Rounding Up

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 Integrow Numeracy Solutions [https://www.integrowmath.org/] Developing Fractions Knowledge [https://www.integrowmath.org/store/67] 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 problem, is super important. And interestingly, she could do 900 plus 300 and figure out that that was 900 and 100 to get 1,000 and then 200 more. So that's additive reasoning, but it's the breaking apart of units and reconstituting them. That's what's really important in the process of solving multiplication and division problems. So that's my big thought about [laughs] that. And the other thing is to not go to patterns too soon. I mean, this is related to what I just said about not thinking that I can just do 3 times 2 and then add 0s and count the 0s because that really doesn't develop. It misses so much in what you can do with units. And so even if some kids do remember that and get the answer right, they're really robbed of the experience that we're trying to give to my daughter of really thinking about, "Well, how can I figure out 40 thirties or 30 forties or 30 twenties?" [laughs] Right now I'm a big advocate of actually doing lots of counting by decade numbers because I feel like it's a way of really enhancing kids' work with larger multiplication. Mike: I've been sitting listening to you talk about this, Amy, and there are multiple things where I'm like, I need to ask her about this. I need to ask her about that. I need to ask about this other thing. So I'm going to ask you a couple of follow-ups. One of the things that is just an observation is the language you used when you were talking about your work with your daughter. When the original task was "30 times 20" and you shifted the language to say "30 twenties," and then you step back even a little bit from there and you said, "Well, what's 30 tens?" This language that you were using, I wonder if you could be explicit about what you think that shift in language accomplishes. Amy: Yeah, I've been also thinking a lot about this, so it's great. Yeah, one of the problems with multiplication notation is that it doesn't make clear anything about what the group is and what the number of groups you have are. And so just saying "30 times 20," I mean, you can think of that as "30 twenties" or I can think of that as "20 thirties," but the language doesn't contain it, so it doesn't refer to the action I might do in thinking about how to actually figure it out. And kids have to bring a lot to the table, then, to really read that into that multiplication notation. It's even more so with fractions. I can say more about that in a second. So I really am advocating with my preservice teachers is that we speak in iterative language with the multiplication. So we try to always say, "I'm talking about 5 sevens," or "I'm talking about 7 fives, 30 forties, 40 thirties." And then of course with the decade numbers, knowing that we can go down to 10 of something and that that's easier to figure out, and then we can build on that. So like 10 twenties and then, "Oh, I'm going to need 3 of those 10 twenties to get to 30 twenties." Mike: Which really to some degree is helping them make meaningful sense of the associative property as well. Amy: Right! Yeah, exactly. It's very mathematically rich. Unfortunately, it's not necessarily worked on [laughs] a lot, I am finding, and I think it's a real missed opportunity. Because I think there's a lot that kids could do with that that would really build strong meanings for multiplication and strong ideas of base ten as well. Mike: Yeah, absolutely. I think one of the things that I've been obsessed with lately is this notion of "nudge" or small-sized shifts in my practice that I can make. Part of what I'd like to mark for the audience is the shift in the language, as you described—30 twenties or 5 sevens—those are moves that a teacher could make to help clarify the fact that units are involved and help students visualize with a bit more clarity what's going on. That feels like something that a teacher could take up and really have an impact on students' understanding. Amy: Yeah, I think so. I think it is something that is reasonable, and what's nice is it also can flow right into fractions because then instead of saying just, "three-fifths," we say, "3 one-fifths, 4 one-fifths, 5 one-fifths, 6 one-fifths, 7 one-fifths." It allows for fractions larger than 1 to have maybe more of an iterative meaning. Not that that's a simple thing at all; that's a whole nother podcast we could do, but [laughs] I've done a lot of research on that. Mike: Well, I think you're hitting on something important, though, Amy, because this notion of, "What is a unit fraction?," it's really, "Four-fifths is a group of 4 one-fifths," right? And that's a critical understanding that I think often floats underneath students' understanding in ways that, if we could make that clearer or help build that understanding, that also has huge ramifications for what comes later in their mathematics learning experience. Amy: Yeah, so I'm a big proponent of iterative language there as well. Mike: You have me thinking about something else too, which is the importance of context and having students deal with