three aspects of teaching and learning: a productive classroom community, student agency, and students’ mathematical practice. So as a starting point, can you unpack that statement for listeners? Todd: Sure. I think all talk moves with intention contribute to advancing all three of those, maybe some more than others. But all can be impactful in this endeavor, and I really think that identifying them or understanding them well upfront is super important. So if you unpack “productive community” first, I think about the word “productive” as an individual word. In different situations, it means a quality or a power of producing, bringing about results, benefits, those types of things. And then if you pair that word “community” alongside, I think about the word “community” as a unified body of individuals, an interacting population. I even like to think about it as joint ownership or participation. When that's present, that's a pretty big deal. So I like to think about those two concepts individually and then also together. So when you think about the “productivity” word and the “community” word and then pairing them well together, is super important. And I think about student agency. Specifically the word “agency” means something pretty powerful that I think we need to have in mind. When you think about it in a way of, like, having the capacity or the condition or state of acting or even exerting some power in your life. I think about students being active in the learning process. I think about engagement and motivation and them owning the learning. I think oftentimes we see that because they feel like they have the capacity to do that and have that agency. So I think about that, that being a thing that we would want in every single classroom so they can be productive contributors later in life as well. So I feel like sometimes there's too many students in classrooms today with underdeveloped agencies. So I think if we can go after agency, that's pretty powerful as well. And when you think about students’ math practice, super important habits of what we want to develop in students. I mean, we're fortunate to have some clarity around those things, those practices, thanks to more than a decade ago when they provided us the standards for mathematical practice. But if you think about the word “practice” alone, it's interesting. I've done some research on this. I think the transitive verb meaning is to do or perform often, customarily or maybe habitually. The transitive verb meaning is to pursue something actively. Or if you think about it with a noun, it's just a usual way of doing something or condition of being proficient through a systematic exercise. So I think all those things are, if we can get kids to develop their math practice in a way it becomes habitual and is really strong within them, it's pretty powerful. So I do think it's important that we start with that. We can't glaze over these three concepts because I think that right now, if you can tie some intentional talk moves to them, I think that it can be a pretty powerful lever to student understanding. Mike: Yeah. You have me thinking about a couple things. One of the first things that jumped out as I was listening to you talk is there's the “what,” which are the talk moves, but you're really exciting the stage with the “why.” Why do we want to do these things? And what I'd like to do is take each one of them in turn. So can we first talk about some of the moves that set up productive community for learners? Todd: Yeah. I think all the moves that are on my mind contribute, but there's probably a couple that I think go after productive community even more so than others. And I would say the “student restates” move, that first move where you're expecting students to repeat or restate in their own words what another student shared, promotes some really special things. I think first it communicates to everyone in the room that “We're going to talk about math in here. We're going to listen to and respectfully consider what others say and think.” It really upholds my expectation as an educator that we're going to interact with and understand the mathematical thinking that's present so that student restates is a great one to get going. And I would also offer the “think, turn, and learn” move is a highly impactful one as well. The general premise here is that you're offering time upfront. Always starting with “think,” you're offering time upfront. And what that should be communicating to students is that “You have something to offer. I'm providing you time to think about it, to organize it, so then you're more apt to share it with either your partner or the community.” It really increases the likelihood that kids have something to contribute. And as you literally turn your body and learn from each other—and those words are intentional, “turn” and “learn”—it opens the door to share, to expand your thinking, to then refine what you're thinking and build to develop both speaking and listening skills that help the community bond become stronger. So in the end it says, "I have something to offer here. I'm valued through my interactions.” And I feel like that there's something that comes out of that process for kids. Mike: You talked about the practice of “think, turn, learn.” And one of the things that jumps out is “think.” Because we've often used language like “turn and talk,” and that's in there with “turn and learn,” but “think” feels really important. I wonder if you could say more about why “think”? Let's just make it explicit. Why “think”? Todd: Sure. No, and I'm not trying to throw shade at “turn and talks” or anything like that, but I do think when we have intention with our moves, they're super impactful relative to other opportunities where maybe we're just not getting the most out of it. So that idea of offering time or providing or ensuring time for kids to think upfront—and depending on the situation, that can be 10 seconds, that can be 30 seconds—where you feel like students have had a chance to internalize what's going on [and] think about what they would say, it puts them in an entirely different mode to build a share with somebody else. I'm often in classrooms, and if we don't provide that think time, you see kids turn and talk to each other, and the first part is them still trying to figure out what should be said. And it just doesn't seem like it's as impactful or as productive during that time as it could be without that “think” first. Mike: Yeah, absolutely. I want to go back to something you said earlier too, when you were describing the value that comes out of restating or rephrasing, having a student do that with another student's thinking. One of the things that struck me is there were points in time when you were talking about that and you were talking about the value for an individual student who's in that spot. Todd: Mm. Mike: But I also heard you come back to it and say, “There's something in this for the group, for the community as well.” And I wonder if you could unpack a little bit: What's in it for the kid when they go through that restating another student’s [idea], or having their [own] idea restated, and then what's in it for the community? Todd: Sure. Well, let's start with the individual, Mike. And I think that with what we know about learning and how much more deeply we learn when we internalize something and reflect on it and actually link it to our past learning and think about what it means to us, is probably the most important thing that comes out of that. So the student that's restating what another student says, they really have to think about what that student said and then internalize it and make sense of it in a way where they can actually say it out to the community again. That's a big deal! So to talk about the impact on the community in that mode, Mike, when you get one or two [ideas], and maybe you ask for a couple more, you now have student thinking in four different forms out in the community rather than, say, one student sharing something and a teacher restating it and moving on. And I just love how those moves together can cause the thinking to linger in the classroom longer for kids. Often when I'm in classrooms, the kids actually learn it more when somebody else says it rather than me. And it kind of ties to that where, like, they just need to hear other kids thinking and start to process that a little bit more on their level. And we get to shore that up too as teachers. We can shore up whatever's missing if we need to later. But I think the depth that comes from thinking about it, putting it out in the community, having more kids think about [it] is pretty powerful. Mike: I think what's cool about that is the idea that there's four or five ideas floating around and how different that is than [when] a kid says something, the teacher restates it and moves on. I might not have made sense of it on the first kid's description or the teacher's description, but when those things linger around, there's a much better chance that I'm going to make sense of it. Todd: Yeah. And I agree, Mike. And what's really important in that process as well is the first move I always talk about is “wait.” You literally have to wait. When the student restates something, we've got to let that sit for a little bit for it to really be something that other kids can grasp onto and then give them time to process what they heard and then ask if someone could restate. At that point, it's causing all this cognition in the brain, and it's making me think about what I understand and what I don't understand about what was said. And it just starts to build and make a huge difference over time. Mike: Yeah. I'm glad you said that because I'm a person who talks to think, but that is not true of a lot of folks. Todd: (laughs) Mike: A lot of people need time to think… Todd: Sure. Mike: …before they talk. And so I think it's really important to recognize that that wait time is really an opportunity for mental space. And if we don't do that, it actually might fall flat. Todd: Totally agree. I'd see it day in and day out in classrooms I'm in, where if we can offer that time to let that concept or thinking permeate across the room for a little bit longer, it's a whole different outcome. Mike: Nice. I'm wondering if we can pivot and talk a little bit about moves that support student agency and their mathematical practice. They really do feel like they're kind of interconnected. Todd: Yeah, I think they are somewhat interconnected as I think about them. And I see agency as like a broader concept, like really that development of capacity to act or have power in a situation. But when you think about math practices—thinking about the standards for mathematical practices—it's a little more specific. So when you think about the math practice of perseverance, I think we have to think about the move [called] wait time that I just talked about. When used with intention, I think it can communicate to kids, “I've got confidence in you. You have something to offer. I believe in you and that you're capable of contributing here.” I just think that we have to think about our use of wait time and the messages that kids get from that and be careful not to squelch their opportunity to grow in those situations. Mike: OK. I have a follow-up. You're making me think about ways to do wait time well and ways to do wait time that might have an unintended consequence. So walk me through a really productive use of wait time—what the language is that the teacher uses or how they manage what can feel uncomfortable for most of us. Todd: Sure. And I will be very upfront that anytime you start to use wait time, if you haven't before, there's going to be some discomfort. (laughs) You think about, if you're a person that always wants to fill that space or feel like you need to because students aren't quite contributing, then you start to shift your practice to cause there to be a little more extended wait time, there's going to be some discomfort that plays out in that situation. So I think honestly, Mike, part of it is having the right question or the right prompt, and setting up the expectation and upholding it over time. I talk a lot with teachers about establishing and maintaining productive community. I think that we have to establish it over time and then maintain it. And what I mean by that is if you start to use wait time, you're establishing that norm in your classroom, is that I'm always going to give you time to think, and that's super important in here because we want to make sure that we get the most out of the experience. The maintaining part of it, I believe, is where we uphold that over time. We don't start to back off if kids don't then share their thinking. We can't always fill that space. And I think sometimes an inappropriate use of wait time is if we do it pretty well, but then we rescue when there's a time that kids aren't sharing something. So I do believe that no matter what classroom you're in, there is always one kid that can give you at least a nugget that you can go with. So I think as much as you can wait and try to draw that out before interjecting is super important. Mike: Yeah. You make me think about a scenario that I encountered a fair amount when I was teaching elementary, which was: I'd ask a question and there were two or three kids who immediately put their hand up. There were quite a few that were still thinking, and it was really uncomfortable for me, but I think also for some of the kids who had their hands up, that I didn't immediately call on them, that I actually waited and let the question marinate… Todd: Yes. Mike: …and the end product was great. I had more kids who had something to say because they had that space. But it was a little uncomfortable, especially for those kids who were like, "Wait, I know it immediately. Why aren't you calling on me? Todd: (laughs) Yes. And I think it's super important what you just shared, Mike, because in our practice, we have to be aware that the day-to-day practices or actions that we enact in our lessons, they're impacting everything from community, agency, practice. All the things that we're talking about today are sometimes just suddenly being impacted either positively or negatively. And I think the scenario you described about your practice is, like, you were intentional about it. You became aware, you realized that there's a handful of kids that I'm probably letting drive the discourse maybe more than I need to. And you're right: You've got other kids in classrooms that I'm in that are really waiting to talk and never have the chance. And I do feel like those are the kids that are going to have a hard time staying caught up with everybody because they're not getting that opportunity to develop some of those habits. Mike: Yeah. It makes me think about when I was a kid as well. I was not the fast kid, right? I was thinking about it, but I was not the first kid with my hand up. You've really got me thinking about how wait time is a real subtle way of saying, “You're not necessarily the most competent person just because you have your hand up first.” There's no added bonus that says, like, "You're the best just because your hand's up first. Everybody can contribute. You might need a little bit of time to process. That's super normal in a math class.” Todd: Yes, it is. And you go back to what we discussed earlier about being a valued contributor in the community, and you think about what those kids feel once they experience that wait time and then their ideas being the ones that drive the discourse or that are highlighted or presented. That's where you draw that in, and if you can have 50% of your kids be the ones that are feeling that, then you got to shoot for 60, and then 70, and so on. But you gotta start to expand the number of kids that talk and share and restate and do all the things around discourse, but wait time is a super powerful tool to do that. Mike: Another thing that you shared when I saw your presentation was the idea that you can pair talk moves in a sequence and that those sequence talk moves can have a powerful impact on kids. And I'm wondering if you can talk a bit about some of the ways that educators can sequence talk moves to have maximum impact. Todd: Sure. Yeah. And I'm not necessarily suggesting that there is an always or a 100% correct way to line them up and sequence them, but I do think there's some [that], if you can go after them in particular instances in your professional practice, I think it's going to change your practice, I think more quickly and more deeply. And the same goes with a lesson. I think right off the bat, we first must wait. We have to start to build that into our practice where we wait. So if we offer a prompt [or] pose a question, let it sit for a second. I always talk about 4 to 6 seconds would be about how long you'd want to just let it sit for a little bit. Then, if you’ve got the right question and the right prompt, I think you could just say, "OK, now I'm going to give you some time to think and then I'm going to have you turn and actually learn with a partner. So I want you to think about the prompt that's on the board. What would you share with your partner?” And literally you give them time to think and then you can turn and learn. So at that point, I think it's important that you're walking about the community, listening in, getting a feel for what's being discussed, because I think at that point you can have a feel for maybe what you might want to go to next, what insight you want to make sure is surfaced that is aligned to the learning goal of the day. That's how you get all that headed in the right direction. So you gotta lean in and figure that out. And I think at that point you could ask someone to share. “OK, who can share what you and your partner talked about?” See what happens; see what you get. You can be strategic if nobody offers. You can just say, "Hey, would you end up sharing? I listened to what you had. Would you mind sharing?" And then I think at that point you could use a “Do you agree or disagree and why?” So here's their thinking on this situation. So I want you to really think about it. Do you agree with what they're sharing or not? And then I'm going to ask you why. Let that sit. Give them some time to think. Let that play out. I think at that point you could offer... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/39597160

Season 4 | Episode 8 – Janet Walkoe & Margaret Walton, Exploring the Seeds of Algebraic Thinking 12/18/2025

Season 4 | Episode 8 – Janet Walkoe & Margaret Walton, Exploring the Seeds of Algebraic Thinking Janet Walkoe & Margaret Walton, Exploring the Seeds of Algebraic Thinking ROUNDING UP: SEASON 4 | EPISODE 8 Algebraic thinking is defined as the ability to use symbols, variables, and mathematical operations to represent and solve problems. This type of reasoning is crucial for a range of disciplines. In this episode, we're talking with Janet Walkoe and Margaret Walton about the seeds of algebraic thinking found in our students' lived experiences and the ways we can draw on them to support student learning. BIOGRAPHIES Margaret Walton joined Towson University’s Department of Mathematics in 2024. She teaches mathematics methods courses to undergraduate preservice teachers and courses about teacher professional development to education graduate students. Her research interests include teacher educator learning and professional development, teacher learning and professional development, and facilitator and teacher noticing. Janet Walkoe is an associate professor in the College of Education at the University of Maryland. Janet's research interests include teacher noticing and teacher responsiveness in the mathematics classroom. She is interested in how teachers attend to and make sense of student thinking and other student resources, including but not limited to student dispositions and students' ways of communicating mathematics. RESOURCES TRANSCRIPT Mike Wallus: Hello, Janet and Margaret, thank you so much for joining us. I'm really excited to talk with you both about the seeds of algebraic thinking. Janet Walkoe: Thanks for having us. We're excited to be here. Margaret Walton: Yeah, thanks so much. Mike: So for listeners, without prayer knowledge, I'm wondering how you would describe the seeds of algebraic thinking. Janet: OK. For a little context, more than a decade ago, my good friend and colleague, [Mariana] Levin—she's at Western Michigan University—she and I used to talk about all of the algebraic thinking we saw our children doing when they were toddlers—this is maybe 10 or more years ago—in their play, and just watching them act in the world. And we started keeping a list of these things we saw. And it grew and grew, and finally we decided to write about this in our 2020 FLM article [ in For the Learning of Mathematics] that introduced the seeds of algebraic thinking idea. Since they were still toddlers, they weren't actually expressing full algebraic conceptions, but they were displaying bits of algebraic thinking that we called “seeds.” And so this idea, these small conceptual resources, grows out of the knowledge and pieces perspective on learning that came out of Berkeley in the nineties, led by Andy diSessa. And generally that's the perspective that knowledge is made up of small cognitive bits rather than larger concepts. So if we're thinking of addition, rather than thinking of it as leveled, maybe at the first level there's knowing how to count and add two groups of numbers. And then maybe at another level we add two negative numbers, and then at another level we could add positives and negatives. So that might be a stage-based way of thinking about it. And instead, if we think about this in terms of little bits of resources that students bring, the idea of combining bunches of things—the idea of like entities or nonlike entities, opposites, positives and negatives, the idea of opposites canceling—all those kinds of things and other such resources to think about addition. It's that perspective that we're going with. And it's not like we master one level and move on to the next. It's more that these pieces are here, available to us. We come to a situation with these resources and call upon them and connect them as it comes up in the context. Mike: I think that feels really intuitive, particularly for anyone who's taught young children. That really brings me back to the days when I was teaching kindergartners and first graders. I want to ask you about something else. You all mentioned several things like this notion of “do, undo” or “closing in” or the idea of “in-betweenness” while we were preparing for this interview. And I'm wondering if you could describe what these things mean in some detail for our audience, and then maybe connect them back with this notion of the seeds of algebraic thinking. Margaret: Yeah, sure. So we would say that these are different seeds of algebraic thinking that kids might activate as they learn math and then also learn more formal algebra. So the first seed, the doing and undoing that you mentioned, is really completing some sort of action or process and then reversing it. So an example might be when a toddler stacks blocks or cups. I have lots of nieces and nephews or friends’ kids who I've seen do this often—all the time, really—when they'll maybe make towers of blocks, stack them up one by one and then sort of unstack them, right? So later this experience might apply to learning about functions, for example, as students plug in values as inputs, that's kind of the doing part, but also solve functions at certain outputs to find the input. So that's kind of one example there. And then you also talked about closing in and in-betweenness, which might both be related to intervals. So closing in is a seed where it's sort of related to getting closer and closer to a desired value. And then in formal algebra, and maybe math leading up to formal algebra, the seed might be activated when students work with inequalities maybe, or maybe ordering fractions. And then the last seed that you mentioned there, in-betweenness, is the idea of being between two things. For example, kids might have experiences with the story of Goldilocks and the Three Bears, and the porridge being too hot, too cold, or just right. So that “just right” is in-between. So these seats might relate to inequalities and the idea that solutions of math problems might be a range of values and not just one. Mike: So part of what's so exciting about this conversation is that the seeds of algebraic thinking really can emerge from children's lived experience, meaning kids are coming with informal prior knowledge that we can access. And I'm wondering if you can describe some examples of children's play, or even everyday tasks, that cultivate these seeds of algebraic thinking. Janet: That's great. So when I think back to the early days when we were thinking about these ideas, one example stands out in my head. I was going to the grocery store with my daughter who was about three at the time, and she just did not like the grocery store at all. And when we were in the car, I told her, “Oh, don't worry, we're just going in for a short bit of time, just a second.” And she sat in the back and said, “Oh, like the capital letter A.” I remember being blown away thinking about all that came together for her to think about that image, just the relationship between time and distance, the amount of time highlighting the instantaneous nature of the time we'd actually be in the store, all kinds of things. And I think in terms of play examples, there were so many. When she was little, she was gifted a play doctor kit. So it was a plastic kit that had a stethoscope and a blood pressure monitor, all these old-school tools. And she would play doctor with her stuffed animals. And she knew that any one of her stuffed animals could be the patient, but it probably wouldn't be a cup. So she had this idea that these could be candidates for patients, and it was this—but only certain things. We refer to this concept as “replacement,” and it's this idea that you can replace whatever this blank box is with any number of things, but maybe those things are limited and maybe that idea comes into play when thinking about variables in formal algebra. Margaret: A couple of other examples just from the seeds that you asked about in the previous question. One might be if you're talking about closing in, games like when kids play things like “you're getting warmer” or “you're getting colder” when they're trying to find a hidden object or you're closing in when tuning an instrument, maybe like a guitar or a violin. And then for in-betweeness, we talked about Goldilocks, but it could be something as simple as, “I'm sitting in between my two parents” or measuring different heights and there's someone who's very tall and someone who's very short, but then there are a bunch of people who also fall in between. So those are some other examples. Mike: You're making me wonder about some of these ideas, these concepts, these habits of mind that these seeds grow into during children's elementary learning experiences. Can we talk about that a bit? Janet: Sure. Thank you for that question. So we think of seeds as a little more general. So rather than a particular seed growing into something or being destined for something, it's more that a seed becomes activated more in a particular context and connections with other seeds get strengthened. So for example, the idea of like or nonlike terms with the positive and negative numbers. Like or nonlike or opposites can come up in so many different contexts. And that's one seed that gets evoked when thinking potentially when thinking about addition. So rather than a seed being planted and growing into things, it's more like there are these seeds, these resources that children collect as they act on the world and experience things. And in particular contexts, certain seeds are evoked and then connected. And then in other contexts, as the context becomes more familiar, maybe they're evoked more often and connected more strongly. And then that becomes something that's connected with that context. And that's how we see children learning as they become more expert in a particular context or situation. Mike: So in some ways it feels almost more like a neural network of sorts. Like the more that these connections are activated, the stronger the connection becomes. Is that a better analogy than this notion of seeds growing? It's more so that there are connections that are made and deepened, for lack of a better way of saying it? Janet: Mm-hmm. And pruned in certain circumstances. We actually struggled a bit with the name because we thought seeds might evoke this, “Here's a seed, it's this particular seed, it grows into this particular concept.” But then we really struggled with other neurons of algebraic thinking. So we tossed around some other potential ideas in it to kind of evoke that image a little better. But yes, that's exactly how I would think about it. Mike: I mean, just to digress a little bit, I think it's an interesting question for you all as you're trying to describe this relationship, because in some respects it does resemble seeds—meaning that the beginnings of this set of ideas are coming out of lived experiences that children have early in their lives. And then those things are connected and deepened—or, as you said, pruned. So it kind of has features of this notion of a seed, but it also has features of a network that is interconnected, which I suspect is probably why it's fairly hard to name that. Janet: Mm-hmm. And it does have—so if you look at, for example, the replacement seed, my daughter playing doctor with her stuffed animals, the replacement seed there. But you can imagine that that seed, it's domain agnostic, so it can come out in grammar. For instance, the ad-libs, a noun goes here, and so it can be any different noun. It's the same idea, different context. And you can see the thread among contexts, even though it's not meaning the same thing or not used in the same way necessarily. Mike: It strikes me that understanding the seeds of algebraic thinking is really a powerful tool for educators. They could, for example, use it as a lens when they're planning instruction or interpreting student reasoning. Can you talk about this, Margaret and Janet? Margaret: Yeah, sure, definitely. So we've seen that teachers who take a seeds lens can be really curious about where student ideas come from. So, for example, when a student talks about a math solution, maybe instead of judging whether the answer is right or wrong, a teacher might actually be more curious about how the student came to that idea. In some of our work, we've seen teachers who have a seeds perspective can look for pieces of a student answer that are productive instead of taking an entire answer as right or wrong. So we think that seeds can really help educators intentionally look for student assets and off of them. And for us, that's students’ informal and lived experiences. Janet: And kind of going along with that, one of the things we really emphasize in our methods courses, and is emphasized in teacher education in general, is this idea of excavating for student ideas and looking at what's good about what the student says and reframing what a student says, not as a misconception, but reframing it as what's positive about this idea. And we think that having this mindset will help teachers do that. Just knowing that these are things students bring to the situation, these potentially productive resources they have. Is it productive in this case? Maybe. If it's not, what could make it more productive? So having teachers look for these kinds of things we found as helpful in classrooms. Mike: I'm going to ask a question right now that I think is perhaps a little bit challenging, but I suspect it might be what people who are listening are wondering, which is: Are there any generalizable instructional moves that might support formal or informal algebraic thinking that you'd like to see elementary teachers integrate into their classroom practice? Margaret: Yeah, I mean, I think, honestly, it's: Listen carefully to kids' ideas with an open mind. So as you listen to what kids are saying, really thinking about why they're saying what they're saying, maybe where that thinking comes from and how you can leverage it in productive ways. Mike: So I want to go back to the analogy of seeds. And I also want to think about this knowing what you said earlier about the fact that some of the analogy about seeds coming early in a child's life or emerging from their lived experiences, that's an important part of thinking about it. But there's also this notion that time and experiences allow some connections to be made and to grow or to be pruned. What I'm thinking about is the gardener. The challenge in education is that the gardener who is working with students in the form of the teacher and they do some cultivation, they might not necessarily be able to kind of see the horizon, see where some of this is going, see what's happening. So if we have a gardener who's cultivating or drawing on some of the seeds of algebraic thinking in their early childhood students and their elementary students, what do you think the impact of trying to draw on the seeds or make those connections can be for children and students in the long run? Janet: I think [there are] a couple of important points there. And first, one is early on in a child's life. Because experiences breed seeds or because seeds come out of experiences, the more experiences children can have, the better. So for example, if you’re in early grades, and you can read a book to a child, they can listen to it, but what else can they do? They could maybe play with toys and act it out. If there's an activity in the book, they could pretend or really do the activity. Maybe it's baking something or maybe it's playing a game. And I think this is advocated in literature on play and early childhood experiences, including Montessori experiences. But the more and varied experiences children can have, the more seeds they'll gain in different experiences. And one thing a teacher can do early on and throughout is look at connections. Look at, “Oh, we did this thing here. Where might it come out here?” If a teacher can identify an important seed, for instance, they can work to strengthen it in different contexts as well. So giving children experiences and then looking for ways to strengthen key ideas through experiences. Mike: One of the challenges of hosting a podcast is that we've got about 20 to 25 minutes to discuss some really big ideas and some powerful practices. And this is one of those times where I really feel that. And I'm wondering, if we have listeners who wanted to continue learning about the ways that they can cultivate the seeds of algebraic thinking, are there particular resources or bodies of research that you would recommend? Janet: So from our particular lab we have a website, and it's , and that's continuing to be built out. The project is funded by NSF [the National Science Foundation], and we're continuing to add resources. We have links to articles. We have links to ways teachers and parents can use seeds. We have links to professional development for teachers. And those will keep getting built out over time. Margaret, do you want to talk about the article? Margaret: Sure, yeah. Janet and I actually just had an article recently come out in Mathematics Teacher: Learning and Teaching from NCTM [National Council of Teachers of Mathematics]. And it's [in] Issue 5, and it's called “Leveraging Early Algebraic Experiences.” So that's definitely another place to check out. And Janet, anything else you want to mention? Janet: I think the website has a lot of resources as well. Mike: So I've read the article and I would encourage anyone to take a look at it. We'll add a link to the article and also a link to the website in the show notes for people who are listening who want to check those things out. I think this is probably a great place to stop. But I want to thank you both so much for joining us. Janet and Margaret, it's really been a pleasure talking with both of you. Janet: Thank you so much, Mike. It's been a pleasure. Margaret: You too. Thanks so much 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 /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/39372550

