Mike Steele, Pacing Discourse-Rich Lessons ROUNDING UP: SEASON 4 | EPISODE 13

As a classroom teacher, pacing lessons was often my Achilles' heel. If my students were sharing their thinking or working on a task, I sometimes struggled to decide when to move on to the next phase of a lesson.

Today we're talking with Mike Steele from Ball State University about several high-leverage practices that educators can use to plan and pace their lessons.

BIOGRAPHY

Mike Steele is a math education researcher focused on teacher knowledge and teacher learning. He is the past president of the Association of Mathematics Teacher Educators, editor in chief of the Mathematics Teacher Educator journal, and member of the NCTM board of directors.

RESOURCES

Journal Article

• "Pacing a Discourse-Rich Lesson: When to Move On"

Books

• 5 Practices for Orchestrating Productive Mathematics Discussions

• The 5 Practices in Practice [Elementary]

• The 5 Practices in Practice [Middle School]

• The 5 Practices in Practice [High School]

• Coaching the 5 Practices

TRANSCRIPT

Mike Wallus: Well, hi, Mike. Welcome to the podcast. I'm excited to talk with you about discourse-rich lessons and what it looks like to pace them.

Mike Steele: Well, I'm excited to talk with you too about this, Mike. This has been a real focus and interest, and I'm so excited that this article grabbed your attention.

Mike Wallus: I suppose the first question I should ask for the audience is: What do you mean when you're talking about a discourse-rich lesson? What does that term mean about the lesson and perhaps also about the role of the teacher?

Mike Steele: Yeah, I think that's a great question to start with. So when we're talking about a discourse-rich lesson, we're talking about one that has some mathematics that's worth talking about in it. So opportunities for thinking, reasoning, problem solving, in-progress thinking that leads to new mathematical understandings. And that kind of implicit in that discourse-rich lesson is student discourse-rich lesson. That we want not just teachers talking about sharing their own thinking about the mathematics, but opportunities for students to share their own thinking, to shape that thinking, to talk with each other, to see each other as intellectual resources in mathematics.

And so to have a lesson like that, you've got to have a number of things in place. You've got to have a mathematical task that's worth talking about. So something that's not just a calculation and we end up at an answer and that the discourse isn't just, "Let me relay to you as a student the steps I took to do this." Because a lot of times when students are just starting to experience discourse-rich lessons, that's kind of mode one that they engage in is, "Let me recite for you the things that I did." But really opportunities to go beyond that and get into the reasoning and the why of the mathematics. And hopefully to explore some approaches or perspectives or representations that they may not have defaulted to in their first run-through or their first experience digging into a mathematical task.

So the task has to have those opportunities and then we have to create learning environments that really foster those opportunities and students as the creators of mathematics and the teacher as the person who's shaping and guiding that discussion in a mathematically productive way.

Mike Wallus: One of the things that struck me is there is likely a problem of practice that you're trying to solve in publishing this article, and I wonder if we could pull the curtain back and have you talk a bit about what was the genesis of this article for you?

Mike Steele: Absolutely. So let me take us back about 20 or 25 years, and I'll take you back to some early work that went on around these sorts of rich tasks and discourse-rich lessons. So a lot of this legacy comes out of research or a project in the late nineties called the Quasar Project that helped identify: What is a rich task? What is a task, as the researchers described it, of high cognitive demand that has those opportunities for thinking and reasoning?

The next question that that line of research brought forward is, "OK, so we know what a task looks like that gives these opportunities. How does this change what teachers do in the classroom? How they plan for lessons, how they make those moment-to-moment decisions as they're engaged in the teaching of that lesson?" Because it's very different than actually when I started teaching middle school in the nineties, where my preparation was: I looked at the content I had for that day, I wrote three example problems I wanted to write on the board that I very carefully got all the steps right and put those up and explained them and answered some questions. "Alright, everybody understand that? OK, great, moving on." And then the students went and reproduced that. That's fine for some procedural things, but if I really wanted them to engage in thinking and reasoning, I had to start changing my whole practice.

So this bubbles up out of the original work of the 5 Practices for Orchestrating Productive Discussions [book] from Peg Smith and Mary Kay Stein. I had the opportunity actually to work with them both in the early two thousands at the University of Pittsburgh. And so as we were working on this five-practices framework that was supposed to help teachers think about, "What does a different conceptualization of planning and teaching look like that really gets us to this discourse-rich classroom environment where students are making sense of and grappling with mathematics and talking to each other in a meaningful way about it?" We worked with teachers around that and the five-practices [framework] is certainly helpful, but then as teachers were working with the five practices and they were anticipating student thinking, they were writing questions that assess and advance student thinking, some of the things that came up were, "OK, what are the moment-to-moment decisions and challenges related to that as we start planning and teaching in this way?"

And a number of common challenges came up. A lot of times when we were using a five-practice lesson, we were doing kind of a launch, explore, share, and discuss sort of format where we've got the teacher who's getting us started on a task, but we're not giving the farm away on that task. We're not saying too much and guiding their thinking. And then we let students have some time individually and in small groups to start messing around with the mathematics, working, talking. And then at some point we're going to call everybody together and we're going to share what the different ways of thinking were. We're going to try to draw that together. Peg Smith likes to talk about this as being more than a show-and-tell. So it's not just, "We stand up, we give our answer, we do that. Great." Next group, doing the same thing, and oftentimes they start to look alike. But there's some really meaningful thinking that goes on in that whole-class discussion.

So one of the really pragmatic concerns here is, "How do I know when to move?" So I've got students working individually, and maybe I gave them 3 minutes to get started. Was that enough? What can I see in the work they're doing? What questions am I going to hear to tell me, "OK, now it's a good moment to move to small groups." And then similarly, when you've got small groups working, they're cranking away on a task. There might be multiple subquestions in that task. What's my cue that we're ready to go on to that whole-class discussion?

We were in so many classrooms where teachers were really working hard to do this work, and this happens to me all the time. I have somehow miscalculated what students are going to be able to do—either how quickly they're going to be able to do it, or I expected them to draw on this piece of prior knowledge and it took us a while to get there, or they've flown through something that I didn't expect them to fly through. So I'm having to make some choice in a moment, saying, "This isn't exactly how I imagined it, so what do I do here?" And frequently with teachers that get caught in that dilemma, the first response is to take control back, [to] say, "OK, you're all struggling with this. Let's come back together and let me show you what you should have figured out here." And it's done with the best of intentions. We need to get some closure on the mathematical ideas. But then it takes us right away from what we were trying to do, which was have our students grapple with the mathematics.

And so we do this lovely polished job of putting that together and maybe students took the important things away from that, that they wanted to, maybe they didn't, but they didn't get all the way they were on their own. So that's really the problem of practice that this helps us to solve is, when we get in those positions of, "OK, I've got to make a call. I've got this much time left. I've got this sort of work that I see going on in the classroom. Am I ready? What can I do next?" That really keeps that ownership of the mathematics with our students but still gives me some ability to orchestrate, to shape that discussion in a way that's mathematically meaningful and that gets at the goals I had for the lesson.

Mike Wallus: Yeah, I appreciated that part of the article and even just hearing you describe that so much, Mike, because you gave words to I think what sat behind the dilemma that I found myself in so often, which was: I was either trying to gauge whether there was enough—and I think the challenge is we're going to get into, what "enough" actually might mean—but given enough time, whether I was confident that there was understanding, how much understanding was necessary. And what that translates into is a lack of clarity around "How do I use my time? How do I gauge when it's worth expending some of the time that I maybe hadn't thought about and when it's worth recognizing that perhaps I didn't need all of that and I'm ready to do something?"

So I think the next question probably should be: Let's talk about "enough." When you talk about knowing if you have enough, say a little bit more about what you mean and perhaps what a teacher might be looking and listening for.

Mike Steele: Absolutely. And I think this is a hidden thread in that five-practices model because we say: "OK, we want that whole-class discussion to still be a site for learning where there are some new ideas that are coming together." So that then backs me up to thinking about the small-group work. I'm putting myself in that mode where I've got six groups working around the classroom. I'm circulating around; I'm asking questions. I of course don't see every single thing at any given moment that the small groups are doing. So I'm getting these little excerpts, these little 2- to 3-minute excerpts, when you stop into a group. So I think when we think about "enough," I want to think about, with that task that I'm doing, with what my mathematical goals are and knowing that we're going to have time on the backend of this whole-class discussion to pull some ideas together, to sharpen some things to clarify some of the mathematics.

Do I have enough mathematical grist for the mill here in what the small groups are doing to be able to then take that and make progress with students' thinking at the center—again, not taking over the thinking myself—to be able to do that work. So, for any given mathematical idea, as I've started thinking about this when I plan lessons using the five-practices model, I am really taking that apart. What's the mathematical nugget that I'm listening for here, that I'm looking for in students' work that tells me: "OK, we've gotten to a point where, if I were to call people together right now and get them thinking about it, that there's more to think about, but we're well on our way."

And also when I'm looking for that, knowing that I'm also not looking at those six groups all at exactly the same time. So, I want to look for those mile markers along the way that tell me we're getting close, but we're not all the way there. Because if I pick one that's, we're pretty much all the way there, that's the first group I come to and I'm going to circulate around to five more. They're going to have run out of interesting things to do, and they're off talking about, thinking about something else.

So, that really becomes the fine line: "What are those little mathematical ideas along the way that are far enough that get us towards our goals, but still we've got a little bit of the journey to go that we're going to go on together?"

Mike Wallus: This is so fascinating. The analogy that's coming together in my mind is almost like you're listening for the ingredients for a conversation that you want to have as a group. So it's not necessarily "Has everyone finished?" And that's your threshold. It's actually "Did I hear this idea starting to bubble up? Did I hear elements of this idea or this strategy start to bubble up? Is there an insight that's percolating in different groups?" And it's the combination of those things that the teacher is listening for, and that's kind of the gauge of enoughness. Is that an accurate analogy?

Mike Steele: It is, and I love that analogy because it reminds me of a favorite in our household as we're relaxing. We love to watch The Great British Baking Show. So, you're watching people take something from ingredients to a finished product. Now as you're watching that 20-minute segment, they're working on their technical challenge and they're all baking the same thing. I don't have to wait until the end of that, where they've presented their finished product, to have a good idea of what's going to happen. As I'm going through, as I'm watching 'em through that baking process, we're at the middle, my wife and I are talking, like, "Ooh, I've got concerns about that one. That one's looking good though." We get an idea of where it's going. So I think the ingredient analogy really lands with me. We don't have to wait. We're looking for those pieces to be able to pull that together, those ingredients. We're not waiting until there's a final product and saying—because then, what is there to say about it? "Oh, look, that looks great. Oh, that one, maybe not exactly what we'd intended." So, it's giving us those ingredients for that whole-class discussion.

Mike Wallus: The other thing that struck me as I was listening to you is: We're not teaching a task; we're teaching a set of ideas or relationships. The task is the vehicle. So, it's perfectly reasonable, it seems, to say, "We're going to pause at this point in the task, or at a place where students might not be entirely finished with the task. And we might have a conversation at that point because we have enough that we can have part of the conversation." And that doesn't mean that they don't go back to the task. But you're really helping me recognize that one of the places where I sometimes get stuck, or got stuck, when I was teaching, is task completion was part of my time marking. And I think really what you're challenging me and other educators to do is to say, "The task is just the vehicle. What's going on? What's percolating around that task as it's happening?" How does that strike you?

Mike Steele: Yeah, absolutely. And it was the same challenge with me and sometimes still is the same challenge with me. (laughs) Yeah, you give this task, and we think about that task as our unit of analysis as a teacher when we're planning. And so we want our students as we're using it to get to the end of it. It's a very natural thing to do.

And let me make this really concrete. If I'm doing a visual pattern task with third graders, we have, I think there's one of the elementary [5 Practices in Practice] book called "Tables & Chairs." So you've got these square tables that have four seats around them, and you're putting a string of tables together and asking kids to get at the generalization. "If you have any number of tables, how many people can you seat?"

And so I think early when I started giving those tasks, I was looking for, "OK, has everybody gotten to the rule? Have they gotten to that generalization? OK, now we can talk about it." And we can talk about the different ways people made sense of that geometrically and those connections, and that's what I want to get out of the whole-class discussion. But we don't even have to get there if groups have a sense of how that pattern is growing, even if they haven't gotten to the formal description of the rule. Because if they've gotten to that point, they've made some sense of the visual. They've made some of those connections. They've parsed that in different ways. That's plenty for me to have a good conversation, that we can come to that rule as a group and we can even come to it in different ways as a group. But it frees me up from being like, "OK, everybody got the rule? Everybody got the rule? Everybody got the rule?" Because that often resulted in, I'd have a couple of groups that maybe had been a little slower getting started and they're still getting there. And then I'm sitting there and I'm talking to them, I'm giving them these terribly leading questions. "Can we just get to the rule? Come on, let's go. You're almost there. We got it. We got it." And that then is, again, me taking over that thinking and not giving them the space for those ideas to breathe.

Mike Wallus: What else is jumping out for me is the ramifications for how thinking this way actually might shift the way that I would plan for teaching, but also how it might shift the way that I'm looking for evidence to assess students' progress during the task. So I wonder if you have situations or maybe some recommendations for: How might a person plan in ways that help them recognize the ways that the task can be a vehicle but also plan for the kind of evidence that they might be looking for along the way? Could you talk a little bit about that?

Mike Steele: Absolutely. So I'll give kind of a multi-layered description of this. When we're using a task that's got multiple solution paths that has these opportunities for diverse thinking, the five-practices framework tells us anticipating student thinking is a critical part of it. So, what are the different solution paths that students can take through it? So, if it's a visual pattern task, they may look at it this way with a visual. They may think about those tables like the tops and the bottoms and then the sides. They may think about the two ends of the tables having different numbers of chairs and the ones in between having a different number of chairs and parsing it that way. And we can develop those. It's actually, for me, quite a lot of fun to develop those fully formed solutions that students can do. And early on when I was enacting lessons like this, I would do that. I'd have those that I was looking for. I'd also think about questions I'd want to ask students who are struggling to get started or maybe were going down a path that may not be mathematically productive and the questions I might ask them to get them on a more mathematically productive path. And I'd go around and I'd look for those solutions, and I'd use that to think about my selecting, my sequencing, my connecting my whole-class discussion. So, great, check. That's layer one.

I think responding to the challenge of what's enough requires us to then take those solution paths apart—both the fully formed ones, maybe the incomplete thinking—and say, "OK, within that solution, what are the things that I want to see and hear that gives me some confidence that we're on this path, even if we're not at the end of this path, and that give me enough to think about?" So, if I think about, I'll go back again to this visual pattern task analogy. If I see groups that are talking about increases, so when we add a table, we're adding two chairs or they're making that distinction between those end tables and the center tables. And I've asked them a couple of questions like: OK, they've done that for 4, they've done that for 5. We may not have done that for 10 or 100 or a generalization, but that might be enough. So, I'm trying to take apart the mathematics and look for those little ideas within it. We've got this idea of a constant rate of change. We've got an idea that the number of tables and the number of chairs have a direct relationship here. So we're setting the stage for that functional thinking, even if, at a third grade level, we're not going to talk about that word. And those might be the important goals that I have for the lesson.

So that's the next phase of what I'm doing. In addition to those fully formed solutions, I'm figuring out: What are the little mathematical ideas in each that I would want to see or hear in my classroom that tell me, "OK, I have a good sense of where they are. I know where this bake's going to turn out 5 minutes from now on the show when they've taken it out of the oven." So, that's I think the next layer of that planning, of trying to figure out how to plan.