measurement division problems specifically as a way to build their understanding. And I know I'm using language right now for the audience that might not be super clear, but I'm wondering if you could talk a little bit about what measurement division means in context and maybe why that would be valuable for students. Amy: Yeah. Right. So, in multiplication and division structures, if we're talking about equal groups, there's always some number of equal groups, some number in the equal group, so a size of the group, and then a total number of items. And so, with measurement division, we know the total number of items, and we know the number of items in a group, but we don't know the number of groups. So my example of, "You've got 36 flowers, and you want to put them in rows of 4" would be a measurement division problem because we know that there are 4 in each row, and we know we have 36, but we don't know how many rows we're going to make. And so those are really nice to pair with work on equal groups multiplication problems because they are very closely related. And for kids, they can become closely related as they solve them and realize, like, "Oh, I can use my multiplication strategies to build up my 4s and find out when I get to 36," and, "Oh, then I do, I know how many rows I've made." So it's highly linked to what we're talking about here. Mike: What I found myself thinking about is that in solving that problem, one of the ways that a kid could do that is they're iterating a set, right? So, potentially, they're iterating a set of 4s multiple times, and then they're finding out how many of those sets of 4 they have, right? So I think part of what you're helping me think about is the way that the structure of a measurement division problem maybe shines a flashlight on this notion of groups and the number in each group, and also some of the ideas you were talking about earlier with units coordination. Amy: Yeah, for sure. And in terms of continuing the theme of using iterative language, then when you get the result of that problem, 9 rows, "Oh, what does that 9 mean?" "Oh, it means 9 fours make 36." So that's a meaning both for 4 times 9 equals 36, as well as 36 divided by 4 equals 9. So it's nice to emphasize that. And yeah, as students build those meanings and have repeated work with that kind of thing, they usually, often—[laughs] we don't know all the mechanisms here—but they usually come to be able to at least make that coordination in their problem-solving activity, and ultimately make it so they can anticipate it, like we're talking about with stage 3. Mike: One of the things that is really helpful is, in the course of this interview, we've talked a lot about what might the behavior of a student at stage 1 or stage 2 or stage 3 not only look like, but what might it mean for how they're thinking. And I think what I'm really appreciating about this, Amy, is there are a few practical things that an educator could do to support students. One is iterative language as we've been talking about. And the other is measurement division, using a particular problem structure like measurement division to shine a light on these parts that we think are really important for kids to attend to if they're in fact going to make some of the shifts that we're hoping for. Amy: Yeah, for sure. And then also exploring the boundaries of what the kids' strategies are and asking for multiple solutions. Because you might see kids, even students at stage 3, that might be counting by 1s, and so you want to [prompt], "Oh, can you solve that another way? Is there another way you can do it?" And so seeing what they see as possible, what they're able to think about is also really important to support units coordination. Mike: Absolutely. Before we close, I typically ask a question about resources or training or learning experiences that would help someone who's listening continue learning or continue to think about how they could take up these ideas in their practice. You, particularly, I know have written some work around this and I also suspect that you might have some recommendations in terms of organizations that can help educators really dig into these ideas if they saw that as something that was important for their growth. Would you be willing to talk a little bit about resources, organizations, or even the types of experience you think support teachers as they're making sense of all of this? Amy: Yeah. Well, yes. I was planning to talk about Integrow at this point because Integrow Numeracy Solutions [https://www.integrowmath.org/] has a lot of great supportive materials for all this kind of work. And everything that I'm talking about is something that is sort of built into much of what they do. For people who are unfamiliar, it's a bit—council, used to be called a council, of people who got together and have really developed materials that are supportive of teachers working one-on-one to support students who might be struggling as well as whole-group instruction all around developing strong number sense. And it's a very well developed set of materials, both for classroom use as well as for teacher development. And we—meaning me and my two coauthors, Andy Norton and Bob Wright—wrote a book in the series for