Rounding Up

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

DeAnn Huinker & Melissa Hedges, Math Trajectories for Young Learners, Part 1 ROUNDING UP: SEASON 4 | EPISODE 14 Research confirms that early mathematics experiences play a more significant role than we once imagined. Studies suggest that specific number competencies in 4-year-olds are strong predictors of fifth grade mathematics success. So what does it look like to provide meaningful mathematical experiences for our youngest learners? Today, we'll explore this question with DeAnn Huinker from UW-Milwaukee and Melissa Hedges from the Milwaukee Public Schools. BIOGRAPHY Dr. DeAnn Huinker is a professor of mathematics education in the Department of Teaching and Learning and directs the University of Wisconsin-Milwaukee Center for Mathematics and Science Education Research. Dr. Huinker teaches courses in mathematics education at the early childhood, elementary, and middle school levels. Dr. Melissa Hedges is a curriculum specialist who supports K–5 and K–8 schools for the Milwaukee Public Schools. RESOURCES Math Trajectories for Young Learners [https://www.nctm.org/Store/Products/Math-Trajectories-for-Young-Learners/] book by DeAnn Huinker and Melissa Hedges Learning Trajectories [https://www.learningtrajectories.org/] website, featuring the work of Doug Clements and Julie Sarama School Readiness and Later Achievement [https://pubmed.ncbi.nlm.nih.gov/18020822/] journal article by Greg Duncan and colleagues Early Math Trajectories: Low‐Income Children's Mathematics Knowledge From Ages 4 to 11 [https://onlinelibrary.wiley.com/doi/abs/10.1111/cdev.12662] journal article by Bethany Rittle-Johnson and colleagues TRANSCRIPT Mike Wallus: Welcome back to the podcast, DeAnn and Melissa. You have both been guests previously. It is a pleasure to have both of you back with us again to discuss your new book, Math Trajectories for Young Learners [https://www.nctm.org/Store/Products/Math-Trajectories-for-Young-Learners/]. Melissa Hedges: Thank you for having us. We're both very excited to be here. DeAnn Huinker: Yes, I concur. Good to see you and be here again. Mike: So DeAnn, I think what I'd like to do is just start with an important grounding question. What's a trajectory? DeAnn: That's exactly where we need to start, right? So as I think about, "What are learning trajectories?," I always envision them as these road maps of children's mathematical development. And what makes them so compelling is that these learning pathways are highly predictable. We can see where children are in their learning, and then we can be more intentional in our teaching when we know where they are currently at. But if I kind of think about the development of learning trajectories, they really are based on weaving together insights from research and practice to give us this clear picture of the typical development of children's learning. And as we always think about these learning trajectories, there are three main components. The first component is a mathematical goal. This is the big ideas of math that children are learning. For example, counting, subitizing, decomposing shapes. The second component of a learning trajectory are developmental progressions. This is really the heart of a trajectory. And the progression lays out a sequence of distinct levels of thinking and reasoning that grow in mathematical sophistication. And then the third component are activities and tasks that align to and support children's movement along that particular trajectory. Now, it's really important that we point out the learning trajectories that we use in our work with teachers and children were developed by Doug Clements and Julie Sarama. So we have taken their trajectories and worked to make them more usable and applicable for teachers in our area. So what Doug and Julie did is they mapped out children's learning starting at birth—when children are just-borns, 1-year-olds, 2-year-olds—and they mapped it out up till about age 8. And right now, last count, they have about 20 learning trajectories. And they're in different topics like number, operations, geometry, and measurement. And we have to put in a plug. They have a wonderful website. It's learningtrajectories.org [http://learningtrajectories.org]. We go there often to learn more about the trajectories and to get ideas for activities and tasks. Now, we're talking about this new book we have on math trajectories for young children. And in the book, we actually take a deep dive into just four of the trajectories. We look at counting, subitizing, composing numbers, and adding and subtracting. So back to your original question: What are they? Learning trajectories are highly predictable roadmaps of children's math learning that we can use to inform and support developmentally appropriate instruction. Mike: That's an incredibly helpful starting point. And I want to ask a follow-up just to get your thinking on the record. I wonder if you have thoughts about how you imagine educators could or should make use of the trajectories. Melissa: This is Melissa. I'll pick up with that question. So I'll piggyback on DeAnn's response and thinking around this highly predictable nature of a trajectory as a way to ground my first comment and that we want to always look at a trajectory as a tool. So it's really meant as an important tool to help us understand where a child is and their thinking right now, and then what those next steps might be to push for some deeper mathematical understanding. So the first thing that when we work with teachers that we like to keep in mind, and one of the things that actually draw teachers to the trajectories is that they're strength-based. So it's not what a child can't do. It's what a child can do right now based off of experience and opportunity that they've had. We also really caution against using our trajectories as a way to kind of pigeonhole kids or rank kids or label kids because what we know is that as children have more experience and opportunity, they grow and they learn and they advance along that trajectory. So really it's a tool that's incredibly powerful when in the hands of a teacher that understands how they work to be able to think about where are the children right now in their classroom