And then as we're in the moment in the classroom, being able to know what we're looking for and listening for. And the listening for me is really, really important. I think when I started doing this and I had a sense of, "What are the mathematical ideas I need to draw on?" I made the mistake of overly looking for those on paper. And if we think about how students make sense of writing things down, and sometimes despite our best efforts, the finality that comes with it: "If I've written it down, I have made it real." And if our thinking is still kind of this in-progress thinking, we may not be ready to write it down. So if I wait for it to be written on the page, I may have waited too long, or longer than I needed to, for everybody to get that idea. So again I want to make sure I listen for key words and phrases. And I might have a couple of questions teed up to help me hear those. And once I've heard those, I'm like, "OK, I am ready to go." And then for me—at least in my early fifties and not having the memory that I did when I was a 22-year-old, fresh-out-of-the-box classroom teacher—I need to have a way of keeping track of that and writing that down. So be it physical, be it digital, I want to say, "OK, I know what I'm listening for, what I'm looking for." And sometimes those may be interchangeable. If it's written on the page, great. If not, if I hear it, that's great too. And then if I've got a pretty good roster of that as I've moved through and say, "OK, I feel like all of my groups or most of my groups are at this point, there we go." I feel confident that when I pull us back together, it's not going to be me asking a question and then that terribly awkward sea of crickets out there. I'm like, "I know you were thinking about stuff; just give it to me. I know you've got this." But it gives me much more confidence that we're going to have that nice transition into a good whole-class discussion.

Mike Wallus: OK. There's a ton of powerful stuff that you just said. So I want to try to mark two things that really jump out for me. One is an observation that I think is important, and then one is a thought that I want to pick your brain around a little bit further.

I think the biggest piece that I heard you say, which as you were talking about, is this notion that I'm waiting for something to appear in written form. And it feels really freeing and it gives me a lot more space to say, "This is something I could hear or I could even see in the way that kids were manipulating materials. That that counts as evidence, and I don't have to literally see it written on a paper in order for me to count that that idea is in the room." I just want to name that for the audience because that feels tremendously important. Because from a practical standpoint, if we're waiting for it to be written, that takes more time. And it doesn't necessarily mean that suddenly it appeared and before when it was just in a child's mind or in the way that they were manipulating something, that it wasn't there. It was there. So I just want to mark that.

The other thing that you had me thinking about is, I know for myself, I've gone through and done some of the anticipation work in the five practices, but what struck me is when my colleagues and I would do that, we often would generate quite a few alternative strategies or ideas. But I feel like what we were looking at is the final outcome, like, "This counting by 1 strategy is what we might see. This decomposing numbers more flexibly is something we might see. This counting on strategy is something we might see." But what we didn't talk about that I think you're advocating for is: What are the moments within that that matter? It's almost like: What in the process of getting to this anticipated strategy is something that is useful or important that counts as one of those ingredients? So I want to run that past you and say, does that follow or am I missing something?

Mike Steele: It does. And I think those two things go together in a really important way because as you're talking about that pivotal moment in student thinking, as they're coming to this new understanding, as they're grappling with that mathematical idea, and thinking about, "What are the implications if we leverage that moment right there to then ask more questions to connect different ways of student thinking as compared to waiting till it's written down?" Because when it's written down, that exciting moment of the new discovery has passed. And so then when we want them to come revisit—"Tell us what you were thinking when you did that."—they're having to rewind and go back and reenact that.

If we have the ability to capture those neurons firing at full throttle in that moment of a new mathematical insight and then use that to build on as a teacher and to really get where we want to go with the lesson, I feel like we're doing the right thing by kids by trying to seize that moment, to leverage it. We always have time to write down what we think we learned later on at the end of the lesson. It's a great task for homework. And that's another thing I love about leaving some things unfinished with a task is, that's just a delightful homework assignment. And the kids love it because they don't feel like I've asked them to do anything new. (laughs) Just write down what you understood about this, and now we're codifying it kind of at a different place in the process.

Mike Wallus: Well, OK, and that makes me think about something else. Because you've helped me recognize that I don't have to wait for a final solution in writing that's fleshed out in order to start a whole-group conversation. But I think what you're saying is, it changes the tone and maybe also the purpose and the impact of that conversation on students. Because if I have a task that I'm midway through and suddenly there's a conversation that helps create some understanding, some aha moments, if my task is unfinished and I had an aha, I probably really want to go back to that and see if I can apply that aha. And that's kind of cool to imagine like a classroom where you have a bunch of kids dying to go back and see if they can figure out how they can put that to use. Now you wouldn't always have to do that, but that strikes me as different than a consolidation conversation where it's kind of like, "Well, everything's finished. What have we learned?" Those are valuable. But I'm just really, I think in love with the possibility that a conversation that doesn't always wait until final solutions creates for learning.

Mike Steele: And when I've seen this done effectively, there are these moments that happen. Mike, they're exactly what you're describing, is that there's an insight that comes up in the whole-class conversation, and you will see people going back to their paper or their tablet that they were doing their original work on and start writing. And we know oftentimes with kids, I remember so many times in my classroom where we're having this discussion, this important point comes up, and everybody's kind of frozen. And I'm like, "No, you should write that down. That's the important thing. Write that down." And when you see it happen organically, it's because something really catalyzed in insight that was important enough that they went back to that work and said, "Oh, I want to capture this."

Mike Wallus: So, I'm wondering if there are habits of mind, habits in planning, or habits in practice that we could distill down. So, how would you unpack the things that a person might do if they're listening and they're like, "I want to do this today," or "I want to do this at my next planning."? Could you talk a little bit about what are the baby steps, so to speak, for a person?

Mike Steele: Yeah, and I think the first one is really about getting into the mathematics and going deep with the mathematics in the task that you're hoping to teach. As somebody who is trained as a secondary math teacher, and early in my career, I was like, "Oh, I know what the math is. I don't need to spend the time on the math." I can't tell you how wrong I was about that. So anticipating those ways of thinking, thinking about where those challenges are, that sort of thing, is absolutely critically important to doing that work. And giving the time and space for that to happen. I mean, it was almost without fail. Every time I shorted myself on the time to think about the mathematics and just popped open my instructional resource and said, "Here we go. Class starts in 5 minutes. Let's get going on this," I'd bump into things that I was like, "Oh, I wish I had thought about that mathematical idea first." Or there'd be a question that would come up that I'd be totally unprepared to answer and I could have been prepared to answer. Now, we're not going to anticipate every way of thinking that students have or every question that they'll have, but I always find that if I've thought through it, I'm probably in a better position to give a meaningful answer to it or ask a good question back in response. And it also frees up my cognitive load to actually spend some time on those questions that I didn't expect rather than trying to make sense of everything as if it's the first time I'm seeing it.

And then along with that, doing this as a group, we used to sit in our PLC sessions and start to solve tasks together and share our thinking about, "OK, what are the mathematical ideas that we're really trying to take apart here?" And there were always insights that didn't occur to me that would occur to somebody else that added to my own thinking. And now in an increasingly digitally connected age, we don't necessarily have to be in the same room with people to do that. We can do that at a distance and still be very effective.

And then the last thing I'll talk about here in terms of getting started is: We are so good as teachers at sharing an interesting task that we found or that we used with our students with our colleagues. "Here's this thing I use in my class. It was great. You're a couple days behind me in the pacing. Maybe you can use this next Tuesday." What we I think are less good at is bringing back the outcomes of that and talking about that. "Here's what students did." I loved it when we had opportunities to gather a group of teachers in the PLC with student work from a task they did and talk about it and see: What did students make sense of? What were the questions that I asked that were helpful, or that maybe weren't helpful, in teaching that lesson. Because we'll share the task, but my goodness, the questions that we came up with to ask students in the moment, those are just as portable from one classroom to another. And we should be thinking about, just like we think about digital archives to share those tasks and those lesson plans—like sharing those questions, sharing that student work—those are the other legs of that stool that are important for really helping us do this work in a meaningful and collaborative way. Because if we don't talk about the outcomes of what students learned, the task could be great, it could be interesting, but so what? What's the important mathematical insights that kids took away from it?

Mike Wallus: Yeah, I'm kind of in love with this notion that in addition to sharing tasks, sharing questions that really generated an impact in the classroom space or sharing moments of insight that led to something that jumped out. It's fascinating to think about taking those ideas and building them into a regular PLC process. It just has so much potential.

Before we close the conversation, I wanted to ask you a question that I ask almost every guest: If someone wanted to learn more about the ideas that you've shared today, what are some of the resources you'd recommend?

Mike Steele: Well, I've talked quite a bit about the work of the 5 Practices for Orchestrating Productive Discussions and that series of books that have been written over the past 15 years on that—the resources that are available online for that, I think, would be a great place to start. I've only scratched the surface at taking you through those five practices—which are actually six practices, because early on we realized that attention to the task we select and the goals for that task is the important "practice zero." In fact, it was a teacher that pointed that out to Peg Smith. And that's the lovely thing. So the reason I've stayed in touch with and helped to develop this work over the years is because when we see teachers taking it up, not only is it meaningful, but the feedback we get from teachers then shapes the next things that we do with it. So there's the original 5 practices book that kind of presents the model, shows some examples of tasks and how you go through the model.

But then in 2019 and 2020, we published a series called The 5 Practices in Practice that, there's a book for each grade band—elementary, middle, and high school. But those were the ones that really aggregated the challenges that we heard from teachers over 10 years of doing this work and started to address those challenges. How do you overcome those things? We also, for each of those books, there's brand-new original video that we took in urban classrooms that illustrated teachers working really effectively with the five practices. I was able to be in the room when we filmed all of the high school classrooms in Milwaukee, Wisconsin, and it was just amazing to see that work.

And then the last piece that I'll suggest to that, which is a book that came out relatively recently in that series. There is a Coaching the 5 Practices book. So if you are a coach, instructional leader who's looking to support a team and a PLC in doing exactly this sort of work that we've been talking about, the Coaching the 5 Practices book is an incredible resource for thinking about how you can structure that work.

Mike Wallus: OK. I have to also ask you, can you give a shout out to the article that you recently wrote and published as well, the title and where people could find it?

Mike Steele: Absolutely. Yes. The article is called "Pacing a Discourse-Rich Lesson: When to Move On," and I authored it alongside an elementary and middle school teacher who provided a reflection on it. It comes from the classroom of a high school teacher, Michael Moore, in Milwaukee, who we filmed for the [5 Practices in Practice] high school book. So I drew from his classroom. And then Kara Benson in Zionsville Community Schools right here in Zionsville, Indiana. And Kelly Agnew who teaches in Muncie Community Schools, which is where Ball State [University] is located. Each provided a reflection from an elementary and middle school standpoint about the ideas in the article. It was published in NCTM'S practitioner journal, Mathematics Teacher: Learning and Teaching PK-12, in the Volume 118, Issue 11, from November of 2025.

Mike Wallus: That's fantastic. And for listeners, just so you know, we're going to put a link to all of the resources that Mike shared.

I think this is probably a good place to stop, Mike. I suspect we could talk for much longer. I just want to thank you, though, for taking the time to join the podcast. It has been an absolute pleasure chatting with you.

Mike Steele: The pleasure has been all mine. As you can tell, I love talking about these ideas, and I was so glad to have the opportunity to share a little bit of this with the audience.

Mike Wallus: 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

Kyndall Thomas, Building a Meaningful Understanding of Properties Through Fact Fluency Tasks ROUNDING UP: SEASON 4 | EPISODE 12

Building fluency with multiplication and division is essential for students in the upper elementary grades. This work also presents opportunities to build students' understanding of the algebraic properties that become increasingly important in secondary mathematics.

In this episode, we're talking with Kyndall Thomas about practical ways educators can support fluency development and build students' understanding of algebraic properties.

BIOGRAPHY

Kyndall Thomas serves as a math interventionist and resource teacher with the Oregon Trail School District, focusing on data-driven support and empowering teachers to spark a love of numbers in their students.

TRANSCRIPT

Mike Wallus: Hi, Kyndall. Welcome to the podcast. I'm really excited to be talking with you today.

Kyndall Thomas: Hi, Mike. Thanks for having me. I'm excited to dive into some math talk with you also.

Mike: Kyndall, tell us a little bit about your background. What brought you to this work?

Kyndall: Yeah. I started in the classroom. I was in upper elementary. I served fifth grade students, and I taught specifically math and science. And then I moved into a more interventionist role where I was a specialist that worked with teachers and also worked with small groups, intervention students. And through that I was able for the first time to really develop an understanding of that mathematical progression that happens at each grade level and the formative things that are introduced at the lower elementary [grades] and then kind of fade out, but still need to be brought back at the upper elementary.

Mike: So I've heard other folks talk about the ways students can learn about the algebraic properties as they're building fluency, but I feel like you've taken this a step further. You have some ideas around how we can use visual models to make those properties visible. And I wonder if you could talk a little bit about what you mean by making properties visible and maybe why you think this is an opportunity that's too good to pass up?

Kyndall: My thought is bringing visual models back into the classroom with our higher upper elementary students so that they can use those models to build a natural immersion of some of the algebraic properties so that they can emerge rather than just be rules that we are teaching. By supporting students' learning through building models with manipulatives, we're able to build a bridge in a student's mind between their experience with those models and then their mental capacity to visualize those models. This is where the opportunity to bring properties to life is too good to pass up.

Mike: OK, so let's get specific. Where would you start? Which of the properties do you see as an opportunity to help students understand as they're building an understanding of fluency?

Kyndall: So, when I begin laying the foundation for understanding of the operations and multiplication and division, I intentionally layer in two other major algebraic properties for discovery: the commutative property and the distributive property. We're not setting our students up for success when we simply introduce these properties as abstract rules to memorize. Strong visual models allow students to discover the why behind the rules. They're able to see these properties in action before I even spend any time naming them.

For example, they get to witness or discover how factors can switch order without changing the product, how grouping affects computation, and how numbers can be broken apart and recombined for efficient counting and solving strategies. By teaching basic facts in this structured and intentional way through the behavior of numbers and the authentic discovery of properties, we're not only building fluency, but we're also developing deep conceptual understanding. Students begin to recognize patterns, understand rules, make connections, and rely on reasoning instead of rote memorization. That approach supports long-term mathematical flexibility, which is exactly what we want our students to be able to do.

Mike: I want to ask you about two particular tools: the number rack and the 10-frame. Tell me a little bit about what's powerful about the way the [10-frame] is set up that helps students make sense of multiplication. What is it about the way it's designed that you love?

Kyndall: The [10-frame] is so powerful because it's set up in our base ten system already. It introduces the tens in a way that is two rows of 5, which is going to lead into properties being identified. So, let me break that up into each individual thing that I love about it.

First, the [10-frame] being broken up into the two rows of 5. That's going to allow students to be able to see that distributive property happening, where we're counting our 5s first and then adding some more into each group. So, when we're seeing a factor like 8 times 2, we're seeing that as two groups of 5 and two groups of 3.

Mike: I think what you're making me remember is how it's difficult to help kids visualize that, right? It's a challenge. You can say "'4 times 4' is the same as '4 times 2 plus 4 times 2,'" but that's still an abstraction of what's happening, right? The visual really brings it to life in a way that—even if you're representing that with an equation and doing a true-false equation where it's 4 times 4 is the same as 4 times 2 plus 4 times 2—that's still at a level of abstraction that's not necessarily accessible for children.