teachers on fractions called Developing Fractions Knowledge [https://www.corwin.com/books/developing-fractions-knowledge-245818]. And that was published—oh my gosh—nine years ago now. So Andy and I are working on a second edition right now, and in that book we address units coordination and talk about its usefulness for teachers. It's mostly, though, a book about fractions and about how units coordination is relevant in trying to support students' fractions knowledge and to help assess students' thinking and also promote their learning. So that is one resource I can recommend on units coordination with a revision coming in the next year [2026]. Mike: That's fantastic. So I'll say for listeners, we'll include a link to Integrow Numeracy Solutions if you want to check out the organization. And Amy will also add a link directly to the book so that if someone wanted to dig in and explore that way they had the option. I think that's probably a great place to stop, although I certainly would love to continue. I want to thank you so much for joining us. It's really been a pleasure talking with you. Amy: Yeah, likewise, Mike. I've really enjoyed it, and I look forward to further conversations. 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 | www.mathlearningcenter.org

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 Math On-A-Stick [https://www.mnstatefair.org/location/math-on-a-stick/], a large-scale family math playspace at the Minnesota State Fair. RESOURCES How Did You Count? A Picture Book [https://www.routledge.com/How-Did-You-Count-Picture-Book/Danielson/p/book/9781032898353] by Christopher Danielson How Many?: A Counting Book [https://www.routledge.com/How-Many-A-Counting-Book/Danielson/p/book/9781625311825?srsltid=AfmBOorIFjAgrjwlQe3nrOiyU5hFKbatwWeYQfXRXn6KBT3xWB1J6L-I] by Christopher Danielson Following Learning blog [https://followinglearning.blogspot.com/] by Simon Gregg Connecting Mathematical Ideas [https://www.heinemann.com/products/connecting-mathematical-ideas-e07818.aspx] 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] Which one doesn't belong? [https://www.mathlearningcenter.org/blog/christopher-danielson-which-one-doesnt-belong-fostering-flexible-reasoning] 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 How Did You Count?[: A Picture Book] [https://www.routledge.com/How-Did-You-Count-Teachers-Guide-with-Picture-Book/Danielson/p/book/9781625312938] 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 How Many?[: A Counting Book] [https://www.routledge.com/How-Many-A-Counting-Book-Teachers-Guide-with-Student-Book/Danielson/p/book/9781625312181], 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, like a thing that has a mathematical purpose. It's communicating part of an idea and that that connects back. Teachers are super skilled at using color to do that, right? So 4 plus 1 might be written in red to match the red circle that goes around here, using not green because of color blindness. They're using blue to do 3 plus 2 in parentheses over here. And teachers might make other choices, right? We might sometimes use color to annotate in the image, but then just black here so that we aren't doing all of that work of corresponding for kids and are asking kids to try to do some of that corresponding work. And we might do it the other way around as well. So annotation as a way of adding, I think, a couple of dimensions to the conversation. And I have to shout out a fabulous teacher who I know through math Twitter. Simon Gregg [https://followinglearning.blogspot.com/] is a teacher in an international school in Toulouse, France. And he has done amazing work with using and producing his own Which one doesn't belong?s, and annotating them and having kids do them; how many?; and then there are a few examples of his work with kids in the teacher guide for How Did You Count? Yeah, he's just a true master at annotation. So go find Simon Gregg on social media if you want to learn some beautiful things about representing kids' ideas in writing. Mike: Love it. So the question that I typically will ask any guest before the close of the interview is: What are some resources that educators might grab onto, be they yours or other work in the field that you think is really powerful that supports the kind of work that we've been talking about? What would you offer to someone who's interested in continuing to learn and maybe to try this out? Christopher: In the teacher guide of How Did You Count?, I make mention of which of the number talks books was most powerful for me. But if you want to take a look at that page in the teacher book and then throw a link in and a shout out to the folks who wrote it. Jo Boaler and Cathleen Humphreys wrote a book called Connecting Mathematical Ideas [https://www.heinemann.com/products/connecting-mathematical-ideas-e07818.aspx]. It's old enough that there are some CD-ROMs in it. I don't know if there's a new edition; I'm sure used ones are available on all the places you buy used books. But the expert work that the teacher Cathy Humphreys does, as described in the book—even if you can't use the CD-ROMS in your computer—expert work at drawing out students' ideas, and then the two collaborating to reflect on that lesson, the connections