Season 4 | Episode 7 - Tutita Casa, Anna Strauss, Jenna Waggoner & Mhret Wondmagegne, Developing Student Agency: The Strategy Showcase 12/04/2025

Season 4 | Episode 7 - Tutita Casa, Anna Strauss, Jenna Waggoner & Mhret Wondmagegne, Developing Student Agency: The Strategy Showcase Tutita Casa, Anna Strauss, Jenna Waggoner & Mhret Wondmagegne, Developing Student Agency: The Strategy Showcase ROUNDING UP: SEASON 4 | EPISODE 7 When students aren't sure how to approach a problem, many of them default to asking the teacher for help. This tendency is one of the central challenges of teaching: walking the fine line between offering support and inadvertently cultivating dependence. In this episode, we're talking with a team of educators about a practice called the strategy showcase, designed to foster collaboration and help students engage with their peers' ideas. BIOGRAPHIES Tutita Casa is an associate professor of elementary mathematics education at the Neag School of Education at the University of Connecticut. Mhret Wondmagegne, Anna Strauss, and Jenna Waggoner are all recent graduates of the University of Connecticut School of Education and early career elementary educators who recently completed their first years of teaching. RESOURCE TRANSCRIPT Mike Wallus: Well, we have a full show today and I want to welcome all of our guests. So Anna, Mhret, Jenna, Tutita, welcome to the podcast. I'm really excited to be talking with you all about the strategy showcase. Jenna Waggoner: Thank you. Tutita Casa: It's our pleasure. Anna Strauss: Thanks. Mhret Wondmagegne: Thank you. Mike: So for listeners who've not read your article, Anna, could you briefly describe a strategy showcase? So what is it and what could it look like in an elementary classroom? Anna: So the main idea of the strategy showcase is to have students' work displayed either on a bulletin board—I know Mhret and Jenna, some of them use posters or whiteboards. It's a place where students can display work that they've either started or that they've completed, and to become a resource for other students to use. It has different strategies that either students identified or you identified that serves as a place for students to go and reference if they need help on a problem or they're stuck, and it's just a good way to have student work up in the classroom and give students confidence to have their work be used as a resource for others. Mike: That was really helpful. I have a picture in my mind of what you're talking about, and I think for a lot of educators that's a really important starting point. Something that really stood out for me in what you said just now, but even in our preparation for the interview, is the idea that this strategy showcase grew out of a common problem of practice that you all and many teachers face. And I'm wondering if we can explore that a little bit. So Tutita, I'm wondering if you could talk about what Anna and Jenna and Mhret were seeing and maybe set the stage for the problem of practice that they were working on and the things that may have led into the design of the strategy showcase. Tutita: Yeah. I had the pleasure of teaching my coauthors when they were master’s students, and a lot of what we talk about in our teacher prep program is how can we get our students to express their own reasoning? And that's been a problem of practice for decades now. The has led that work. And to me, [what] I see is that idea of letting go and really being curious about where students are coming from. So that reasoning is really theirs. So the question is what can teachers do? And I think at the core of that is really trying to find out what might be limiting students in that work. And so Anna, Jenna, and Mhret, one of the issues that they kept bringing back to our university classroom is just being bothered by the fact that their students across the elementary grades were just lacking the confidence, and they knew that their students were more than capable. Mike: Jenna, I wonder if you could talk a little bit about, what did that actually look like? I'm trying to imagine what that lack of confidence translated into. What you were seeing potentially or what you and Anna and Mhret were seeing in classrooms that led you to this work. Jenna: Yeah, I know definitely we were reflecting, we were all in upper elementary, but we were also across grade levels anywhere from fourth to fifth grade all the way to sixth and seventh. And across all of those places, when we would give students especially a word problem or something that didn't feel like it had one definite answer or one way to solve it or something that could be more open-ended, we a lot of times saw students either looking to teachers. “I'm not sure what to do. Can you help me?” Or just sitting there looking at the problem and not even approaching it or putting something on their paper, or trying to think, “What do I know?” A lot of times if they didn't feel like there was one concrete approach to start the problem, they would shut down and feel like they weren't doing what they were supposed to or they didn't know what the right way to solve it was. And then that felt like kind of a halting thing to them. So we would see a lot of hesitancy and not that courage to just kind of be productively struggling. They wanted to either feel like there was something to do or they would kind of wait for teacher guidance on what to do. Mike: So we're doing this interview and I can see Jenna and the audience who's listening, obviously Jenna, they can't see you, but when you said “the right way,” you used a set of air quotes around that. And I'm wondering if you or Anna or Mhret would like to talk about this notion of the right way and how when students imagined there was a right way, that had an effect on what you saw in the classroom. Jenna: I think it can be definitely, even if you're working on a concept like multiplication or division, whatever they've been currently learning, depending on how they're presented instruction, if they're shown one way how to do something but they don't understand it, they feel like that's how they're supposed to understand to solve the problem. But if it doesn't make sense for them or they can't see how it connects to the problem and the overall concept, if they don't understand the concept for multiplication, but they've been taught one strategy that they don't understand, they feel like they don't know how to approach it. So I think a lot of it comes down to they're not being taught how to understand the concept, but they're more just being given one direct way to do something. And if that doesn't make sense to them or they don't understand the concepts through that, then they have a really difficult time of being able to approach something independently. Mike: Mhret, I think Jenna offered a really nice segue here because you all were dealing with this question of confidence and with kids who, when they didn't see a clear path or they didn't see something that they could replicate, just got stuck, or for lack of a better word, they kind of turned to the teacher or imagined that that was the next step. And I was really excited about the fact that you all had designed some really specific features into the strategy showcase that addressed that problem of practice. So I'm wondering if you could just talk about the particular features or the practices that you all thought were important in setting up the strategy showcase and trying to take up this practice of a strategy showcase. Mhret: Yeah, so we had three components in this strategy showcase. The first one, we saw it being really important, being open-ended tasks, and that combats what Jenna was saying of “the right way.” The questions that we asked didn't ask them to use a specific strategy. It was open-ended in a way that it asked them if they agreed or disagreed with a way that someone found an answer, and it just was open to see whatever came to their mind and how they wanted to start the task. So that was very important as being the first component. And the second one was the student work displayed, which Anna was talking about earlier. The root of this being we want students' confidence to grow and have their voices heard. And so their work being displayed was very important—not teacher work or not an example being given to them, but what they had in their mind. And so we did that intentionally with having their names covered up in the beginning because we didn't want the focus to be on who did it, but just seeing their work displayed—being worth it to be displayed and to learn from—and so their names were covered up in the beginning and it was on one side of the board. And then the third component was the students’ co-identified strategies. So that's when after they have displayed their individual work, we would come up as a group and talk about what similarities did we see, what differences in what the students have used. And they start naming strategies out of that. They start giving names to the strategies that they see their peers using, and we co-identify and create this strategy that they are owning. So those are the three important components. Mike: OK. Wow. There's a lot there. And I want to spend a little bit of time digging into each one of these and I'm going to invite all four of you to feel free to jump in and just let us know who's talking so that everybody has a sense of that. I wonder if you could talk about this whole idea that, when you say open-ended tasks, I think that's really important because it's important that we build a common definition. So when you all describe open-ended tasks, let's make sure that we're talking the same language. What does that mean? And Tutita, I wonder if you want to just jump in on that one. Tutita: Sure. Yeah. An open-ended task, as it suggests, it's not a direct line where, for example, you can prompt students to say, “You must use ‘blank’ strategy to solve this particular problem.” To me, it's just mathematical. That's what a really good rich problem is, is that it really allows for that problem solving, that reasoning. You want to be able to showcase and really gauge where your students are. Which, as a side benefit, is really beneficial to teachers because you can formatively assess where they're even starting with a problem and what approaches they try, which might not work out at first—which is OK, that's part of the reasoning process—and they might try something else. So what's in their toolbox and what tool do they reach for first and how do they use it? Mike: I want to name another one that really jumped out for me. I really—this was a big deal that everybody's strategy goes up. And Anna, I wonder if you can talk about the value and the importance of everybody's strategy going up. Why did that matter so much? Anna: I think it really helps, the main thing, for confidence. I had a lot of students who in the beginning of starting the strategy showcase would start kind of like at least with a couple ideas, maybe a drawing, maybe they outlined all of the numbers, and it helps to see all of the strategies because even if you are a student who started out with maybe one simple idea and didn't get too far in the problem, seeing up on the board maybe, “Oh, I have the same beginning as someone else who got farther into the problem.” And really using that to be like, “I can start a problem and I can start with different ideas, and it's something that can potentially lead to a solution.” So there is a lot of value in having all of the work that everyone did because even something that is just the beginning of a solution, someone can jump in and be like, “Oh, I love the way that you outlined that,” or “You picked those numbers first to work on. Let's see what we can use from the way that you started the problem to begin to work on a solution.” So in that way, everyone's voice and everyone's decisions have value. And even if you just start off with something small, it can lead to something that can grow into a bigger solution. Mike: Mhret, can I ask you about another feature that you mentioned? You talked about the importance, at least initially, of having names removed from the work. And I wonder if you could just expand on why that was important and maybe just the practical ways that you managed withholding the names, at least for some of the time when the strategy showcase was being set up. Can you talk about both of those please? Mhret: Yes, yeah. I think all three of us when we were implementing this, we—all kids are different. Some of them are very eager to share their work and have their name on it. But we had those kids that maybe they just started with a picture or whatever it may be. And so we saw their nerves with that, and we didn't want that to just mask that whole experience. And so it was very important for us that everybody felt safe. And later we'll talk about group norms and how we made it a safe space for everyone to try different strategies. But I think not having their names attached to it helped them focus not on who did it, but just the process of reasoning and doing the work. And so we did that practically I think in different ways, but I just use tape, masking tape to cover up their names. I know some of—I think maybe Jenna, you wrote their names on the back of the paper instead of the front. But I think a way to not make the name the focus is very important. And then hopefully by the end of it, our hope is that they would gain more confidence and want to name their strategy and say that that is who did it. Mike: I want to ask a follow up about this because it feels like one of the things that this very simple, but I think really important, idea of withholding who created the strategy or who did the work. I mean, I think I can say during my time in classrooms when I was teaching, there are kids that classmates kind of saw as really competent or strong in math. And I also know that there were kids who didn't think they were good at math or perhaps their classmates didn't think were good at math. And it feels like by withholding the names that would have a real impact on the extent to which work would be considered as valuable. Because you don't know who created it, you're really looking at the work as opposed to looking at who did the work and then deciding whether it's worth taking up. Did you see any effects like that as you were doing this? Jenna: This is Jenna. I was going to say, I know for me, even once the names were removed, you would still see kids sometimes want to be like, “Oh, who did this?” You could tell they still are almost very fixated on that idea of who is doing the work. So I think by removing it, it still was definitely good too. With time, they started to less focus on “Who did this?” And like you said, it's more taking ownership if they feel comfortable later down the road. But sometimes you would have, several students would choose one approach, kind of what they've seen in classrooms, and then you might have a few other slightly different, of maybe drawing a picture or using division and connecting it to multiplication. And then you never wanted those kids to feel like what they were doing was wrong. Even if they chose the wrong operation, there was still value in seeing how that was connected to the problem or why they got confused. So we never wanted one or two students also to feel individually focused on if maybe what they did initially—not [that it] wasn't correct, but maybe was leading them in the wrong direction, but still had value to understand why they chose to do that. So I think just helping, again, all the strategies work that they did feel valuable and not having any one particular person feel like they were being focused on when we were reflecting on what we put up on display. Mike: I want to go back to one other thing that, Mhret, you mentioned, and I'm going to invite any of you, again, to jump in and talk about this, but this whole idea that part of the prompting that you did when you invited kids to examine the strategies was this question of do you agree or do you disagree? And I think that's a really interesting way to kind of initiate students' reflections. I wonder if you can talk about why this idea of, “Do you agree or do you disagree” was something that you chose to engage with when you were prompting kids? And again, any of you all are welcome to jump in and address this, Anna: It's Anna. I think one of the reasons that we chose to [have them] agree or disagree is because students are starting to look for different ways to address the problem at hand. Instead of being like, “I need to find this final number” or “I need to find this final solution,” it's kind of looking [at], “How did this person go about solving the problem? What did they use?” And it gives them more of an opportunity to really think about what they would do and how what they're looking at helps in any way. Jenna: And then this is Jenna. I was also going to add on that I think by being “agree or disagree” versus being like, “yes, I got the same answer,” and I feel like the conversation just kind of ends at that point. But they could even be like, “I agree with the solution that was reached, but I would've solved it this way, or my approach was different.” So I think by having “agree or disagree,” it wasn't just focusing on, “yes, this is the correct number, this is the correct solution,” and more focused on, again, that approach and the different strategies that could be used to reach one specific solution that was the answer or the correct thing that you're looking for. Tutita: And this is Tutita, and I agree with all of that. And I can't help but going back just to the word “strategy,” which really reflects students' reasoning, their problem solving, argumentation. It’s really not a noun; it's a verb. It's a very active process. And sometimes we, as teachers, we're so excited to have our students get the right answer that we forget the fun in mathematics is trying to figure it out. And I can't help but think of an analogy. So many people love to watch sports. I know Jenna's a huge UConn women's basketball… Jenna: Woohoo! Tutita: …fan, big time. Or if you're into football, whatever it might be, that there's always that goal. You're trying to get as many more points, and as many as you can, more points than the other team. And there are a lot of different strategies to get there, but we appreciate the fact that the team is trying to move forward and individuals are trying to move forward. So it's that idea with the strategy, we need to as teachers really open up that space to allow that to come out and progressively—in the end, we're moving forward even though within a particular time frame, it might not look like we are quite yet. I like the word “yet.” But it's really giving students the time that they need to figure it out themselves to deepen their understanding. Mike: Well, I will say as a former Twin Cities resident, I've watched Paige Bueckers for a long time, and… Tutita: There we go. Mike: …in addition to being a great shooter, she's a pretty darn good passer and moves the ball. And in some ways that kind of connects with what you all are doing with kids, which is that—moving ideas around a space is really not that different from moving the ball in basketball. And that you have the same goal in scoring a basket or reaching understanding, but it's the exchange that are actually the things that sometimes makes that happen. Jenna: I love it. Thank you. Tutita: Nice job. Mike: Mhret, I wanted to go back to this notion that you were talking about, which is co-naming the strategies as you were going through and reflecting on them. I wonder if you could talk a little bit about, what does co-naming mean and why was it important as a part of the process? Mhret: Mm-hmm. Yeah. So, I think the idea of co-naming and co-identifying the strategies was important. Just to add on to the idea, we wanted... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/39186995

Season 4 | Episode 6 - Christy Pettis & Terry Wyberg, The Case for Choral Counting with Fractions 11/20/2025

Season 4 | Episode 6 - Christy Pettis & Terry Wyberg, The Case for Choral Counting with Fractions Christy Pettis & Terry Wyberg, The Case for Choral Counting with Fractions ROUNDING UP: SEASON 4 | EPISODE 6 How can educators help students recognize similarities in the way whole numbers and fractions behave? And are there ways educators can build on students' understanding of whole numbers to support their understanding of fractions? The answer from today's guests is an emphatic yes. Today we're talking with Terry Wyberg and Christy Pettis about the ways choral counting can support students' understanding of fractions. BIOGRAPHIES Terry Wyberg is a senior lecturer in the Department of Curriculum and Instruction at the University of Minnesota. His interests include teacher education and development, exploring how teachers’ content knowledge is related to their teaching approaches. Christy Pettis is an assistant professor of teacher education at the University of Wisconsin-River Falls. RESOURCES by Megan L. Franke, Elham Kazemi, and Angela Chan Turrou by The Math Learning Center TRANSCRIPT Mike Wallus: Welcome to the podcast, Terry and Christy. I'm excited to talk with you both today. Christy Pettis: Thanks for having us. Terry Wyberg: Thank you. Mike: So, for listeners who don't have prior knowledge, I'm wondering if we could just offer them some background. I'm wondering if one of you could briefly describe the choral counting routine. So, how does it work? How would you describe the roles of the teacher and the students when they're engaging with this routine? Christy: Yeah, so I can describe it. The way that we usually would say is that it's a whole-class routine for, often done in kind of the middle grades. The teachers and the students are going to count aloud by a particular number. So maybe you're going to start at 5 and skip-count by 10s or start at 24 and skip-count by 100 or start at two-thirds and skip-count by two-thirds. So you're going to start at some number, and you're going to skip-count by some number. And the students are all saying those numbers aloud. And while the students are saying them, the teacher is writing those numbers on the board, creating essentially what looks like an array of numbers. And then at certain points along with that talk, the teacher will stop and ask students to look at the numbers and talk about things they're noticing. And they'll kind of unpack some of that. Often they'll make predictions about things. They'll come next, continue the count to see where those go. Mike: So you already pivoted to my next question, which was to ask if you could share an example of a choral count with the audience. And I'm happy to play the part of a student if you'd like me to. Christy: So I think it helps a little bit to hear what it would sound like. So let's start at 3 and skip-count by 3s. The way that I would usually tell my teachers to start this out is I like to call it the runway. So usually I would write the first three numbers. So I would write “3, 6, 9” on the board, and then I would say, “OK, so today we're going to start at 3 and we're going to skip-count by 3s. Give me a thumbs-up or give me the number 2 when you know the next two numbers in that count.” So I'm just giving students a little time to kind of think about what those next two things are before we start the count together. And then when I see most people kind of have those next two numbers, then we're going to start at that 3 and we're going to skip-count together. Are you ready? Mike: I am. Christy: OK. So we're going to go 3… Mike & Christy: 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36. Christy: Keep going. Mike & Christy: 39, 42, 45, 48, 51. Christy: Let's stop there. So we would go for a while like that until we have an array of numbers on the board. In this case, I might've been recording them, like where there were five in each row. So it would be 3, 6, 9, 12, 15 would be the first row, and the second row would say 18, 21, 24, 27, 30, and so on. So we would go that far and then I would stop and I would say to the class, “OK, take a minute, let your brains take it in. Give me a number 1 when your brain notices one thing. Show me 2 if your brain notices two things, 3 if your brain notices three things.” And just let students have a moment to just take it in and think about what they notice. And once we've seen them have some time, then I would say, “Turn and talk to your neighbor, and tell them some things that you notice.” So they would do that. They would talk back and forth. And then I would usually warm-call someone from that and say something like, “Terry, why don't you tell me what you and Mike talked about?” So Terry, do you have something that you would notice? Terry: Yeah, I noticed that the last column goes up by 15, Christy: The last column goes up by 15. OK, so you're saying that you see this 15, 30, 45? Terry: Yes. Christy: In that last column. And you're thinking that 15 plus 15 is 30 and 30 plus 15 is 45. Is that right? Terry: Yes. Christy: Yeah. And so then usually what I would say to the students is say, “OK, so if you also noticed that last column is increasing by 15, give me a ‘me too’ sign. And if you didn't notice it, show an ‘open mind’ sign.” So I like to give everybody something they can do. And then we'd say, “Let's hear from somebody else. So how about you, Mike? What's something that you would notice?” Mike: So one of the things that I was noticing is that there's patterns in the digits that are in the ones place. And I can definitely see that because the first number 3 [is] in the first row. In the next row, the first number is 18 and the 8 is in the ones place. And then when I look at the next row, 33 is the first number in that row, and there's a 3 again. So I see this column pattern of 3 in the ones place, 8 in the ones place, 3 in the ones place, 8 in the ones place. And it looks like that same kind of a number, a different number. The same number is repeating again, where there's kind of like a number and then another number. And then it repeats in that kind of double, like two numbers and then it repeats the same two numbers. Christy: So, what I would say in that one is try to revoice it, and I'd probably be gesturing, where I’d do this. But I'd say, “OK, so Mike's noticing in this ones place, in this first column, he's saying he notices it's ‘3, 8, 3, 8.’ And then in other columns he's noticing that they do something similar. So the next column, or whatever, is like ‘6, 1, 6, 1’ in the ones place. Why don't you give, again, give me a ‘me too’ [sign] if you also noticed that pattern or an ‘open mind’ [sign] if you didn't.” So, that's what we would do. So, we would let people share some things. We would get a bunch of noticings while students are noticing those things. I would be, like I said, revoicing and annotating on the board. So typically I would revoice it and point it out with gestures, and then I would annotate that to take a record of this thing that they've noticed on the board. Once we've gotten several students’ noticings on the board, then we're going to stop and we're going to unpack some of those. So I might do something like, “Oh, so Terry noticed this really interesting thing where he said that the last column increases by 15 because he saw 15, 30, 45, and he recognized that. I'm wondering if the other columns do something like that too. Do they also increase by the same kind of number? Hmm, why don't you take a minute and look at it and then turn and talk to your neighbor and see what you notice.” And we're going to get them to notice then that these other ones also increase by 15. So if that hadn't already come out, I could use it as a press move to go in and unpack that one further. And then we would ask the question, in this case, “Why do they always increase by 15?” And we might then use that question and that conversation to go and talk about Mike's observation, and to say, like, “Huh, I wonder if we could use what we just noticed here to figure out about why this idea that [the numbers in the] ones places are going back and forth between 3, 8, 3, 8. I wonder if that has something to do with this.” Right? So we might use them to unpack it. They'll notice these patterns. And while the students were talking about these things, I'd be taking opportunities to both orient them to each other with linking moves to say, “Hey, what do you notice? What can you add on to what Mike said, or could you revoice it?” And also to annotate those things to make them available for conversation. Mike: There was a lot in your description, Christy, and I think that provides a useful way to understand what's happening because there's the choice of numbers, there's the choice of how big the array is when you're recording initially, there are the moves that the teacher's making. What you've set up is a really cool conversation that comes forward. We did this with whole numbers just now, and I'm wondering if we could take a step forward and think about, OK, if we're imagining a choral count with fractions, what would that look and sound like? Christy: Yeah, so one of the ones I really like to do is to do these ones that are just straight multiples, like start at 3 and skip-count by 3s. And then to either that same day or the very next day—so very, very close in time in proximity—do one where we're going to do something similar but with fractions. So one of my favorites is for the parallel of the whole number of skip-counting by 3s is we'll start at 3 fourths and we'll skip-count by 3 fourths. And when we write those numbers, we're not going to put them in simplest form; we're just going to write 3 fourths, 6 fourths, 9 fourths. So in this case, I would probably set it up in the exact same very parallel structure to that other one that we just did with the whole numbers. And I would put the numbers 3 fourths, 6 fourths, 9 fourths on the board. I would say, “OK, here's our first numbers. We're going to start starting at 4 fourths. We're going to skip-count by 3 fourths. And give me a thumbs-up or the show me a 2 when you know the next two numbers.” And then we would skip-count them together, and we would write them on the board. And so we'd end up—and in this case I would probably arrange them again in five columns just to have them and be a parallel structure to that one that we did before with the whole numbers. So it would look like 3 fourths, 6 fourths, 9 fourths, 12 fourths, 15 fourths on the first row. And then the next row, I would say 18 fourths, 21 fourths, 24 fourths, 27 fourths, 30 fourths. And again, I'd probably go all the way up until I got to 51 fourths before we'd stop and we'd look for patterns. Mike: So I think what's cool about that—it was unsaid, but it kind of implied—is that you're making a choice there. So that students had just had this experience where they were counting in increments of 3, and 3, 6, 9, 12, 15, and then you start another row and you get to 30, and in this case, 3 fourths, 6 fourths, 9 fourths, 12 fourths, 15 fourths. So they are likely to notice that there's something similar that's going on here. And I suspect that's on purpose. Christy: Right, that's precisely the thing that we want right here is to be able to say that fractions aren't something entirely new, something that you—just very different than anything that you've ever seen before in numbers. But to allow them to have an opportunity to really see the ways that numerators enumerate, they act like the counting numbers that they've always known, and the denominator names, and tells you what you're counting. And so it's just a nice space where, when they can see these in these parallel ways and experience counting with fractions, they have this opportunity to see some of the ways that both fraction notation works, what it's talking about, and also how the different parts of the fraction relate to things they already know with whole numbers. Mike: Well, let's dig into that a little bit more. So the question I was going to ask Terry was: Can we talk a bit more about the ways the choral counting routine can help students make sense of the mathematics of fractions? So what are some of the ideas or the features of fractions that you found choral counting really allows you to draw out and make sense of with students? Terry: Well, we know from our work with the rational number project how important language is when kids are developing an understanding of the role of the numerator and the denominator. And the choral counts really just show, like what Christy was just saying, how the numerator just enumerates and changes just like whole numbers. And then the denominator stays the same and names something. And so it's been a really good opportunity to develop language together as a class. Christy: Yeah. I think that something that's really important in these ones that you get to see when you have them. So when they're doing that language, they're also—a really important part of a choral count is that it's not just that they're hearing those things, they're also seeing the notation on the board. And because of the way that we're both making this choice to repeatedly add the same amount, right? So we're creating something that's going to have a pattern that's going to have some mathematical relationships we can really unpack. But they're also seeing the notation on there that's arranged in a very intentional way to allow them to see those patterns in rows and columns as they get to talk about them. So because those things are there, we're creating this chance now, right? So they see both the numerator and denominator. If we're doing them in parallel to things with whole numbers, they can see how both fractions are alike, things that they know with whole numbers, but also how some things are different. And instead of it being something that we're just telling them as rules, it invites them to make these observations. So in the example that I just gave you of the skip-counting, starting at 3 fourths and skip-counting by 3 fourths, every time I have done this, someone always observes that the right-hand column, they will always say it goes up by 15. And what they're observing right there is they're paying attention to the numerator and thinking, “Well, I don't really need to talk about the denominator,” and it buys me this opportunity as a teacher to say, “Yes, I see that too. I see that these 15 fourths and then you get another, then you get 30 fourths and you get 45 fourths. And I see in those numerators that 15, 30, 45—just like we had with the whole numbers—and here's how I would write that as a mathematician: I would write 15 fourths plus 15 fourths equals 30 fourths.” Because I'm trying to be clear about what I'm counting right now. So instead of telling it like it's a rule that you have to remember, you have to keep the same denominators when you're going to add, it instead becomes something where we get to talk about it. It's just something that we get to be clear about. And that in fractions, we also do this other piece where we both enumerate and we name, and we keep track of that when we write things down to be clear. And so it usually invites this very nice parallel conversation and opportunity just to set up the idea that when we're doing things like adding and thinking about them, that we're trying to be clear and we're trying to communicate something in the same way that we always have been. Mike: Well, Terry, it strikes me that this does set the foundation for some important things, correct? Terry: Yeah, it sets the foundation for adding and subtracting fractions and how that numerator counts things and the denominator tells you the size of the pieces. It also sets up multiplication. The last column, we can think of it as 5 groups of 3 fourths. And the next number underneath there might be 10 groups of 3 fourths. And as we start to describe or record what students' noticings are, we get a chance to highlight those features of adding fractions, subtracting fractions, multiplying fractions. Mike: We've played around the edges of a big idea here. And one of the things that I want to bring back is something we talked about when we were preparing for the interview. This idea that learners of any age, generally speaking, they want to make use of their understanding of the way that whole numbers work as they're learning about fractions. And I'm wondering if one or both of you want to say a little bit more about this. Terry: I think a mistake that we made previously in fraction teaching is we kind of stayed under 1. We just stayed and worked within 0 and 1 and we didn't go past it. And if you're going to make 1 a benchmark or 2 a benchmark or any whole number a benchmark, when you're counting by 3 fourths or 2 thirds or whatever, you have to go past it. So what choral counting has allowed us to do is to really get past these benchmarks, and kids saw patterns around those benchmarks, and they see them. And then I think we also saw a whole-number thinking get in the way. So if you ask, for example, somebody to compare 3 seventeenths and 3 twenty-thirds, they might say that 3 twenty-thirds are bigger because 23 is bigger than 17. And instead of embracing their whole-number knowledge, we kind of moved away from it. And so I think now with the choral counting, they're seeing that fractions behave like whole numbers. They can leverage that knowledge, and instead of trying to make it go away, they're using it as an asset. Mike: So the parallel that I'm drawing is, if you're trying to teach kids about the structure of numbers in whole number, if you can yourself to thinking about the whole numbers between 0 and 10, and you never worked in the teens or larger numbers, that structure's really hard to see. Am I thinking about that properly? Terry: Yes, you are. Christy: I think there's two things here to highlight. So one of them that I think Terry would say more about here is just the idea that, around the idea of benchmarks. So you're right that there's things that come out as the patterns and notation that happen because of how we write them. And when we're talking about place value notation, we really need to get into tens and really into hundreds before a lot of those things become really available to us as something we talk about, that structure of how 10 plays a special role. In fractions, a very parallel idea of these things that become friendly to us because of the notation and things we know, whole numbers act very much like that. When we're talking about rational numbers, right? So they become these nice benchmarks because they're really friendly to us, there's things that we know about them, so when we can get to them, they help us. And the choral count that we were just talking about, there's something that's a little bit different that's happening though because we're not highlighting the whole numbers in the way that we're choosing to count right there. So we're not—we're using those, I guess, improper fractions. In that case, what we're doing is we're allowing students to have an opportunity to play with this idea, the numerator and denominator or the numerator is the piece that's acting like whole numbers that they know. So when Terry was first talking about how oftentimes when we first teach fractions and we were thinking about them, we were think a lot about the denominator. The denominator is something that's new that we're putting in with fractions that we weren't ever doing before with whole numbers. And we have that denominator. We focus a lot on like, “Look, you could take a unit and you can cut it up and you can cut it up in eight pieces, and those are called eighths, or you could cut it up in 10 pieces, and those are called... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/39074290