Kyndall: And as we're talking through this, if I see students and they're working on four groups of 3 and they're seeing those 3s as a double fact plus one more group, I'm on the board writing out the equation, and I'm using the parentheses as that introduction to what this looks like abstractly. They're building it, and they're building those visuals both with their hands and with their minds, and then I'm bringing it to life in the equation on the board.

Mike: So, I think what I see in my mind as I hear you describe that is, you have kids with a set of materials. You're doing, for lack of a better word, a translation into a more abstract version of that, and you're helping kids connect the physical materials that they have in front of them to that abstraction and really kind of drawing the connection between the two. Am I getting that right?

Kyndall: Yeah. As the students are doing the physical work of math, I'm translating it into its own language up on the board. Absolutely.

Mike: I think what's clear to me from this conversation is the way that the tools can illuminate the property, and I think this also helps me think about what my role is as a teacher in terms of building a bridge to an abstraction. Do you actually feel like there's a point where you do introduce the formal language of it? And if you do, how do you decide when?

Kyndall: So, the vocabulary kind of comes after the concept has been discovered. But I don't like to introduce the vocabulary first as a rote memorization tool because that has no meaning to it.

Mike: I think if I were to summarize this, you're giving them a physical experience with the properties. You're translating that into an abstraction. And then once they've got an experience that they can hang those ideas on top of, then you might decide to introduce the formal language to them at some point.

Kyndall: Yeah, absolutely.

Mike: So, just as a refresher, for folks who might teach upper elementary and don't have a lot of lived experiences with the number rack—be it the ten or twenty or the hundred—can you describe a little bit about the structure, and maybe what about the structure in particular is important?

Kyndall: The structure of a number rack has rows, and each row has 10 beads in it. And typically those beads are divided into two sets of 5: five red beads and five white beads. Then we typically move into a number rack that has two rows so that we're working within 20.

Now, my thought is to take that [to] our third, fourth, and fifth grade, our upper elementary students, and use the hundreds rekenrek [i.e., number rack], where now we have 10 rows and we have 10 beads in each row—still split up into five red [beads] and five white—so that we can use that to teach things. If we're looking at the zero property, students are starting to notice that the rows represent the groups—the rows with the beads on it, that's one group. And so, if we're building zero groups of 3, we don't have a group that we can access to put three beads in. If we're looking at it with the commutative property, students are able to say, "One group of 3. We have one row and we're putting three beads in it."

But what happens when we switch those factors? Now we're utilizing three of our rows, but we're only sliding over one bead. The number rack is also so important when we get to the distributive property because of the way that they have separated those colors. So when we're looking at a factor like 7 times 6—seven groups of 6—then we're gonna be accessing seven rows with six beads in each. That is already set up in the structure of the tool to have five red beads and one white bead showing seven groups of 5 and seven groups of 1 put together.

Mike: That is super powerful. One of the things that really jumped out that I want to mark is: If I treat the rows like the groups and then I treat the beads like the number of things in each group, I can model one group with three inside of it, or I can model three groups with one inside of it, and I can really make the difference between those things clear, but also [I can make] the way that the product is still the same clear, right? So, I've got an actual physical model that helps kids understand what was often a rule that was just like 1 times 3 is the same as 3 times 1, because it is. But you're actually saying this is a tool that helps us make meaning of that.

The other thing that jumps out from what you said is: If I'm doing 6 times 5 or 6 times 7 and I push over six [beads], and six looks like five red, one white, I'm automatically set up to make sense of the distributive property because the visual helps me see it. Am I getting that right?

Kyndall: Yes, except let me correct you on that last one. You said "6 times 5," and you said, "If I slide over six," Now, six is our group number. We have to be deliberate; that's six groups of 5. So, we're grabbing our groups first, but absolutely, yes. That is the key structure there. [laughs] That's the idea.

Mike: This is why this would've been very helpful for a young Mike Wallus.

Kyndall: [laughs]

Mike: Well, before we go, are there any resources that you'd recommend to a listener that have either informed your thinking or that might help someone take what you've been talking about and put these ideas into action?

Kyndall: Yeah. I've been putting this practice into play here at my own district and tracking its progress for a while now. After seeing the success in my own halls here in Sandy, [Oregon,] I've started to reach out and work with other educators on purposeful tool use and mathematical progression.

If it resonates with you, whether you're in the classroom or in a leadership role, I would genuinely love to connect and learn alongside you. You're always welcome to reach out to me directly at [email protected]. I anticipate more conversations in collaboration, and I'd love to bring them to life through trainings moving forward. I believe that when teachers are confident in their own understanding, they build that same confidence in students.

Mike: I think that's a great place to stop. Kyndall, thank you so much. It has really been a pleasure talking with you and learning from you.

Kyndall: Thank you so much for having me. It's been fun.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.

© 2026 The Math Learning Center | www.mathlearningcenter.org

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

Developing Fractions Knowledge 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 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. 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

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, a large-scale family math playspace at the Minnesota State Fair.

RESOURCES

How Did You Count? A Picture Book by Christopher Danielson

How Many?: A Counting Book by Christopher Danielson

Following Learning blog by Simon Gregg

Connecting Mathematical Ideas 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? 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] 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], 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 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. 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

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

Standards for Mathematical Practice by William McCallum

5 Practices for Orchestrating Productive Mathematics Discussions by Margaret (Peg) Smith and Mary Kay Stein

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 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 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, 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

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"

"Seeds of Algebraic Thinking: Towards a Research Agenda"

NOTICE Lab

"Leveraging Early Algebraic Experiences"

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" 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, 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

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

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 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

Christy Pettis & Terry Wyberg, The Case for Choral Counting with Fractions ROUNDING UP: SEASON 4 | EPISODE 6

How can educators help students recognize similarities in the way whole numbers and fractions behave? And are there ways educators can build on students' understanding of whole numbers to support their understanding of fractions?

The answer from today's guests is an emphatic yes. Today we're talking with Terry Wyberg and Christy Pettis about the ways choral counting can support students' understanding of fractions.

BIOGRAPHIES

Terry Wyberg is a senior lecturer in the Department of Curriculum and Instruction at the University of Minnesota. His interests include teacher education and development, exploring how teachers' content knowledge is related to their teaching approaches.

Christy Pettis is an assistant professor of teacher education at the University of Wisconsin-River Falls.

RESOURCES

Choral Counting & Counting Collections: Transforming the PreK-5 Math Classroom by Megan L. Franke, Elham Kazemi, and Angela Chan Turrou

Teacher Education by Design

Number Chart app by The Math Learning Center

TRANSCRIPT

Mike Wallus: Welcome to the podcast, Terry and Christy. I'm excited to talk with you both today.

Christy Pettis: Thanks for having us.

Terry Wyberg: Thank you.

Mike: So, for listeners who don't have prior knowledge, I'm wondering if we could just offer them some background. I'm wondering if one of you could briefly describe the choral counting routine. So, how does it work? How would you describe the roles of the teacher and the students when they're engaging with this routine?

Christy: Yeah, so I can describe it. The way that we usually would say is that it's a whole-class routine for, often done in kind of the middle grades. The teachers and the students are going to count aloud by a particular number. So maybe you're going to start at 5 and skip-count by 10s or start at 24 and skip-count by 100 or start at two-thirds and skip-count by two-thirds.

So you're going to start at some number, and you're going to skip-count by some number. And the students are all saying those numbers aloud. And while the students are saying them, the teacher is writing those numbers on the board, creating essentially what looks like an array of numbers. And then at certain points along with that talk, the teacher will stop and ask students to look at the numbers and talk about things they're noticing. And they'll kind of unpack some of that. Often they'll make predictions about things. They'll come next, continue the count to see where those go.

Mike: So you already pivoted to my next question, which was to ask if you could share an example of a choral count with the audience. And I'm happy to play the part of a student if you'd like me to.

Christy: So I think it helps a little bit to hear what it would sound like. So let's start at 3 and skip-count by 3s. The way that I would usually tell my teachers to start this out is I like to call it the runway. So usually I would write the first three numbers. So I would write "3, 6, 9" on the board, and then I would say, "OK, so today we're going to start at 3 and we're going to skip-count by 3s. Give me a thumbs-up or give me the number 2 when you know the next two numbers in that count." So I'm just giving students a little time to kind of think about what those next two things are before we start the count together. And then when I see most people kind of have those next two numbers, then we're going to start at that 3 and we're going to skip-count together.

Are you ready?

Mike: I am.

Christy: OK. So we're going to go 3…

Mike & Christy: 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36.

Christy: Keep going.

Mike & Christy: 39, 42, 45, 48, 51.

Christy: Let's stop there.

So we would go for a while like that until we have an array of numbers on the board. In this case, I might've been recording them, like where there were five in each row. So it would be 3, 6, 9, 12, 15 would be the first row, and the second row would say 18, 21, 24, 27, 30, and so on. So we would go that far and then I would stop and I would say to the class, "OK, take a minute, let your brains take it in. Give me a number 1 when your brain notices one thing. Show me 2 if your brain notices two things, 3 if your brain notices three things." And just let students have a moment to just take it in and think about what they notice.

And once we've seen them have some time, then I would say, "Turn and talk to your neighbor, and tell them some things that you notice." So they would do that. They would talk back and forth. And then I would usually warm-call someone from that and say something like, "Terry, why don't you tell me what you and Mike talked about?" So Terry, do you have something that you would notice?

Terry: Yeah, I noticed that the last column goes up by 15,

Christy: The last column goes up by 15. OK, so you're saying that you see this 15, 30, 45?

Terry: Yes.

Christy: In that last column. And you're thinking that 15 plus 15 is 30 and 30 plus 15 is 45. Is that right?

Terry: Yes.

Christy: Yeah. And so then usually what I would say to the students is say, "OK, so if you also noticed that last column is increasing by 15, give me a 'me too' sign. And if you didn't notice it, show an 'open mind' sign." So I like to give everybody something they can do. And then we'd say, "Let's hear from somebody else. So how about you, Mike? What's something that you would notice?"

Mike: So one of the things that I was noticing is that there's patterns in the digits that are in the ones place. And I can definitely see that because the first number 3 [is] in the first row. In the next row, the first number is 18 and the 8 is in the ones place. And then when I look at the next row, 33 is the first number in that row, and there's a 3 again. So I see this column pattern of 3 in the ones place, 8 in the ones place, 3 in the ones place, 8 in the ones place. And it looks like that same kind of a number, a different number. The same number is repeating again, where there's kind of like a number and then another number. And then it repeats in that kind of double, like two numbers and then it repeats the same two numbers.

Christy: So, what I would say in that one is try to revoice it, and I'd probably be gesturing, where I'd do this. But I'd say, "OK, so Mike's noticing in this ones place, in this first column, he's saying he notices it's '3, 8, 3, 8.' And then in other columns he's noticing that they do something similar. So the next column, or whatever, is like '6, 1, 6, 1' in the ones place. Why don't you give, again, give me a 'me too' [sign] if you also noticed that pattern or an 'open mind' [sign] if you didn't."

So, that's what we would do. So, we would let people share some things. We would get a bunch of noticings while students are noticing those things. I would be, like I said, revoicing and annotating on the board. So typically I would revoice it and point it out with gestures, and then I would annotate that to take a record of this thing that they've noticed on the board.

Once we've gotten several students' noticings on the board, then we're going to stop and we're going to unpack some of those. So I might do something like, "Oh, so Terry noticed this really interesting thing where he said that the last column increases by 15 because he saw 15, 30, 45, and he recognized that. I'm wondering if the other columns do something like that too. Do they also increase by the same kind of number? Hmm, why don't you take a minute and look at it and then turn and talk to your neighbor and see what you notice." And we're going to get them to notice then that these other ones also increase by 15. So if that hadn't already come out, I could use it as a press move to go in and unpack that one further.

And then we would ask the question, in this case, "Why do they always increase by 15?" And we might then use that question and that conversation to go and talk about Mike's observation, and to say, like, "Huh, I wonder if we could use what we just noticed here to figure out about why this idea that [the numbers in the] ones places are going back and forth between 3, 8, 3, 8. I wonder if that has something to do with this." Right? So we might use them to unpack it. They'll notice these patterns. And while the students were talking about these things, I'd be taking opportunities to both orient them to each other with linking moves to say, "Hey, what do you notice? What can you add on to what Mike said, or could you revoice it?" And also to annotate those things to make them available for conversation.

Mike: There was a lot in your description, Christy, and I think that provides a useful way to understand what's happening because there's the choice of numbers, there's the choice of how big the array is when you're recording initially, there are the moves that the teacher's making. What you've set up is a really cool conversation that comes forward. We did this with whole numbers just now, and I'm wondering if we could take a step forward and think about, OK, if we're imagining a choral count with fractions, what would that look and sound like?

Christy: Yeah, so one of the ones I really like to do is to do these ones that are just straight multiples, like start at 3 and skip-count by 3s. And then to either that same day or the very next day—so very, very close in time in proximity—do one where we're going to do something similar but with fractions. So one of my favorites is for the parallel of the whole number of skip-counting by 3s is we'll start at 3 fourths and we'll skip-count by 3 fourths. And when we write those numbers, we're not going to put them in simplest form; we're just going to write 3 fourths, 6 fourths, 9 fourths. So in this case, I would probably set it up in the exact same very parallel structure to that other one that we just did with the whole numbers. And I would put the numbers 3 fourths, 6 fourths, 9 fourths on the board.

I would say, "OK, here's our first numbers. We're going to start starting at 4 fourths. We're going to skip-count by 3 fourths. And give me a thumbs-up or the show me a 2 when you know the next two numbers." And then we would skip-count them together, and we would write them on the board. And so we'd end up—and in this case I would probably arrange them again in five columns just to have them and be a parallel structure to that one that we did before with the whole numbers. So it would look like 3 fourths, 6 fourths, 9 fourths, 12 fourths, 15 fourths on the first row. And then the next row, I would say 18 fourths, 21 fourths, 24 fourths, 27 fourths, 30 fourths. And again, I'd probably go all the way up until I got to 51 fourths before we'd stop and we'd look for patterns.

Mike: So I think what's cool about that—it was unsaid, but it kind of implied—is that you're making a choice there. So that students had just had this experience where they were counting in increments of 3, and 3, 6, 9, 12, 15, and then you start another row and you get to 30, and in this case, 3 fourths, 6 fourths, 9 fourths, 12 fourths, 15 fourths. So they are likely to notice that there's something similar that's going on here. And I suspect that's on purpose.

Christy: Right, that's precisely the thing that we want right here is to be able to say that fractions aren't something entirely new, something that you—just very different than anything that you've ever seen before in numbers. But to allow them to have an opportunity to really see the ways that numerators enumerate, they act like the counting numbers that they've always known, and the denominator names, and tells you what you're counting. And so it's just a nice space where, when they can see these in these parallel ways and experience counting with fractions, they have this opportunity to see some of the ways that both fraction notation works, what it's talking about, and also how the different parts of the fraction relate to things they already know with whole numbers.

Mike: Well, let's dig into that a little bit more. So the question I was going to ask Terry was: Can we talk a bit more about the ways the choral counting routine can help students make sense of the mathematics of fractions? So what are some of the ideas or the features of fractions that you found choral counting really allows you to draw out and make sense of with students?

Terry: Well, we know from our work with the rational number project how important language is when kids are developing an understanding of the role of the numerator and the denominator. And the choral counts really just show, like what Christy was just saying, how the numerator just enumerates and changes just like whole numbers. And then the denominator stays the same and names something. And so it's been a really good opportunity to develop language together as a class.