they were drawing. It's been a while since I read it, but I imagine the annotations have got to come up. Fabulous resources for thinking about how these ideas pertain to middle school classrooms, but absolutely stuff that we can learn as college teachers or as elementary teachers on either side of that bridge from arithmetic to algebra. Mike: So for listeners, just so you know, we're going to add links to the resources that Christopher referred to in all of our show notes for folks' convenience. Christopher, I think this is probably a good place to stop. Thank you so much for joining us. It's absolutely been a pleasure chatting with you. Christopher: Yeah. Thank you for the invitation, for your thoughtful prep work and support of both the small and the larger projects along the way. I appreciate that. I appreciate all of you at Bridges and The Math Learning Center. You do fabulous work. 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 | www.mathlearningcenter.org

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 Talk Moves with Intention for Math Learning Center [https://drive.google.com/file/d/1RAwGc2Wz7KqKbifpojJyXf8ywUhvxAiA/view] Standards for Mathematical Practice by William McCallum [https://static.pdesas.org/content/documents/M3-grouping_mathematical_practices.pdf] 5 Practices for Orchestrating Productive Mathematics Discussions by Margaret (Peg) Smith and Mary Kay Stein [https://www.corwin.com/books/5-practices-262956] 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 talk moves [https://drive.google.com/file/d/1RAwGc2Wz7KqKbifpojJyXf8ywUhvxAiA/view] 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 the work of Dr. [William] McCallum and his team [https://static.pdesas.org/content/documents/M3-grouping_mathematical_practices.pdf] 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 the floor to whoever wants to argue about that and try to convince the community that they agree or disagree and why. And then I think, even, (laughs) I guess to keep going, Mike, I think you could at that point use the "tell us more," when that student's offering the reasoning on why they agree or disagree, and you don't feel like it's enough or maybe there's other kids in the room not quite understanding where they're going. "OK. So tell us a little bit more. Keep going." And offer that space and time for them to do that. So yeah, there's several ways that you can sequence them, but I really think you have to figure out the learning goal, be intentional about the discourse and how you can get it headed in the right direction and also slow it down enough that there's some depth to it as well. Mike: We had a guest on [Rounding Up] earlier this season, and he was talking about the importance of "agree or disagree." He called it "pick a side," but I think the idea is the same. Todd: Oh yeah. Yeah, same concept. Mike: And I wonder if you could talk about, what is it about agree [or] disagree that you think is particularly powerful for kids? Todd: Sure, Mike. Do you agree [or] disagree? It does make you take a stand. Like you have to understand the situation well enough to be able to say, "Hey, I agree with this thinking because..." fill in the blank. I think it puts you in a position where you've got to weigh everything that's playing out in the discourse and then actually understand it well enough to be able to then communicate about it. Your approach may be different than the thinking that was shared, but if you can understand it well enough and then state whether you agree or disagree and why, that's some pretty deep understanding. I mean, there's some high value in that if you can get to that point. Mike: Absolutely. I get the sense that a fair number of these talk moves might start to feel pretty organic. They might happen almost like muscle memory when an educator starts to use them, but you really have me thinking about planning for talk moves. Do you have any guidance for an educator who might be trying to think about, "Hey, I want to purposefully integrate some of these moves into my practice." What would it look like to plan for that? Todd: Sure. I think first of all, when you talk about muscle memory, that's a great way to put it. Some of these moves may not be strong in a professional practice for a person right now, but the more you get to using them and trying them out and implementing them and seeing what they'll do for productive community agency math practice, you're going to start to develop a level of growth in your practice that I think is going to be tremendous. But as you think about being intentional with them as you plan a lesson and go after a particular learning goal for a lesson, the one thing that comes to mind for me is really Dr. Peg Smith's work around the five practices and orchestrating discussions [https://www.corwin.com/books/5-practices-262956], right? You think about anticipating and then selecting and sequencing and connecting for sure all come to mind in that. So I guess I would go as far as saying, as we prepare for what students may say or do, we can intentionally think about the moves that might be most impactful in different scenarios. For instance, let's just say [there's a] a third grade student; you're working on a model to represent