Season 4 | Episode 5 - Ramsey Merritt, Improving Students’ Turn & Talk Experience 11/06/2025

Season 4 | Episode 5 - Ramsey Merritt, Improving Students’ Turn & Talk Experience Ramsey Merritt, Improving Students’ Turn & Talk Experience ROUNDING UP: SEASON 4 | EPISODE 5 Most educators know what a turn and talk is—but are your students excited to do them? In this episode, we put turn and talks under a microscope. We'll talk with Ramsey Merritt from the Harvard Graduate School of Education about ways to revamp and better scaffold turn and talks to ensure your students are having productive mathematical discussions. BIOGRAPHY Ramsey Merritt is a lecturer in education at Brandeis University and the director of leadership development for Reading (MA) Public Schools. He has taught and coached at every level of the U.S. school system in both public and independent schools from New York to California. Ramsey also runs an instructional leadership consulting firm, Instructional Success Partners, LLC. Prior to his career in education, he worked in a variety of roles at the New York Times. He is currently completing his doctorate in education leadership at Harvard Graduate School of Education. Ramsey's book, , will be released in spring 2026. TRANSCRIPT Mike Wallus: Welcome to the podcast, Ramsey. So great to have you on. Ramsey Merritt: It is my pleasure. Thank you so much for having me. Mike: So turn and talk's been around for a while now, and I guess I'd call it ubiquitous at this point. When I visit classrooms, I see turn and talks happen often with quite mixed results. And I wanted to start with this question: At the broadest level, what's the promise of a turn and talk? When strategically done well, what's it good for? Ramsey: I think at the broadest level, we want students talking about their thinking and we also want them listening to other students’ thinking and ideally being open to reflect, ask questions, and maybe even change their minds on their own thinking or add a new strategy to their thinking. That's at the broadest level. I think if we were to zoom in a little bit, I think turn and talks are great for idea generation. When you are entering a new concept or a new lesson or a new unit, I think they're great for comparing strategies. They're obviously great for building listening skills with the caveat that you put structures in place for them, which I'm sure we'll talk about later. And building critical-thinking and questioning skills as well. I think I've also seen turn and talks broadly categorized into engagement, and it's interesting when I read that because to me I think about engagement as the teacher's responsibility and what the teacher needs to do no matter what the pedagogical tool is. So no matter whether it's a turn and talk or something else, engagement is what the teacher needs to craft and create a moment. And I think a lot of what we'll probably talk about today is about crafting moments for the turn and talk. In other words, how to engage students in a turn and talk, but not that a turn and talk is automatically engagement. Mike: I love that, and I think the language that you've used around crafting is really important. And it gets to the heart of what I was excited about in this conversation because a turn and talk is a tool, but there is an art and a craft to designing its implementation that really can make or break the tool itself. Ramsey: Yeah. If we look back a little bit as to where turn and talk came from, I sort of tried to dig into the papers on this. And what I found was that it seems as if turn and talks may have been a sort of spinoff of the think-pair-share, which has been around a little bit longer. And what's interesting in looking into this is, I think that turn and talks were originally positioned as a sort of cousin of think-pair-share that can be more spontaneous and more in the moment. And I think what has happened is we've lost the “think” part. So we've run with it, and we've said, “This is great,” but we forgot that students still need time to think before they turn and talk. And so what I see a lot is, it gets to be somewhat too spontaneous, and certain students are not prepared to just jump into conversations. And we have to take a step back and sort of think about that. Mike: That really leads into my next question quite well because I have to confess that when I've attended presentations, there are points in time when I've been asked to turn and talk when I can tell you I had not a lot of interest nor a lot of clarity about what I should do. And then there were other points where I couldn't wait to start that conversation. And I think this is the craft and it's also the place where we should probably think about, “What are the pitfalls that can derail or have a turn and talk kind of lose the value that's possible?” How would you talk about that? Ramsey: Yeah, it is funny that we as adults have that reaction when people say, “Turn and talk.” The three big ones that I see the most, and I should sort of say here, I've probably been in 75 to 100 buildings and triple or quadruple that for classrooms. So I've seen a lot of turn and talks, just like you said. And the three big ones for me, I'll start with the one that I see less frequently but still see it enough to cringe and want to tell you about it. And it's what I call the “stall” turn and talk. So it's where teachers will sometimes use it to buy themselves a little time. I have literally heard teachers say something along the lines of, “OK, turn and talk to your neighbor while I go grab something off the printer.” But the two biggest ones I think lead to turn and talk failure are a lack of specificity. And in that same vein too, what are you actually asking them to discuss? So there's a bit of vagueness in the prompting, so that's one of the big ones. The other big one for me is, and it seems so simple, and I think most elementary teachers are very good at using an engaging voice. They've learned what tone does for students and what signals tone sends to them about, “Is now the time to engage? Should I be excited?” But I so often see the turn and talk launched unenthusiastically, and that leads to an engagement deficit. And that's what you're starting out with if you don't have a good launch: Students are already sort of against you because you haven't made them excited to talk. Mike: I mean those things resonate. And I have to say there are some of them that I cringe because I've been guilty of doing, definitely the first thing when I've been unprepared. But I think these two that you just shared, they really go to this question of how intentionally I am thinking about building that sense of engagement and also digging into the features that make a turn and talk effective and engaging. So let's talk about the features that make turn and talks effective and engaging for students. I've heard you talk about the importance of picking the right moment for a turn and talk. So what's that mean? Ramsey: So for me, I break it down into three key elements. And one of them, as you say, is the timing. And this might actually be the most important element, and it goes back to the origin story, is: If you ask a question, and say you haven't planned a turn and talk, but you ask a question to a whole group and you see 12 hands shoot up, that is an ideal moment for a turn and talk. You automatically know that students are interested in this topic. So I think that's the sort of origin story, is: Instead of whipping around the room and asking all 12 students—because especially at the elementary level, if students don't get their chance to share, they are very disappointed. So I've also seen these moments drag out far too long. So it's kind of a good way to get everyone's voice heard. Maybe they're not saying it out to the whole group, but they get to have everyone's voice heard. And also you're buying into the engagement that's already there. So that would be the more spontaneous version, but you can plan in your lesson planning to time a turn and talk at a specific moment if you know your students well enough that you know can get them engaged in. And so that leads to one of the other points is the launch itself. So then you're really thinking about, “OK, I think this could be an interesting moment for students. Let me think a little bit deeper about what the hook is.” Almost every teacher knows what a hook is, but they typically think about the hook at the very top of their lesson. And they don't necessarily think about, “How do I hook students in to every part of my lesson?” And maybe it's not a full 1-minute launch, maybe it's not a full hook, but you’ve got to reengage students, especially now in this day and time, we're seeing students with increasingly smaller attention spans. So it's important to think about how you're launching every single piece of your lesson. And then the third one, which goes against that origin story that I may or may not even be right about, but it goes against that sort of spontaneous nature of turn and talks, is: I think the best turn and talks are usually planned out in advance. So for me it's planning, timing, and launching. Those are my elements to success when I'm coaching teachers on doing a turn and talk. Mike: Another question that I wanted to unpack is: Talk about what. The turn and talk is a vehicle, but there's also content, right? So I'm wondering about that. And then I'm also wondering are there prompts or particular types of questions that educators can use that are more interesting and engaging, and they help draw students in and build that engagement experience you were talking about? Ramsey: Yeah, and it's funny you say, “Talk about what” because that's actually feedback that I've given to teachers, when I say, “How did that go for you?” And they go, “Well, it went OK.” And I say, “Well, what did you ask them to talk about? Talk about what is important to think about in that planning process.” So I hate to throw something big out there, but I would actually argue that at this point, we have seen the turn and talk sort of devolve into something that is stigmatized that often is vague. So what if instead of calling everything a turn and talk, you had specific types of turn and talks in your classroom. And these would take a little time to routinize; students would have to get used to them. But one idea I had is: What if you just called one “pick a side”? Pick a side, it tells the students right away what they need to do; it's extremely specific. So you're giving them one or two or—well not one, you're giving them two or three strategies, and you're telling them, “You have to pick one of these. And you're going to be explaining to your partner your rationale as to why you think that strategy works best or most efficiently.” Or maybe it's an error analysis kind of thing. Maybe you plant one n as wrong, one n as right. And then you still ask them, “Pick a side here. Who do you agree with?” And then you also get a check for understanding because the students around the room who are picking the wrong one, you're picking up data on what they understand about the topic. Another one you can do is, you could just call it “justify your thinking.” Justify your thinking. So that just simply says to them, “I have to explain to the person next to me why I'm thinking the way that I'm thinking about this prompt or this problem.” So that could also be a “help their thinking.” So maybe you put up someone's thinking on the board that is half baked, and now their job is to help that person. So that's a sort of deeper knowledge kind of thing too. And then the last one is we can turn the “What do you notice? What do you wonder?” [activity] into a routine that is very similar to a turn and talk, where both people have an opportunity to share what they're wondering or what they're noticing. But I think no matter what you call them, no matter how you routinize them, I think it's important to be more specific than “turn and talk.” Mike: You use the word routinized. It's making me think a lot about why we find routines to have value, right? Because once you teach a particular routine, kids know what it is to do said routine. They know what it is to show up when you're doing Which one doesn't belong? They know the role that they play. And I think part of what really jumps out is: If you had a series of more granular turn and talk experiences that you were trying to cultivate, kids actually have a sense of what it is to do a turn and talk if you are helping thinking, or if you are agreeing or disagreeing, or whatever the choice might be. Ramsey: That's right. For me, everything, even when I'm working with middle and high school teachers, I say, “The more that you can put structures in place that remove those sort of barriers for thinking, the better off you're going to be.” And so we could talk more too about how to differentiate and scaffold turn and talk. Sometimes that gets forgotten as well. But I think the other piece I would love to point out here is around—you're right, turn and talk is so ubiquitous. And what that means, what I've seen in schools, if I've seen, I'll go into a school and I might watch four different teachers teach the same lesson and the turn and talk will look and feel differently in each room. So the other advantage to being more specific is that if a student—let's say they went to, because even in an elementary school you might go to a specialist, you might go to art class. And that teacher might use a turn and talk. And what happens is they sort of get this general idea around the turn and talk and then they come into your room with whatever the turn and talk was in the last class or however the teacher used it last year. So to me there's also a benefit in personalizing it to your room as well so that you can get rid of some of that stigma if it wasn't going well for the student before, especially if you then go in and scaffold it. Mike: Let's talk a little bit about those scaffolds and maybe dig in a little bit deeper to some of the different kinds of routinized turn and talks. I'm wondering if you wanted to unpack anything in particular that you think would really be important for a teacher to think about as they're trying to take up the ideas that we've been discussing. Ramsey: And one of the simplest ones to implement is the Partner A, Partner B routine. I think maybe many of your listeners will be like, “Yeah, I use that.” But one of the pieces that's really important there is that you really hold students accountable to honoring Partner A's time. So when Partner A is speaking, Partner B needs to be trying to make—you know, not everybody can do the eye contact thing, but there are some things that you can recommend and suggest for them. Maybe they have something to take notes on. So this could be having whiteboards at your rug, it could be clipboards, it could be that they have a turn and talk thought-catcher notebook or folder. And it doesn't matter what it is, but not everyone has the same processing skills. So we think about turn and talk sometimes as spontaneous, but we're forgetting that 12 students raised their hand and they were eager. What about the other 12 or 15? If they didn't raise their hand, it could be that they're shy but they have something on their mind. But it also could be that you just threw out a prompt and they haven't fully processed it yet. We know kids process things at different times and at different speeds. So incorporating in that—maybe it's even a minute up top. Everybody's taking their silent and solo minute to think about this prompt. Then Partner A is going to go. It's about equity and voice across the room. It's about encouraging listening, it's about giving think time. Mike: Well, I want to stop and mark a couple things. What occurs to me is that in some ways a podcast interview like this is one long turn and talk in the sense that you and I are both listening and talking with one another. And as you were talking, one of the things I realized is I didn't have a piece of paper with me. And what you were saying really connects deeply because even if it's just jotting down a word or two to help me remember that was a salient point or this is something that I want to follow up on, that's really critical. Otherwise, it really can feel like it can evaporate and then you're left not being able to explore something that might've been really important. I think the other thing that jumps out is the way that this notion of having a notepad or something to jot is actually a way to not necessarily just privilege spoken communication. That if I'm going to process or if I'm going to try to participate, having something like that might actually open up space for a kid whose favorite thing to do isn't to talk and process as they're talking. Does that make sense? Ramsey: Totally. I had a student in a program I was working with this summer who was 13 years old but was selectively mute. And the student teachers who were working in this room wanted to still be able to do a turn and talk. And they had her still partner with people, but she wrote down sentences and she literally held up her whiteboard and then the other student responded to the sentence that she wrote down on her whiteboard. So that's real. And to your other point about being able to jot down so you can remember—yeah, we have to remember we're talking about six-, seven-, eight-, nine-year olds. We're fully functioning adults and we still need to jot things down. So imagine when your brain is not even fully developed. We can't expect them to remember something from when they haven't been allowed to interrupt the other. And so I think going on now what you're saying is, that then makes me think about the Partner A, Partner B thing could also sort of tamper down the excitement a little bit if you make another student wait. So you also have to think about maybe that time in between, you might need to reengage. That's my own thinking right now, evolving as we're talking. Mike: So in some ways this is a nice segue to something else that you really made me think about. When we were preparing for this interview, much of what I was thinking about is the role of the teacher in finding the moment, as you said, where you can build excitement and build engagement, or thinking about the kind of prompts that have a specificity and how that could impact the substance of what kids are talking about. But what really jumped out from our conversation is that there's also a receptive side of turn and talk, meaning that there are people who are talking, but we also don't want the other person to just be passive. What does it look like to support the listening side of turn and talk? And I would love it if you would talk about the kinds of things you think it's important for educators to think about when they're thinking about that side of turn and talk. Ramsey: I would say don't forget about sentence starters that have to do with listening. So often when we're scaffolding, we're thinking about, “How do I get them to share out? How do I get them to be able to address this prompt?” But one of the easiest scaffolds that I've heard for listening—and it works very, very well—is, “What I heard you say is, blank.” And so then the receptive student knows that a—tells them they have to be listening pretty carefully because they're about to be asked to repeat what the other person said. And this is an age-old elementary school sort of piece of pedagogy, is a call and response situation. But then we want to give them a stem that allows them maybe to ask a question. So it's, “What I heard you say was, blank. What I'm wondering is, blank.” So that takes it to the next thinking level. But again, it's about being really specific and very intentional... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/38852870

Season 4 | Episode 4 - Pam Harris, Exploring the Power & Purpose of Number Strings 10/23/2025