Christy: Yeah. I think that something that's really important in these ones that you get to see when you have them. So when they're doing that language, they're also—a really important part of a choral count is that it's not just that they're hearing those things, they're also seeing the notation on the board. And because of the way that we're both making this choice to repeatedly add the same amount, right? So we're creating something that's going to have a pattern that's going to have some mathematical relationships we can really unpack. But they're also seeing the notation on there that's arranged in a very intentional way to allow them to see those patterns in rows and columns as they get to talk about them.

So because those things are there, we're creating this chance now, right? So they see both the numerator and denominator. If we're doing them in parallel to things with whole numbers, they can see how both fractions are alike, things that they know with whole numbers, but also how some things are different. And instead of it being something that we're just telling them as rules, it invites them to make these observations.

So in the example that I just gave you of the skip-counting, starting at 3 fourths and skip-counting by 3 fourths, every time I have done this, someone always observes that the right-hand column, they will always say it goes up by 15. And what they're observing right there is they're paying attention to the numerator and thinking, "Well, I don't really need to talk about the denominator," and it buys me this opportunity as a teacher to say, "Yes, I see that too. I see that these 15 fourths and then you get another, then you get 30 fourths and you get 45 fourths. And I see in those numerators that 15, 30, 45—just like we had with the whole numbers—and here's how I would write that as a mathematician: I would write 15 fourths plus 15 fourths equals 30 fourths." Because I'm trying to be clear about what I'm counting right now. So instead of telling it like it's a rule that you have to remember, you have to keep the same denominators when you're going to add, it instead becomes something where we get to talk about it. It's just something that we get to be clear about. And that in fractions, we also do this other piece where we both enumerate and we name, and we keep track of that when we write things down to be clear. And so it usually invites this very nice parallel conversation and opportunity just to set up the idea that when we're doing things like adding and thinking about them, that we're trying to be clear and we're trying to communicate something in the same way that we always have been.

Mike: Well, Terry, it strikes me that this does set the foundation for some important things, correct?

Terry: Yeah, it sets the foundation for adding and subtracting fractions and how that numerator counts things and the denominator tells you the size of the pieces.

It also sets up multiplication. The last column, we can think of it as 5 groups of 3 fourths. And the next number underneath there might be 10 groups of 3 fourths. And as we start to describe or record what students' noticings are, we get a chance to highlight those features of adding fractions, subtracting fractions, multiplying fractions.

Mike: We've played around the edges of a big idea here. And one of the things that I want to bring back is something we talked about when we were preparing for the interview. This idea that learners of any age, generally speaking, they want to make use of their understanding of the way that whole numbers work as they're learning about fractions. And I'm wondering if one or both of you want to say a little bit more about this.

Terry: I think a mistake that we made previously in fraction teaching is we kind of stayed under 1. We just stayed and worked within 0 and 1 and we didn't go past it. And if you're going to make 1 a benchmark or 2 a benchmark or any whole number a benchmark, when you're counting by 3 fourths or 2 thirds or whatever, you have to go past it. So what choral counting has allowed us to do is to really get past these benchmarks, and kids saw patterns around those benchmarks, and they see them.

And then I think we also saw a whole-number thinking get in the way. So if you ask, for example, somebody to compare 3 seventeenths and 3 twenty-thirds, they might say that 3 twenty-thirds are bigger because 23 is bigger than 17. And instead of embracing their whole-number knowledge, we kind of moved away from it. And so I think now with the choral counting, they're seeing that fractions behave like whole numbers. They can leverage that knowledge, and instead of trying to make it go away, they're using it as an asset.

Mike: So the parallel that I'm drawing is, if you're trying to teach kids about the structure of numbers in whole number, if you can yourself to thinking about the whole numbers between 0 and 10, and you never worked in the teens or larger numbers, that structure's really hard to see. Am I thinking about that properly?

Terry: Yes, you are.

Christy: I think there's two things here to highlight.

So one of them that I think Terry would say more about here is just the idea that, around the idea of benchmarks. So you're right that there's things that come out as the patterns and notation that happen because of how we write them. And when we're talking about place value notation, we really need to get into tens and really into hundreds before a lot of those things become really available to us as something we talk about, that structure of how 10 plays a special role.

In fractions, a very parallel idea of these things that become friendly to us because of the notation and things we know, whole numbers act very much like that. When we're talking about rational numbers, right? So they become these nice benchmarks because they're really friendly to us, there's things that we know about them, so when we can get to them, they help us.

And the choral count that we were just talking about, there's something that's a little bit different that's happening though because we're not highlighting the whole numbers in the way that we're choosing to count right there. So we're not—we're using those, I guess, improper fractions. In that case, what we're doing is we're allowing students to have an opportunity to play with this idea, the numerator and denominator or the numerator is the piece that's acting like whole numbers that they know. So when Terry was first talking about how oftentimes when we first teach fractions and we were thinking about them, we were think a lot about the denominator. The denominator is something that's new that we're putting in with fractions that we weren't ever doing before with whole numbers. And we have that denominator. We focus a lot on like, "Look, you could take a unit and you can cut it up and you can cut it up in eight pieces, and those are called eighths, or you could cut it up in 10 pieces, and those are called tenths."

And we focus a lot on that because it's something that's new. But the thing that allows them to bridge from whole numbers is the thing that's the same as whole numbers. That's the numerator. And so when we want them to have chances to be able to make those connections back to the things they know and see that yes, there is something here that's new, it's the denominator, but connecting back to the things they know from whole numbers, we really do need to focus some on the numerator and letting them have a chance to play with what the numerator is, to see how it's acting, and to do things. It's not very interesting to say—to look at a bunch of things and say, like, "2 thirds plus 4 thirds equals 6 thirds," right? Because they'll just start to say, "Well, you can ignore the denominator." But when you play with it and counting and doing things like we was talking about—setting up a whole-number count and a fraction count in parallel to each other—now they get to notice things like that. [It] invites them to say things like, "Oh, so adding 15 in the whole numbers is kind of adding 15 fourths in the fourths." So they get to say this because you've kind of set it up as low-hanging fruit for them, but it's allowing them really to play with that notion of the numerator and a common denominator setting.

And then later we can do other kinds of things that let them play with the denominator and what that means in those kinds of pieces. So one of the things I really like about choral counts and choral counts with fractions is it's setting up this space where the numerator becomes something that's interesting and something worth talking about in some way to be able to draw parallels and allow them to see it. And then of course, equivalency starts to come into play too. We can talk about how things like 12 fourths is equivalent to 3 wholes, and then we get to see where those play their role inside of this count too. But it's just something that I really like about choral counting with fractions that I think comes out here. And it's not quite the idea of benchmarks, but it is important.

Mike: Well, let's talk a little bit about equivalency then. Terry. I'm wondering if you could say a little bit about how this routine can potentially set up a conversation around ideas related to equivalency.

Terry: We could do this choral count—instead of just writing improper fractions all the way through, we could write them with mixed numbers. And as you start writing mixed numbers, the pattern becomes "3 fourths, 1 and a half, 2 and a quarter," and we can start bringing in equivalent fractions. And you still do the same five columns and make parallel connections between the whole numbers, the fractions that are written as improper fractions and the fractions with mixed numbers. And so you get many conversations about equivalencies. And this has happened almost every time I do a choral count with fractions is, the kids will comment that they stop thinking. They go, "I'm just writing these numbers down." Part of it is they're seeing equivalency, but they're also seeing patterns and letting the patterns take over for them. And we think that's a good thing rather than a bad thing. It's not that they're stopped thinking, they're just, they're just—

Christy: They're experiencing the moment that patterns start to help, that pattern recognition starts to become an aid in their ability to make predictions. All of a sudden you can feel it kick online.

So if you said it in the context, then what happens is even in the mixed-number version or in the improper-number version, that students will then have a way of talking about that 12 fourths is equivalent to 3, and then you're going to see that whole-number diagonal sort of pop in, and then you'll see those other ones, even in the original version of it.

Terry: Yeah, as we started to play around with this and talk with people, we started using the context of sandwiches, fourths of sandwiches. And so when they would start looking at that, the sandwiches gave them language around wholes. So the equivalence that they saw, they had language to talk about. That's 12 fourths of a sandwich, which would be 3 full sandwiches. And then we started using paper strips with the choral counts and putting paper strips on each piece so kids could see that when it fills up they can see a full sandwich. And so we get both equivalencies, we get language, we get connections between images, symbols, and context.

Mike: One of the questions that I've been asking folks is: At the broadest level, regardless of the number being counted or whether it's a whole number or a rational number, what do you think the choral counting routine is good for?

Christy: So I would say that I think of these routines, like a choral count or a number talk or other routines like that that you would be doing frequently in a classroom, they really serve as a way of building mathematical language. So they serve as a language routine. And then one of the things that's really important about it is that it's not just that there's skip-counting, but that count. So you're hearing the way that patterns happen in language, but they're seeing it at the same time. And then they're having chances, once that static set of representations on the board, those visuals of the numbers has been created and set up in this structured way, it's allowing them to unpack those things. So they get to first engage in language and hearing it in this multimodal way. So they hear it and they see it, but then they get to unpack it and they get to engage in language in this other way where they get to say, "Well, here's things that stand out to me."

So they make these observations and they will do it using informal language. And then it's buying the teacher an opportunity then to not only highlight that, but then to also help formalize that language. So they might say, "Oh, I saw a column goes up by 5." And I would get to say, "Oh, so you're saying that you add each time to this column, and here's how a mathematician would write that." And we would write that with those symbols. And so now they're getting chances to see how their ideas are mathematical ideas and they're being expressed using the language and tools of math. "Here's the way you said it; here's what your brain was thinking about. And here's what that looks like when a mathematician writes it." So they're getting this chance to see this very deeply authentic way and just also buying this opportunity not only to do it for yourself, but then to take up ideas of others. "Oh, who else saw this column?" Or, "Do you think that we could extend that? Do you think it's anywhere else?" And they get to then immediately pick up that language and practice it and try it. So I look at these as a really important opportunity, not just for building curiosity around mathematics, but for building language.

Mike: Let's shift a little bit to teacher moves, to teacher practice, which I think y'all were kind of already doing there when you were talking about opportunities. What are some of the teacher moves that you think are really critical to bringing choral counting with fractions particularly to life?

Terry: I think just using the strips to help them visualize it, and it gave them some language. I think the context of sandwiches, or whatever it happens to be, gives them some ways to name what the unit is. We found starting with that runway, it really helps to have something that they can start to kind of take off and start the counting routine. We also found that the move where you ask them, "What do you notice? What patterns do you notice?," we really reserve for three and a half rows. So we try to go three full rows and a half and it gives everybody a chance to see something. If I go and do it too quick, I find that I don't get everybody participating in that, noticing as well, as doing three and a half rows. It just seems to be a magic part of the array is about three and a half rows in.

Mike: I want to restate and mark a couple things that you said, Terry. One is this notion of a runway that you want to give kids. And that functions as a way to help them start to think about, again, "What might come next?" And then I really wanted to pause and talk about this idea of, you want to go at least three rows, or at least—is it three or three and a half?

Terry: Three and a half.

Christy: When you have three of something, then you can start to use patterns. You need at least those three for even to think there could be a pattern. So when you get those, at least three of them, and they have that pattern to do—and like Terry was saying, when you have a partial row, then what happens is those predictions can come from two directions. You could keep going in the row, so you could keep going horizontally, or you could come down a column. And so now it kind of invites people to do things in more than one way when you stop mid-row.

Mike: So let me ask a follow-up question. When a teacher stops or pauses the count, what are some of the first things you'd love to see them do to spark some of the pattern recognition or the pattern seeking that you just talked about?

Christy: Teacher moves?

Mike: Yeah.

Christy: OK. So we do get to work with preservice teachers all the time. So this is one of my favorite parts of this piece of it. So what do you do as a teacher that you want? So we're going to want an array up there that has enough, at least three of things in some different ways people can start to see some patterns.

You can also, when you do one of these counts, you'll hear the moment—what Terry described earlier as "stop thinking." You can hear a moment where people, it just gets easier to start, the pattern starts to help you find what comes next, and you'll hear it. The voices will get louder and more confident as you do it. So you want a little of that. Once you're into that kind of space, then you can stop. You know because you've just heard them get a little more confident that their brains are going. So you're kind of looking for that moment. Then you're going to stop in there again partway through a row so that you've got a little bit of runway in both directions. So they can keep going horizontally, they can come down vertically. And you say, "OK," and you're going to give them now a moment to think. And so that stopping for a second before they just talk, creating space for people to formulate some language, to notice some things is really, really important.

So we're going to create some thinking space, but we know there's some thinking happening, so you just give them a way to do it. Our favorite way to do it is to, instead of just doing a thumbs-up and thumbs-down in front of the chest, we just do a silent count at the chest rather than hands going up. We just keep those hands out of the air, and I say, "Give me a 1 at your chest"—so a silent number 1 right at your chest—"when you've noticed one thing. And if you notice two things, give me a 2. And if you notice three things, give me a 3." They will absolutely extrapolate from there. And you'll definitely see some very anxious person who definitely wants to say something with a 10 at their chest. But what you're doing at that moment is you're buying people time to think, and you're buying yourself as a teacher some insight into where they are. So you now get to look out and you can see who's kind of taking a while for that 1 to come up and who has immediately five things, and other things.

And you can use that along with your knowledge of the students now to think about how you want to bring people into that discussion. Somebody with 10 things, they do not need to be the first person you call on. They are desperate to share something, and they will share something no matter when you call on them. So you want to use this information now to be able to get yourself some ideas of, like, "OK, I want to make sure that I'm creating equitable experiences, that I want to bring a lot of voices in." And so the first thing we do is we have now a sense of that because we just watched, we gave ourselves away into some of the thinking that's happening. And then we're going to partner that immediately with a turn and talk. So first they're going to think and then they're going to have a chance to practice that language in a partnership. And then, again, you're buying yourself a chance to listen into those conversations and to know that they have something to share. And to bring it in, I will pretty much always make that a warm call. I won't say, "Who wants to share?" I will say, "Terry or Mike, let's hear." And then I won't just say, "Terry, what was your idea?" I would say, "Terry, tell me something that either you or Mike shared that you noticed." So we'll give a choice. So now they've got a couple ways in. You know they just said something. So you're creating this space where you're really lowering the temperature of how nerve-racking it is to share something. They have something to say, and they have something to do. So I want all of those moves.

And then I kind of alluded to it when we were doing the practice one, but the other one I really like is to have all-class gestures so that everyone constantly has a way they need to engage and listen. And so I like to use ones not just the "me too" gesture, but we do the "open mind" gesture as well so that everyone has one of the two. Either it's something that you were thinking or they've just opened your mind to a new idea. And it looks, we use it kind of like an open book at your forehead. So, the best way I can describe it to you, you put both hands at your forehead and you touch them like they're opening up, opening doors. And so everyone does one of those, right? And then as a teacher, you now have some more information because you could say, "Oh, Terry, you just said that was open mind. You hadn't noticed it. Well, tell us something different you noticed." So you get that choice of what you're doing. So you're going to use these things as a teacher to not just get ideas out but to really be able to pull people in ways they've sort of communicated something to you that they have something to share.

So I love it for all the ways we get to practice these teacher moves that don't just then work in just this choral count, but that do a really great job in all these other spaces that we want to work on with students too, in terms of equitably and creating talk, orienting students to one another, asking them to listen to and build on each other's ideas.