multiplication. The student draws a model to represent a multiplication scenario. You can plan to project or show the model and then simply use the think, turn, and learn move. You can show the student thinking, ask students to not talk upfront. We need to give people individual time to think. And then I want you to think about what you see and then I want you to turn and learn with your partner about that. So I think at that point, with that move being used, you're going to get a lot of discourse around whether the model is an appropriate representation or not. I think there's going to be depth in what kids take away from the experience. And you can go back to that as like, OK, so if I know that a kid says this and just says, "Well, it looks like there's this many rows and this many arrays," [then] the tell us more move is a great one there. "Tell us more. What do you mean by that?" Then they have to extend to give more depth to their thinking and then refine it a little bit more. So I think as you think about the learning goal, there's certainly ways that you can think about any of these talk moves, and in a way where you want to make sure that the right move is being used to get you closer to the goal of sense being made and such. So yeah. Mike: I want to come back to something that you touched on in the beginning, but it feels like a through line, which is: These talk moves are about building engagement and math, but really they're about so much more. What do you see as the long-term payoff for kids who experience this type of a learning experience? Todd: Well, (laughs) it often feels counterintuitive when I'm in schools and talking about these things because I think we've shifted into a mode where professional learning communities are so honed in on that exact math content standard, what do we want kids to know, be able to do? How will we know? What are we going to do when they don't? And I really believe that the more I'm in these situations, Mike, I'm understanding that we can't shift that learning like we want to until we deal with some of these that—I call them more general pedagogy practices, like discourse and talk moves with intention. Those are more general practices, not a content-specific practice teaching kids how to find a common denominator so they can add fractions and such. But I really think if we can get at some of these general pedagogy things to build up community, agency, math practice—all those things that I think will transcend time—I often talk about it, we're going after something bigger than just the priority standards or the most important standards within our state. We're going after things that are deeper, bigger, pay off more later in life than we may even realize that we're experiencing in the moment. Mike: Yeah. I mean, things like flexibility, problem solving, citizenship are all pieces that really jump out when I listen to you talk about that, Todd. Todd: Sure. No, you think about—and I mean, as we were talking about productive community earlier, I always offer them: Is there anything different you would want in your classroom or your school? You think about the words "productive" and "community," can we all come together and think about things in a way where we're contributing, we're all valued, we're producing together. And that's not something that I think we spend a lot of time talking about in schools now that it's so specific to content and how kids do on state assessments and such, but these things transcend all that. Mike: Absolutely. We're at that point in time where I could probably keep chatting with you about this for hours, but you are a busy school educator and you gotta get out of here. I'm wondering though, if you could leave listeners with a thought or a question or maybe a nudge related to their practice, what would you share? Todd: I first would say I just think it's important to always be reflecting on whose talk is driving the experience. As you think about everything we've talked about today, Mike, is the student talk driving it? Is their reasoning driving it or is it ours? And I think understanding that these talk moves with intention and what they go after and using them consistently with intention, it just starts to shift the balance to favor more student-to-student discourse. And I think it presents as, in turn, more developed community, agency and math practice. And I just think that you get more out of that than [a] high quantity of teacher-to-student discourse or student-to-teacher discourse. So I always offer [to] just pick out a move, try it for a week, find a wing person, collaborate around that, share ideas. How'd it go? What were your barriers? What did you see happening? Just these small shifts I think can create some big opportunities for people down the road. Mike: I think that's a great place to stop. Thank you so much for joining us, Todd. It's really been a pleasure chatting. Todd: Mine as well. Thank you, Mike. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2026 The Math Learning Center | www.mathlearningcenter.org