Season 4 | Episode 4 - Pam Harris, Exploring the Power & Purpose of Number Strings Pam Harris, Exploring the Power & Purpose of Number Strings ROUNDING UP: SEASON 4 | EPISODE 4 I've struggled when I have a new strategy I want my students to consider and despite my best efforts, it just doesn't surface organically. While I didn't want to just tell my students what to do, I wasn't sure how to move forward. Then I discovered number strings. Today, we're talking with Pam Harris about the ways number strings enable teachers to introduce new strategies while maintaining opportunities for students to discover important relationships. BIOGRAPHY Pam Harris, founder and CEO of Math is Figure-out-able™, is a mom, a former high school math teacher, a university lecturer, an author, and a mathematics teacher educator. Pam believes real math is thinking mathematically, not just mimicking what a teacher does. Pam helps leaders and teachers to make the shift that supports students to learn real math. RESOURCES by Catherine Fosnot and Maarten Dolk by the National Council of Teachers of Mathematics from Grade 5, Unit 3, Module 1, Session 1 (BES login required) by Pamela Weber Harris and Cameron Harris TRANSCRIPT Mike Wallus: Welcome to the podcast, Pam. I'm really excited to talk with you today. Pam Harris: Thanks, Mike. I'm super glad to be on. Thanks for having me. Mike: Absolutely. So before we jump in, I want to offer a quick note to listeners. The routine we're going to talk about today goes by several different names in the field. Some folks, including Pam, refer to this routine as “problem strings,” and other folks, including some folks at The Math Learning Center, refer to them as “number strings.” For the sake of consistency, we'll use the term “strings” during our conversation today. And Pam, with that said, I'm wondering if for listeners, without prior knowledge, could you briefly describe strings? How are they designed? How are they intended to work? Pam: Yeah, if I could tell you just a little of my history. When I was a secondary math teacher and I dove into research, I got really curious: How can we do the mental actions that I was seeing my son and other people use that weren't the remote memorizing and mimicking I'd gotten used to? I ran into the work of Cathy Fosnot and Maarten Dolk, and [their book] Young Mathematicians at Work, and they had pulled from the Netherlands strings. They called them “strings.” And they were a series of problems that were in a certain order. The order mattered, the relationship between the problems mattered, and maybe the most important part that I saw was I saw students thinking about the problems and using what they learned and saw and heard from their classmates in one problem, starting to let that impact their work on the next problem. And then they would see that thinking made visible and the conversation between it and then it would impact how they thought about the next problem. And as I saw those students literally learn before my eyes, I was like, “This is unbelievable!” And honestly, at the very beginning, I didn't really even parse out what was different between maybe one of Fosnot's rich tasks versus her strings versus just a conversation with students. I was just so enthralled with the learning because what I was seeing were the kind of mental actions that I was intrigued with. I was seeing them not only happen live but grow live, develop, like they were getting stronger and more sophisticated because of the series of the order the problems were in, because of that sequence of problems. That was unbelievable. And I was so excited about that that I began to dive in and get more clear on: What is a string of problems? The reason I call them “problem strings” is I'm K–12. So I will have data strings and geometry strings and—pick one—trig strings, like strings with functions in algebra. But for the purposes of this podcast, there's strings of problems with numbers in them. Mike: So I have a question, but I think I just want to make an observation first. The way you described that moment where students are taking advantage of the things that they made sense of in one problem and then the next part of the string offers them the opportunity to use that and to see a set of relationships. I vividly remember the first time I watched someone facilitate a string and feeling that same way, of this routine really offers kids an opportunity to take what they've made sense of and immediately apply it. And I think that is something that I cannot say about all the routines that I've seen, but it was really so clear. I just really resonate with that experience of, what will this do for children? Pam: Yeah, and if I can offer an additional word in there, it influences their work. We're taking the major relationships, the major mathematical strategies, and we're high-dosing kids with them. So we give them a problem, maybe a problem or two, that has a major relationship involved. And then, like you said, we give them the next one, and now they can notice the pattern, what they learned in the first one or the first couple, and they can let it influence. They have the opportunity for it to nudge them to go, “Hmm. Well, I saw what just happened there. I wonder if it could be useful here. I'm going to tinker with that. I'm going to play with that relationship a little bit.” And then we do it again. So in a way, we're taking the relationships that I think, for whatever reason, some of us can wander through life and we could run into the mathematical patterns that are all around us in the low dose that they are all around us, but many of us don't pick up on that low dose and connect them and make relationships and then let it influence when we do another problem. We need a higher dose. I needed a higher dose of those major patterns. I think most kids do. Problem strings or number strings are so brilliant because of that sequence and the way that the problems are purposely one after the other. Give students the opportunity to, like you said, apply what they've been learning instantly [snaps]. And then not just then, but on the next problem and then sometimes in a particular structure we might then say, “Mm, based on what you've been seeing, what could you do on this last problem?” And we might make that last problem even a little bit further away from the pattern, a little bit more sophisticated, a little more difficult, a little less lockstep, a little bit more where they have to think outside the box but still could apply that important relationship. Mike: So I have two thoughts, Pam, as I listen to you talk. One is that for both of us, there's a really clear payoff for children that we've seen in the way that strings are designed and the way that teachers can use them to influence students' thinking and also help kids build a recognition or high-dose a set of relationships that are really important. The interesting thing is, I taught kindergarten through second grade for most of my teaching career, and you've run the gamut. You've done this in middle school and high school. So I think one of the things that might be helpful is to share a few examples of what a string could look like at a couple different grade levels. Are you OK to share a few? Pam: You bet. Can I tack on one quick thing before I do? Mike: Absolutely. Pam: You mentioned that the payoff is huge for children. I'm going to also suggest that one of the things that makes strings really unique and powerful in teaching is the payoff for adults. Because let's just be clear, most of us—now, not all, but most of us, I think—had a similar experience to me that we were in classrooms where the teacher said, “Do this thing.” That's the definition of math is for you to rote memorize these disconnected facts and mimic these procedures. And for whatever reason, many of us just believed that and we did it. Some people didn't. Some of us played with relationships and everything. Regardless, we all kind of had the same learning experience where we may have taken at different places, but we still saw the teacher say, “Do these things. Rote memorize. Mimic.” And so as we now say to ourselves, “Whoa, I've just seen how cool this can be for students, and we want to affect our practice.” We want to take what we do, do something—we now believe this could be really helpful, like you said, for children, but doing that's not trivial. But strings make it easier. Strings are, I think, a fantastic differentiated kind of task for teachers because a teacher who's very new to thinking and using relationships and teaching math a different way than they were taught can dive in and do a problem string. Learn right along with your students. A veteran teacher, an expert teacher who's really working on their teacher moves and really owns the landscape of learning and all the things still uses problem strings because they're so powerful. Like, anybody across the gamut can use strings—I just said problem strings, sorry—number strengths—[laughs] strings, all of us no matter where we are in our teaching journey can get a lot out of strings. Mike: So with all that said, let's jump in. Let's talk about some examples across the elementary span. Pam: Nice. So I'm going to take a young learner, not our youngest, but a young learner. I might ask a question like, “What is 8 plus 10?” And then if they're super young learners, I expect some students might know that 10 plus a single digit is a teen, but I might expect many of the students to actually say “8, 9, 10, 11, 12,” or “10, 11,” and they might count by ones given—maybe from the larger, maybe from the whatever. But anyway, we're going to kind of do that. I'm going to get that answer from them. I'm going to write on the board, “8 plus 10 is 18,” and then I would have done some number line work before this, but then I'm going to represent on the board: 8 plus 10, jump of 10, that's 18. And then the next problem's going to be something like 8 plus 9. And I'm going to say, “Go ahead and solve it any way you want, but I wonder—maybe you could use the first problem, maybe not.” I'm just going to lightly suggest that you consider what's on the board. Let them do whatever they do. I'm going to expect some students to still be counting. Some students are going to be like, “Oh, well I can think about 9 plus 8 counting by ones.” I think by 8—”maybe I can think about 8 plus 8. Maybe I can think about 9 plus 9.” Some students are going to be using relationships, some are counting. Kids are over the map. When I get an answer, they're all saying, like, 17. Then I'm going to say, “Did anybody use the first problem to help? You didn't have to, but did anybody?” Then I'm going to grab that kid. And if no one did, I'm going to say, “Could you?” and pause. Now, if no one sparks at that moment, then I'm not going to make a big deal of it. I'll just go, “Hmm, OK, alright,” and I'll do the next problem. And the next problem might be something like, “What's 5 plus 10?” Again, same thing, we're going to get 15. I'm going to draw it on the board. Oh, I should have mentioned: When we got to the 8 plus 9, right underneath that 8, jump, 10 land on 18, I'm going to draw an 8 jump 9, shorter jump. I'm going to have these lined up, land on the 17. Then I might just step back and go, “Hmm. Like 17, that's almost where the 18 was.” Now if kids have noticed, if somebody used that first problem, then I'm going to say, “Well, tell us about that.” “Well, miss, we added 10 and that was 18, but now we're adding 1 less, so it's got to be 1 less.” And we go, “Well, is 17 one less than 18? Huh, sure enough.” Then I give the next set of problems. That might be 5 plus 10 and then 5 plus 9, and then I might do 7 plus 10. Maybe I'll do 9 next. 9 plus 10 and then 9 plus 9. Then I might end that string. The next problem, the last problem might be, “What is 7 plus 9?” Now notice I didn't give the helper. So in this case I might go, “Hey, I've kind of gave you plus 10. A lot of you use that to do plus 9. I gave you plus 10. Some of you use that to do plus 9, I gave you plus 10. Some of you used that plus 9. For this one, I'm not giving you a helper. I wonder if you could come up with your own helper.” Now brilliantly, what we've done is say to students, “You've been using what I have up here, or not, but could you actually think, ‘What is the pattern that's happening?’ and create your own helper?” Now that's meta. Right? Now we're thinking about our thinking. I'm encouraging that pattern recognition in a different way. I'm asking kids, “What would you create?” We're going to share that helper. I'm not even having them solve the problem. They're just creating that helper and then we can move from there. So that's an example of a young string that actually can grow up. So now I can be in a second grade class and I could ask a similar [question]: “Could you use something that's adding a bit too much to back up?” But I could do that with bigger numbers. So I could start with that 8 plus 10, 8 plus 9, but then the next pair might be 34 plus 10, 34 plus 9. But then the next pair might be 48 plus 20 and 48 plus 19. And the last problem of that string might be something like 26 plus 18. Mike: So in those cases, there's this mental scaffolding that you're creating. And I just want to mark this. I have a good friend who used to tell me that part of teaching mathematics is you can lead the horse to water, you can show them the water, they can look at it, but darn it, do not push their head in the water. And I think what he meant by that is “You can't force it,” right? But you're not doing that with a string. You're creating a set of opportunities for kids to notice. You're doing all kinds of implicit things to make structure available for kids to attend to—and yet you're still allowing them the ability to use the strategies that they have. We might really want them to notice that, and that's beautiful about a string, but you're not forcing. And I think it's worth saying that because I could imagine that's a place where folks might have questions, like, “If the kids don't do the thing that I'm hoping that they would do, what should I do?” Pam: Yeah, that's a great question. Let me give you another example. And in that example I'll talk about that. So especially as the kids get older, I'm going to use the same kind of relationship. It's maybe easier for people to hang on to if I stay with the same sort of relationship. So I might say, “Hey everybody. 7 times 8. That's a fact I'm noticing most of us just don't have [snaps] at our fingertips. Let's just work on that. What do you know?” I might get a couple of strategies for kids to think about 7 times 8. We all agree it's 56. Then I might say, “What's 70 times 8?” And then let kids think about that. Now, this would be the first time I do that, but if we've dealt with scaling times 10 at all, if I have 10 times the number of whatever the things is, then often kids will say, “Well, I've got 10 times 7 is 70, so then 10 times 56 is 560.” And then the next problem might be, “I wonder if you could think about 69 times 8. If we've got 70 eights, can I use that to help me think about 69 eights?” And I'm saying that in a very specific way to help ping on prior knowledge. So then I might do something similar. Well, let's pick another often missed facts, I don't know, 6 times 9. And then we could share some strategies on how kids are thinking about that. We all agree it's 54. And then I might say, “Well, could you think about 6 times 90?” I'm going to talk about scaling up again. So that would be 540. Now I'm going really fast. But then I might say, “Could we use that to help us think about 6 times 89?” I don’t know if you noticed, but I sort of swapped. I'm not thinking about 90 sixes to 89 sixes. Now I'm thinking about 6 nineties to help me think about 6 eighty-nines. So that's a little bit of a—we have to decide how we're going to deal with that. I'll kind of mess around with that. And then I might have what we call that clunker problem at the end. “Notice that I've had a helper: 7 times 8, 70 times 8. A lot of you use that to help you think about 69 times 8. Then I had a helper: 6 times 9, 6 times 90. A lot of you use that to help you think about 6 times 89. What if I don't give you those helpers? What if I had something like”—now I'm making this up off the cuff here, like—“9 times 69. 9 times 69. Could you use relationships we just did?” Now notice, Mike, I might've had kids solving all those problems using an algorithm. They might've been punching their calculator, but now I'm asking the question, “Could you come up with these helper problems?” Notice how I'm now inviting you into a different space. It's not about getting an answer. I'm inviting you into, “What are the patterns that we've been establishing here?” And so what would be those two problems that would be like the patterns we've just been using? That's almost like saying when you're out in the world and you hit a problem, could you say to yourself, “Hmm, I don't know that one, but what do I know? What do I know that could help me get there?” And that's math-ing. Mike: So, you could have had a kid say, “Well, I'm not sure about how—I don't know the answer to that, but I could do 9 times 60, right?” Or “I could do 10 times”—I'm thinking—“10 times 69.” Correct? Pam: Yes, yes. In fact, when I gave that clunker problem, 9 times 69, I said to myself, “Oh, I shouldn't have said 9 because now you could go either direction.” You could either “over” either way. To find 9 I can do 10, or to find 69 I can do 70. And then I thought, “Ah, we'll go with it because you can go either way.” So I might want to focus it, but I might not. And this is a moment where a novice could just throw it out there and then almost be surprised. “Whoa, they could go either direction.” And an expert could plan, and be like, “Is this the moment where I want lots of different ways to go? Or do I want to focus, narrow it a little bit more, be a little bit more explicit?” It's not that I'm telling kids, but I'm having an explicit goal. So I'm maybe narrowing the field a little bit. And maybe the problem could have been 7 times 69, then I wouldn't have gotten that other “over,” not the 10 to get 9. Does that make sense? Mike: It absolutely does. What you really have me thinking about is NCTM’s [National Council of Teachers of Mathematics’] definition of “fluency,” which is “accuracy, efficiency, and flexibility.” And the flexibility that I hear coming out of the kinds of things that kids might do with a string, it's exciting to imagine that that's one of the outcomes you could get from engaging with strings. Pam: Absolutely. Because if you're stuck teaching memorizing algorithms, there's no flexibility, like none, like zilch. But if you're doing strings like this, kids have a brilliant flexibility. And one of the conversations I'd want to have here, Mike, is if a kid came up with 10 times 69 to help with 9 times 69, and a different kid came up with 9 times 70 to help with 9 times 69, I would want to just have a brief conversation: “Which one of those do you like better, class, and why?” Not that one is better than the other, but just to have the comparison conversation. So the kids go, “Huh, I have access to both of those. Well, I wonder when I'm walking down the street, I have to answer that one: Which one do I want my brain to gravitate towards next time?” And that's mathematical behavior. That's mathematical disposition to do one of the strands of proficiency. We want that productive disposition where... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/38735015

Season 4 | Episode 3 - Kim Montague—I Have, You Need: The Utility Player of Instructional Routines 10/09/2025

Season 4 | Episode 3 - Kim Montague—I Have, You Need: The Utility Player of Instructional Routines Kim Montague, I Have, You Need: The Utility Player of Instructional Routines ROUNDING UP: SEASON 4 | EPISODE 3 In sports, a utility player is someone who can play multiple positions competently, providing flexibility and adaptability. From my perspective, the routine I have, you need may just be the utility player of classroom routines. Today we're talking with Kim Montague about I have, you need and the ways it can be used to support everything from fact fluency to an understanding of algebraic properties. BIOGRAPHY Kim Montague is a podcast cohost and content lead at Math is Figure-out-able™. She has also been a teacher for grades 3–5, an instructional coach, a workshop presenter, and a curriculum developer. Kim loves visiting classrooms and believes that when you know your content and know your kids, real learning occurs. RESOURCES (Download) TRANSCRIPT Mike Wallus: Welcome to the podcast, Kim. I am really excited to talk with you today. So let me do a little bit of grounding. For listeners without prior knowledge, I'm wondering if you could briefly describe the I have, you need routine. How does it work, and how would you describe the roles that the teacher and the student play? Kim Montague: Thanks for having me, Mike. I'm excited to be here. I think it's an important routine. So for those people who have never heard of I have, you need, it is a super simple routine that came from a desire that I had for students to become more fluent with partners of ten, hundred, thousand. And so it simply works as a call-and-response. Often I start with a context, and I might say, “Hey, we're going to pretend that we have 10 of something, and if I have 7 of them, how many would you need so that together we have those 10?” And so it's often prosed as a missing addend. With older students, obviously, I'm going to have some higher numbers, but it's very call-and-response. It's playful. It’s game-like. I'll lob out a question, wait for students to respond. I'm choosing the numbers, so it's a teacher-driven purposeful number sequence, and then students figure out the missing number. I often will introduce a private signal so that kids have enough wait time to think about their answer and then I'll signal everyone to give their response. Mike: OK, so there's a lot to unpack there. I cannot wait to do it. One of the questions I've been asking folks about routines this season is just, at the broadest level, regardless of the numbers that the educator selects, how would you describe what you think I have, you need is good for? What's the routine good for? How can an educator think about its purpose or its value? You mentioned fluency. Maybe say a little bit more about that and if there's anything else that you think it's particularly good for. Kim: So I think one of the things that is really fantastic about I have, you need is that it's really simple. It's a simple-to-introduce, simple-to-facilitate routine, and it's great for so many different grade levels and so many different areas of content. And I think that's true for lots of routines. Teachers don't have time to reintroduce something brand new every single day. So when you find a routine that you can exchange pieces of content, that's really helpful. It's short, and it can be done anywhere. And like I said, it builds fluency, which is a hot topic and something that's important. So I can build fluency with partners of ten, partners of a hundred, partners of thousand, partners of one. I can build complementary numbers for angle measure and fractions. Lots of different areas depending on the grade that you're teaching and what you're trying to focus on. Mike: So one of the things that jumped out for me is the extent to which this can reveal structure. When we're talking about fluency, in some ways that's code for the idea that a lot of our combinations we're having kids think about—the structure of ten or a hundred or a thousand or, in the case of fractions, one whole and its equivalence. Does that make sense? Kim: Yeah, absolutely. So we have a really cool place value system. And I think that we give a lot of opportunities, maybe to place label, but we don't give a lot of opportunities to experience the structure of number. And so there are some very nice structures within partners of ten that then repeat themselves, in a way, within partners of a hundred and partners of a thousand and partners of one, like I mentioned. And if kids really deeply understand the way numbers form and the way they are fitting together, we can make use of those ideas and those experiences within other things like addition, subtraction. So this routine is not simply about, “Can you name a partner number?,” but it's laying foundation in a fun experience that kids then are gaining fluency that is going to be applied to other work that they're doing. Mike: I love that, and I think it's a great segue. My next question was going to be, “Could we talk a little bit about different sequences that you might use at different grade levels?” Kim: Sure. So younger students, especially in first grade, we're making a lot of use out of partners of 10 and working on owning those relationships. But then once students understand partners of 10, or when they're messing with partners of 10, the teacher can help make connections moving from partners of 10 to partners of 100 or partners of 20. So if you know that 9 plus 1 is 10, then there's some work to be done to help students understand that 9 tens and 1 ten makes 10 tens or 100. You can also use—capitalize on the idea of “9 and 1 makes 10” to understand that within 20, there are 2 tens. And so if you say “9” and I say “1,” and then you say “19,” and I say “1,” that work can help sharpen the idea that there's a ten within 20 and there's some tens within 30. So when we do partners of ten, it's a foundation, but we've got to be looking for opportunities to connect it to other relationships. I think that one of the things that's so great I have, you need is that we keep it game-like, but there's so many extensions, so many different directions that you can go, and we want teachers to purposefully record and draw out these relationships with their students. There's a bit to it where it's a call-and-response oral, but I think as we'll talk about further, there's a lot of nuance to number choice and there's a lot of nuance and when to record to help capitalize on those relationships. Mike: So I think the next best thing we could do is listen to a clip. I've got a clip of you working with a student, and I'm wondering if you could set the stage for what we're about to hear. Kim: Yeah, one of my very favorite things to do is to sit down with students and interview and kind of poke around in their head a little bit to find out where they currently are with the things that they're working on and where they can sharpen some content and where to take them next. So this is me sitting down with a student, Lanaya, who I didn't know very well, but I thought, let me start off by playing I have, you need with you, because that gives me a lot of insight into your number development. So this is me sitting down with her and saying, let's just play this game that I'd like to introduce to you. Kim (teacher): Oh, can I do one more thing with you? Can I play a game that I love? Lanaya (student): Sure. Kim (teacher): OK, one more game. It's called I have, you need. And so it's a pretty simple game, actually. It just helps me think about or hear what kids are thinking. So it just is simply, if I say a number, you tell me how much more to get to 100. So if I have 50, you would say you need… Lanaya (student): 50. Kim (teacher): …so that together we would have 100. What if I said 92? Lanaya (student): 8. Kim (teacher): What if I said 75? Lanaya (student): Um…25. Kim (teacher): How do you know that one? Lanaya (student): Because it's 30 to 70, so I just like minus 5 more. Kim (teacher): Oh, cool. What if I said 64? Lanaya (student): Um…36. Kim (teacher): What if I said 27? Lanaya (student): Um…27…8—no, 72? No, 73. Kim (teacher): I don't remember what I said. [laughs] Did I say…? Lanaya (student): 27, I think. Kim (teacher): 27. So then you said 73, is that what you said? And you were about to say 80-something. Why were you going to say 80-something? Lanaya (student): Because 20 is like 80, like it’s the other half, but I just had to take away more. Kim (teacher): Perfect. I see. Three more. What if I said 32? Lanaya (student): Um…68. Kim (teacher): What if I said 68? Lanaya (student): 32. Kim (teacher): [laughs] What if I said 79? Lanaya (student): Um…21. Kim (teacher): How do you know that one? Lanaya (student): Because…wait, wait, what was that one? Kim (teacher): What if I said 79? Lanaya (student): 79. Because 70 plus 30 is 100, but then I have to take away 9 more because the other half is 1, so yeah. Kim (teacher): Oh, you want to do it a little harder? Are you willing? Maybe I'll ask you that. Are you willing? Lanaya (student): Sure. Kim (teacher): OK. What if I said now our total is 1,000? What if I said 850? Lanaya (student): Um…250? Kim (teacher): How do you know? Lanaya (student): Or, actually, that'd be 150. Kim (teacher): How do you know? Lanaya (student): Because, um…uh…800 plus 200 is 1,000. And so I would just have to take—what was the number again? Kim (teacher): 850. Lanaya (student): I would have to add 50—er, have to minus 50 to that number. Kim (teacher): Um, 640. Lanaya (student): Uh, thir—360. Kim (teacher): What about 545? Lanaya (student): 400…uh, you said 549? Kim (teacher): 545, I think is what I said. Lanaya (student): Um…that'd be 465. Kim (teacher): How do you know? Lanaya (student): Because the—I just took away the number of each one. So this is 5 to make 10, and then this is 6 to make 10, and then it's 5 again, I think, or no, it would be 465, right? Kim (teacher): 465. Lanaya (student): I don't… Kim (teacher): Not sure about that one. There's a lot of 5s in there. What if I give you another one? What if I said seven hundred and thirty…721? Lanaya (student): Uh, that'd be… Kim (teacher): If it helps to write it down, so you can see it, go ahead. Lanaya (student): 389, I think? Kim (teacher): Ah, OK. Because you wanna—you’re making a 10 in the… Lanaya (student): Yeah. Kim (teacher): …hundreds and a 10 in the middle and a 10 at the end. Lanaya (student): Yeah. Kim (teacher): Interesting. Mike: Wow. So there is a lot to unpack in that clip. Kim: There is, yeah. Mike: I want to ask you to pull the curtain back on this a little bit. Let's start with this question: As you were thinking about the sequence of numbers, what was going through your mind as the person who's facilitating? Kim: Yeah, so as I said, I don't really know Lanaya much at this point, so I'm kind of guessing in the beginning, and I just want her comfortable with the routine, and I'm going to give her maybe what I think might be a simple entry. So I asked [her about] 50 and then I asked [about] 92. Just gives a chance to see kind of where she is. Is she comfortable with those size of numbers? You'll notice that I did 50 and 92 and then I did 75. 75, often, if—I might hear a student talk about quarters with 75, and she didn't, but I did ask her her strategy, and throughout she uses the same strategy, which is interesting. But I changed the number choices up and you'll see—if you were to write down the numbers that I did— [I] kind of backed away from the higher numbers. I went to 64 and then 27 and then 32. So getting further and further away from the target number. If I have students who are counting a lot, then it becomes cumbersome for them to count and they might be nudged away from accounting strategy into something a little bit more sophisticated. At one point I asked her [about] 32, and then I asked her [about] the turnaround of that, 68. Just checking to see what she knows about the commutative property. Eventually I moved into 1,000. And I mentioned earlier that [with] young students, you start with 10 and maybe combinations of 100, multiples of 10. But I didn't mention that with older grades, we might do hundreds by 1 or thousands by multiples of 100 and then by 5s. So I did that with Lanaya. She seemed to feel very comfortable with the two-digit numbers, and I thought, “Well, let's take it to the thousands.” But if you notice, I did 850, 640, some multiples of 10 still. She seemed comfortable with those, but [she] is still using the strategy of, “Let me go a little bit over. Let me add all the hundreds I need and then make adjustments.” Mike: Mm-hmm. Kim: And so then I decided to do 545 and see what happened in that moment because at that point she's having to readjust more than one digit. Mike: Yep. Kim: And when I said the number 545, I thought, “Oh man, this is a poor choice because there's a lot of 5s and 4s.” And so when she kind of maybe fumbled a little bit, I thought, “Is this because I did a poor number choice and there are lots of 4s and 5s, or is it because she's using a particular strategy that is a little more cumbersome?” So I gave her a final problem of 721, and again, that was a little bit more to adjust. So in that moment, I thought, “OK, I know where we need to work. And I need to work with her on some different strategies that aren't always about making tens.” Because as she gets larger numbers or she's getting numbers that are by 1s, that becomes less sophisticated. It becomes more cumbersome. It becomes more adjustment than you maybe are even able to hold. It's not about holding it in your head. We could have been writing some things down and we did towards the end. But it's just a lot of adjustment to make, and the strategies that she's using really aren't going to be ones that help later in addition or in subtraction. So it's just kind of playing with number, and she's pretty strong with what she's working on, but there is some work to do there that I would want to do with her. Mike: It was fascinating because as I was attending to the choices you were making and what she was doing and the back and forth, I found myself thinking a bit about this notion of fluency, that part of it is the ability to be efficient, but also to be flexible at the same time. And I really connect that with what you said because she had a strategy that was working for her, but you also made a move to kind of say, “Let's see what happens if we give a set of numbers where that becomes more cumbersome.” And it kind of exposed— there's this space where, again, as you said, “Now I know where we need to work.” So it's a bit like a formative assessment too. Kim: Yeah, yeah. Interviewing students, like I said, is my very favorite thing to do. And it's tough because we want kids to be successful, which is a great goal, but I think it's often unfortunate that we leave students with a strategy that we think, “Oh, that's great. They have a strategy and it works for them,” but we aren't really thinking about the long game. We're not thinking about, “Will this thing that they're doing support their needs as the size of the numbers increase, as the type of the numbers change?” And we want them to have choice. And again, I have, you need is fantastic because within this game, this simple routine, you can share strategies. There's a handful of strategies that kids generally use, and in the routine in the game, we get to talk about those strategies. So we have a student who's using the kind of same strategy over and over and it stops working because it's less sophisticated, it's less efficient, it's more cumbersome. Then in the routine, we get to expose other strategies that they can try on and see what works for them based on the numbers that they're being given. Mike: You made me think about something that, I'm not sure how you could even put my finger on why, but sometimes people are wonky about this notion that students should have a choice of their strategies. In some ways, it makes me think that what you're really suggesting is part of this work around flexibility is building options, right? You're not trapped in a strategy if suddenly the numbers don't make it something that's efficient. You have options, and I think that really jumps out when you think about what happened with Lanaya, but just generally what you're trying to build when you're using this routine. Kim: Yeah, I mean we are big fans of building relationships, so that strategies are natural outcomes. And I think if you are new to numeracy or you didn't grow up playing with number, it can feel like, “I'm just going to offer multiple and kids have to own them all, and now there's too many things and they don't know how to pick.” But when we really focus on relationship in number, then we strengthen those relationships like in a routine with I have you, need. I grew up messing with number, and the strategies don't feel like a bunch of new things I have to memorize. I've strengthened partners of ten and hundred and thousand, and I understand doubles, and I understand the fact that you can add a little too much and back up. And so those relationships just get used in the way that I solve problems, and that's what we want for kids. Mike: I love that. We've spent a fair amount of time talking about this connection between building fluency and helping kids see and make use of structure. I'm also really taken by some of the properties that jump out of this routine. They're not formal, meaning they come up organically, and I found myself thinking a lot about algebraic reasoning or setting kids up for algebra. Could you just talk a little bit about some of that part of the work? Kim: I think that when we want kids to own and use properties, one way to go about it is to say, “Today we're going to talk about the commutative property.” And you define it and you verbalize it and you write it down. You might make a poster. But more organically is the opportunity to use it and then name it as it's occurring. So in the routine, if I say “68” and she says “32” and then I say “32” and she says “68,” then we are absolutely using the idea of “68 plus something is 100” and then “32 plus something is 100.” There is something natural about you just [knowing] it's the other addend. In some of the other strategies that we develop through I have, you need, it's about breaking apart numbers in such a way that they are reassociating. And so when that happens for students, then we can name it afterward and say, “Oh, that's just this thing.” And whether we name the property to students or not, it's more important that they're using them. And so we put it in a game, we put it in a form that we just say, “Oh, that's just where you're breaking apart numbers and finding friendly addends to go together.” And I think it's really more important that teachers really understand the strategies that work so that they invite students to participate in experiences where they're using them. Mike: Yeah, I mean, what hits me about that is there's something about making use of a relationship, fleshing it out through this process of I have, you need, and then at the end coming back and saying, “Oh, we have a formal name for that.” That's different than saying, “Here's the thing, here's the definition. Remember the definition, remember the name.” It just works so much more smoothly and sensibly because I've been able to apply that relationship and it feels like it's inside of me now. I have an understanding and now I've just attached a name to that thing. That just feels really, really different. Kim: Yeah, I mean, if we give students the right experiences,... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/38413265

Season 4 | Episode 2 - Dr. Sue Looney - Same but Different: Encouraging Students to Think Flexibly 09/18/2025