Terry: When you first start doing this, you want to just stop and listen. So I think some of my mistakes early on was trying to annotate too quickly. But I found that a really good teacher move is just to listen. And I get to listen when they're think-pair-sharing, I get a chance to listen when they're just thinking together, I get a chance to listen when they describe it to the whole class. And then I get to think about how I'm going to write and record what they said so that it amplifies what they're saying to the whole class. And that's the annotation piece. And getting better at annotating is practicing what you're going to write first and then they always say something a little different than what you anticipate, but you've already practiced. So you can get your colors down, you can get how you're going to write it without overlapping too much with your annotations.

Mike: I think that feels like a really important point for someone who is listening to the podcast and thinking about their own practice. Because if I examine my own places where I sometimes jump before I need to, it often is to take in some ideas but maybe not enough and then start to immediately annotate. And I'm really drawn to this idea that there's something to, I want to listen enough to kind of hear the body of ideas that are coming out of the group before I get to annotation. Is that a fair kind of summary of the piece that you think is really important about that?

Terry: Yes. And as I'm getting better with it, I'm listening more and then writing after I think I know what they're saying. And I check with them as I'm writing.

Mike: So you started to already go to my next question, which is about annotation. I heard you mention color, so I'm curious: What are some of the ideas about annotation that you think are particularly important when you are doing it in the context of a choral count?

Christy: Well, yeah, I think a choral count. So color helps just to distinguish different ideas. So that's a useful tool for that piece of it. What we typically want, people will notice patterns usually in lines. And so you're going to get vertical lines and horizontal lines, but you'll also get diagonals. That's usually where those will be. And they will also notice things that are recognizable. So like the 15, 30, 45 being a number sequence that is a well-known one is typically wouldn't going to be the first one we notice. Another one that happens along a diagonal, and the examples we gave will be 12, 24, 36, it comes on a diagonal. People will often notice it because it's there. So then what you want is you're going to want to draw in those lines that help draw students' eyes, other students' eyes, not the ones who are seeing it, but the ones who weren't seeing it to that space so they can start to see that pattern too. So you're going to use a little bit of lines or underlining that sort of thing. These definitely do over time get messier and messier as you add more stuff to them. So color helps just distinguish some of those pieces.

And then what you want is to leave yourself some room to write things. So if you have fractions, for example, you're going to need some space between things because fractions take up a little bit more room to write. And you definitely want to be able to write "plus 15 fourths," not just, "plus 15." And so you need to make sure you're leaving yourself enough room and practicing and thinking. You also have to leave enough room for if you want to continue the count, because one of the beautiful things you get to do here is to make predictions once you've noticed patterns. And so you're going to probably want to ask at some point, "Well, what number do you think comes in some box further down the road?" So you need to leave yourself enough room then to continue that count to get there.

So it's just some of the things you have to kind of think about as a teacher as you do it, and then as you annotate, so you're kind of thinking about trying to keep [the numbers] pretty straight so that those lines are available to students and then maybe drawing them in so students can see them. And then probably off to the side writing things like, if there's addition or multiplication sentences that are coming out of it, you probably want to leave yourself some room to be able to sometimes write those. In a fraction one, which Terry talked about a little bit, because equivalency is something that's available now where we can talk about, for example, the really common one that would come out in our example would be that 12 fourths is equivalent to 3 wholes. Somehow you're going to have to ask this question of, "Well, why is that? Where could we see it?" And so in that case, usually we would draw the picture of the sandwiches, which will be rectangles all cut up in the same way. So not like grilled cheese sandwiches in fourth, but like a subway sandwich in fourths. And then you're going to need some space to be able to draw those above it and below it.

So again, you're kind of thinking about what's going to make this visible to students in a way that's meaningful to them. So you're going to need some space to be left for those things. What I find is that I typically end up having to write some things, and then sometimes after the new idea comes in, I might have to erase a little bit of what's there to make some more room for the writing. But I would say with fractions, it's going to be important to think about leaving enough space between, because you're probably going to need a little bit of pictures sometimes to help make sense of that equivalency. That's a really useful one. And leaving enough space for the notation itself, it takes a little bit of room.

Mike: Every time I do a podcast, I get to this point where I say to the guest or guests, "We could probably talk for an hour or more, and we're out of time." So I want to extend the offer that I often share with guests, which is if someone wanted to keep learning about choral counting or more generally about some of the ideas about fractions that we're talking about, are there any particular resources that the two of you would recommend?

Terry: We started our work with the Choral Counting & Counting Collections book by Megan Franke[, Elham Kazemi, and Angela Chan Turrou], and it really is transformational, both routines.

Christy: And it has fractions and decimals and ideas in it too. So you can see it across many things. Well, it's just, even just big numbers, small numbers, all kinds of different things. So teachers at different grade levels could use it.

The Teacher Education by Design [website], at tedd.org, has a beautiful unit on counting collections for teachers. So if you're interested in learning more about it, it has videos, it has planning guides, things like that to really help you get started.

Terry: And we found you just have to do them. And so as we just started to do them, writing it on paper was really helpful. And then The Math Learning Center has an app that you can use—the Number Chart app—and you can write [the choral counts] in so many different ways and check your timing out. And it's been a very helpful tool in preparing for quality choral counts with fractions and whole numbers.

Mike: I think that's a great place to stop.

Christy and Terry, I want to thank you both so much for joining us. It has really just absolutely been a pleasure chatting with you both.

Christy: So much fun getting to talk to you.

Terry: Thank you.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.

© 2025 The Math Learning Center | www.mathlearningcenter.org

Ramsey Merritt, Improving Students' Turn & Talk Experience ROUNDING UP: SEASON 4 | EPISODE 5

Most educators know what a turn and talk is—but are your students excited to do them?

In this episode, we put turn and talks under a microscope. We'll talk with Ramsey Merritt from the Harvard Graduate School of Education about ways to revamp and better scaffold turn and talks to ensure your students are having productive mathematical discussions.

BIOGRAPHY

Ramsey Merritt is a lecturer in education at Brandeis University and the director of leadership development for Reading (MA) Public Schools. He has taught and coached at every level of the U.S. school system in both public and independent schools from New York to California. Ramsey also runs an instructional leadership consulting firm, Instructional Success Partners, LLC. Prior to his career in education, he worked in a variety of roles at the New York Times. He is currently completing his doctorate in education leadership at Harvard Graduate School of Education. Ramsey's book, Diving Deeper with Upper Elementary Math, will be released in spring 2026.

TRANSCRIPT

Mike Wallus: Welcome to the podcast, Ramsey. So great to have you on.

Ramsey Merritt: It is my pleasure. Thank you so much for having me.

Mike: So turn and talk's been around for a while now, and I guess I'd call it ubiquitous at this point. When I visit classrooms, I see turn and talks happen often with quite mixed results. And I wanted to start with this question: At the broadest level, what's the promise of a turn and talk? When strategically done well, what's it good for?

Ramsey: I think at the broadest level, we want students talking about their thinking and we also want them listening to other students' thinking and ideally being open to reflect, ask questions, and maybe even change their minds on their own thinking or add a new strategy to their thinking. That's at the broadest level.

I think if we were to zoom in a little bit, I think turn and talks are great for idea generation. When you are entering a new concept or a new lesson or a new unit, I think they're great for comparing strategies. They're obviously great for building listening skills with the caveat that you put structures in place for them, which I'm sure we'll talk about later. And building critical-thinking and questioning skills as well.

I think I've also seen turn and talks broadly categorized into engagement, and it's interesting when I read that because to me I think about engagement as the teacher's responsibility and what the teacher needs to do no matter what the pedagogical tool is. So no matter whether it's a turn and talk or something else, engagement is what the teacher needs to craft and create a moment. And I think a lot of what we'll probably talk about today is about crafting moments for the turn and talk. In other words, how to engage students in a turn and talk, but not that a turn and talk is automatically engagement.

Mike: I love that, and I think the language that you've used around crafting is really important. And it gets to the heart of what I was excited about in this conversation because a turn and talk is a tool, but there is an art and a craft to designing its implementation that really can make or break the tool itself.

Ramsey: Yeah. If we look back a little bit as to where turn and talk came from, I sort of tried to dig into the papers on this. And what I found was that it seems as if turn and talks may have been a sort of spinoff of the think-pair-share, which has been around a little bit longer. And what's interesting in looking into this is, I think that turn and talks were originally positioned as a sort of cousin of think-pair-share that can be more spontaneous and more in the moment. And I think what has happened is we've lost the "think" part. So we've run with it, and we've said, "This is great," but we forgot that students still need time to think before they turn and talk. And so what I see a lot is, it gets to be somewhat too spontaneous, and certain students are not prepared to just jump into conversations. And we have to take a step back and sort of think about that.

Mike: That really leads into my next question quite well because I have to confess that when I've attended presentations, there are points in time when I've been asked to turn and talk when I can tell you I had not a lot of interest nor a lot of clarity about what I should do. And then there were other points where I couldn't wait to start that conversation. And I think this is the craft and it's also the place where we should probably think about, "What are the pitfalls that can derail or have a turn and talk kind of lose the value that's possible?" How would you talk about that?

Ramsey: Yeah, it is funny that we as adults have that reaction when people say, "Turn and talk."

The three big ones that I see the most, and I should sort of say here, I've probably been in 75 to 100 buildings and triple or quadruple that for classrooms. So I've seen a lot of turn and talks, just like you said. And the three big ones for me, I'll start with the one that I see less frequently but still see it enough to cringe and want to tell you about it. And it's what I call the "stall" turn and talk. So it's where teachers will sometimes use it to buy themselves a little time. I have literally heard teachers say something along the lines of, "OK, turn and talk to your neighbor while I go grab something off the printer."

But the two biggest ones I think lead to turn and talk failure are a lack of specificity. And in that same vein too, what are you actually asking them to discuss? So there's a bit of vagueness in the prompting, so that's one of the big ones.

The other big one for me is, and it seems so simple, and I think most elementary teachers are very good at using an engaging voice. They've learned what tone does for students and what signals tone sends to them about, "Is now the time to engage? Should I be excited?" But I so often see the turn and talk launched unenthusiastically, and that leads to an engagement deficit. And that's what you're starting out with if you don't have a good launch: Students are already sort of against you because you haven't made them excited to talk.

Mike: I mean those things resonate. And I have to say there are some of them that I cringe because I've been guilty of doing, definitely the first thing when I've been unprepared. But I think these two that you just shared, they really go to this question of how intentionally I am thinking about building that sense of engagement and also digging into the features that make a turn and talk effective and engaging.

So let's talk about the features that make turn and talks effective and engaging for students. I've heard you talk about the importance of picking the right moment for a turn and talk. So what's that mean?

Ramsey: So for me, I break it down into three key elements. And one of them, as you say, is the timing. And this might actually be the most important element, and it goes back to the origin story, is: If you ask a question, and say you haven't planned a turn and talk, but you ask a question to a whole group and you see 12 hands shoot up, that is an ideal moment for a turn and talk. You automatically know that students are interested in this topic. So I think that's the sort of origin story, is: Instead of whipping around the room and asking all 12 students—because especially at the elementary level, if students don't get their chance to share, they are very disappointed. So I've also seen these moments drag out far too long. So it's kind of a good way to get everyone's voice heard. Maybe they're not saying it out to the whole group, but they get to have everyone's voice heard. And also you're buying into the engagement that's already there. So that would be the more spontaneous version, but you can plan in your lesson planning to time a turn and talk at a specific moment if you know your students well enough that you know can get them engaged in.

And so that leads to one of the other points is the launch itself. So then you're really thinking about, "OK, I think this could be an interesting moment for students. Let me think a little bit deeper about what the hook is." Almost every teacher knows what a hook is, but they typically think about the hook at the very top of their lesson. And they don't necessarily think about, "How do I hook students in to every part of my lesson?" And maybe it's not a full 1-minute launch, maybe it's not a full hook, but you've got to reengage students, especially now in this day and time, we're seeing students with increasingly smaller attention spans. So it's important to think about how you're launching every single piece of your lesson.

And then the third one, which goes against that origin story that I may or may not even be right about, but it goes against that sort of spontaneous nature of turn and talks, is: I think the best turn and talks are usually planned out in advance.

So for me it's planning, timing, and launching. Those are my elements to success when I'm coaching teachers on doing a turn and talk.

Mike: Another question that I wanted to unpack is: Talk about what. The turn and talk is a vehicle, but there's also content, right? So I'm wondering about that. And then I'm also wondering are there prompts or particular types of questions that educators can use that are more interesting and engaging, and they help draw students in and build that engagement experience you were talking about?

Ramsey: Yeah, and it's funny you say, "Talk about what" because that's actually feedback that I've given to teachers, when I say, "How did that go for you?" And they go, "Well, it went OK." And I say, "Well, what did you ask them to talk about? Talk about what is important to think about in that planning process." So I hate to throw something big out there, but I would actually argue that at this point, we have seen the turn and talk sort of devolve into something that is stigmatized that often is vague.

So what if instead of calling everything a turn and talk, you had specific types of turn and talks in your classroom. And these would take a little time to routinize; students would have to get used to them. But one idea I had is: What if you just called one "pick a side"? Pick a side, it tells the students right away what they need to do; it's extremely specific. So you're giving them one or two or—well not one, you're giving them two or three strategies, and you're telling them, "You have to pick one of these. And you're going to be explaining to your partner your rationale as to why you think that strategy works best or most efficiently." Or maybe it's an error analysis kind of thing. Maybe you plant one n as wrong, one n as right. And then you still ask them, "Pick a side here. Who do you agree with?" And then you also get a check for understanding because the students around the room who are picking the wrong one, you're picking up data on what they understand about the topic.

Another one you can do is, you could just call it "justify your thinking." Justify your thinking. So that just simply says to them, "I have to explain to the person next to me why I'm thinking the way that I'm thinking about this prompt or this problem."

So that could also be a "help their thinking." So maybe you put up someone's thinking on the board that is half baked, and now their job is to help that person. So that's a sort of deeper knowledge kind of thing too.

And then the last one is we can turn the "What do you notice? What do you wonder?" [activity] into a routine that is very similar to a turn and talk, where both people have an opportunity to share what they're wondering or what they're noticing.

But I think no matter what you call them, no matter how you routinize them, I think it's important to be more specific than "turn and talk."

Mike: You use the word routinized. It's making me think a lot about why we find routines to have value, right? Because once you teach a particular routine, kids know what it is to do said routine. They know what it is to show up when you're doing Which one doesn't belong? They know the role that they play. And I think part of what really jumps out is: If you had a series of more granular turn and talk experiences that you were trying to cultivate, kids actually have a sense of what it is to do a turn and talk if you are helping thinking, or if you are agreeing or disagreeing, or whatever the choice might be.

Ramsey: That's right. For me, everything, even when I'm working with middle and high school teachers, I say, "The more that you can put structures in place that remove those sort of barriers for thinking, the better off you're going to be." And so we could talk more too about how to differentiate and scaffold turn and talk. Sometimes that gets forgotten as well.

But I think the other piece I would love to point out here is around—you're right, turn and talk is so ubiquitous. And what that means, what I've seen in schools, if I've seen, I'll go into a school and I might watch four different teachers teach the same lesson and the turn and talk will look and feel differently in each room. So the other advantage to being more specific is that if a student—let's say they went to, because even in an elementary school you might go to a specialist, you might go to art class. And that teacher might use a turn and talk. And what happens is they sort of get this general idea around the turn and talk and then they come into your room with whatever the turn and talk was in the last class or however the teacher used it last year. So to me there's also a benefit in personalizing it to your room as well so that you can get rid of some of that stigma if it wasn't going well for the student before, especially if you then go in and scaffold it.