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 "Seeds of Algebraic Thinking: a Knowledge in Pieces Perspective on the Development of Algebraic Thinking" [https://link.springer.com/article/10.1007/s11858-022-01374-2] "Seeds of Algebraic Thinking: Towards a Research Agenda" [https://flm-journal.org/Articles/2A926DEBF042F89C466253F106FD0C.pdf] NOTICE Lab [http://notice-lab.com] "Leveraging Early Algebraic Experiences" [https://pubs.nctm.org/view/journals/mtlt/118/5/article-p357.xml] 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 ["Seeds of Algebraic Thinking: Towards a Research Agenda" [https://flm-journal.org/Articles/2A926DEBF042F89C466253F106FD0C.pdf] 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 notice-lab.com [http://notice-lab.com], 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

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 National Council of Teachers of Mathematics [https://www.nctm.org/] 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 National Council of Teachers of Mathematics [https://www.nctm.org/] 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 it all to be about the students and their voice, and it's their strategy and they're discussing and coming up with everything. And we know of the standard names of strategies like standard algorithm or whatever, but I think it gave them an extra confidence when it was like, "Oh, we want to call it—" I forgot the different names that they would come up with for strategies. Jenna: I think they had said maybe "stacking numbers," something like that. They would put their own words. It wasn't standard algorithm, but like, "We're going to stack the numbers on top of each other," I think was maybe one they had said. Mhret: Mm-hmm. So I think it added to that collaboration within the group that they were in and also just them owning their strategy. And so, yeah. Mike: That leads really nicely into my next question. And Anna, this is one I was going to pose to you, but everyone else is certainly welcome to contribute. I'm wondering if you could talk a little bit about what happened when you all started to implement this strategy showcase in your classroom. So what impacts did you see on students' efficacy, their confidence, the ways that they collaborated? Could you talk a little bit about that? Anna: So I think one of the biggest things that I saw that I was very proud of was there was less of a need for me to become part of the conversation as the teacher because students were more confident to build off of each other's ideas instead of me having to jump in and be like, "Alright, what do we think about what this person did?" Students, because their work became more anonymous and because everyone was kind of working together and had different strategies, they were more open to discussing with each other or working off of each other's ideas because it wasn't just, "I don't know how to do this strategy." It was working together to really put the pieces together and come to a final agree or disagree. So it really helped me almost figure out where students are, and it brought the confidence into the students without me having to step in and really officiate the conversation. So that was the really big thing that I saw at least in some of my groups, was that huge confidence and more communication happening. Mhret: Yeah. This is Mhret. I think it was very exciting too, like Anna was saying, that—them getting excited about their work, and everything up on the board is their work. And so seeing them with a sticky note, trying to find the similarities and differences between strategies, and getting excited about what someone is doing, I think that was a very good experience and feeling for me because of the confidence that I saw grow through the process of the kids, but also the collaboration of, "It's OK to use what other people know to build upon the things that I need to build upon." And so I think it just increased collaboration, which I think is really important when we talk about reasoning and strategies. Mike: Which actually brings me to my next question, and Jenna, I was wondering if you could talk a little bit about: What did you see in the ways that students were reasoning around the mathematics or engaging in problem solving? Jenna: Yeah, I know one specific example that stood out was—again, that initial thing of when we gave a student a problem, they would look to the teacher and a little bit later on in the process when giving a problem, we had done putting the strategies up, we'd cocreated the names, and then they were trying a similar problem independently. And one of my students right off the bat had that initial reaction that we would've seen a few weeks ago of being like, "I don't know what to do." And she put a question mark on the paper. So I gave her a minute and then she looked at me and I said, "Look at this strategy. Look at what you and your classmates have done to come together." And then she got a little redirection, but it wasn't me telling her what to do. And from there I stepped away and let her just reference that tool that was being displayed. And from there, she was able to show her work, she was able to choose a strategy she wanted to do, and she was able to give her answer of whether she agreed or disagreed on what she had seen. So I think it was just again, that moment of realizing that what I needed to step in and do was a lot smaller than it had previously