Season 4 | Episode 2 - Dr. Sue Looney - Same but Different: Encouraging Students to Think Flexibly Sue Looney, Same but Different: Encouraging Students to Think Flexibly ROUNDING UP: SEASON 4 | EPISODE 2 Sometimes students struggle in math because they fail to make connections. For too many students, every concept feels like its own entity without any connection to the larger network of mathematical ideas. On the podcast today, we’re talking with Dr. Sue Looney about the powerful same and different routine. We explore the ways that teachers can use this routine to help students identify connections and foster flexible reasoning. BIOGRAPHY Sue Looney holds a doctorate in curriculum and instruction with a specialty in mathematics from Boston University. Sue is particularly interested in our most vulnerable and underrepresented populations and supporting the teachers that, day in and day out, serve these students with compassion, enthusiasm, and kindness. RESOURCES TRANSCRIPT Mike Wallus: Students sometimes struggle in math because they fail to make connections. For too many students, every concept feels like its own entity without any connection to the larger network of mathematical ideas. Today we're talking with Sue Looney about a powerful routine called same but different and the ways teachers can use it to help students identify connections and foster flexible reasoning. Well, hi, Sue. Welcome to the podcast. I'm so excited to be talking with you today. Sue Looney: Hi Mike. Thank you so much. I am thrilled too. I've been really looking forward to this. Mike: Well, for listeners who don't have prior knowledge, I'm wondering if we could start by having you offer a description of the same but different routine. Sue: Absolutely. So the same but different routine is a classroom routine that takes two images or numbers or words and puts them next to each other and asks students to describe how they are the same but different. It's based in a language learning routine but applied to the math classroom. Mike: I think that's a great segue because what I wanted to ask is: At the broadest level—regardless of the numbers or the content or the image or images that educators select—how would you explain what [the] same but different [routine] is good for? Maybe put another way: How should a teacher think about its purpose or its value? Sue: Great question. I think a good analogy is to imagine you're in your ELA— your English language arts—classroom and you were asked to compare and contrast two characters in a novel. So the foundations of the routine really sit there. And what it's good for is to help our brains think categorically and relationally. So, in mathematics in particular, there's a lot of overlap between concepts and we're trying to develop this relational understanding of concepts so that they sort of build and grow on one another. And when we ask ourselves that question—“How are these two things the same but different?”—it helps us put things into categories and understand that sometimes there's overlap, so there's gray space. So it helps us move from black and white thinking into this understanding of grayscale thinking. And if I just zoom out a little bit, if I could, Mike—when we zoom out into that grayscale area, we're developing flexibility of thought, which is so important in all aspects of our lives. We need to be nimble on our feet, we need to be ready for what's coming. And it might not be black or white, it might actually be a little bit of both. So that's the power of the routine and its roots come in exploring executive functioning and language acquisition. And so we just layer that on top of mathematics and it's pure gold. Mike: When we were preparing for this podcast, you shared several really lovely examples of how an educator might use same but different to draw out ideas that involve things like place value, geometry, equivalent fractions, and that's just a few. So I'm wondering if you might share a few examples from different grade levels with our listeners, perhaps at some different grade levels. Sue: Sure. So starting out, we can start with place value. It really sort of pops when we look in that topic area. So when we think about place value, we have a base ten number system, and our numbers are based on this idea that 10 of one makes one group of the next. And so, using same but different, we can help young learners make sense of that system. So, for example, we could look at an image that shows a 10-stick. So maybe that's made out of Unifix cubes. There's one 10-stick a—stick of 10—with three extras next to it and next to that are 13 separate cubes. And then we ask, “How are they the same but different?” And so helping children develop that idea that while I have 1 ten in that collection, I also have 10 ones. Mike: That is so amazing because I will say as a former kindergarten and first grade teacher, that notion of something being a unit of 1 composed of smaller units is such a big deal. And we can talk about that so much, but the way that I can visualize this in my mind with the stick of 10 and the 3, and then the 13 individuals—what jumps out is that it invites the students to notice that as opposed to me as the teacher feeling like I need to offer some kind of perfect description that suddenly the light bulb goes off for kids. Does that make sense? Sue: It does. And I love that description of it. So what we do is we invite the students to add their own understanding and their own language around a pretty complex idea. And they're invited in because it seems so simple: “How are these the same but different?” “What do you notice?” And so it's a pretty complex idea, and we gloss over it. Sometimes we think our students understand that and they really don't. Mike: Is there another example that you want to share? Sue: Yeah, I love the fraction example. So equivalence—when I learned about this routine, the first thing that came to mind for me when I layered it from thinking about language into mathematics was, “Oh my gosh, it's equivalent fractions.” So if I were to ask listeners to think about—put a picture in your head of one-half, and imagine in your mind's eye what that looks like. And then if I said to you, “OK, well now I want you to imagine two-fourths. What does that look like?” And chances are those pictures are not the same. Mike, when you imagine, did you picture the same thing or did you picture different things? Mike: They were actually fairly different. Sue: Yeah. So when we think about one-half as two fourths, and we tell kids those are the same—yes and no, right? They have the same value that, if we were looking at a collection of M&M’S or Skittles or something, maybe half of them are green, and if we make four groups, [then] two-fourths are green. But contextually it could really vary. And so helping children make sense of equivalence is a perfect example of how we can ask the question, same but different. So we just show two pictures. One picture is one-half and one picture is two-fourths, and we use the same colors, the same shapes, sort of the same topic, but we group them a little differently and we have that conversation with kids to help make sense of equivalence. Mike: So I want to shift because we've spent a fair amount of time right now describing two instances where you could take a concept like equivalent fractions or place value and you could design a set of images within the same but different routine and do some work around that. But you also talked with me, as we were preparing, about different scenarios where same but different could be a helpful tool. So what I remember is you mentioned three discrete instances: this notion of concepts that connect; things learned in pairs; and common misconceptions—or, as I've heard you describe them, naive conceptions. Can you talk about each of those briefly? Sue: Sure. As I talk about this routine to people, I really want educators to be able to find the opportunities—on their own, authentically—as opportunities arise. So we should think about each of these as an opportunity. So I'll start with concepts that connect. When you're teaching something new, it's good practice to connect it to, “What do I already know?” So maybe I'm in a third grade classroom, and I want to start thinking about multiplication. And so I might want to connect repeated addition to multiplication. So we could look at 2 plus 2 plus 2 next to 2 times 3. And it can be an expression, these don't always have to be images. And a fun thing to look at might be to find out, “Where do I see 3 and 2 plus 2 plus 2?” So what's happening here with factors? What is happening with the operations? And then of course they both yield the same answer of 6. So concepts that connect are particularly powerful for helping children build from where they know, which is the most powerful place for us to be. Mike: Love that. Sue: Great. The next one is things that are learned in pairs. So there's all sorts of things that come in pairs and can be confusing. And we teach kids all sorts of weird tricks and poems to tell themselves and whatever to keep stuff straight. And another approach could be to—let's get right in there, to where it's confusing. So for example, if we think about area and perimeter, those are two ideas that are frequently confusing for children. And we often focus on, “Well, this is how they're different.” But what if we put up an image, let's say it's a rectangle, but [it] wouldn't have to be. And we've got some dimensions on there. We're going to think about the area of one and then the perimeter on the other. What is the same though, right? Because where the confusion is happening. So just telling students, “Well, perimeter’s around the outside, so think of ‘P’ for ‘pen’ or something like that, and area’s on the inside.” What if we looked at, “Well, what's the same about these two things?” We're using those same dimensions, we've got the same shape, we're measuring in both of those. And let students tell you what is the same and then focus on that critical thing that's different, which ultimately leads to understanding formula for finding both of those things. But we've got to start at that concept level and link it to scenarios that make sense for kids. Mike: Before we move on to talking about misconceptions, or naive conceptions, I want to mark that point: this idea that confusion for children might actually arise from the fact that there are some things that are the same as opposed to a misunderstanding of what's different. I really think that's an important question that an educator could consider when they're thinking about making this bridging step between one concept or another or the fact that kids have learned how whole numbers behave and also how fractions might behave. That there actually might be some things that are similar about that that caused the confusion, particularly on the front end of exploration, as opposed to, “They just don't understand the difference.” Sue: And what happens there is then we aid in memory because we've developed these aha moments and painted a more detailed picture of our understanding in our mind's eye. And so it's going to really help children to remember those things as opposed to these mnemonic tricks that we give kids that may work, but it doesn't mean they understand it. Mike: Absolutely. Well, let's talk about naive conceptions and the ways that same and [different] can work with those. Sue: So, I want to kick it up to maybe middle school, and I was thinking about what example might be good here, and I want to talk about exponents. So if we have 2 raised to the third power, the most common naive conception would be, like, “Oh, I just multiply that. It's just 2 times 3.” So let's talk about that. So if I am working on exponents, I notice a lot of my students are doing that, let's put it right up on the board: “Two rays to the third power [and] 2 times 3. How are these the same but different?” And the conversation’s a bit like that last example, “Well, let's pay attention to what's the same here.” But noticing something that a lot of children have not quite developed clearly and then putting it up there against where we want them to go and then helping them—I like that you use the word “bridge”—helping them bridge their way over there through this conversation is especially powerful. Mike: I think the other thing that jumps out for me as you were describing that example with exponents is that, in some ways, what's happening there when you have an example like “2 times 3” next to “2 to the third power” is you're actually inviting kids to tell you, “This is what I know about multiplication.” So you're not just disregarding it or saying, “We're through with that.” It's in the exploration that those ideas come out, and you can say to kids, “You are right. That is how multiplication functions. And I can see why that would lead you to think this way.” And it's a flow that's different. It doesn't disregard kids' thinking. It actually acknowledges it. And that feels subtle, but really important. Sue: I really love shining a light on that. So it allows us to operate from a strength perspective. So here's what I know, and let's build from there. So it absolutely draws out in the discussion what it is that children know about the concepts that we put in front of them. Mike: So I want to shift now and talk about enacting same but different. I know that you've developed a protocol for facilitating the same but different routine, and I'm wondering if you could talk us through the protocol, Sue. How should a teacher think about their role during same but different? And are there particular teacher moves that you think are particularly important? Sue: Sure. So the protocol I've worked out goes through five steps, and it's really nice to just kind of think about them succinctly. And all of them have embedded within them particular teacher moves. They are all based on research of how children learn mathematics and engage in meaningful conversation with one another. So step 1 is to look. So if I'm using this routine with 3- and 4-year-olds, and I'm putting a picture in front of them, learning that to be a good observer, we've got to have eyes on what it is we're looking at. So I have examples of counting, asking a 4-year-old, “How many things do I have in front of me?” And they're counting away without even looking at the stuff. So teaching the skill of observation. Step 1 is look. Step 2 is silent think time. And this is so critically important. So giving everybody the time to get their thoughts together. If we allow hands to go in the air right away, it makes others that haven't had that processing time to figure it out shut down quite often. And we all think at different speeds with different tasks all the time, all day long. So, we just honor that everyone's going to have generally about 60 seconds in which to silently think, and we give students a sentence frame at that time to help them. Because, again, this is a language-based learning routine. So we would maybe put on the board or practice saying out loud, “I'd like you to think about: ‘They are the same because blank; they are different because blank.’” And that silent think time is just so important for allowing access and equitable opportunities in the classrooms. Mike: The way that you described the importance of giving kids that space, it seems like it's a little bit of a two-for-one because we're also kind of pushing back on this notion that to be good at math, you have to have your hand in the air first, and if you don't have your hand in the air first or close to first, your idea may be less valuable. So I just wanted to shine a light on the different ways that that seems important for children, both in the task that they're engaging with and also in the culture that you're trying to build around mathematics. Sue: I think it's really important. And if we even zoom out further just in life, we should think before we speak. We should take a moment. We should get our thoughts together. We should formulate what it is that we want to say. And learning how to be thoughtful and giving the luxury of what we're just going to all think for 60 seconds. And guess what? If you had an idea quickly, maybe you have another one. How else are they the same but different? So we just keep that culture that we're fostering, like you mentioned, we just sort of grow that within this routine. Mike: I think it's very safe to say that the world might be a better place if we all took 60 seconds to think about [laughs] what we wanted to say sometimes. Sue: Yes, yes. So as teachers, we can start teaching that and we can teach kids to advocate for that. “I just need a moment to get my thoughts together.” All right, so the third step is the turn and talk. And it's so important and it's such an easy move. It might be my favorite part. So during that time, we get to have both an experience with expressive language and receptive language—every single person. So as opposed to hands in the air and I'm playing ball with you, Mike, and you raise your hand and you get to speak and we're having a good time. When I do a turn and talk, everybody has an opportunity to speak. And so taking the thoughts that are in their head and expressing them is a big deal. And if we think about our multilingual learners, our young learners, even our older learners, and it's just a brand new concept that I've never talked about before. And then on the other side, the receptive learning. So you are hearing from someone else and you're getting that opportunity of perspective taking. Maybe they notice something you hadn't noticed, which is likely to happen to somebody within that discussion. “Wow, I never thought about it that way.” So the turn and talk is really critical. And the teacher's role during this is so much fun because we are walking around and we're listening. And I started walking around with a notebook. So I tell students, “While you are talking, I'm going to collect your thinking.” And so I'm already imagining where this is going next. And so I'm on the ground if we're sitting on the rug, I'm leaning over, I'm collecting thoughts, I'm noticing patterns, I'm noticing where I want to go next as the facilitator of the conversation that's going to happen whole group. So that's the third component, turn and talk. The fourth component is the share. So if I've walked around and gathered student thinking, I could say, “Who would like to share their thinking?” and just throw it out there. But I could instead say—let's say we're doing the same but different with squares and rectangles. And I could say, “Hmm, I noticed a lot of you talking about the length of the sides. Is there anyone that was talking about the lengths of the sides that would like to share what either you or your partner said?” So I know that I want to steer it in that direction. I know a lot of people talked about that, so let's get that in the air. But the share is really important because these little conversations have been happening. Now we want to make it public for everybody, and we're calling on maybe three or four students. We're not trying to get around to everybody. We're probably hopefully not going to [be] drawing Popsicle sticks and going random. At this point, students have had the opportunity to talk, to listen, to prepare. They've had a sentence stem. So let's see who would like to share and get those important ideas out. Mike: I think what strikes me is there's some subtlety to what's happening there because you are naming some themes that you heard. And as you do that, and you name that, kids can say, “That's me,” or, “I thought about that,” or, “My... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/38135260

Season 4 | Episode 1 - Dr. Christopher Danielson, Which One Doesn’t Belong Routine 09/04/2025

Season 4 | Episode 1 - Dr. Christopher Danielson, Which One Doesn’t Belong Routine Christopher Danielson, Which One Doesn’t Belong? Routine: Fostering Flexible Reasoning ROUNDING UP: SEASON 4 | EPISODE 1 The idea of comparing items and looking for similarities and differences has been explored by many math educators. Christopher Danielson has taken this idea to new heights. Inspired by the Sesame Street song “One of These Things (Is Not Like the Others),” Christopher wrote the book Which One Doesn't Belong? In this episode, we'll ask Christopher about the routine of the same name and the features that make it such a powerful learning experience for students. BIOGRAPHY Christopher Danielson started teaching in 1994 in the Saint Paul (MN) Public Schools. He earned his PhD in mathematics education from Michigan State University in 2005 and taught at the college level for 10 years after that. Christopher is the author of Which One Doesn’t Belong?, How Many?, and How Did You Count? Christopher also founded Math On-A-Stick, a large-scale family math playspace at the Minnesota State Fair. RESOURCES by Christopher Danielson by Margaret (Peg) Smith & Mary Kay Stein by Christopher Danielson by Christopher Danielson TRANSCRIPT Mike Wallus: The idea of comparing items and looking for similarities and differences has been explored by many math educators. That said, Christopher Danielson has taken this idea to new heights. Inspired by Sesame Street’s [song] “One of These Things (Is Not Like the Others),” Christopher wrote the book Which One Doesn't Belong? In this episode, we'll ask Christopher about the Which one doesn't belong? routine and the features that make it such a powerful learning experience for students. Well, welcome to the podcast, Christopher. I'm excited to be talking with you today. Christopher Danielson: Thank you for the invitation. Delightful to be invited. Mike: I would love to chat a little bit about the routine Which one doesn't belong? So, I'll ask a question that I often will ask folks, which is: If I'm a listener, and I don't have prior knowledge of that routine, how would you describe it for someone? Christopher: Yeah. Sesame Street, back in the day, had a routine called Which one doesn't belong? There was a little song that went along with it. And for me, the iconic Sesame Street image is [this:] Grover is on the stairs up to the brownstone on the Sesame Street set, and there are four circles drawn in a 2-by-2 grid in chalk on the wall. And there are a few of the adults and a couple of the puppets sitting around, and they're asking Grover and singing the song, “Which One of Them Doesn't Belong?” There are four circles. Three of them are large and one is small—or maybe it's the other way around, I don't remember. So, there's one right answer, and Grover is thinking really hard—"think real hard” is part of the song. They're singing to him. He's under kind of a lot of pressure to come up with which one doesn't belong and fortunately, Grover succeeds. Grover's a hero. But what we're wanting kids to attend to there is size. There are three things that are the same size. All of them are the same shape, three that are the same size, one that has a different size. They're wanting to attend to size. Lovely. This one doesn't belong because it is a different size, just like my underwear doesn't belong in my socks drawer because it has a different function. I mean, it's not—for me there is, we could talk a little bit about this in a moment. The belonging is in that mathematical and everyday sense of objects and whether they belong. So, that's the Sesame Street version. Through a long chain of math educators, I came across a sort of tradition that had been flying along under the radar of rethinking that, with the idea being that instead of there being one property to attend to, we're going to have a rich set of shapes that have rich and interesting relationships with each other. And so Which one doesn't belong? depends on which property you're attending to. So, the first page of the book that I published, called Which One Doesn't Belong?, has four shapes on it. One is an equilateral triangle standing on a vertex. One is a square standing on a vertex. One is a rhombus, a nonsquare rhombus standing on its vertex, and it's not colored in. All the other shapes are colored in. And then there is the same nonsquare thrombus colored in, resting on a side. So, all sort of simple shapes that offer simple introductory properties, but different people are going to notice different things. Some kids will hone in on that. The one in the lower left doesn't belong because it's not colored in. Other kids will say, “Well, I'm counting the number of sides or the number of corners. And so, the triangle doesn't belong because all the others have four and it has three.” Others will think about angle measure, they'll choose a square. Others will think about orientation. I've been taken to task by a couple of people about this. Kindergartners are still thinking about orientation as one of the properties. So, the shape that is in the lower right on that first page is a rhombus resting on a side instead of on a vertex. And kids will describe it as “the one that feels like it's leaning over” or that “has a flat bottom” or “it's pointing up and to the right” and all the others are pointing straight up and down. So that's the routine. And then things, as with “How Did You Count?” as with “How Many?” As you page your way through the book, things get more sophisticated. And for me, the entry was a geometry book because when my kids were small, we had sort of these simplistic shapes books, but really rich narrative stories in picture books that we could read. And it was always a bummer to me that we'd read these rich stories about characters interacting. We'd see how their interactions, their conflicts relate to our own lives, and then we'd get to the math books, and it would be like, “triangle: always equilateral, always on a side.” “Square: never a square on the rectangle page.” Rectangle gets a different page from square. And so, we understand culturally that children can deal with and are interested in and find fascinating and imaginative rich narratives, but we don't understand as a culture that children also have rich math minds. So, for a long time I wanted there to be a better shapes book, and there are some better shapes books. They're not all like that, but they're almost all like that. And so, I had this idea after watching one of my colleagues here in Minnesota, Terry Wyberg. This routine, he was doing it with fractions, but about a week later I thought to myself, “Hey, wait a minute, what if I took Terry's idea about there not being one right answer, but any of the four could be, and combine that with my wish for a better shapes book?” And along came Which One Doesn't Belong? as a shapes book. So, there's a square and a rectangle on the same page. There are shapes with curvy sides and shapes with straight sides on the same page, and kids have to wrestle with or often do wrestle with: What does it mean to be a vertex or a corner? A lot of really rich ideas can come out of some well-chosen, simple examples. I chose to do it in the field of geometry, but there are lots of other mathematical objects as well as nonmathematical objects you could apply the same mathematical thinking to. Mike: So, I think you have implicitly answered the question that I'm going to ask. If you were to say at the broadest level, regardless of whether you're using shapes, numbers, images—whatever the content is that an educator selects to put into the 2-by-2, that is structurally the way that Which one doesn't belong? is set up—what's it good for? What should a teacher think about in terms of “This will help me or will help my students…,” fill in the blank. How do you think about the value that comes out of this Which one doesn't belong? structure and experience? Christopher: Multidimensional for me. I don't know if I'll remember to say all of the dimensions, so I'll just try to mention a couple that I think are important. One is that I'm going to make you a promise that whatever mathematical ideas you bring to this classroom during this routine are going to be valued. The measure of what's right, what counts as a right answer here, is going to be what's true—not what I thought of when I was setting up this set. I think there is a lot of power in making that promise and then in holding that promise. It is really, really easy—all of us have been there as teachers—[to] make an instructional promise to kids, [but] then there comes a time where it either inadvertently or we make a decision to break that promise. I think there's a lot of costs to that. I know from my own experience as a learner, from my own experiences as a teacher, that there can be a high cost to that. So valuing ideas, I think this is a space. I love having Which one doesn't belong? as a time that we can set aside for the measure of “what's right is what's true.” So, when children are making claims about this one in the upper right doesn't belong, I want you to for a moment try to think like that person, even if you disagree that that's important. And so, teachers have to play that role also. Where that comes up a lot is in, especially when I'm talking with adults, if I'm talking to parents about Which one doesn't belong?, often parents who don't identify as math people or who explicitly identify as nonmath people, will say, “That one in the lower left, it's not colored in. But I don't think that really counts.” In that moment, kids are less likely to make that apology, but adults will make that apology all the time. And in that moment, I have to both bring the adult in as a mathematical thinker but also model for them: What does it look like when their kid chooses something that the parent doesn't think counts? So, for me, the real thing that Which one doesn't belong? is doing is teaching children, giving children practice and expertise—therefore learning—about a particular mathematical practice, which is abstraction. That when we look at these sets of shapes, there are lots of properties. And so, we have to for a moment, just think about number of sides. And if we do that, then the triangle doesn't belong because of the other four. But as soon as we shift the property and say, “Well, let's think about angle measures,” then the ways that we're going to sort those shapes, the relationships that they have with each other, changes. And that's true with all mathematical objects. And you can do that kind of mathematical thinking with non-mathematical objects. One of my favorite Which one doesn't belong? sets is: There's a doughnut, a chocolate doughnut; there's a coffee cup, one of those speckled blue camping metal coffee cups; there's half a hamburger bun with a bunch of seeds on top; and then there is a square everything bagel. And so, as kids start thinking about that, they're like, “Well, if we're thinking about holes, the hamburger bun doesn't have a hole. If we're thinking about speckling, the chocolate doughnut isn't speckled. If we're thinking about whether it's an edible substance, the coffee cup is not edible.” And so that's that same abstraction. If we pay attention to just this one property, that forces a sort. If we pay attention to a different property, we're going to get a different sort. And that's one of the practices of mathematicians on a regular basis. So regular that often when we're doing mathematics, we don't even notice that we're doing it. We don't notice that we're asking kids to ignore all the other properties of the number 2 except for its evenness right now. If you do that, then 2 and 4 are like each other. But if we're supposed to be paying attention to primality as to a prime number, then 2 and 4 are not like each other. All mathematical objects, all mathematicians have to do that kind of sort on the objects that they're working with. I had a college algebra class at the community college while I was working on Which One Doesn't Belong?, and so, I was test-driving this with graphs and my students. I can still see Rosalie in the middle of the room—a room full of 45 adults ranging from 17 to 52, and I'm this 45-year-old college instructor—and we have three parabolas and one absolute value function. So, a parabola is “y equals x squared.” It's that nice curving swooping thing that goes up at one end down to a nice bowl and then up again. There was one that's upside down. I think there was one pointing sideways. And then an absolute value function is the same idea, except it's two lines coming together to make a bowl, sort of a very sharp bowl, instead of being curved. And we got this lovely Which one doesn't belong?, right? So, we've got this lovely collection of them. And Rosalie, her eyebrows are getting more and more knitted as this conversation goes on. So finally, she raises her hand. I call on her, and she says, “Mr. Danielson, I get that all of these things are true about these, but which ones matter?” Which is a fabulous question that within itself holds a lot of tensions that Rosalie is used to being in math class and being told what things she's supposed to pay attention to. And so, in some ways it's sort of disturbing to have me up there, and I get that, up there in front of the classroom valuing all these different ways of viewing these graphs because she's like, “Which one is going to matter when you ask me this question about something on an exam? Which ones matter?” But truly, the only intellectually honest answer to her question is, “Well, it depends. Are we paying attention to direction of concavity? Then the one that's pointing sideways doesn't count.” Any one of these is, it depends on whether you're studying algebra or whether you're studying geometry or topology. And I did give her, I think—I hope—what was a satisfying answer after giving her the true but not very satisfying answer of “It depends,” which is something like, “Well, in the work we're about to do with absolute value functions, the direction that they open up and how steeply they open up are going to be the things that we're really attending to, and we're not going to be attending as much to how they are or are not like parabolas. But seeing how they have some properties in common with these parabolas is probably going to be really useful for us. Mike: That actually makes me think of, one, a statement of what I think is really powerful about this. And then, two, a pair of questions that I think are related. It really struck me—Rosalie's question—how different the experience of engaging with a Which one doesn't belong? is from what people have traditionally considered math tasks where there is in fact an answer, right? There's something that the teacher's like, “Yep, that's the thing.” Even if it's perhaps obscured by the task at first, ultimately, oftentimes there is a thing and a Which one doesn't belong? is a very, very different type of experience. So that really does lead me to two questions. One is: What is important to think about when you're facilitating a Which one doesn't belong? experience? And then, maybe even the better question to start with is: What's important to think about when you're planning for that experience? Christopher: Facilitating is going to be about making a promise to kids. That measure of “what's right is what's true.” I'm interested in the various ways that you're thinking and doing all the kind of work that we discussed but now in this context of geometry, or in my case in the college algebra classroom, in the context of algebraic representations. Planning. I have been so deeply influenced by and her colleagues and the five practices for facilitating mathematical conversations. And in particular, I think in planning for these conversations, planning a set—when I'm deciding what shapes are going to go in the set, or how I'm going to arrange the eggs in the egg carton, or how many half avocados am I going to put on the cutting board—I'm anticipating one of those practices: What is it that kids are likely to do with this? And if I can't anticipate anything interesting that they're going to do with it, then either my imagination isn't good enough, and I better go try it out with kids or my imagination is absolutely good enough and it's just kind of a junky thing that's not going to take me anywhere, and I should abandon it. So over time, I've gotten so much better at that anticipating work because I have learned, I've become much more expert at what kids are likely to see. But I also always get surprised. In a sufficiently large group of kids, somebody will notice something or have some way of articulating differences among the shapes, even these simple shapes on the first page, that I haven't encountered before. And I get to file that away again for next time. That's learning that gets fed back into the machine, both for the next time I'm going to work with a group of kids, but also for the next time I'm sitting down to design an experience. Mike: You have me thinking about something else, which is what closure might look like in an experience like this. Because I'm struck by the fact that there might be some really intentional choices of the items in the Which one doesn't belong? So, the four items that end up being there, [they] may be designed to drive a conversation around a set of properties or a set of relationships—and yet at the same time be open enough to allow lots of kids to be right in the things that they're noticing. And so, if I've got a Which one doesn't belong? that kind of is intended to draw out some ideas or have kids notice some of those ideas and articulate them, what does closure look like? Because I could imagine you don't know what you're going to get necessarily from kids when you put a Which one doesn't belong? in front of them. So, how do you think about different ways that a routine or experience like this might close for a teacher and for students? Christopher: Yeah, I think one of the best roles that a teacher can play at the end of a Which one doesn't belong? conversation is going back and summarizing the various properties that kids attended to. Because as they're being presented and maybe annotated, we're noticing them sort of one by one. And we might not have a moment to set them aside. It might take a minute for a kid to draw out their ideas about the orientation of this shape. And it might take a little bit and some clarification with another kid about how they were counting sides. They might not have great words for “sides” or “corners,” and [instead they use] gestures, and we're all trying to figure things out. And so, by the time we figured that out, we've forgotten about the orientation answer that we had before. So I think a really powerful move, one of many that are in teachers' toolkits, is to come back and say, “All right, so we looked at these four shapes, and what we noticed is that if you're paying attention to how this thing is sitting on the page, to its orientation, which direction it's pointing, then this one didn't belong, and Susie gave us that answer. And then another thing you might pay attention to, another property could be the number of sides. If you're paying attention to the number of sides the triangle doesn't belong, and we got that one from Brent, right?” And so run through some of the various properties. Also, noticing along the way that there were two reasons to pick the triangle as the one that doesn't belong. It might be the sides, and it might be, you might have some other reason for picking it that isn't the number of sides. For kindergartners, the number of... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/37781230