Mike: Let's talk a little bit about those scaffolds and maybe dig in a little bit deeper to some of the different kinds of routinized turn and talks. I'm wondering if you wanted to unpack anything in particular that you think would really be important for a teacher to think about as they're trying to take up the ideas that we've been discussing.

Ramsey: And one of the simplest ones to implement is the Partner A, Partner B routine. I think maybe many of your listeners will be like, "Yeah, I use that." But one of the pieces that's really important there is that you really hold students accountable to honoring Partner A's time. So when Partner A is speaking, Partner B needs to be trying to make—you know, not everybody can do the eye contact thing, but there are some things that you can recommend and suggest for them. Maybe they have something to take notes on. So this could be having whiteboards at your rug, it could be clipboards, it could be that they have a turn and talk thought-catcher notebook or folder.

And it doesn't matter what it is, but not everyone has the same processing skills. So we think about turn and talk sometimes as spontaneous, but we're forgetting that 12 students raised their hand and they were eager. What about the other 12 or 15? If they didn't raise their hand, it could be that they're shy but they have something on their mind. But it also could be that you just threw out a prompt and they haven't fully processed it yet. We know kids process things at different times and at different speeds. So incorporating in that—maybe it's even a minute up top. Everybody's taking their silent and solo minute to think about this prompt. Then Partner A is going to go. It's about equity and voice across the room. It's about encouraging listening, it's about giving think time.

Mike: Well, I want to stop and mark a couple things.

What occurs to me is that in some ways a podcast interview like this is one long turn and talk in the sense that you and I are both listening and talking with one another. And as you were talking, one of the things I realized is I didn't have a piece of paper with me. And what you were saying really connects deeply because even if it's just jotting down a word or two to help me remember that was a salient point or this is something that I want to follow up on, that's really critical. Otherwise, it really can feel like it can evaporate and then you're left not being able to explore something that might've been really important.

I think the other thing that jumps out is the way that this notion of having a notepad or something to jot is actually a way to not necessarily just privilege spoken communication. That if I'm going to process or if I'm going to try to participate, having something like that might actually open up space for a kid whose favorite thing to do isn't to talk and process as they're talking. Does that make sense?

Ramsey: Totally. I had a student in a program I was working with this summer who was 13 years old but was selectively mute. And the student teachers who were working in this room wanted to still be able to do a turn and talk. And they had her still partner with people, but she wrote down sentences and she literally held up her whiteboard and then the other student responded to the sentence that she wrote down on her whiteboard. So that's real.

And to your other point about being able to jot down so you can remember—yeah, we have to remember we're talking about six-, seven-, eight-, nine-year olds. We're fully functioning adults and we still need to jot things down. So imagine when your brain is not even fully developed. We can't expect them to remember something from when they haven't been allowed to interrupt the other. And so I think going on now what you're saying is, that then makes me think about the Partner A, Partner B thing could also sort of tamper down the excitement a little bit if you make another student wait. So you also have to think about maybe that time in between, you might need to reengage. That's my own thinking right now, evolving as we're talking.

Mike: So in some ways this is a nice segue to something else that you really made me think about. When we were preparing for this interview, much of what I was thinking about is the role of the teacher in finding the moment, as you said, where you can build excitement and build engagement, or thinking about the kind of prompts that have a specificity and how that could impact the substance of what kids are talking about. But what really jumped out from our conversation is that there's also a receptive side of turn and talk, meaning that there are people who are talking, but we also don't want the other person to just be passive. What does it look like to support the listening side of turn and talk? And I would love it if you would talk about the kinds of things you think it's important for educators to think about when they're thinking about that side of turn and talk.

Ramsey: I would say don't forget about sentence starters that have to do with listening. So often when we're scaffolding, we're thinking about, "How do I get them to share out? How do I get them to be able to address this prompt?"

But one of the easiest scaffolds that I've heard for listening—and it works very, very well—is, "What I heard you say is, blank." And so then the receptive student knows that a—tells them they have to be listening pretty carefully because they're about to be asked to repeat what the other person said. And this is an age-old elementary school sort of piece of pedagogy, is a call and response situation. But then we want to give them a stem that allows them maybe to ask a question. So it's, "What I heard you say was, blank. What I'm wondering is, blank." So that takes it to the next thinking level. But again, it's about being really specific and very intentional with your students and saying, "When it's Partner B's turn, you must lead with, 'What I heard you say is,' and only then can you get to your thinking or asking questions."

Mike: That's huge. I think particularly when you think about the fact that there may be status issues between Partner A and Partner B. If Partner A is seen as or sees themselves as someone who's good at math and that's less true for Partner B, the likelihood of actually listening in a productive way seems like it's in danger at the very least. So I see these as tools that really do, one, build a level of accountability responsibility, but also level the playing field when it comes to things like status between two students.

Ramsey: I would agree with that, yeah.

I think, too, we always want to be mixing our groups. I think sometimes you get, when I think about those sort of people or those students who—you can walk into any classroom and you right away can look around the room, if you've seen enough math teaching, you can see the students who have the most confidence in math.

So another piece to sort of leveling that field is making sure that your turn and talks are not always built on skill or high-level conceptual understanding. So that's where it might be helpful to have a more low-floor task, like a What do you notice? What do you wonder? But using the turn and talk routine of that. So it gives people more of a chance to get involved even when they don't have the highest level. It's kind of like the same idea with a Which one doesn't belong? [task] or a typical number talk. But, so you as the teachers have to be thinking about, "OK, yesterday we did one that was comparing two people's strategies, and I know that some of my students didn't quite understand either one of them. So today, in order to rebuild some of that confidence, I might do a version of a turn and talk that is much more open to different kinds of thinking."

Mike: You started to go there in this last conversation we had about supporting the receptive side of turn and talk. I did want to ask if we can go a little bit deeper and think about tools like anchor charts. And you already mentioned sentence prompts, but sentence frames. To what extent do you feel like those can be helpful in building the kinds of habits we're talking about, and do you have any thoughts about those or any other resources that you think are important scaffolds?

Ramsey: Yeah. I have seen some really, really wonderful teachers bring in such a simple way of activating an anchor chart and that is especially—it's easier to do an inquiry-based learning, but I think you can do it in any kind of classroom—is, when a student presents their thinking early on in a unit, and let's say we're talking about comparing fractions. And they say, "This is how I compared fractions," and you're annotating and you're charting it up for them as the teacher, you can call that strategy, "Maya's strategy." And so now it has a little bit more stickiness for both the students and for you. Now you know that there's a specific mathematical name for that strategy, but the students don't necessarily need to know that. You could put it in parentheses if you want. But I have seen that be really effective, and I've actually heard other students go, "I'm going to use Maya's strategy for this one," and able to then look and reference it.

I think what happens sometimes with the anchor charts is, we still live in a sort of Pinterest world, and some people want those anchor charts to be beautiful, but they're not actually useful because it was drawn up perfectly and it's lovely and it's pretty, but the students don't have a real connection to it. So the other piece to that is the cocreation of the anchor chart. So it's not just naming the student; it's also going through it step by step. Maybe they're leading through it, maybe you're guiding it. Maybe you're asking probing questions. Maybe you throw in a turn and talk in the middle of that sort of exploration. And then students have a connection to that piece of paper. Anchor charts that have been created during your prep period, I guarantee you will have very little effect. So that's how I feel about those.

I also love, I call them like mini anchor charts, but they sit on tables. In recent years I've seen more and more, especially in elementary classrooms—and I've encouraged them at the middle school and the high school level—of putting in a little, I don't really know the best way to describe it for listeners, but it sits on the tabletop, and it's almost like a placard holder. And inside of that you put a mini version of an anchor chart that sits at the students' tables. So if you're doing turn and talks at their desks, and they're sitting in desks of four, and that's right there in front of them with some sentence starters or maybe your very specific routines—pick a side!—and then you have the three steps to picking aside underneath. If that's sitting on the table right in front of them, they are much more likely to reference it than if it's on the wall across the room. That gets a little trickier if you're down at the rug if you're doing turn and talks down at the rug, but hey, you can get a slightly bigger one and stick a few down on the rug around them too if you really need to.

Mike: I love that. That seems powerful and yet imminently practical.

Ramsey: I've seen it work.

Mike: Well, this happens to me every time I do a podcast. I have a lovely conversation, and we get close to the end of it, and I find myself asking: For listeners, what recommendations do you have for people who either want to learn more or would like to get started implementing some of the ideas we discussed today?

Ramsey: Sure. I mean the biggest one that I tell both new teachers and veterans when you're looking to sort of improve on your practice is to go watch someone else teach. So it's as simple as asking a colleague, "Hey, do you know anybody who does this really well?" In fact, I've led some [professional development trainings] at schools where I've said, "Who in the room is great at this?" And a few people will throw their hands up, and I go, "Great. Instead of me explaining it, I'm going to have you tell us why you're so successful at that." So the easiest one is to go watch someone who has this down.

But for some of the things that I've mentioned, I would think about not biting off too much. So if you are someone who your turn and talks, you readily admit that they're not specific, they're fully routinized, and they don't go well for you, I would not recommend putting in four new routines tomorrow, the A/B partner thing, and making the anchor charts for the tables all at once.

What I always say is try one thing and also be transparent with the students. It goes a really long way, even with seven-year-olds, when you say, "Alright guys, we're going to do a new version of the turn and talk today because I've noticed that some of you have not been able to share as much as I would like you to. So we're going to try this, which is for me, I hope it allows both people to share and afterwards you can let me know how that felt." Students really appreciate that gesture, and I think that's really important if you are going to try something new to sort of be transparent about it. Oftentimes when teachers implement something new, it can feel like, not a punishment, but it's almost like a, "Ooh, why is she changing this up on us?" So letting them know also creates a warmer space too, and it shows them that you're learning, you're growing.

Mike: I love that, and I think that's a great place to stop.

Ramsey, thank you so much. It has really been a pleasure talking with you.

Ramsey: Thank you. Like you said, I could do it all day, so I really appreciate it. I wish everyone out there well, and thanks again.

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

Pam Harris, Exploring the Power & Purpose of Number Strings ROUNDING UP: SEASON 4 | EPISODE 4

I've struggled when I have a new strategy I want my students to consider and despite my best efforts, it just doesn't surface organically. While I didn't want to just tell my students what to do, I wasn't sure how to move forward. Then I discovered number strings.

Today, we're talking with Pam Harris about the ways number strings enable teachers to introduce new strategies while maintaining opportunities for students to discover important relationships.

BIOGRAPHY

Pam Harris, founder and CEO of Math is Figure-out-able™, is a mom, a former high school math teacher, a university lecturer, an author, and a mathematics teacher educator. Pam believes real math is thinking mathematically, not just mimicking what a teacher does. Pam helps leaders and teachers to make the shift that supports students to learn real math.

RESOURCES

Young Mathematicians at Work by Catherine Fosnot and Maarten Dolk

Procedural fluency in mathematics: Reasoning and decision-making, not rote application of procedures position by the National Council of Teachers of Mathematics

Bridges number string example from Grade 5, Unit 3, Module 1, Session 1 (BES login required)

Developing Mathematical Reasoning: Avoiding the Trap of Algorithms by Pamela Weber Harris and Cameron Harris

Math is Figure-out-able!™ Problem Strings

TRANSCRIPT

Mike Wallus: Welcome to the podcast, Pam. I'm really excited to talk with you today.

Pam Harris: Thanks, Mike. I'm super glad to be on. Thanks for having me.

Mike: Absolutely.

So before we jump in, I want to offer a quick note to listeners. The routine we're going to talk about today goes by several different names in the field. Some folks, including Pam, refer to this routine as "problem strings," and other folks, including some folks at The Math Learning Center, refer to them as "number strings." For the sake of consistency, we'll use the term "strings" during our conversation today.

And Pam, with that said, I'm wondering if for listeners, without prior knowledge, could you briefly describe strings? How are they designed? How are they intended to work?

Pam: Yeah, if I could tell you just a little of my history. When I was a secondary math teacher and I dove into research, I got really curious: How can we do the mental actions that I was seeing my son and other people use that weren't the remote memorizing and mimicking I'd gotten used to?

I ran into the work of Cathy Fosnot and Maarten Dolk, and [their book] Young Mathematicians at Work, and they had pulled from the Netherlands strings. They called them "strings." And they were a series of problems that were in a certain order. The order mattered, the relationship between the problems mattered, and maybe the most important part that I saw was I saw students thinking about the problems and using what they learned and saw and heard from their classmates in one problem, starting to let that impact their work on the next problem. And then they would see that thinking made visible and the conversation between it and then it would impact how they thought about the next problem.

And as I saw those students literally learn before my eyes, I was like, "This is unbelievable!" And honestly, at the very beginning, I didn't really even parse out what was different between maybe one of Fosnot's rich tasks versus her strings versus just a conversation with students. I was just so enthralled with the learning because what I was seeing were the kind of mental actions that I was intrigued with. I was seeing them not only happen live but grow live, develop, like they were getting stronger and more sophisticated because of the series of the order the problems were in, because of that sequence of problems. That was unbelievable. And I was so excited about that that I began to dive in and get more clear on: What is a string of problems?

The reason I call them "problem strings" is I'm K–12. So I will have data strings and geometry strings and—pick one—trig strings, like strings with functions in algebra. But for the purposes of this podcast, there's strings of problems with numbers in them.

Mike: So I have a question, but I think I just want to make an observation first.

The way you described that moment where students are taking advantage of the things that they made sense of in one problem and then the next part of the string offers them the opportunity to use that and to see a set of relationships. I vividly remember the first time I watched someone facilitate a string and feeling that same way, of this routine really offers kids an opportunity to take what they've made sense of and immediately apply it. And I think that is something that I cannot say about all the routines that I've seen, but it was really so clear. I just really resonate with that experience of, what will this do for children?

Pam: Yeah, and if I can offer an additional word in there, it influences their work. We're taking the major relationships, the major mathematical strategies, and we're high-dosing kids with them. So we give them a problem, maybe a problem or two, that has a major relationship involved. And then, like you said, we give them the next one, and now they can notice the pattern, what they learned in the first one or the first couple, and they can let it influence. They have the opportunity for it to nudge them to go, "Hmm. Well, I saw what just happened there. I wonder if it could be useful here. I'm going to tinker with that. I'm going to play with that relationship a little bit." And then we do it again. So in a way, we're taking the relationships that I think, for whatever reason, some of us can wander through life and we could run into the mathematical patterns that are all around us in the low dose that they are all around us, but many of us don't pick up on that low dose and connect them and make relationships and then let it influence when we do another problem.

We need a higher dose. I needed a higher dose of those major patterns. I think most kids do. Problem strings or number strings are so brilliant because of that sequence and the way that the problems are purposely one after the other. Give students the opportunity to, like you said, apply what they've been learning instantly [snaps]. And then not just then, but on the next problem and then sometimes in a particular structure we might then say, "Mm, based on what you've been seeing, what could you do on this last problem?" And we might make that last problem even a little bit further away from the pattern, a little bit more sophisticated, a little more difficult, a little less lockstep, a little bit more where they have to think outside the box but still could apply that important relationship.

Mike: So I have two thoughts, Pam, as I listen to you talk.

One is that for both of us, there's a really clear payoff for children that we've seen in the way that strings are designed and the way that teachers can use them to influence students' thinking and also help kids build a recognition or high-dose a set of relationships that are really important.

The interesting thing is, I taught kindergarten through second grade for most of my teaching career, and you've run the gamut. You've done this in middle school and high school. So I think one of the things that might be helpful is to share a few examples of what a string could look like at a couple different grade levels. Are you OK to share a few?