been, and she could use this tool that we had created together and that she had created with her peers to help her answer that question. Anna: I think to add onto that, it's Anna, there was a huge spike in efficiency as well because all these different strategies were being discovered and brought to light and put onto the strategy showcase. Maybe if we're talking about multiplication, if some student had repeated addition in the beginning and they're repeatedly adding numbers together to find a multiplication product, they're realizing, "Oh my goodness, I can do this so much more efficiently if I use this person's strategy or if I try this one instead." And it gives them the confidence to try different things. Instead of getting stuck in the rut of saying, "This is my strategy and this is the way that I'm going to do it," they became a little more explorative, and they wanted to try different things out or maybe draw a picture and use that resource to differentiate their math experience. Mike: I want to mark something here that seems meaningful, which is this whole notion that you saw this spike. But the part that I'm really contemplating is when you said kids were less attached to, "This is my strategy" and more willing to adopt some of the ideas that they saw coming out of the group. That feels really, really significant, both in terms of how we want kids to engage in problem solving and also in terms of efficacy. That really I think is one to ponder for folks who are listening to the podcast, is the effect on students' ability to be more flexible in adopting ideas that may not have been theirs to begin with. Thank you for sharing that. Anna. I wonder if you could also spend a bit of time talking about some of the ways that you held onto or preserve the insights and the strategies that emerged during a showcase. Are there artifacts or ways that a teacher might save what came from a strategy showcase for future reference? Anna: So, I think the biggest thing as a takeaway and something to hold onto as a teacher who uses the strategy showcase is the ability to take a step back and allow students to utilize the resources that they created. And I think something that I used is I had a lot of intervention time and time where students were able to work in small groups and work together in teams and that sort of thing, keeping their strategies and utilizing them in groups. Remember when this person brought up this strategy, maybe we can build off of that and really utilizing their work and carrying it through instead of just putting it up and taking it down and putting up another one. Really bringing it through. And any student work is valuable. Anything that a student can bring to the table that can be used in the future, like holding onto that and re-giving them that confidence. "Remember when this person brought up that we can use a picture to help solve this problem?" Bringing that back in and recycling those ideas and bringing back in not just something that the teacher came up with, but what another student came up with, really helps any student's confidence in the classroom. Mike: So I want to ask a question, and Tutita and Mhret, I'm hoping you all can weigh in on this. If an educator wanted to implement the strategy showcase in their classroom, I want to explore a bit about how we could help them get started. And Tutita, I think I want to start with you and just say from a foundational perspective of building the understanding that helps support something like a strategy showcase, what do you think is important? Tutita: I actually think there are two critical things. The first is considering the social aspect and just building off of what Anna was saying is, if you've listened carefully, she's really honoring the individual. So instead of saying, "Look," that there was this paper up there—as teachers, we have a lot on our walls—it's actually naming the student and honoring that student, even though it's something that as a teacher, you're like, "Yes, someone said it! I want them to actually think more about that." But it's so much more powerful by giving students the credit for the thinking that they're doing to continue to advance that. And all that starts with assuming that students can. And oftentimes at the elementary level, we tend to overlook that. They're so cute—especially those kindergartens, pre-K, kindergarten—but it's amazing what they can do. So if you start with assuming that they can and waiting for their response, then following up and nurturing that, I think you as teachers will get so much more from our students and starting with that confidence. And that brings me to the next point that I think listeners who teach in the upper elementary grades or maybe middle school or high school might be like, "Oh, this sounds great. I'll start with them." But I want to caution that those students might be even more reticent because they might think that to be a good math student, you're supposed to know the answer, you're supposed to know it quickly, and there's one strategy you're supposed to use. And so, in fact, I would argue that probably those really cute pre-K and kindergartners will probably be more open because if anyone has asked a primary student to explain what