Season 3 | Episode 17 - Understanding the Role of Language in Math Classrooms - Guest: William Zahner 05/08/2025

Season 3 | Episode 17 - Understanding the Role of Language in Math Classrooms - Guest: William Zahner William Zahner, Understanding the Role of Language in Math Classrooms ROUNDING UP: SEASON 3 | EPISODE 17 How can educators understand the relationship between language and the mathematical concepts and skills students engage with in their classrooms? And how might educators think about the mathematical demands and the language demands of tasks when planning their instruction? In this episode, we discuss these questions with Bill Zahner, director of the Center for Research in Mathematics and Science Education at San Diego State University. BIOGRAPHY Bill Zahner is a professor in the mathematics department at San Diego State University and the director of the Center for Research in Mathematics and Science Education. Zahner's research is focused on improving mathematics learning for all students, especially multilingual students who are classified as English Learners and students from historically marginalized communities that are underrepresented in STEM fields. RESOURCES [BES login required] TRANSCRIPT Mike Wallus: How can educators understand the way that language interacts with the mathematical concepts and skills their students are learning? And how can educators focus on the mathematics of a task without losing sight of its language demands as their planning for instruction? We'll examine these topics with our guest, Bill Zahner, director of the Center for Research in Mathematics and Science Education at San Diego State University. Welcome to the podcast, Bill. Thank you for joining us today. Bill Zahner: Oh, thanks. I'm glad to be here. Mike: So, I'd like to start by asking you to address a few ideas that often surface in conversations around multilingual learners and mathematics. The first is the notion that math is universal, and it's detached from language. What, if anything, is wrong with this idea and what impact might an idea like that have on the ways that we try to support multilingual learners? Bill: Yeah, thanks for that. That's a great question because I think we have a common-sense and strongly held idea that math is math no matter where you are and who you are. And of course, the example that's always given is something like 2 plus 2 equals 4, no matter who you are or where you are. And that is true, I guess [in] the sense that 2 plus 2 is 4, unless you're in base 3 or something. But that is not necessarily what mathematics in its fullness is. And when we think about what mathematics broadly is, mathematics is a way of thinking and a way of reasoning and a way of using various tools to make sense of the world or to engage with those tools [in] their own right. And oftentimes, that is deeply embedded with language. Probably the most straightforward example is anytime I ask someone to justify or explain what they're thinking in mathematics. I'm immediately bringing in language into that case. And we all know the old funny examples where a kid is asked to show their thinking and they draw a diagram of themselves with a thought bubble on a math problem. And that's a really good case where I think a teacher can say, “OK, clearly that was not what I had in mind when I said, ‘Show your thinking.’” And instead, the demand or the request was for a student to show their reasoning or their thought process, typically in words or in a combination of words and pictures and equations. And so, there's where I see this idea that math is detached from language is something of a myth; that there's actually a lot of [language in] mathematics. And the interesting part of mathematics is often deeply entwined with language. So, that's my first response and thought about that. And if you look at our Common Core State Standards for Mathematics, especially those standards for mathematical practice, you see all sorts of connections to communication and to language interspersed throughout those standards. So, “create viable arguments,” that's a language practice. And even “attend to precision,” which most of us tend to think of as, “round appropriately.” But when you actually read the standard itself, it's really about mathematical communication and definitions and using those definitions with precision. So again, that's an example, bringing it right back into the school mathematics domain where language and mathematics are somewhat inseparable from my perspective here. Mike: That's really helpful. So, the second idea that I often hear is, “The best way to support multilingual learners is by focusing on facts or procedures,” and that language comes later, for lack of a better way of saying it. And it seems like this is connected to that first notion, but I wanted to ask the question again: What, if anything, is wrong with this idea that a focus on facts or procedures with language coming after the fact? What impact do you suspect that that would have on the way that we support multilingual learners? Bill: So, that's a great question, too, because there's a grain of truth, right? Both of these questions have simultaneously a grain of truth and simultaneously a fundamental problem in them. So, the grain of truth—and an experience that I've heard from many folks who learned mathematics in a second language—was that they felt more competent in mathematics than they did in say, a literature class, where the only activity was engaging with texts or engaging with words because there was a connection to the numbers and to symbols that were familiar. So, on one level, I think that this idea of focusing on facts or procedures comes out of this observation that sometimes an emergent multilingual student feels most comfortable in that context, in that setting. But then the second part of the answer goes back to this first idea that really what we're trying to teach students in school mathematics now is not simply, or only, how to apply procedures to really big numbers or to know your times tables fast. I think we have a much more ambitious goal when it comes to teaching and learning mathematics. That includes explaining, justifying, modeling, using mathematics to analyze the world and so on. And so, those practices are deeply tied with language and deeply tied with using communication. And so, if we want to develop those, well, the best way to do that is to develop them, to think about, “What are the scaffolds? What are the supports that we need to integrate into our lessons or into our designs to make that possible?” And so, that might be the takeaway there, is that if you simply look at mathematics as calculations, then this could be true. But I think our vision of mathematics is much broader than that, and that's where I see this potential. Mike: That's really clarifying. I think the way that you unpack that is if you view mathematics as simply a set of procedures or calculations, maybe? But I would agree with you. What we want for students is actually so much more than that. One of the things that I heard you say when we were preparing for this interview is that at the elementary level, learning mathematics is a deeply social endeavor. Tell us a little bit about what you mean by that, Bill. Bill: Sure. So, mathematics itself, maybe as a premise, is a social activity. It's created by humans as a way of engaging with the world and a way of reasoning. So, the learning of mathematics is also social in the sense that we're giving students an introduction to this way of engaging in the world. Using numbers and quantities and shapes in order to make sense of our environment. And when I think about learning mathematics, I think that we are not simply downloading knowledge and sticking it into our heads. And in the modern day where artificial intelligence and computers can do almost every calculation that we can imagine—although your AI may do it incorrectly, just as a fair warning [laughs]—but in the modern day, the actual answer is not what we're so focused on. It's actually the process and the reasoning and the modeling and justification of those choices. And so, when I think about learning mathematics as learning to use these language tools, learning to use these ways of communication, how do we learn to communicate? We learn to communicate by engaging with other people, by engaging with the ideas and the minds and the feelings and so on of the folks around us, whether it's the teacher and the student, the student and the student, the whole class and the teacher. That's where I really see the power. And most of us who have learned, I think can attest to the fact that even when we're engaging with a text, really fundamentally we're engaging with something that was created by somebody else. So, fundamentally, even when you're sitting by yourself doing a math word problem or doing calculations, someone has given that to you and you think that that's important enough to do, right? So, from that stance, I see all of teaching and learning mathematics is social. And maybe one of our goals in mathematics classrooms, beyond memorizing the times tables, is learning to communicate with other people, learning to be participants in this activity with other folks. Mike: One of the things that strikes me about what you were saying, Bill, is there's this kind of virtuous cycle, right? That by engaging with language and having the social aspect of it, you're actually also deepening the opportunity for students to make sense of the math. You're building the scaffolds that help kids communicate their ideas as opposed to removing or stripping out the language. That's the context in some ways that helps them filter and make sense. You could either be in a vicious cycle, which comes from removing the language, or a virtuous cycle. And it seems a little counterintuitive because I think people perceive language as the thing that is holding kids back as opposed to the thing that might actually help them move forward and make sense. Bill: Yeah. And actually that's one of the really interesting pieces that we've looked at in my research and the broader research is this question of, “What makes mathematics linguistically complex?” is a complicated question. And so sometimes we think of things like looking at the word count as a way to say, “If there are fewer words, it's less complex, and if there are more words, it's more complex.” But that's not totally true. And similarly, “If there's no context, it's easier or more accessible, and if there is a context, then it's less accessible.” And I don't see these as binary choices. I see these as happening on a somewhat complicated terrain where we want to think about, “How do these words or these contexts add to student understanding or potentially impede [it]?” And that's where I think this social aspect of learning mathematics—as you described, it could be a virtuous cycle so that we can use language in order to engage in the process of learning language. Or, the vicious cycle is, you withhold all language and then get frustrated when students can't apply their mathematics. That’s maybe the most stereotypical answer: “My kids can do this, but as soon as they get a word problem, they can't do it.” And it's like, “Well, did you give them opportunities to learn how to do this? [laughs] Or is this the first time?” Because that would explain a lot. Mike: Well, it's an interesting question, too, because I think what sits behind that in some ways is the idea that you're kind of going to reach a point, or students might reach a point, where they're “ready” for word problems. Bill: Right. Mike: And I think what we're really saying is it's actually through engaging with word problems that you build your proficiency, your skillset that actually allows you to become a stronger mathematician. Bill: Mm-hmm. Right. Exactly. And it's a daily practice, right? It's not something that you just hold off to the end of the unit, and then you have the word problems, but it's part of the process of learning. And thinking about how you integrate and support that. That's the key question that I really wrestle with. Not trivial, but I think that's the key and the most important part of this. Mike: Well, I think that's actually a really good segue because I wanted to shift and talk about some of the concrete or productive ways that educators can support multilingual learners. And in preparing for this conversation, one of the things that I've heard you stress is this notion of a consistent context. So, can you just talk a little bit more about what you mean by that and how educators can use that when they're looking at their lessons or when they're writing lessons or looking at the curriculum that they're using? Bill: Absolutely. So, in our past work, we engaged in some cycles of design research with teachers looking at their mathematics curriculum and opportunities to engage multilingual learners in communication and reasoning in the classroom. And one of the surprising things that we found—just by looking at a couple of standard textbooks—was a surprising number of contexts were introduced that are all related to the same concept. So, the concept would be something like rate of change or ratio, and then the contexts, there would be a half dozen of them in the same section of the book. Now, this was, I should say, at a secondary level, so not quite where most of the Bridges work is happening. But I think it's an interesting lesson for us that we took away from this. Actually, at the elementary level, Kathryn Chval has made the same observation. What we realized was that contexts are not good or bad by themselves. In fact, they can be highly supportive of student reasoning or they can get in the way. And it's how they are used and introduced. And so, the other way we thought about this was: When you introduce a context, you want to make sure that that context is one that you give sufficient time for the students to understand and to engage with; that is relatable, that everyone has access to it; not something that's just completely unrelated to students' experiences. And then you can really leverage that relatable, understandable context for multiple problems and iterations and opportunities to go deeper and deeper. To give a concrete example of that, when we were looking at this ratio and rate of change, we went all the way back to one of the fundamental contexts that's been studied for a long time, which is motion and speed and distance and time. And that seemed like a really important topic because we know that that starts all the way back in elementary school and continues through college-level physics and beyond. So, it was a rich context. It was also something that was accessible in the sense that we could do things like act out story problems or reenact a race that's described in a story problem. And so, the students themselves had access to the context in a deep way. And then, last, that context was one that we could come back to again and again, so we could do variations [of] that context on that story. And I think there's lots of examples of materials out there that start off with a core context and build it out. I’m thinking of some of the Bridges materials, even on the counting and the multiplication. I think there's stories of the insects and their legs and wings and counting and multiplying. And that's a really nice example of—it's accessible, you can go find insects almost anywhere you are. Kids like it. [Laughs] They enjoy thinking about insects and other icky, creepy-crawly things. And then you can take that and run with it in lots of different ways, right? Counting, multiplication, division ratio, and so on. Mike: This last bit of our conversation has me thinking about what it might look like to plan a lesson for a class or a group of multilingual learners. And I know that it's important that I think about mathematical demands as well as the language demands of a given task. Can you unpack why it's important to set math and language development learning goals for a task, or a set of tasks, and what are the opportunities that come along with that, if I'm thinking about both of those things during my planning? Bill: Yeah, that's a great question. And I want to mark the shift, right? We've gone from thinking about the demands to thinking about the goals, and where we're going to go next. And so, when I think about integrating mathematical goals—mathematical learning goals and language learning goals—I often go back to these ideas that we call the practices, or these standards that are about how you engage in mathematics. And then I think about linking those back to the content itself. And so, there's kind of a two-piece element to that. And so, when we're setting our goals and lesson planning, at least here in the great state of California, sometimes we'll have these templates that have, “What standard are you addressing?,” [Laughs] “What language standard are you addressing?,” “What ELD standard are you addressing?,” “What SEL standard are you addressing?” And I've seen sometimes teachers approach that as a checkbox, right? Tick, tick, tick, tick, tick. But I see that as a missed opportunity—if you just look at this like you're plugging things in—because as we started with talking about how learning mathematics is deeply social and integrated with language, that we can integrate the mathematical goals and the language goals in a lesson. And I think really good materials should be suggesting that to the teacher. You shouldn't be doing this yourself every day from scratch. But I think really high-quality materials will say, “Here's the mathematical goal, and here's an associated language goal,” whether it's productive or receptive functions of language. “And here's how the language goal connects the mathematical goal.” Now, just to get really concrete, if we're talking about an example of reasoning with ratios—so I was going back to that—then it might be generalized, the relationship between distance and time. And that the ratio of distance and time gives you this quantity called speed, and that different combinations of distance and time can lead to the same speed. And so, explain and justify and show using words, pictures, diagrams. So, that would be a language goal, but it's also very much a mathematical goal. And I guess I see the mathematical content, the practices, and the language really braided together in these goals. And that I think is the ideal, and at least from our work, has been most powerful and productive for students. Mike: This is off script, but I'm going to ask it, and you can pass if you want to. Bill: Mm-hmm. Mike: I wonder if you could just share a little bit about what the impact of those [kinds] of practices that you described [have been]—have you seen what that impact looks like? Either for an educator who has made the step and is doing that integration or for students who are in a classroom where an educator is purposely thinking about that level of integration? Bill: Yeah, I can talk a little bit about that. In our research, we have tried to measure the effects of some of these efforts. It is a difficult thing to measure because it's not just a simple true-false test question type of thing that you can give a multiple-choice test for. But one of the ways that we've looked for the impact [of] these types of intentional designs is by looking at patterns of student participation in classroom discussions and seeing who is accessing the floor of the discussion and how. And then looking at other results, like giving an assessment, but deeper than looking at the outcome, the binary correct versus incorrect. Also looking at the quality of the explanation that's provided. So, how [do] you justify an answer? Does the student provide a deeper or a more mathematically complete... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/36412605

Season 3 | Episode 16 - Assessment as a Shared Journey: Cultivating Partnerships with Families and Caregivers - Guest: Tisha Jones 04/17/2025

Season 3 | Episode 16 - Assessment as a Shared Journey: Cultivating Partnerships with Families and Caregivers - Guest: Tisha Jones 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 /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/36087650

Season 3 | Episode 15 – What If I Don’t Understand Their Thinking? - Guest: Ryan Flessner 04/03/2025

Season 3 | Episode 15 – What If I Don’t Understand Their Thinking? - Guest: Ryan Flessner 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 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... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/35963375

Season 3 | Episode 14 – Supporting Neurodiverse Students in Elementary Mathematics Classrooms - Guest: Dr. Cathery Yeh 03/20/2025

Season 3 | Episode 14 – Supporting Neurodiverse Students in Elementary Mathematics Classrooms - Guest: Dr. Cathery Yeh 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... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/35703725

Season 3 | Episode 13 – Assessment in the Early Years - Guest: Shelly Scheafer 03/06/2025

Season 3 | Episode 13 – Assessment in the Early Years - Guest: Shelly Scheafer Assessment in the Early Years Guest: Shelly Scheafer ROUNDING UP: SEASON 3 | EPISODE 13 Mike: How is the work of assessing young children different from assessing students in upper elementary grades or in grades six through 12? And what actions can we take to ensure we understand our youngest learners' thinking? Today we're talking with Shelly Schafer, senior manager of Content Development with the math Learning Center about the ways educators can understand and advance the mathematical thinking of our youngest learners. Welcome to the podcast, Shelly. Thank you so much for joining us today. Shelly: Thank you, Mike, for having me. Mike: So I'd like to start with this question. What makes the work of assessing younger children, particularly students in grades K through two, different from assessing students in upper elementary grades or even beyond? Shelly: Wow, there's a lot to that question, Mike. I think there's some obvious things. So effective assessment of our youngest learners is different because obviously our pre-K, our first, even our second grade students are developmentally different from fourth and fifth graders. So when we think about assessing these early primary students, we need to use appropriate assessment methods that match their stage of development. For example, when we think of typical paper pencil assessments and how we often ask students to show their thinking with pictures, numbers and words. Our youngest learners, our just starting to connect symbolic representations to mathematical ideas, let alone put letters together to make words. So we need to take into consideration that primary students are in the early stages of development with their language, their reading, and their writing skills. And this makes it challenging for them to fully articulate, write, sketch, any of their mathematical thinking. So we often find that with young children, interviews can be really helpful, but even then there's some drawbacks. Some children find it challenging to show in the moment what they know. Others just aren't fully engaged or interested because you've called them over from something that they're busy doing, or maybe they're not yet comfortable with the setting or even the person doing the interview. So when we work within children, we need to recognize all of these little peculiarities that come with working with that age. We also need to understand that their mathematical development is fluid, it's continually evolving, and this is why they often or some may respond differently to the same prompter question, especially if the setting of the context has changed. We may find that a kindergarten student who counts to nine on Monday may count to 69 or even a hundred later in the week, depending on what's going on in their mind at the time. So this means that assessment with young children needs to be frequent, formative, and ongoing. So we're not necessarily waiting for the end of the unit to see, aha, did they get this? What do we do? We're looking at their work all of the time. And fortunately, some of the best assessments on young children are the observations in their natural setting, like times when maybe they're playing a math game or working with a center activity or even during just your classroom routines. And it's these authentic situations that we can look at as assessments help us capture a more accurate picture of their abilities because we not only get to hear what they say or see what they write on paper, we get to watch them in action. We get to see what they do when they're engaged in small group activities or playing games with friends. Mike: So I want to go back to something you said and the way that you said it. You were talking about watching or noticing what students can do and you really emphasize the words. Do talk a little bit about what you were trying to convey with that Shelly Shelly: Young children are doers when they work on a math pass, they show their thinking and their actions with finger formations and objects. And we can see if a student has one-to-one correspondence when they're counting, if they group their objects, how they line 'em up, do they tag them? Do they move them as they count them? They may not always have the verbal skills to articulate their thinking, but we can also attend to things like head nodding, finger counting, and even how they cluster or match objects. I'm going to give you an example. Let's say that I'm watching some early first graders and they're solving the expression six plus seven, and the first student picks up a number rack, and if you're not familiar with a number rack, it's a tool with two rows of beads. And on the first row there are five red beads and five white beads. And on the second row there's five red beads and five white beads. And the students solving six plus seven begins by pushing over five red beads in one push and then one more bead on the top row. And then they do the same thing for the seven. They push over five red beads and two white beads, and they haven't said a word to me, I'm just watching their actions and I'm already able to tell, hmm, that student could subitize a group of five because I saw 'em push over all five beads in one push. And that they know that six is composed of five in one and seven is composed of five and two, and they haven't said a word. I'm just watching what they're doing. And then I might watch the student and I see 'em pause, nothing's being said, but I start to notice this slight little head nodding. And then they say 13, and they give me the answer and they're really pleased. I didn't get a lot of language from them, but boy did I get a lot from watching how they solve that problem. And I want to contrast that observation with a student who might be solving the same expression, six plus seven, and they might go, Hmm, six. And then they start popping up one finger at a time while counting 7, 8, 9, 10, 11, 12, 13. And when they get seven fingers held up, they say 13. Again, they've approached that problem quite differently, but again, I get that information that they understood the equation they were able to count on starting with six, and they kept track of their count with their fingers and they knew to stop when seven fingers were raised. And I might even have a different student that might start talking to me and they say, well, six plus six is 12, and seven is one more than six. So the answer is 13. And if this were being done on a paper pencil as an assessment item or they were answering on some kind of a device, all I would know about my students is that they were able to get the correct answer. I wouldn't really know a lot about how they got the answer, what skills do they have? What was their thinking? And there's not a lot that I can work with to plan my instruction. Does that kind of make sense? Mike: Absolutely. I think the way that you described this, attending to behaviors, to gestures, to the way that kids are interacting with manipulatives, the self-talk that's happening, it makes a ton of sense. And I think for me, when I think back to my own practice, I wish I could wind the clock back because I think I was attending a lot to what kids were saying and sometimes their written communication. And there was a lot that I could have also taken in if I was attending to those things in a little bit more depth. It also strikes me that this might feel a bit overwhelming for an educator. How could an educator know what they're looking for? Shelly: I do think it can feel overwhelming at first, but as teachers begin to make informal observations, really listening and watching students actions as part of just their daily practice, something that they're doing on a normal basis, they start to develop these kind of intuitive understandings of how children learn, what to expect them to do, what they might say next if they see a certain action. And after several years, let's say, of teaching kindergarten, if you've been a kindergarten teacher for four or 5, 6, 20 plus years, you start to notice these patterns of behavior, things that five and six year olds seem to say and think and do on a fairly consistent basis. And that kind of helps you know what you're looking at. And fortunately, we have several researchers that have been, let's say, kid watching for 40 years, and they have identified stages through which most children pass as they develop their counting skills or maybe strategies for solving addition and subtraction problems. And these stages are laid out as progressions of thinking or actions that students exhibit as they develop understanding over periods of time. And listeners might know these as learning progressions or learning trajectories. And these are ways to convey an idea of concept in little bits of understanding. So when I was sharing the thinking and actions of three students solving six plus seven listeners familiar with cognitively guided instruction, CGI, they might've recognized the sequence of strategies that children go through when they're solving addition and subtraction problems. So in my first student, they didn't say anything but gave me an answer was using direct modeling. We saw them push over five and one beads for six and then five and two beads for seven, and then kind of pause at their model. And I could tell with their head nodding that they were counting quietly in their head counting all the beads to get the answer. And that's one of those first stages that we see and recognize with direct modeling. And that gives me information on what I might do with a student coming next time. I might work on the second strategy that I conveyed with my second student where they were able to count on, they started with that six, and then they counted seven more using their fingers to keep track of their count and got the answer. And then that third kind of level in that progression as we're moving of understanding was shown with my third student when they were able to use a derived fact strategy. The student said, oh, well I know that six plus six is 12. I knew my double fact, and then I use that relationship of knowing that seven is one more than six. And so that's kind of how we move kids through. And so when I'm watching them, I can kind of pinpoint where they are and where they might go next, and I can also think about what I might do. And so it's this knowledge of development and progressions and how children, number concepts that can help teachers recognize the skills as they emerge, as they begin to see them with their students, and they can use those to guide their instruction for that student or look at the class overall and plan their instruction or think about more open-ended kinds of questions that they can ask that recognize these different levels that students are working with. Mike: As a K one teacher, I remember that I spent a lot of my time tracking students with things like checklists. So I note if students had or didn't have a skill, and as I hear you talk, that feels fairly oversimplified when we think about this idea of developmental progressions. How do you suggest that teachers approach capturing evidence of student learning, Shelly? Shelly: Well, we have to really think about assessment and children's learning is something that is ongoing and involving, and if we do, it becomes part of what we can do every day. We can look for opportunities to observe students' skills in authentic settings. Maybe it's something that we're having them write down on their whiteboard, or maybe it's something where they're showing the answer with finger formations or we're giving a thumbs up or a thumbs down to check in on their understanding. We might not be checking on every student, but we're capturing a few and we can take note because we're doing this on a daily basis of who we want to check in with, what do we want to see? We can also do a little more formal planning when we draw from what we're going to do already in our lesson. Let's say for example that our lesson today includes a.talk or a number talk, something that we're going to write down. We're going to record student thinking. And during the lesson, the teacher's going to be busy facilitating the discussion, recording the student's thinking and making all of those notes. But if we write the child's name, honor their thinking and give it that caption on that public record at the end of the lesson, we can capture a picture, just use an iPad quickly, take a picture of that student's thinking, and then we can record that where we're keeping track of our students. So we have, okay, another moment in time. And it's this collection of evidence that we keep growing. We can also by capturing these public records note, whose voice and thinking we're elevating in the classroom. So it gives us how are they thinking and who are we listening to and making sure that we're spreading that out. And hearing everyone, I think like you mentioned checklists that you use. Mike: I did, Shelly: Yeah. And even checklists can play a role in observation and assessments when they have a focus and a way to capture students' thinking. One of the things we did in third edition is we designed additional tools for gathering and recording information during workplaces. That's a routine where students are playing games and or engaged with partners doing some sort of a math activity. And we designed these based on what we might see students do at these different games and activities. And we didn't necessarily think about this is something you're going to do with every student or even in one day because these are spanned out over a period of four to six weeks where that they can go to these games and we might see the students go to these activities multiple times. And so let's say that kindergarten students are playing something like the game beat you to 10 where they're spinning a spinner, they're counting cubes, and they're trying to race their partner to collect 10 cubes. And with an activity like that, I might want to focus on students who I still want to see, do they have one-to-one correspondence? Are they developing cardinality? Are they able to count out a set? And those might be kinds of skills that you might've had typically on a checklist, right, Mike for kindergarten. But I could use this activity to gather that note and make any comment. So just for those kids I'm looking at, or maybe first graders are playing a game like sort the sum where they're drawing two different dominoes and they're supposed to find how many they have in all. So with a game like that, I might focus on what are their strategies? Are they counting all the dots? Are they counting on from one set of the dots on one side and then counting on the other? Are they starting with the greater number or the most dots? Are they starting with the one always on the left? Or I might even see they might instantly recognize some of those. So I might know the skills that I want to look for with those games and be making notes, which kind of feels checklist, but I can target that time to do it on students. I want that information by thinking ahead of time, what can I get by watching observing these students at these games? I mean, as you know, young children love it. Older children love it when the teacher goes over and wants to watch them play or even better wants to engage in the gameplay with them. But I can still use that as an assessment. Mike: That's helpful, Shelly, for a couple of reasons. One of the things that you said was really powerful is thinking about not just the assessment tools that might be within your curriculum, but looking at the task itself that you're going to have students engage with, be it a game or a project or some kind of activity and really thinking, what can I get from this as a person who's trying to make sense of students' thinking? And I think my checklist suddenly feels really different when I've got a clear vision of what can I get from this task or this game that students are playing and looking for evidence of that versus feeling like I was pulling kids over one-on-one, which I think I would still do because there's some depth that I might want to capture, but it changes the way that I think about what I might do and also what I might get out of a task that really resonates for me. The other thing that you made me think about is the extent to which I remember thinking is I need to make sure if a student has got it or not. Got it. What you're making me think can really come out of this experience of observing students when they're working on a task or with a partner is that I can gather more evidence about the application of that idea. I can see the extent to which students are doing something like counting on in the context of a game or a task, and that adds to the evidence that I gather in a one-on-one interview with them. But it gives me a chance to see, is this way of thinking something that students are applying in different contexts or did it just happen at that one particular moment in time when I was with 'em? So that really helps me think about how those two different ways of assessing students be it one-on-one or observing them and seeing what's happening, support one another. Shelly: And I think you also made me think it really hit on this idea that students, like I said, their learning is evolving over time and it might change with the context so that they show us that they know something in one context with these numbers or this scenario, but they don't necessarily always see that it applies across the board. They don't make generalizations. That's something that we really have to work with students to develop. And they're also young children. Think about how quickly a three-year-old and a four-year-old change the same five to six, six to seven, I mean, they're evolving all the time. And so we want to get this information for them on a regular basis. A unit of instruction may be a month or more long, and a lot can happen in that time. So we want to make sure that we continue to check in with them and help them to develop if needed or that we advance them, we nudge them along, we challenge them with maybe a question, will that apply to every number? So a student discovers when we add one to every number, it's like saying the next number, so six and one more, seven and eight and one more is nine. And you can challenge them. Ooh, does that always work? What if the number was 22? What if it was 132? Would it always work? So when you're checking in with kids, you have those opportunities to keep them thinking, to help them grow. Mike: I want to pick up on something that we haven't necessarily said aloud, but I'd like to explore it. Looking at young students' work from an asset-based perspective, particularly with younger students, I've had points in time where there felt like so much that I needed to teach them, and sometimes I felt myself focusing on what they couldn't do. Looking back, I wish I had thought about my work as noticing the assets, the strategies, the ways of thinking that they were accumulating. Are there practices you think support an asset-based approach to assessment with young learners? Shelly: I think probably the biggest thing we can do is broaden our thinking about assessment. The National Council of Teachers of Mathematics wrote in catalyzing change in early childhood and elementary mathematics that the primary purpose of assessment is to gather evidence of children's thinking, understanding and reasoning to inform both instructional decisions and student in teaching learning. If we consider assessments and observations as tools to inform our instruction, we need to pay attention to the details of the child's thinking. And when we're paying attention to the details, what the child is bringing to the table, what they can do, that's where our focus goes. So the question becomes, what is the... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/35427155