Pam: You bet. Can I tack on one quick thing before I do?

Mike: Absolutely.

Pam: You mentioned that the payoff is huge for children. I'm going to also suggest that one of the things that makes strings really unique and powerful in teaching is the payoff for adults. Because let's just be clear, most of us—now, not all, but most of us, I think—had a similar experience to me that we were in classrooms where the teacher said, "Do this thing." That's the definition of math is for you to rote memorize these disconnected facts and mimic these procedures. And for whatever reason, many of us just believed that and we did it. Some people didn't. Some of us played with relationships and everything. Regardless, we all kind of had the same learning experience where we may have taken at different places, but we still saw the teacher say, "Do these things. Rote memorize. Mimic."

And so as we now say to ourselves, "Whoa, I've just seen how cool this can be for students, and we want to affect our practice." We want to take what we do, do something—we now believe this could be really helpful, like you said, for children, but doing that's not trivial. But strings make it easier. Strings are, I think, a fantastic differentiated kind of task for teachers because a teacher who's very new to thinking and using relationships and teaching math a different way than they were taught can dive in and do a problem string. Learn right along with your students. A veteran teacher, an expert teacher who's really working on their teacher moves and really owns the landscape of learning and all the things still uses problem strings because they're so powerful. Like, anybody across the gamut can use strings—I just said problem strings, sorry—number strengths—[laughs] strings, all of us no matter where we are in our teaching journey can get a lot out of strings.

Mike: So with all that said, let's jump in. Let's talk about some examples across the elementary span.

Pam: Nice. So I'm going to take a young learner, not our youngest, but a young learner. I might ask a question like, "What is 8 plus 10?" And then if they're super young learners, I expect some students might know that 10 plus a single digit is a teen, but I might expect many of the students to actually say "8, 9, 10, 11, 12," or "10, 11," and they might count by ones given—maybe from the larger, maybe from the whatever. But anyway, we're going to kind of do that. I'm going to get that answer from them. I'm going to write on the board, "8 plus 10 is 18," and then I would have done some number line work before this, but then I'm going to represent on the board: 8 plus 10, jump of 10, that's 18. And then the next problem's going to be something like 8 plus 9. And I'm going to say, "Go ahead and solve it any way you want, but I wonder—maybe you could use the first problem, maybe not." I'm just going to lightly suggest that you consider what's on the board. Let them do whatever they do. I'm going to expect some students to still be counting. Some students are going to be like, "Oh, well I can think about 9 plus 8 counting by ones." I think by 8—"maybe I can think about 8 plus 8. Maybe I can think about 9 plus 9." Some students are going to be using relationships, some are counting. Kids are over the map.

When I get an answer, they're all saying, like, 17. Then I'm going to say, "Did anybody use the first problem to help? You didn't have to, but did anybody?" Then I'm going to grab that kid. And if no one did, I'm going to say, "Could you?" and pause.

Now, if no one sparks at that moment, then I'm not going to make a big deal of it. I'll just go, "Hmm, OK, alright," and I'll do the next problem. And the next problem might be something like, "What's 5 plus 10?" Again, same thing, we're going to get 15. I'm going to draw it on the board.

Oh, I should have mentioned: When we got to the 8 plus 9, right underneath that 8, jump, 10 land on 18, I'm going to draw an 8 jump 9, shorter jump. I'm going to have these lined up, land on the 17. Then I might just step back and go, "Hmm. Like 17, that's almost where the 18 was." Now if kids have noticed, if somebody used that first problem, then I'm going to say, "Well, tell us about that." "Well, miss, we added 10 and that was 18, but now we're adding 1 less, so it's got to be 1 less." And we go, "Well, is 17 one less than 18? Huh, sure enough."

Then I give the next set of problems. That might be 5 plus 10 and then 5 plus 9, and then I might do 7 plus 10. Maybe I'll do 9 next. 9 plus 10 and then 9 plus 9. Then I might end that string. The next problem, the last problem might be, "What is 7 plus 9?" Now notice I didn't give the helper. So in this case I might go, "Hey, I've kind of gave you plus 10. A lot of you use that to do plus 9. I gave you plus 10. Some of you use that to do plus 9, I gave you plus 10. Some of you used that plus 9. For this one, I'm not giving you a helper. I wonder if you could come up with your own helper."

Now brilliantly, what we've done is say to students, "You've been using what I have up here, or not, but could you actually think, 'What is the pattern that's happening?' and create your own helper?" Now that's meta. Right? Now we're thinking about our thinking. I'm encouraging that pattern recognition in a different way. I'm asking kids, "What would you create?" We're going to share that helper. I'm not even having them solve the problem. They're just creating that helper and then we can move from there.

So that's an example of a young string that actually can grow up. So now I can be in a second grade class and I could ask a similar [question]: "Could you use something that's adding a bit too much to back up?" But I could do that with bigger numbers. So I could start with that 8 plus 10, 8 plus 9, but then the next pair might be 34 plus 10, 34 plus 9. But then the next pair might be 48 plus 20 and 48 plus 19. And the last problem of that string might be something like 26 plus 18.

Mike: So in those cases, there's this mental scaffolding that you're creating. And I just want to mark this. I have a good friend who used to tell me that part of teaching mathematics is you can lead the horse to water, you can show them the water, they can look at it, but darn it, do not push their head in the water. And I think what he meant by that is "You can't force it," right?

But you're not doing that with a string. You're creating a set of opportunities for kids to notice. You're doing all kinds of implicit things to make structure available for kids to attend to—and yet you're still allowing them the ability to use the strategies that they have. We might really want them to notice that, and that's beautiful about a string, but you're not forcing. And I think it's worth saying that because I could imagine that's a place where folks might have questions, like, "If the kids don't do the thing that I'm hoping that they would do, what should I do?"

Pam: Yeah, that's a great question. Let me give you another example. And in that example I'll talk about that.

So especially as the kids get older, I'm going to use the same kind of relationship. It's maybe easier for people to hang on to if I stay with the same sort of relationship. So I might say, "Hey everybody. 7 times 8. That's a fact I'm noticing most of us just don't have [snaps] at our fingertips. Let's just work on that. What do you know?" I might get a couple of strategies for kids to think about 7 times 8. We all agree it's 56.

Then I might say, "What's 70 times 8?" And then let kids think about that. Now, this would be the first time I do that, but if we've dealt with scaling times 10 at all, if I have 10 times the number of whatever the things is, then often kids will say, "Well, I've got 10 times 7 is 70, so then 10 times 56 is 560." And then the next problem might be, "I wonder if you could think about 69 times 8. If we've got 70 eights, can I use that to help me think about 69 eights?" And I'm saying that in a very specific way to help ping on prior knowledge. So then I might do something similar. Well, let's pick another often missed facts, I don't know, 6 times 9. And then we could share some strategies on how kids are thinking about that. We all agree it's 54. And then I might say, "Well, could you think about 6 times 90?" I'm going to talk about scaling up again. So that would be 540. Now I'm going really fast. But then I might say, "Could we use that to help us think about 6 times 89?" I don't know if you noticed, but I sort of swapped. I'm not thinking about 90 sixes to 89 sixes. Now I'm thinking about 6 nineties to help me think about 6 eighty-nines. So that's a little bit of a—we have to decide how we're going to deal with that. I'll kind of mess around with that. And then I might have what we call that clunker problem at the end. "Notice that I've had a helper: 7 times 8, 70 times 8. A lot of you use that to help you think about 69 times 8. Then I had a helper: 6 times 9, 6 times 90. A lot of you use that to help you think about 6 times 89. What if I don't give you those helpers? What if I had something like"—now I'm making this up off the cuff here, like—"9 times 69. 9 times 69. Could you use relationships we just did?"

Now notice, Mike, I might've had kids solving all those problems using an algorithm. They might've been punching their calculator, but now I'm asking the question, "Could you come up with these helper problems?" Notice how I'm now inviting you into a different space. It's not about getting an answer. I'm inviting you into, "What are the patterns that we've been establishing here?" And so what would be those two problems that would be like the patterns we've just been using? That's almost like saying when you're out in the world and you hit a problem, could you say to yourself, "Hmm, I don't know that one, but what do I know? What do I know that could help me get there?" And that's math-ing.

Mike: So, you could have had a kid say, "Well, I'm not sure about how—I don't know the answer to that, but I could do 9 times 60, right?" Or "I could do 10 times"—I'm thinking—"10 times 69." Correct?

Pam: Yes, yes. In fact, when I gave that clunker problem, 9 times 69, I said to myself, "Oh, I shouldn't have said 9 because now you could go either direction." You could either "over" either way. To find 9 I can do 10, or to find 69 I can do 70. And then I thought, "Ah, we'll go with it because you can go either way." So I might want to focus it, but I might not. And this is a moment where a novice could just throw it out there and then almost be surprised. "Whoa, they could go either direction." And an expert could plan, and be like, "Is this the moment where I want lots of different ways to go? Or do I want to focus, narrow it a little bit more, be a little bit more explicit?" It's not that I'm telling kids, but I'm having an explicit goal. So I'm maybe narrowing the field a little bit. And maybe the problem could have been 7 times 69, then I wouldn't have gotten that other "over," not the 10 to get 9. Does that make sense?

Mike: It absolutely does. What you really have me thinking about is NCTM's [National Council of Teachers of Mathematics'] definition of "fluency," which is "accuracy, efficiency, and flexibility." And the flexibility that I hear coming out of the kinds of things that kids might do with a string, it's exciting to imagine that that's one of the outcomes you could get from engaging with strings.

Pam: Absolutely. Because if you're stuck teaching memorizing algorithms, there's no flexibility, like none, like zilch. But if you're doing strings like this, kids have a brilliant flexibility. And one of the conversations I'd want to have here, Mike, is if a kid came up with 10 times 69 to help with 9 times 69, and a different kid came up with 9 times 70 to help with 9 times 69, I would want to just have a brief conversation: "Which one of those do you like better, class, and why?" Not that one is better than the other, but just to have the comparison conversation. So the kids go, "Huh, I have access to both of those. Well, I wonder when I'm walking down the street, I have to answer that one: Which one do I want my brain to gravitate towards next time?" And that's mathematical behavior. That's mathematical disposition to do one of the strands of proficiency. We want that productive disposition where kids are thinking to themselves, "I own relationships. I just got to pick a good one here to—what's the best one I could find here?" And try that one, then try that one. "Ah, I'll go with this one today."

Mike: I love that.

As we were talking, I wanted to ask you about the design of the string, and you started to use some language like "helper problems" and "the clunker." And I think that's really the nod to the kinds of features that you would want to design into a string. Could you talk about either a teacher who's designing their own string—what are some of the features?—or a teacher who's looking at a string that they might find in a book that you've written or that they might find in, say, the Bridges curriculum? What are some of the different problems along the way that really kind of inform the structure?

Pam: So you might find it interesting that over time, we've identified that there's at least five major structures to strings, and the one that I just did with you is kind of the easiest one to facilitate. It's the easiest one to understand where it's going, and it's the helper-clunker structure. So the helper-clunker structure is all about, "I'm going to give you a helper problem that we expect all kids can kind of hang on." They have some facility with, enough that everybody has access to. Then we give you a clunker that you could use that helper to inform how you could solve that clunker problem. In the first string I did with you, I did a helper, clunker, helper, clunker, helper, clunker, clunker. And the second one we did, I did helper, helper, clunker, helper, helper, clunker, clunker. So you can mix and match kind of helpers and clunkers in that, but there are other major structures of strings.

If you're new to strings, I would dive in and do a lot of helper-clunker strings first. But I would also suggest—I didn't create my own strings for a long time. I did prewritten [ones by] Cathy Fosnot from the Netherlands, from the Freudenthal Institute. I was doing their strings to get a feel for the mathematical relationships for the structure of a string. I would watch videos of teachers doing it so I could get an idea of, "Oh, that move right there made all the difference. I see how you just invited kids in, not demand what they do." The idea of when to have paper and pencil and when not, and just lots of different things can come up that if you're having to write the string as well, create the string, that could feel insurmountable.

So I would invite anybody out listening that's like, "Whoa, this seems kind of complicated," feel free to facilitate someone else's prewritten strings. Now I like mine. I think mine are pretty good. I think Bridges has some pretty good ones. But I think you'd really gain a lot from facilitating prewritten strings.

Can I make one quick differentiation that I'm running into more and more? So I have had some sharp people say to me, "Hey, sometimes you have extra problems in your string. Why do you have extra problems in your string?" And I'll say—well, at first I said, "What do you mean?" Because I didn't know what they were talking about. Are you telling me my string's bad? Why are you dogging my string? But what they meant was, they thought a string was the process a kid—or the steps, the relationships a kid used to solve the last problem. Does that make sense?

Mike: It does.

Pam: And they were like, "You did a lot of work to just get that one answer down there." And I'm like, "No, no, no, no, no, no. A problem string or a number string, a string is an instructional routine. It is a lesson structure. It's a way of teaching. It's not a record of the relationships a kid used to solve a problem." In fact, a teacher just asked—we run a challenge three times a year. It's free. I get on and just teach. One of the questions that was asked was, "How do we help our kids write their own strings?" And I was like, "Oh, no, kids don't write strings. Kids solve problems using relationships." And so I think what the teachers were saying was, "Oh, I could use that relationship to help me get this one. Oh, and then I can use that to solve the problem." As if, then, the lesson's structure, the instructional routine of a string was then what we want kids to do is use what they know to logic their way through using mathematical relationships and connections to get answers and to solve problems. That record is not a string, that record is a record of their work. Does that make sense, how there's a little difference there?

Mike: It totally does, but I think that's a good distinction. And frankly, that's a misunderstanding that I had when I first started working with strings as well. It took me a while to realize that the point of a string is to unveil a set of relationships and then allow kids to take them up and use them. And really it's about making these relationships or these problem solving strategies sticky, right? You want them to stick. We could go back to what you said. We're trying to high-dose a set of relationships that are going to help kids with strategies, not only in this particular string, but across the mathematical work they're doing in their school life.

Pam: Yes, very well said. So for example, we did an addition "over" relationship in the addition string that I talked through, and then we did a multiplication "over" set of relationships and multiplication. We can do the same thing with subtraction. We could have a subtraction string where the helper problem is to subtract a bit too much. So something like 42 minus 20, and then the next problem could be 42 minus 19. And we're using that: I'm going to subtract a bit too much and then how do you adjust? And hoo, after you've been thinking about addition "over," subtraction "over" is quite tricky. You're like, "Wait, why are we adding what we're subtracting?" And it's not about teaching kids a series of steps. It's really helping them reason. "Well, if I give you—if you owe me 19 bucks and I give you a $20 bill, what are we going to do?" "Oh, you've got to give me 1 back." Now that's a little harder today because kids don't mess around with money. So we might have to do something that feels like they can—or help them feel money. That's my personal preference. Let's do it with money and help them feel money.

So one of the things I think is unique to my work is as I dove in and started facilitating other people's strings and really building my mathematical relationships and connections, I began to realize that many teachers I worked with, myself included, thought, "Whoa, there's just this uncountable, innumerable wide universe of all the relationships that are out there, and there's so many strategies, and anything goes, and they're all of equal value." And I began to realize, "No, no, no, there's only a small set of major relationships that lead to a small set of major strategies." And if we can get those down, kids can solve any problem that's reasonable to solve without a calculator, but in the process, building their brains to reason mathematically. And that's really our goal, is to build kids' brains to reason mathematically. And in the process we're getting answers. Answers aren't our goal. We'll get answers, sure. But our goal is to get them to build that small set of relationships because that small set of strategies now sets them free to logic their way through problems. And bam, we've got kids math-ing using the mental actions of math-ing.