they have down on paper, 83 minutes later, the story will be done. And so it might take time. You have to start with that belief and just really going with where your class and individuals are socially. Some of them might not care that you use their name. Others might, and that might take time. So taking the time and finding different ways to stay with that belief and make sure that you're transferring it to students once they have it. As you can hear, a lot of what my coauthors mentioned, then they take it from there. But you have to start with that belief at the beginning that elementary students can. Mike: Mhret, I wonder if you'd be willing to pick up on that, because I find myself thinking that the belief aspect of this is absolutely critical, and then there's the work that a teacher does to build a set of norms or routines that actually bring that belief to life, not only for yourself but for students. I wonder if you could talk about some of the ways that a teacher might set up norms, set up routines, maybe even just set up their classroom in ways that support the showcase. Mhret: Yeah. So practically, I think for the strategy showcase, an important aspect is finding a space that's accessible to students because we wanted them to be going back to it to use it as a resource. So some of us used a poster board, a whiteboard, but a vertical space in the room where students can go and see their work up I think is really important so that the classroom can feel like theirs. And then we also did a group norm during our first meeting with the kids where we co-constructed group norms with the kids of like, "What does it look like to disagree with one another?" "If you see a strategy that you haven't used, how can you be kind with our words and how we talk about different strategies that we see up there?" I think that's really important for all grades in elementary because some kids can be quick to their opinions or comments, and then providing resources that students can use to share their idea or have their idea on paper I think is important. If that's sticky notes, a blank piece of paper, pencils, just practical things like that where students have access to resources where they can be thinking through their ideas. And then, yeah, I think just constantly affirming their ideas that, as a teacher, I think—I teach second grade this year and [they are] very different from the fourth graders that I student taught—but I think just knowing that every kid can do it. They are able, they have a lot in their mind. And I think affirming what you see and building their confidence does a lot for them. And so I think always being positive in what you see and starting with what you see them doing and not the mistakes or problems that are not important. Mike: Jenna, before we go, I wanted to ask you one final question. I wonder if you could talk about the resources that you drew on when you were developing the strategy showcase. Are there any particular recommendations you would have for someone who's listening to the podcast and wants to learn a little bit more about the practices or the foundations that would be important? Or anything else that you think it would be worth someone reading if they wanted to try to take up your ideas? Jenna: I know, in general, when we were developing this project—a lot of it again came from our seminar class that we did at UConn with Tutita—and we had a lot of great resources that she provided us. But I know one thing that we would see a lot that we referenced throughout our article is the National Council of Teachers of Mathematics. I think it's just really important that when you're building ideas to, one, look at research and projects that other people are doing to see connections that you can build on from your own classroom, and then also talking with your colleagues. A lot of this came from us talking and seeing what we saw in our classrooms and commonalities that we realized that we're in very different districts, we're in very different grades and what classrooms look like. Some of us were helping, pushing into a general ed classroom. Some of us were taking kids for small groups. But even across all those differences, there were so many similarities that we saw rooted in how kids approach problems or how kids thought about math. So I think also it's just really important to talk with the people that you work with and see how can you best support the students. And I think that was one really important thing for us, that collaboration along with the research that's already out there that people have done. Mike: Well, I think this is a good place to stop, but I just want to say thank you again. I really appreciate the way that you unpack the features of the strategy showcase, the way that you brought it to life in this interview. And I'm really hopeful that for folks who are listening, we've offered a spark and other people will start to take up some of the ideas and the features that you described. Thanks so much to all of you for joining us. It really has been a pleasure talking with all of you. Jenna: Thank you. Anna: Thank you Mhret: Thank you. Tutita: Thank you so much. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center | www.mathlearningcenter.org