Season 3 | Episode 12 – Inside Out: Examining the Meaning and Purpose of our Questions - Guest: Dr. Victoria Jacobs 02/20/2025

Season 3 | Episode 12 – Inside Out: Examining the Meaning and Purpose of our Questions - Guest: Dr. Victoria Jacobs Dr. Victoria Jacobs, Examining the Meaning and Purpose of our Questions ROUNDING UP: SEASON 3 | EPISODE 12 Mike (00:03): The questions educators ask their students matter. They can have a profound impact on students' thinking and the shape of their mathematical identities. Today we're examining different types of questions, their purpose and the meaning students make of them. Joining us for this conversation is Dr. Vicki Jacobs from the University of North Carolina Greensboro. Welcome to the podcast, Vicki. I'm really excited to talk with you today. Vicki (00:33): Thanks so much for having me. I'm excited to be here. Mike (00:36): So you've been examining the ways that educators use questioning to explore the details of students' thinking. And I wonder if we could start by having you share what drew you to the topic. Vicki (00:47): For me, it all starts with children's thinking because it's absolutely fascinating, but it's also mathematically rich. And so a core part of good math instruction is when teachers elicit children's ideas and then build instruction based on that. And so questioning obviously plays a big role in that, but it's hard. It's hard to do that well in the moment. So I found questioning to explore children's thinking to be a worthwhile thing to spend time thinking about and working on. Mike (01:17): Well, let's dig into the ideas that have emerged from that work. How can teachers think about the types of questions that they might ask their students? Vicki (01:24): Happy to share. But before I talk about what I've learned about questioning, I really need to acknowledge some of the many people that have helped me learn about questioning over the years. And I want to give a particular shout out to the teachers and researchers in the wonderful cognitively guided instruction or CGI community as well as my long-term research collaborators at San Diego State University. And more recently, Susan Sen. This work isn't done alone, but what have we learned about teacher questioning across a variety of projects? I'll share two big ideas and the first relates to the goals of questioning and the second addresses more directly the types of questions teachers might ask. So let's start with the goals of questioning because there are lots of reasons teachers might ask questions in math classrooms. And one common way to think about the goal of questioning is that we need to direct children to particular strategies during problem solving. (02:23): So if children are stuck or they're headed down a wrong path, we can use questions to redirect them so that they can get to correct answers with particular strategies. Sometimes that may be okay, but when we only do that, we're missing a big opportunity to tap into children's sense-making. Another way to think about the goal of questioning is that we're trying to explore children's thinking during problem solving. So think about a math task where multiple strategies are encouraged and children can approach problem solving in any way that makes sense to. So we can then ask questions that are designed to reveal how children are thinking about the problem solving, not just how well they're executing our strategies. And we can ask these questions when children are stuck, but also when they solve problems correctly. So this shift in the purpose of questioning is huge. And I want to share a quote from a teacher that I think captures the enormity of this shift. (03:26): She's a fifth grade teacher, and what she said was the biggest thing I learned from the professional development was not asking questions to get them to the answers so that I could move them up a strategy, but to understand their thinking. That literally changed my world. It changed everything. So I love this quote because it shows how transformative this shift can be because when teachers become curious about how children are thinking about problem solving, they give children more space to problem solve in multiple ways, and then they can question to understand and support children's ideas. And these types of questions are great because they increase learning opportunities for both children and teachers. So children get more opportunities to learn how to talk math in a way that's meaningful to them because they're talking about their own ideas and they also get to clarify what they did think more about important math that's embedded in their strategies and sometimes to even self-correct. And then as teachers, these types of questions give us a window into children's understandings, and that helps us determine our next steps. Questioning can have a different and powerful purpose when we shift from directing children toward particular strategies to exploring their mathematical thinking. Mike (04:54): I keep going back to the quote that you shared, and I think the details of the why and kind of the difference in the experience for students really jump out. But I'm really compelled by what that teacher said to you about how it changes everything. And I wonder if we could just linger there for a moment and you could talk about some of the things that you've seen happen for educators who have that kind of aha moment in the same way that that teacher did, how that impacts the work that they're doing with children or how they see themselves as an educator. Vicki (05:28): That's a great question. I think it's freeing in some way because it changes how educators think about what their next steps are. Every teacher has lots of pressures from standards and sometimes pacing guides and grade level teams that are working on the same page, all sorts of things that are a big part of teaching. But it puts the focus back on children and children's thinking and that my next steps should then come from there. And so in some ways, I think it gives a clearer direction for how to navigate all those various pressures that teachers have. Mike (06:14): I love that. Let's talk about part two. Vicki (06:17): Sure. So if we have the goal of questioning to explore children's thinking, how do we decide what questions to ask? So first of all, there's never a best question. There are many questioning frameworks out there that can provide lots of ideas, but what we've found is that the most productive questions always start with what children say and do. So that means I can't plan all my questions in advance, and instead I have to pay close attention to what children are saying and doing during problem solving. And to help us with that, we found a distinction between inside questions and outside questions. And that distinction has been really useful to us and also usable even during instruction. So inside questions are questions that explore details that are part of inside children's current strategies. And outside questions are questions that focus on strategies or representations that are not what children have done and may even be linked to how we as teachers are thinking about problem solving. (07:26): So I promised an example, and this is from our recent research project on teaching and learning about fractions. And we asked teachers to think about a child's written strategy for a fraction story problem. And the problem was that there are six children equally sharing four pancakes, and they need to figure out how much pancake each child can get. So we're going to talk about Joy's strategy for solving this problem. She is a fourth grader who solved the problem successfully, but in a complex and rather unconventional way. So I'm going to describe her strategy as a reminder. We have six children sharing four pancakes. So she drew the four pancakes. She split the first three pancakes into fourths and distributed the pieces to the six children, and that works out to two fourths for each child. But now she has a problem because she has one pancake left and fourths aren't going to work anymore because that's not enough pieces for her six children. (08:23): So she split the pancake first into eighths and then into 20 fourths and distributed those pieces. So each child ends up receiving two fourths, one eighth and one 24th. And when you put all those amounts together, they equal the correct amount of two thirds pancake per child. But Joy left her answer in pieces as two fourths, one eighth and one 24th, and she wrote those fractions in words rather than using symbols. Okay, so there's a lot going on in this strategy. And the specific strategy doesn't matter so much for our conversation, but the situation does. Here we have a child who has successfully solved the problem, but how she solved it and how she represented her answer are different than what we as adults typically do. So we ask teachers to think about what kind of follow-up conversation would you want to have with joy? (09:23): What types of questions would you want to ask her? And there were these two main questioning approaches, what we call inside questioning and outside questioning. So let's start with outside questioning. These teachers focused on improving Joy's strategy. So they ask follow-up questions like, is there another way you can share the four pancakes with six children? Or is your strategy the most efficient way you could share the pancakes? Or is there a way to cut bigger servings that would be more efficient? So given the complexity of Joy's strategy, we can appreciate these teachers' goals of helping joy move to a more efficient strategy. But all of these questions are pushing her to use a different strategy. So we considered them outside questions because they were outside of her current strategy. And outside questions can sometimes be productive, but they tend to get overused. And when we use them a lot, they can communicate to kids that what they're actually doing was wrong and that it needs fixing. (10:29): So let's think about the other approach of inside questioning. These teachers started by exploring what Joy had done in all of its complexity. And they ask a variety of questions. Usually it started with a general question, can you tell me what you did? But then they zoomed in on some of the many details. So for examples, they've asked how she split the pancakes. They offered questions like, why did you split the first three pancakes into four pieces? Or Tell me about the last pancake. That was the one that she split into eights and 20 fourths. Or they might ask about how she knew how to name each of the fractional amounts, especially the one 24th, because that's something that many children might've struggled with. And then there were questions about a variety of other details. Some of them are hard to explain without showing you a picture of the strategy, but the point is that the teachers took seriously what Joy had done and elevated it to the focus of the conversation. So Joy had a chance to share her reasoning and reflect on it, and the teachers could better understand Joy's approach to problem solving. So we found this distinction between inside and outside questioning to be useful to teachers and even in the midst of instruction because teachers can quickly check in with themselves. Am I asking an inside question or an outside question? Mike (11:49): Well, I have so many questions about inside and outside questions, but I want to linger on inside questions. What I found myself thinking is that for the learner, there are benefits for building number sense or conceptual understanding. The other thing that strikes me is that inside questions are also an opportunity to support students' math identity. And I wonder if that's something that you've seen in your work with teachers and with students. Vicki (12:14): Absolutely. I love your question. One of my favorite things about inside questions is that children see that their ideas are being taken seriously. And that's so empowering. It helps children believe that they can do math and that they are in charge of their mathematical thinking. I'll share a short story that was memorable for me, and this was from a while ago when I was in graduate school. So I was working on a research project and we were conducting problem solving interviews with young children. And our job was to document their strategies. So if we could see exactly what they did, we were told to write down the strategy and move on. But if we needed to clarify something, we could ask follow up questions. I was working with a first grader who had just spent a really long time solving a story problem. He had solved it successfully, and he had done that by joining many, many, many unifix cubes into a very long train. (13:10): And then he had counted them by ones multiple times. So he had been successful. I could tell exactly what he had done. So I started to move on to the next problem. So this young child looked at me a little incredulous and simply asked, don't you want to know how I did it? And he had come from a class where his math thinking was valued, and talking about children's thinking was a regular part of what they did. So he couldn't quite understand why this adult was not interested in how he had thought about the problem. Well, I was a little embarrassed and of course backtracked and listened to his full explanation. But the interaction stuck with me because it showed me how empowering it was for children to truly be listened to as math thinkers. And I think that's something we want for all children. Mike (14:00): The other thing that's hitting me in that story and in the story of joy is mea culpa. I am a person who has lived in the cult of efficiency where I looked at a student's work and my initial thought was, how do I nip the edges of this to get to more efficiency? But I really am struck by it how different the idea of asking the student to explain their thinking or the why behind it. I find myself thinking about joy, and it appears that she was intent on making sure that there were equal shares for each person. So there's ways that she could build to a different level of efficiency. But I think recognizing that there's something here that is really important to note about how and why she chose that, that would feel really meaningful as a learner. Vicki (14:44): I agree. I think what I like about inside questions is that they encourage us to, that children's thinking makes sense, even if it's different than how we think about it. It's our job to figure out how it makes sense. And then to build from there. Mike (15:03): Can you just say more about that? That feels like kind of a revelation. Vicki (15:08): Well, if we start with how kids are thinking and we take that seriously and we make that the center of the conversation, then we're acknowledging to the student and to ourselves that the child has something meaningful to bring to this conversation. And so we need to figure out how the child is thinking all the kind of kernels of mathematical strength in that thinking. And then yes, we can build from there, but we start with where they are as opposed to how we might solve the problem. Mike (15:49): If you were to offer educators a universal inside question or a few sentence frames for inside questions, is it possible to construct something like that that's generic or do you have other advice for us? Vicki (16:02): So that's a nice trick question. I wish it were that easy. I don't really think there are any universal inside questions. Perhaps the only universal one I can think of is something like, how did you solve this problem? It's a great general open-ended question. That's a good starter question in most situations. But the really powerful questions generally come from noticing mathematically important details in children's strategies. So a sentence stem that has been helpful in our work is, I noticed blank, so I wonder blank. Obviously questions don't have to be phrased exactly like this, but the idea is that we pick something that the child has done in their strategy and ask a question about the child's thinking behind that strategy detail. And that keeps us honest because the question absolutely has to begin with something in the child's strategy rather than inadvertently kind of slipping into our strategy. Mike (17:04): Vicki, what do you think about the purpose of outside questions? Are there circumstances where we would want to ask our students an outside question? Vicki (17:12): Absolutely. Sometimes we need to push children's thinking or share particular ideas, and that's okay. It's not that all outside questions are bad, it's just that we tend to overuse them and we could use them at more productive times. And by that I mean that we generally want to understand children's thinking before nudging their thinking forward with outside questions. So let's go back to the earlier example of Joy. Who was solving that problem about six children sharing four pancakes. And we had the two groups of teachers that had the different approaches to follow up questioning. There was the outside questioning that immediately zeroed in on improving Joy's strategy and the inside questioning that spent time exploring Joy's reasoning behind her strategy. So I'm thinking of two specific teachers right now. One generally took the outside questioning approach and the other inside questioning approach. And what was interesting about this pair was that they both asked the same outside question, could Joy partition the pancakes in a different way? (18:19): But they asked this question at different times and the timing really matters. So the teacher who took an outside questioning approach wanted to begin her conversation that way. She wanted to ask Joy, could she partition in a different way? But in contrast, the teacher who took an inside questioning approach wanted to ask Joy lots of questions about the details of her existing strategy, and then posed this very same question at the end to see if Joy had some new ideas for partitioning after their conversation about her existing strategy. And that feels really different to children. So the exact same question can send children different messages when outside questions are posed. First they communicate to children that what they did was wrong and needs fixing. But when outside questions are posed after a conversation about their thinking, it communicates a puzzle or a problem to be solved. (19:17): And children often are better equipped to consider this new problem having thoroughly discussed their own strategy. So I guess when I think about outside questions, I think of timing and amount. We generally want to start with inside questions, and we want most of our questions to be inside questions, but some outside questions can be productive. It's just that we overuse them. I want to mention one other thing about outside questions, and I think we often need fewer outside questions than we think we do, as long as we have space for children to learn from other children's thinking. So think about a typical lesson structure like launch, explore, discuss where children solve problems independently. And then the lesson concludes with a whole class discussion where children share their strategies and reflect on their problem solving. Will these sharing sessions serve as natural outside questions? Because children get to think about strategies that are outside of their own, but in a way that doesn't point to their own strategy as lacking in some way. So outside questions definitely have a place we just need to think about when we ask them and how many of them are really necessary. Mike (20:34): That is really helpful. I find myself thinking about my own process when I'm working on a problem, be it mathematical or organizational or what have you. When someone asks me to talk about how I've thought about it, engaging in that process in some ways primes me, right? Because I've gotten clearer on my own thinking. I suspect that the person who's asking me the question is also clearer on that, which allows them to ask a... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/35287750

Season 3 | Episode 11 – Affirming Students’ Mathematics Identities - Guest: Dr. Karisma Morton 02/06/2025

Season 3 | Episode 11 – Affirming Students’ Mathematics Identities - Guest: Dr. Karisma Morton 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 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 . 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 , 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. 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, . 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 | /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/35074890

Season 3 | Episode 10 – Building Productive Partnerships - Guests: Sue Kim and Myuriel von Aspen 01/23/2025

Season 3 | Episode 10 – Building Productive Partnerships - Guests: Sue Kim and Myuriel von Aspen 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 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... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/34945260

Season 3 | Episode 9 – Breaking the Cycle of Math Trauma - Guest: Dr. Kasi Allen 01/09/2025