Mike: Absolutely. You made me think about the fact that there's a set of relationships that I can apply when I'm working with numbers Under 20. There's a set of relationships, that same set of relationships, I can apply and make use of when I'm working with multidigit numbers, when I'm working with decimals, when I'm working with fractions. It's really the relationships that we want to expose and then generalize and recognize this notion of going over or getting strategically to a friendly number and then going after that or getting to a friendly number and then going back from that. That's a really powerful strategy, regardless of whether you're talking about 8 and 3 or whether you're talking about adding unit fractions together. Strings allow us to help kids see how that idea translates across different types of numbers.

Pam: And it's not trivial when you change a type of number or the number gets bigger. It's not trivial for kids to take this "over" strategy and to be thinking about something like 2,467 plus 1,995—and I know I just threw a bunch of numbers out, on purpose. It's not trivial for them to go, "What do I know about those numbers? Can I use some of these relationships I've been thinking about?" Well, 2,467, that's not really close to a friendly number. Well, 1,995 is. Bam. Let's just add 2,000. Oh, sweet. And then you just got to back up 5. It's not trivial for them to consider, "What do I know about these two numbers, and are they close to something that I could use?" That's the necessary work of building place value and magnitude and reasonableness. We've not known how to do that, so in some curriculum we create our whole extra unit that's all about place value reasonableness. Now we have kids that are learning to rote memorize, how to estimate by round. I mean there's all this crazy stuff that we add on when instead we could actually use strings to help kids build that stuff naturally kind of ingrained as we are learning something else.

Can I just say one other thing that we did in my new book? Developing Mathematical Reasoning: Avoiding the Trap of Algorithms. So I actually wrote it with my son, who is maybe the biggest impetus to me diving into the research and figuring out all of this math-ing and what it means. He said, as we were writing, he said, "I think we could make the point that algorithms don't help you learn a new algorithm." If you learn the addition algorithm and you get good at it and you can do all the addition and columns and all the whatever, and then when you learn the subtraction algorithm, it's a whole new thing. All of a sudden it's a new world, and you're doing different—it looks the same at the beginning. You line those numbers still up and you're still working on that same first column, but boy, you're doing all sorts—now you're crossing stuff out. You're not just little ones, and what? Algorithms don't necessarily help you learn the next algorithm. It's a whole new experience. Strategies are synergistic. If you learn a strategy, that helps you learn the next set of relationships, which then refines to become a new strategy. I think that's really helpful to know, that we can—strategies build on each other. There's synergy involved. Algorithms, you got to learn a new one every time.

Mike: And it turns out that memorizing the dictionary of mathematics is fairly challenging.

Pam: Indeed [laughs], indeed. I tried hard to memorize that. Yeah.

Mike: You said something to me when we were preparing for this podcast that I really have not been able to get out of my mind, and I'm going to try to approximate what you said. You said that during the string, as the teacher and the students are engaging with it, you want students' mental energy primarily to go into reasoning. And I wonder if you could just explicitly say, for you at least, what does that mean and what might that look like on a practical level?

Pam: So I wonder if you're referring to when teachers will say, "Do we have students write? Do we not have them write?" And I will suggest: "It depends. It's not if they write; it's what they write that's important."

What do I mean by that? What I mean is if we give kids paper and pencil, there is a chance that they're going to be like, "Oh, thou shalt get an answer. I'm going to write these down and mimic something that I learned last year." And put their mental energy either into mimicking steps or writing stuff down. They might even try to copy what you've been representing strategies on the board. And their mental effort either goes into mimicking, or it might go into copying.

What I want to do is free students up [so] that their mental energy is, how are you reasoning? What relationships are you using? What's occurring to you? What's front and center and sort of occurring? Because we're high-dosing you with patterns, we're expecting those to start happening, and I'm going to be saying things, giving that helper problem. "Oh, that's occurring to you? It's almost like it's your idea—even though I just gave you the helper problem!" It's letting those ideas bubble up and percolate naturally and then we can use those to our advantage. So that's what I mean when [I say] I want mental energy into "Hmm, what do I know, and how can I use what I know to logic my way through this problem?" And that's math-ing. Those are the mental actions of mathematicians, and that's where I want kids' mental energy.

Mike: So I want to pull this string a little bit further. Pun 100% intended there. Apologies to listeners.

What I find myself thinking about is there've got to be some do's and don'ts for how to facilitate a string that support the kind of reasoning and experience that you've been talking about. I wonder if you could talk about what you've learned about what you want to do as a facilitator when you're working with a string and maybe what you don't want to do.

Pam: Yeah, absolutely. So a good thing to keep in mind is you want to keep a string snappy. You don't want a lot of dead space. You don't want to put—one of the things that we see novice, well, even sometimes not-novice, teachers do, that's not very helpful, is they will put the same weight on all the problems.

So I'll just use the example 8 plus 10, 8 plus 9, they'll—well, let me do a higher one. 7 times 8, 70 times 8. They'll say, "OK, you guys, 7 times 8. Let's really work on that. That's super hard." And kids are like, "It's 56." Maybe they have to do a little bit of reasoning to get it, because it is an often missed fact, but I don't want to land on it, especially—what was the one we did before? 34 plus 10. I don't want to be like, "OK, guys, phew." If the last problem on my string is 26 plus 18, I don't want to spend a ton of time. "All right, everybody really put all your mental energy in 36 plus 10" or whatever I said. Or, let's do the 7 times 8 one again. So, "OK, everybody, 7 times 8, how are you guys thinking about that?" Often we're missing it. I might put some time into sharing some strategies that kids use to come up with 7 times 8 because we know it's often missed. But then when I do 70 times 8, if I'm doing this string, kids should have some facility with times 10. I'm not going to be like, "OK. Alright, you guys, let's see what your strategies are. Right? Everybody ready? You better write something down on your paper. Take your time, tell your neighbor how…." Like, it's times 10. So you don't want to put the same weight—as in emphasis and time, wait time—either one on the problems that are kind of the gimmes, we're pretty sure everybody's got this one. Let's move on and apply it now in the next one. So there's one thing. Keep it snappy. If no one has a sense of what the patterns are, it's probably not the right problem string. Just bail on it, bail on it. You're like, "Let me rethink that. Let me kind of see what's going on." If, on the other hand, everybody's just like, "Well, duh, it's this" and "duh, it's that," then it's also probably not the right string. You probably want to up the ante somehow.

So one of the things that we did in our problem string books is we would give you a lesson and give you what we call the main string, and we would write up that and some sample dialogs and what the board could look like when you're done and lots of help. But then we would give you two echo strings. Here are two strings that get at the same relationships with about the same kind of numbers, but they're different and it will give you two extra experiences to kind of hang there if you're like, "Mm, I think my kids need some more with exactly this." But we also then gave you two next-step strings that sort of up the ante. These are just little steps that are just a little bit more to crunch on before you go to the next lesson that's a bit of a step up, that's now going to help everybody increase. Maybe the numbers got a little bit harder. Maybe we're shifting strategy. Maybe we're going to use a different model. I might do the first set of strings on an area model if I'm doing multiplication. I might do the next set of strings in a ratio table. And I want kids to get used to both of those.

When we switch up from the 8 string to the next string, kind of think about only switching one thing. Don't up the numbers, change the model, and change the strategy at the same time. Keep two of those constant. Stay with the same model, maybe up the numbers, stay with the same strategy. Maybe if you're going to change strategies, you might back up the numbers a little bit, stick with the model for a minute before you switch the model before you go up the numbers. So those are three things to consider. Kind of—only change up one of them at a time or kids are going to be like, "Wait, what?" Kids will get higher dosed with the pattern you want them to see better if you only switch one thing at a time.

Mike: Part of what you had me thinking was it's helpful, whether you're constructing your own string or whether you're looking at a string that's in a textbook or a set of materials, it's still helpful to think about, "What are the variables at play here?" I really appreciated the notion that they're not all created equal. There are times where you want to pause and linger a little bit that you don't need to spend that exact same amount of time on every clunker and every helper. There's a critical problem that you really want to invest some time in at one point in the string. And I appreciated the way you described, you're playing with the size of the number or the complexity of the number, the shift in the model, and then being able to look at those kinds of things and say, "What all is changing?" Because like you said, we're trying to kind of walk this line of creating a space of discovery where we haven't suddenly turned the volume up to 11 and made it really go from like, "Oh, we discovered this thing, now we're at full complexity," and yet we don't want to have it turned down to, "It's not even discovery because it's so obvious that I knew it immediately. There's not really anything even to talk about."

Pam: Nice. Yeah, and I would say we want to be right on the edge of kids' own proximal development, right on the edge. Right on the edge where they have to grapple with what's happening. And I love the word "grapple." I've been in martial arts for quite a while, and grappling makes you stronger. I think sometimes people hear the word "struggle" and they're like, "Why would you ever want kids to struggle?" I don't know that I've met anybody that ever hears the word "grapple" as a negative thing. When you "grapple," you get stronger. You learn. So I want kids right on that edge where they are grappling and succeeding. They're getting stronger. They're not just like, "Let me just have you guess what's in my head." You're off in the field and, "Sure hope you figure out math, guys, today." It's not that kind of discovery that people think it is. It really is: "Let me put you in a place where you can use what you know to notice maybe a new pattern and use it maybe in a new way. And poof! Now you own those relationships, and let's build on that." And it continues to go from there.

When you just said—the equal weight thing, let me just, if I can—there's another, so I mentioned that there's at least five structures of problem strings. Let me just mention one other one that we like, to give you an example of how the weight could change in a string. So if I have an equivalent structure, an equivalent structure looks like: I give a problem, and an example of that might be 15 times 18. Now I'm not going to give a helper; I'm just going to give 15 times 18. If I'm going to do this string, we would have developed a few strategies before now. Kids would have some partial products going on. I would probably hope they would have an "over," I would've done partial products over and probably, what I call "5 is half a 10."

So for 15 times 18, they could use any one of those. They could break those up. They could think about twenty 15s to get rid of the extra two to have 18, 15. So in that case, I'm going to go find a partial product, an "over" and a "5 is half a 10," and I'm going to model those. And I'm going to go, "Alright, everybody clear? Everybody clear on this answer?" Then the next problem I give—so notice that we just spent some time on that, unlike those helper clunker strings where the first problem was like a gimme, nobody needed to spend time on that. That was going to help us with the next one.

In this case, this one's a bit of a clunker. We're starting with one that kids are having to dive in, chew on. Then I give the next problem: 30 times 9. So I had 15 times 18 now 30 times 9. Now kids get a chance to go, "Oh, that's not too bad. That's just 3 times 9 times 10. So that's 270. Wait, that was the answer to the first problem. That was probably just coincidence. Or was it?" And now especially if I have represented that 15 times 18, one of those strategies with an area model with an open array, now when I draw the 30 by 9, I will purposely say, "OK, we have the 15 by 18 up here. That's what that looked like. Mm, I'll just use that to kind of make sure the 30 by 9 looks like it should. How could I use the 15 by 18? Oh, I could double the 15? OK, well here's the 15. I'm going to double that. Alright, there's the 30. Well, how about the 9? Oh, I could half? You think I should half? OK. Well I guess half of 18. That's 9."

So I've just helped them. I've brought out, because I'm inviting them to help me draw it on the board. They're thinking about, "Oh, I just half that side, double that side. Did we lose any area? Oh, maybe that's why the products are the same. The areas of those two rectangles are the same. Ha!" And then I give the next problem. Now I give another kind of clunker problem and then I give its equivalent. And again, we just sort of notice: "Did it happen again?" And then I might give another one and then I might end the string with something like 3.5 times—I'm thinking off the cuff here, 16. So 3.5 times 16. Kids might say, "Well, I could double 3.5 to get 7 and I could half the 16 to get 8, and now I'm landing on 7 times 8." And that's another way to think about 3.5 times 16. Anyway, so, equivalent structure is also a brilliant structure that we use primarily when we're trying to teach kids what I call the most sophisticated of all of the strategies. So like in addition, give and take, I think, is the most sophisticated addition. In subtraction, constant difference. In multiplication, there's a few of them. There's doubling and having, I call it flexible factoring to develop those strategies. We often use the equivalent structure, like what's happening here? So there's just a little bit more about structure.

Mike: There's a bit of a persona that I've noticed that you take on when you're facilitating a string. I'm wondering if you can talk about that or if you could maybe explain a little bit because I've heard it a couple different times, and it makes me want to lean in as a person who's listening to you. And I suspect that's part of its intent when it comes to facilitating a string. Can you talk about this?

Pam: So I wonder if what you're referring to, sometimes people will say, "You're just pretending you don't know what we're talking about." And I will say, "No, no, I'm actually intensely interested in what you're thinking. I know the answer, but I'm intensely interested in what you're thinking." So I'm trying to say things like, "I wonder." "I wonder if there's something up here you could use to help. I don't know. Maybe not. Mm. What kind of clunker could—or helper could you write for this clunker?"

So I don't know if that's what you're referring to, but I'm trying to exude curiosity and belief that what you are thinking about is worth hearing about. And I'm intensely interested in how you're thinking about the problem and there's something worth talking about here. Is that kind of what you're referring to?

Mike: Absolutely.

OK. We're at the point in the podcast that always happens, which is: I would love to continue talking with you, and I suspect there are people who are listening who would love for us to keep talking. We're at the end of our time. What resources would you recommend people think about if they really want to take a deeper dive into understanding strings, how they're constructed, what it looks like to facilitate them. Perhaps they're a coach and they're thinking about, "How might I apply this set of ideas to educators who are working with kindergartners and first graders, and yet I also coach teachers who are working in middle school and high school." What kind of resources or guidance would you offer to folks?

Pam: So the easiest way to dive in immediately would be my brand-new book from Corwin. It's called Developing Mathematical Reasoning: Avoiding the Trap of Algorithms. There's a section in there all about strings. We also do a walk-through where you get to feel a problem string in a K–2 class and a 3–5 [class]. And well, what we really did was counting strategies, additive reasoning, multiplicative reasoning, proportional reasoning, and functional reasoning. So there's a chapter in there where you go through a functional reasoning problem string. So you get to feel: What is it like to have a string with real kids? What's on the board? What are kids saying? And then we link to videos of those. So from the book, you can go and see those, live, with real kids, expert teachers, like facilitating good strings. If anybody's middle school, middle school coaches: I've got building powerful numeracy and lessons and activities for building powerful numeracy. Half of the books are all problem strings, so lots of good resources.

If you'd like to see them live, you could go to mathisfigureoutable.com/ps, and we have videos there that you can watch of problem strings happening.

If I could mention just one more, when we did the K–12, Developing Mathematical Reasoning, Avoiding the Trap of Algorithms, that we will now have grade band companion books coming out in the fall of '25. The K–2 book will come out in the spring of '26. The [grades] 3–5 book will come out in the fall of '26. The 6–8 book will come out and then six months after that, the 9–12 companion book will come out. And those are what to do to build reasoning, lots of problem strings and other tasks, rich tasks and other instructional routines to really dive in and help your students reason like math-y people reason because we are all math-y people.

Mike: I think that's a great place to stop. Pam, thank you so much for joining us. It's been a pleasure talking with you.

Pam: Mike, it was a pleasure to be on. Thanks so much.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.

© 2025 The Math Learning Center | www.mathlearningcenter.org