Season 3 | Episode 9 – Breaking the Cycle of Math Trauma - Guest: Dr. Kasi Allen Dr. Kasi Allen, Breaking the Cycle of Math Trauma ROUNDING UP: SEASON 3 | EPISODE 9 If you are an educator, you’ve likely heard people say things like “I’m a math person.” While this may make you cringe, if you dig a bit deeper, many people can identify specific experiences that convinced them that this was true. In fact, some of you might secretly wonder if you are a math person as well. Today we’re talking with Dr. Kasi Allen about math trauma: what it is and how educators can take steps to address it. BIOGRAPHY Kasi Allen serves as the vice president of learning and impact at The Ford Family Foundation. She holds a PhD degree in educational policy and a bachelor’s degree in mathematics and its history, both from Stanford University. RESOURCES TRANSCRIPT Mike Wallus: If you're an educator, I'm almost certain you've heard people say things like, “I am not a math person.” While this may make you cringe, if you dig a bit deeper, many of those folks can identify specific experiences that convinced them that this was true. In fact, some of you might secretly wonder if you're actually a math person. Today we're talking with Dr. Kasi Allen about math trauma: what it is and how educators can take steps to address it. Well, hello, Kasi. Welcome to the podcast. Kasi Allen: Hi, Mike. Thanks for having me. Great to be here. Mike: I wonder if we could start by talking about what drew you to the topic of math trauma in the first place? Kasi: Really good question. You know, I've been curious about this topic for almost as long as I can remember, especially about how people's different relationships with math seem to affect their lives and how that starts at a very early age. I think it was around fourth grade for me probably, that I became aware of how much I liked math and how much my best friend and my sister had an absolutely opposite relationship with it—even though we were attending the same school, same teachers, and so on. And I really wanted to understand why that was happening. And honestly, I think that's what made me want to become a high school math teacher. I was convinced I could do it in a way that maybe wouldn't hurt people as much. Or it might even make them like it and feel like they could do anything that they wanted to do. But it wasn't until many years later, as a professor of education, when I was teaching teachers how to teach math, that this topic really resurfaced for me [in] a whole new way among my family, among my friends. And if you're somebody who's taught math, you're the math emergency person. And so, I had collected over the years stories of people's not-so-awesome experiences with math. But it was when I was asked to teach an algebra for elementary teachers course, that was actually the students’ idea. And the idea of this course was that we'd help preservice elementary teachers get a better window into how the math they were teaching was planting the seeds for how people might access algebra later. On the very first day, the first year I taught this class, there were three sections. I passed out the syllabus; in all three sections, the same thing happened. Somebody either started crying in a way that needed consoling by another peer, or they got up and left, or both. And I was just pretty dismayed. I hadn't spoken a word. The syllabi were just sitting on the table. And it really made me want to go after this in a new way. I mean, something—it just made me feel like something different was happening here. This was not the math anxiety that everybody talked about when I was younger. This was definitely different, and it became my passion project: trying to figure how we disrupt that cycle. Mike: Well, I think that's a good segue because I've heard you say that the term “math anxiety” centers this as a problem that's within the person. And that in fact, this isn't about the person. Instead, it's about the experience, something that's happened to people that's causing this type of reaction. Do I have that right, Kasi? Kasi: One hundred percent. And I think this is really important. When I grew up and when I became a teacher, I think that was an era when there was a lot of focus on math anxiety, the prevalence of math anxiety. Sheila Tobias wrote the famous book Overcoming Math Anxiety. This was especially a problem among women. There were dozens of books. And there were a number of problems with that work at the time, and that most of the research people were citing was taking place outside of math education. The work was all really before the field of neuroscience was actually a thing. Lots of deficit thinking that something is wrong with the person who is suffering this anxiety. And most of these books were very self-helpy. And so, not only is there something wrong with you, but you need to fix it yourself. So, it really centers all these negative emotions around math on the person that's experiencing the pain, that something's wrong with them. Whereas math trauma really shifts the focus to say, “No, no, no. This reaction, this emotional reaction, nobody's born that way.” Right? This came from a place, from an experience. And so, math trauma is saying, “No, there's been some series of events, maybe a set of circumstances, that this individual began to see as harmful or threatening, and that it's having long-lasting adverse effects. And that those long-lasting effects, this kind of triggering that starts to happen, is really beginning to affect that person's functioning, their sense of well-being when they're in the presence, in this case, of mathematics.” And I think the thing about trauma is just that. And I have to say in the early days of my doing this research, I was honestly a little bit hesitant to use that word because I didn't want to devalue some of the horrific experiences that people have experienced in times of war, witnessing the murder of a parent or something. But it's about the brain. It's how the brain is responding to the situation. And what I think we know now, even more than when I started this work, is that there is simply trauma [in] everyday life. There are things that we experience that cause our brains to be triggered. And math is unfortunately this subject in school that we require nearly every year of a young person's life. And there are things about the way it's been taught over time that can be humiliating, ridiculing; that can cause people to have just some really negative experiences that then they carry with them into the next year. And so that's really the shift. The shift is instead of labeling somebody as math anxious—“Oh, you poor thing, you better fix yourself”—it's like, “No, we have some prevalence of math trauma, and we've got to figure out how people's experiences with math are causing this kind of a reaction in their bodies and brains.” Mike: I want to take this a little bit further before we start to talk about causes and solutions. This idea that you mentioned of feeling under threat, it made me think that when we're talking about trauma, we are talking about a physiological response. Something is happening within the brain that's being manifested in the body. And I wonder if you could talk just a little bit about what happens to people experiencing trauma? What does that feel like in their body? Kasi: So, this is really important and our brains have evolved over time. We have this incredible processing capacity, and it's coupled with a very powerful filter called the amygdala. And the amygdala [has been] there from eons ago to protect us. It's the filter that says, “Hey, do not provide access to that powerful processor unless I'm safe, unless my needs are met. Otherwise, I gotta focus on being well over here.” So, we're not going to give access to that higher-order thinking unless we're safe. And this is really important because modern imaging has given us really new insights into how we learn and how our body is reacting when our brain gets fired in this way. And so, when somebody is experiencing math trauma, you know it. They sweat. Their face turns red. They cry. Their body and brain are telling them, “Get out. Get away from this thing. It will hurt you.” And I just feel like that is so important for us to remember because the amygdala also becomes increasingly sensitive to repeat negativity. So, it's one thing that you have a bad day in math, or you maybe have a teacher that makes you feel not great about yourself. But day after day, week after week, year after year, that messaging can start to make the amygdala hypersensitive to these sorts of situations. Is that what you were getting at with your question? Mike: It is. And I think you really hit on something. There's this idea of repeat negativity causing increased sensitivity, I think has real ramifications for classroom culture or the importance of the way that I show up as an educator. It's making me think a lot about culture and norms related to math in schools. I'm starting to wonder about the type of traumatizing traditions that we've had in math education that might contribute to this type of experience. What does that make you think? Kasi: Oh, for sure. Unfortunately, I think the list is a little long of the things that we may have been doing completely inadvertently. Everybody wants their students to have a great experience, and I actually think our practices have evolved. But culturally, I think there are some things about math that contribute to these “traumatizing traditions,” is what I've called them. Before we go there, I do want to say just one other thing about this trauma piece, and that is that we've learned about some things about trauma in childhood. And a lot of the trauma in childhood is about not a single life-altering event. But childhood trauma is often about these things that happened repeatedly where a child was being ridiculed, being treated cruelly. And it's about that repetition that is really seeding that trauma so deeply and that sense that they can't stop it, that they don't have control to stop the thing that is causing them pain or suffering. So, I just wanted to make sure that I tagged that because I think there is something about what we've learned about the different forms of childhood trauma that's especially salient in this situation. And so, I'll tie it to your question, which is, think about some of the things we've done in math historically. We don't do them in every place, but the ability grouping that has happened over time, it seems to go in and out of fashion. When a kid is told they're in the lower class, “Oh, this is something you're not good [at]—the slower math.” We often use speed to measure understanding, and so smarter is not faster. And there's some great quotes, Einstein among them. So that's a thing. When you gotta do it right now, it has to be one-hundred-percent right. It has to be superfast. We've often prioritized individual work over collaboration. So, you're all alone in this. In fact, if you're working with others, somehow that's cheating as opposed to collaborating. We teach kids tricks rather than teaching them how to think. And I think we deprive kids of the opportunity to have an idea. It's really hard to get excited about something where all you're doing is reproducing—reproducing something that somebody else thought of as quickly as possible and [it] needs to be one-hundred-percent [accurate]. You don't get to bring your own spin to it. And so, we focus on answers rather than people's reasoning behind the answers. That can be something that happens as well. And I think one of the things that's always gotten me is that there's only one way. Not only is there only one right answer, but there's only one way to get there, which also contributes to this idea of having to absorb somebody else's thinking rather than actualizing your own. And I absolutely know that most teachers are working to not do as much of these things in their math classrooms. And I want to be sure in having this conversation that—you know, I'm a lover of education and teachers, I taught teachers for many years. This is not about the teachers so much as the sort of culture of math and math education that we were all brought up in. And we've got to figure out how to make math something more so that kids can see themselves in it. And that it's not something that happens in a vacuum and is this performance course rather than a class where you get to solve cool problems that no one knows the exact answer to, or there's the exact right way, or that you get to get your own questions answered. Things you wonder about. That it's a chance to explore. So, I mean, ultimately, I think we just know that there's a lot of negativity that happens around math, and we accept it. And that is perhaps the most traumatizing tradition of all because that kind of repeat negativity we know affects the amygdala. It affects people's ability to access math in the long run. So, we gotta have neutral or better. Mike: So, in the field of psychology, there's this notion of generational trauma, and it's passed from generation to generation. And you're making me wonder if we're facing something similar when it comes to the field of math education. I'm wondering what you think educators might be able to do to reclaim math for themselves, especially if they're a person who potentially does have a traumatic mathematics experience and maybe some of the ways that they might create a different type of experience for their students. Kasi: Yeah, let's talk about each of those. I'm going to talk about one, the multigenerational piece, and then let's talk about how we can help ourselves and our students. One is, I think it's really very possible that that's what we're looking at in terms of math trauma. Culturally, I think we've known for a while that this is happening, with respect to math, that—you know, I've had parents come to back-to-school night and tell me that they're just not a math family. And even jokingly say, “Oh, we're all bad at math, don't be too hard on us,” and all the other things. And so, kids inherit that. And it's very common for kids to have the same attitude towards math that their parents do and also that their teachers do. And that's where I think in my mind, I really want to help every elementary teacher fall in love with math because if we look at the data, I think of any undergraduate major, it's those who major in education who report the highest rates of math anxiety and math trauma. And so, when you think about folks who feel that way about math, then being in charge of teaching it to kids in the early years, that's a lot to carry. And so, we want to give those teachers and anyone who has had this experience with math an opportunity to reclaim, regroup. And in my experience, what I've found is actually simply shifting the location of the problem is a really strong first step. When people understand that they actually aren't broken, that the feelings that they have about math don't reflect some sort of flaw in them as a human, but that it's a result of something they've experienced, a lot is unlocked. And most folks that I have worked with over my time working on this issue, they know. They know exactly the moment. They know the set of experiences that led to the reactions that they feel in their body. They can name it, and with actually fairly startling detail. So, in my teaching—and I think this is something anybody can do—is they would write a “mathography.” What is the story of your life through a math lens? What has been the story of your relationship with math over the course of your life and what windows does that give you into the places where you might need to heal? We've never had more tools to go back and sort of relearn areas of math that we thought we couldn't learn. And so often the trauma points are as math becomes more abstract. So many people have something that happened around fractions or multidigit multiplication and division. When we started—we get letters involved in math. I had somebody say, “Math was great as long as it was numbers. Then we got letters involved, and it was terrible.” And so, if people can locate, “This is where I had the problem. It's not me. I can go back and relearn some things.” I feel like that's a lot of the healing, and that, in fact, if I'm a teacher or if I'm a parent, I love my kids, whether they're my children or my students, and I'm going to work on me so that they have a better experience than I had. And I've found so many teachers embrace that idea and go to work. So, some of the things that can happen in classrooms that I think fall from this is that, first of all, the recognition that emotional safety, you can't have cognition and problem solving without it. If you have kids in your classroom who have had these negative experiences in math, you're going to need to help them unpack those and level set in order to move on. And “mathography” is also a good tool for that. Some people use breathing. Making sure that when you encounter kids that are exhibiting math anxiety, that you help them localize the problem outside of them. No one is born with math anxiety. It's the math of school that creates it. And if we ignore it, it's just going to get worse. So, some people feel like they can kind of smooth it over. I think we need to give kids the tools to unpack it and move beyond it. But it's so widespread, and I've encountered teachers who were afraid to go there. It's like the Pandora's box. My advice to them is that if you'll open the box and heal what's inside, the teaching becomes much easier. Whereas if you don't, you're fighting that uphill battle all the time. You know, students will feel more safe in classrooms where mistakes are opportunities to learn; where they're not a bad thing and where they see each other as resources, where they are not alone, and where they can collaborate and really take responsibility for each other's learning. So, some of the most powerful classrooms I've seen where there were a lot of kids who had very negative experiences with math, a teacher had succeeded in creating this learning environment, this community of learners where all the kids seem to recognize that somebody would have a good day, someone else would have a not good day, but it would be their turn for a good day a few days from now. [chuckles] So, we're all just going to take care of each other as we go. I think some things that teachers can keep a particular eye on is being sure that kids are given authentic work to do in math. It's really easy to start giving kids what we've called busywork, but work that really isn't engaging their brain. And it turns out that that boredom cycle triggers the negativity cycle, which can actually get your amygdala operating in a way that is not as far from trauma as we might all like to think. And so, while it isn't the same kind of math trauma that we're talking about here, it does affect the amygdala. And so that's something we should be aware of. And so, this is something—I think kids should learn about their brains in school. I don't know if it's the math teacher's job. But if they haven't learned about their brains yet, when you get them, I would recommend teaching kids about their brains, teaching them strategies for when they feel that kind of shutdown, that headache, like “I can't think.” Because most of the time, they actually can't. And they need to have some kind of reset. Another tip, just in terms of disrupting that trauma cycle in the classroom, is that by the time kids get to be third, fourth grade and up, they know who is good at math, or they've labeled each other. You know, “Who's good at math? Who's struggled?” Even if they are not tracked and sorted, they've... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/34743550

Season 3 | Episode 8 – Helping Our Students Build a Meaningful Understanding of Geometry - Guest: Dr. Rebecca Ambrose 12/19/2024

Season 3 | Episode 8 – Helping Our Students Build a Meaningful Understanding of Geometry - Guest: Dr. Rebecca Ambrose Dr. Rebecca Ambrose, Helping Our Students Build a Meaningful Understanding of Geometry 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 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 , 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 . 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... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/34401460

Season 3 | Episode 7 – How You Say It Matters: Teacher Language Choices That Support Number Sense - Guest: Dr. James Brickwedde 12/05/2024

Season 3 | Episode 7 – How You Say It Matters: Teacher Language Choices That Support Number Sense - Guest: Dr. James Brickwedde Dr. James Brickwedde, How You Say It Matters: Teacher Language Choices That Support Number Sense ROUNDING UP: SEASON 3 | EPISODE 7 Carry the 1. Add a 0. Cross multiply. All of these are phrases that educators heard when they were growing up. This language is so ingrained that many educators use it without even thinking. But what’s the long-term impact of language like this on the development of our students’ number sense? Today, we’re talking with Dr. James Brickwedde about the impact of language and the ways educators can use it to cultivate their students’ number sense. BIOGRAPHIES James Brickwedde is the director of the Project for Elementary Mathematics. He served on the faculty of Hamline University’s School of Education & Leadership from 2011–2021, supporting teacher candidates in their content and pedagogy coursework in elementary mathematics. RESOURCES TRANSCRIPT Mike Wallus: Carry the 1, add a 0, cross multiply. All of these are phrases that educators heard when they were growing up. This language is so ingrained, we often use it without even thinking. But what's the long-term impact of language like this on our students’ number sense? Today we're talking with Dr. James Brickwedde about the impact of language and the ways educators can use it to cultivate their students’ number sense. Welcome to the podcast, James. I'm excited to be talking with you today. James Brickwedde: Glad to be here. Mike: Well, I want to start with something that you said as we were preparing for this podcast. You described how an educator’s language can play a critical role in helping students think in value rather than digits. And I'm wondering if you can start by explaining what you mean when you say that. James: Well, thinking first of primary students—so, kindergarten, second grade, that age bracket—kindergartners, in particular, come to school thinking that numbers are just piles of ones. They're trying to figure out the standard order. They're trying to figure out cardinality. There are a lot of those initial counting principles that lead to strong number sense that they are trying to integrate neurologically. And so, one of the goals of kindergarten, first grade, and above is to build the solid quantity sense—number sense—of how one number is relative to the next number in terms of its size, magnitude, et cetera. And then as you get beyond 10 and you start dealing with the place value components that are inherent behind our multidigit numbers, it's important for teachers to really think carefully of the language that they're using so that, neurologically, students are connecting the value that goes with the quantities that they're after. So, helping the brain to understand that 23 can be thought of not only as that pile of ones, but I can decompose it into a pile of 20 ones and three ones, and eventually that 20 can be organized into two groups of 10. And so, using manipulatives, tracking your language so that when somebody asks, “How do I write 23?” it's not a 2 and a 3 that you put together, which is what a lot of young children think is happening. But rather, they realize that there's the 20 and the 3. Mike: So, you're making me think about the words in the number sequence that we use to describe quantities. And I wonder about the types of tasks or the language that can help children build a meaningful understanding of whole numbers, like say, 11 or 23. James: The English language is not as kind to our learners [laughs] as other languages around the world are when it comes to multidigit numbers. We have in English 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And when we get beyond 10, we have this unique word called “eleven” and another unique word called “twelve.” And so, they really are words capturing collections of ones really then capturing any sort of tens and ones relationship. There's been a lot of wonderful documentation around the Chinese-based languages. So, that would be Chinese, Japanese, Korean, Vietnamese, Hmong follows the similar language patterns, where when they get after 10, it literally translates as “10, 1,” “10, 2.” When they get to 20, it's “2, 10”—”2, 10, 1,” “2, 10, 2.” And so, the place value language is inherent in the words that they are saying to describe the quantities. The teen numbers, when you get to 13, a lot of young children try to write 13 as “3, 1” because they're trying to follow the language patterns of other numbers where you start left to right. And so, they're bringing meaning to something, which of course is not the social convention. So, the teens are all screwed up in terms of English. Spanish does begin to do some regularizing when they get to 16 because of the name “diez y seis,” so “ten, six.” But prior to that you have, again, sort of more unique names that either don't follow the order of how you write the number or they're unique like 11 and 12 is. Somali is another interesting language in that—and I apologize to anybody who is fluent in that language because I'm hoping I'm going to articulate it correctly—I believe that there, when they get into the teens, it's “1 and 10,” “2 and 10,” is the literal translation. So, while it may not be the “10, 1” sort of order, it still is giving … the fact that there's ten-ness there as you go. So, for the classrooms that I have been in and out of—both [in] my own classroom years ago as well as the ones I still go in and out of now—I try to encourage teachers to tap the language assets that are among their students so that they can use them to think about the English numbers, the English language, that can help them wire that brain so that the various representations—the manipulatives, expanded notation cards or dice, the numbers that I write, how I break the numbers apart, say that 23 is equal to 20 plus 3—all of those models that you're using, and the language that you use to back it up with, is consistent so that, neurologically, those pathways are deeply organized. Piaget, in his learning theory, talks about young children—this is sort of the 10 years and younger—can only really think about one attribute at a time. So that if you start operating on multidigit numbers, and I'm using digitized language, I'm asking that kindergartner, first [grader], second grader to think of two things at the same time. I'm, say, moving a 1 while I also mean 10. What you find, therefore, is when I start scratching the surface of kids who were really procedural-bound, that they really are not reflecting on the values of how they've decomposed the numbers or are reconfiguring the numbers. They're just doing digit manipulation. They may be getting a correct answer, they may be very fast with it, but they've lost track of what values they're tracking. There's been a lot of research on kids’ development of multidigit operations, and it's inherent in that research about students following—the students who are more fluid with it talk in values rather than in digits. And that's the piece that has always caught my attention as a teacher and helped transform how I talked with kids with it. And now as a professional development supporter of teachers, I'm trying to encourage them to incorporate in their practice. Mike: So, I want to hang on to this theme that we're starting to talk about. I'm thinking a lot about the very digit-based language that as a child I learned for adding and subtracting multidigit numbers. So, phrases like, “Carry the 1” or “Borrow something from the 6.” Those were really commonplace. And in many ways, they were tied to this standard algorithm, where a number was stacked on top of another number. And they really obscured the meaning of addition and subtraction. I wonder if we can walk through what it might sound like or what other models might draw out some of the value-based language that we want to model for kids and also that we want kids to eventually adopt when they're operating on numbers. James: A task that I give adults, whether they are parents that I’m out doing a family math night with or my teacher candidates that I have worked with, I have them just build 54 and 38, say, with base ten blocks. And then I say, “How would you quickly add them?” And invariably everybody grabs the tens before they move to the ones. Now your upbringing, my upbringing is the same and still in many classrooms: Students are directed only to start with the ones place. And if you get a new 10, you have to borrow and you have to do all of this exchange kinds of things. But the research shows when school gets out of the way [chuckles] and students and adults are operating on more of their natural number sense, people start with the larger and then move to the smaller. And this has been found around the world. This is not just unique to US classrooms that have been working this way. If, in the standard algorithms—which really grew out of accounting procedures that needed to save space in ledger books out of the 18th, 19th centuries—they are efficient, space-saving means to be able to accurately compute. But in today's world, technology takes over a lot of that bookkeeping type of thing. An analogy I like to make is, in today's world, Bob Cratchit out of [A] Christmas Carol, Charles Dickens’s character, doesn't have a job because technology has taken over everything that he was in charge of. So, in order for Bob Cratchit to have a job, [laughs] he does need to know how to compute. But he really needs to think in values. So, what I try to encourage educators to loosen up their practice is to say, “If I'm adding 54 plus 38, so if you keep those two numbers in your mind, [chuckles] if I start with the ones and I add 4 and 8, I can get 12.” There's no reason, if I'm working in a vertical format, to not put 12 fully under the line down below, particularly when kids are first learning how to add. But then language-wise, when they go to the tens place, they're adding 50 and 30 to get 80, and the 80 goes under the 12. Now, many teachers will know that's partial sums. That's not the standard algorithm. That is the standard algorithm. The difference between the shortcut of carrying digits is only a space-saving version of partial sums. Once you go to partial sums in a formatting piece, and you're having kids watch their language—and that's a phrase I use constantly in my classrooms—is, it's not a 5 and 3 that you are working with, it's a 50 and a 30. So when you move to the language of value, you allow kids to initially, at least, get well-grounded in the partial sums formatting of their work, the algebra of the connectivity property pops out, the number sense of how I am building the quantities, how I'm adding another 10 to the 80, and then the 2, all of that begins to more fully fall into place. There are some of the longitudinal studies that have come out that students who were using more of the partial sums approach for addition, their place value knowledge fell into place sooner than the students who only did the standard algorithm and used the digitized language. So, I don't mind if a student starts in the ones place, but I want them to watch their language. So, if they're going to put down a 2, they're not carrying a 1—because I'll challenge them on that—is “What did you do to the 12 to just isolate the 2? What's left?” “Oh, you have a 10 up there and the 10 plus the 50 plus the 30 gives me 90.” So, the internal script that they are verbalizing is different than the internal digitized script that you and I and many students still learn today in classrooms around the country. So, that's where the language and the values and the number sense all begin to gel together. And when you get to subtraction, there's a whole other set of language things. So, when I taught first grade and a student would say, “Well, you can't take 8 from 4,” if I still use that 54 and 38 numbers as a reference here, my challenge to them is, “Who said?” Now, my students are in Minnesota. So, Minnesota is at a cultural advantage of knowing what happens in wintertime when temperatures drop below 0. [laughs] And so, I usually have as a representation model in my room, a number line that’s swept around the edges of the room, that started from negative 35 and went to 185. And so, there are kids who've been puzzling about those other numbers on the other side of 0. And so, somebody pops up and says, “Well, you'll get a negative number.” “What do you mean?” And then they whip around and start pointing at that number line and being able to say, “Well, if you're at 4 and you count back 8, you'll be at negative 4.” So, I am not expecting first graders to be able to master the idea of negative integers, but I want them to know the door is open. And there are some students in late first grade and certainly in second grade who start using partial differences where they begin to consciously use … the idea of negative integers. However, there [are] other students, given that same scenario, who think going into the negative numbers is too much of The Twilight Zone. [laughs] They'll say, “Well, I have 4 and I need 8. I don't have enough to take 8 from 4.” And another phrase I ask them is, “Well, what are you short?” And that actually brings us back to the accounting reference point of sort of debit-credit language of, “I'm short 4.” “Well, if you're short 4, we’ll just write ‘minus 4.’” But if they already have subtracted 30 from 50 and have 20, then the question becomes, “Where are you going to get that 4 from?” “Well, you have 20 cookies sitting on that plate there. I'm going to get that 4 out of the 20.” So again, the language around some of these strategies in subtractions shifts kids to think with alternative strategies and algorithms compared to the American standard algorithm that predominates US education. Mike: I think what's interesting about what you just said too is you're making me think about an article. I believe it was “Rules That Expire.” And what strikes me is that this whole notion that you can't take 8 away from 4 is actually a rule that expires once kids do begin to work in integers. And what you're suggesting about subtraction is, “Let's not do that. Let's use language to help them make meaning of, “Well, what if?” As a former Minnesotan, I can definitely validate that when it's 4 degrees outside and the temperature drops 8 degrees, kids can look at a thermometer and that context helps them understand. I suppose if you're a person listening to this in Southern California or Arizona, that might feel a little bit odd. But I would say that I have seen first graders do the same thing. James: And if you are more international travelers, as soon as, say, people in Southern California or southern Arizona step across into Mexico, everything is in Celsius. If those of us in the northern plains go into Canada, everything is in Celsius. And so, you see negative numbers sooner [laughs] than we do in Fahrenheit, but that's another story. Mike: This is a place where I want to talk a little bit about multiplication, particularly this idea of multiplying by 10. Because I personally learned a fairly procedural understanding of what it is to multiply by 10 or 100 or 1,000. And the language of “add a 0” was the language that was my internal script. And for a long time when I was teaching, that was the language that I passed along. You're making me wonder how we could actually help kids build a more meaningful understanding of multiplying by 10 or multiplying by powers of 10. James: I have spent a lot of time with my own research as well as working with teachers about what is practical in the classroom, in terms of their approach to this. First of all, and I've alluded to this earlier, when you start talking in values, et cetera, and allow multiple strategies to emerge with students, the underlying algebraic properties, the properties of operations begin to come to the surface. So, one of the properties is the zero property, [laughs], right? What happens when you add a number to 0 or a 0 to a number? I'm now going to shift more towards a third-grade scenario here. When a student needs to multiply four groups of 30. “I want 30 four times,” if you're using the times language. And they'd say, “Well, I know 3 times 4 is 12 and then I just add a 0.” And that's where I as a teacher reply, “Well, I thought 12 plus 0 is still 12. How could you make it 120?” And they’d say, “Well, because I put it there.” So, I begin to try to create some cognitive dissonance [laughs] over what they're trying to describe, and I do stop and say this to kids: “I see that you recognize a pattern that's happening there. But I want us to explore, and I want you to describe why does that pattern work mathematically?” So, with addition and subtraction, kids learn that they need to decompose the numbers to work on them more readily and efficiently. Same thing when it comes to multiplication. I have to decompose the numbers somehow. So if, for the moment, you come back to, if you can visualize the numbers four groups of 36. Kids would say, “Well, yeah, I have to decompose the 36 into 30 plus 6.” But by them now exploring how to multiply four groups of 30 without being additive and just adding above, which is an early stage to it. But as they become more abstract and thinking more in multiples, I want them to explore the fact that they are decomposing the 30 into factors. Now, factors isn't necessarily a third-grade standard, right? But I want students to understand that that's how they are breaking that number apart. So, I'm left with 4 times 3 times 10. And if they've explored, in this case, the associate of property of multiplication, “Oh, I did that. So, I want to do 4 times 3 because that's easy. I know that. But now I have 12 times 10.” And how can you justify what 12 times 10 is? And that's where students who are starting to move in this place quickly say, “Well, I know 10 tens are 100 and 2 tens are 20, so it's 120.” They can explain it. The explanation sometimes comes longer than the fact that they are able to calculate it in their heads, but the pathway to understanding why it should be in the hundreds is because I have a 10 times a ten there. So that when the numbers now begin to increase to a double digit times a double digit—so now let's make it 42 groups of 36. And I now am faced with, first of all, estimating how large might my number be? If I've gotten students grounded in being able to pull out the factors of 10, I know that I have a double digit times a double digit, I have a factor of 10, a factor of 10. My answer's going to be in the hundreds. How high in the hundreds? In this case with the 42 and 36, 1,200. Because if I grab the largest partial product, then I know my answer is at least above 1,200 or one thousand, two hundred. Again, this is a language issue. It's breaking things into factors of 10 so that the powers of 10 are operated on. So that when I get deeper into fourth grade, and it's a two digit times a three digit, I know that I'm going to have a ten times a hundred. So, my answer's at least going to be up in the thousands. I can grab that information and use it both from an estimation point of view, but also, strategically, to multiply the first partial product or however you are decomposing the number. Because you don't have to always break everything down into their place value components. That's another story and requires a visual [laughs] work to explain that. But going back to your question, the “add the 0,” or as I have heard, some teachers say, “Just append the 0,” they think that that's going to solve the mathematical issue. No, that doesn't. That's still masking why the pattern... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/34147105

Season 3 | Episode 5 - Building Asset-Focused Professional Learning Communities - Guests: Summer Pettigrew and Megan Williams 11/26/2024

Season 3 | Episode 5 - Building Asset-Focused Professional Learning Communities - Guests: Summer Pettigrew and Megan Williams Summer Pettigrew and Megan Williams, Building Asset-Focused Professional Learning Communities ROUNDING UP: SEASON 3 | EPISODE 5 Professional learning communities have been around for a long time, in many different iterations. But what does it look like to schedule and structure professional learning communities that 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 County Public Schools about building asset-focused professional learning communities. BIOGRAPHIES Summer Pettigrew serves as an instructional coach at Springfield Elementary School in Charleston, South Carolina. Megan Williams serves as principal at Springfield Elementary School in Charleston, South Carolina. RESOURCES TRANSCRIPT 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 [STEM]. 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. Megan: Mm-hmm, 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 [period] and just not being one-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, right, 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. 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 flash back to, Megan, what you said earlier about [how] 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 one-hundred-percent right. Summer has teachers sometimes [take] 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? Meghan: It seems right. Summer: 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 sort 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 nonjudgmental, nonconfrontational 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 reengage 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,... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/33606657

Season 3 | Episode 6 – Nurturing Mathematical Curiosity: Supporting Mathematical Argumentation in the Early Grades - Guests: Drs. Jody Guarino and Chepina Rumsey 11/21/2024

Season 3 | Episode 6 – Nurturing Mathematical Curiosity: Supporting Mathematical Argumentation in the Early Grades - Guests: Drs. Jody Guarino and Chepina Rumsey Drs. Jody Guarino and Chepina Rumsey, Nurturing Mathematical Curiosity: Supporting Mathematical Argumentation in the Early Grades ROUNDING UP: SEASON 3 | EPISODE 6 Argumentation, justification, conjecture. All of these are practices we hope to cultivate, but they may not be practices we associate with kindergartners, first-, or 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. BIOGRAPHIES Chepina Rumsey, PhD, is an associate professor of mathematics education at the University of Northern Iowa (UNI). Jody Guarino is currently a mathematics coordinator at the Orange County Department of Education and a lecturer at the University of California, Irvine. RESOURCES . Teaching Children Mathematics, 25(4), 208–217. TRANSCRIPT 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 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 take away 2 versus 2 take away 5.” So, just this idea of, “Now [that] 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 have 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 discrete point in time to get a discrete 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 [classroom], the teacher was Rachael Gildea, and she had led a choral count with her first graders. And they were counting by [10s] but starting with 8. So, like, “8, 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,” Racheal 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 Rachael walked over to the girls, and I followed her, and they were using base ten 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, “8, 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 ten blocks, to make sense of the conjecture that came from the choral count. And I thought it was beautiful. And so, I did choral 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 lesson—she used that conjecture in the next lesson to bring it together. It fit 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 [have] 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 too 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 [words] “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 choral 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... /episode/index/show/213226b3-06ca-4dc4-ae8f-d21fa1edb913/id/33811367

Rounding Up

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