and what can they do to advance them. And I think the other point that I would emphasize other than what moves children along is experience and opportunity. Children are going to be all over on the trajectory—that's been our experience—and they're in the same classroom. And it's not that some can't and some won't and some can; it's just some need more experience and some need more opportunity. So it's really opened up the door many ways to view a more equitable approach to mathematics instruction. The other thing that I would say is, and DeAnn and I had big conversations about this when we were first using the trajectories, is: Do we look at the ages? So the trajectories that Clements and Sarama develop do have age markers on them. And we were a bit back and forth on, "Do we use them?," "Do we not?," knowing that mathematical growth is meant to be viewed through a developmental lens. So we had them on and then we had them off and then we shared them with teachers and many of our projects and the teachers were like, "No, no, no, put the ages back on. Trust us. We'll use them well." (laughs) And so the ages are back onto the trajectories. And what we've noticed is that they really do help us understand how to take either intentional steps forward or intentional steps back, depending on what kids are showing us on that trajectory. The other spot that I would maybe put a plugin for on where we could use a trajectory and what would be an appropriate use for it would be for our special educators out there and to really start to use them to support clear, measurable IEP goals grounded in a developmental progress. So that's kind of what our rule of thumb would be around a "should" and "shouldn't" with the trajectories. Mike: That's really helpful. You mentioned the notion of experiences and opportunities being critical. So I wanted to take perhaps a bit of a detour and talk about what research tells us about the impact of early mathematics experiences, what impact that has on children. I wonder if you could share some of the research that you cite in the book with our listeners. DeAnn: Sure. This is DeAnn, and in the book we cite research throughout all of the chapters and aligned to all of the different trajectories. But as we think about our work, there really are a few studies that we anchor in, always, as we think about children's learning. And the research evidence is really clear that early mathematics matters. The math that children learn in these early years in prekindergarten, kindergarten, first grade—I mean, we're talking 4-, 5-, 6-year-olds, 7-year-olds—that their math learning is really more important than a lot of people think it is. OK? So as we think about these kind of anchor studies that we look at, one of the major studies in this area is from Greg Duncan and his colleagues [https://pubmed.ncbi.nlm.nih.gov/18020822/], and there was a study published in 2007. And what they did is they examined data from thousands of children drawing information from six large-scale studies, and they found that the math knowledge and abilities of 4- and 5-year-olds was the strongest predictor of later achievement. I mean, 4- and 5-year-olds, that's just as they're starting school. Mike: Wow. DeAnn: Yeah. One of the surprising findings was that they found early math knowledge and abilities was a stronger predictor than social emotional skills, stronger than family background, and stronger than family income. That it was the math knowledge that was predictive. Mike: That's incredible. DeAnn: Yes. A couple other surprising things from this study was that early math was a stronger predictor than early reading. Now, we know reading is really important, and we know reading gets a lot of emphasis in the early grades, but math is a stronger predictor than reading. And then one last thing I'll say about this study is that early math not only predicts later math achievement, it also predicts later reading achievement. So that is always a surprise as we share that information with teachers, that early math seems to matter as much and perhaps more than early reading abilities. There's a couple other studies I'll share with you as well. So there's this body of research that talks about [how] early math is very predictive of later learning, but we're teachers, we're educators. We like to know, "Well, what math seems to be most important?" So there was a study in 2016 [https://onlinelibrary.wiley.com/doi/abs/10.1111/cdev.12662] that looked at children's math learning in prekindergarten, 4-year-olds, and then looked at their learning again back in fifth grade. And what was unique about this study is they looked closely at what specific math topics seemed to matter the most. And what they found was that advanced number competencies were the strongest predictors of later achievement. Now, what are advanced number competencies? So these are the three that really stood out as being important. One was being able to count a set of objects with cardinality. So in other words, counting things, not just being able to recite a count sequence, no. So not verbal rote counting, but actually counting things, putting those numbers to objects. Another thing that they found [that] was really important was being able to count forward from any number. So if I said, "Start at 7 and keep counting," "Start at 23 and keep counting," that that was predictive of later learning. And the reason for that is when kids can count forward from a number, it helps them understand the structure of the number system, something we're always working on. And then the third thing that they found as part of advanced number competencies was conceptual subitizing. Now, what that is, is being able to see a number such as 5 as composed of subgroups, like 5 being composed of 4 and 1 or 3 and 2. So subitizing is being able to see the parts of a number, and that was really important for these 4-year-olds to begin working on for later learning. All right. One more, Mike, that I can share? Mike: Fire away! Yes. DeAnn: OK. So this last area of research that I want to share is actually really important as we think about the work of teachers in kindergarten and first grade in particular. So what these researchers did is they looked at children's learning at the beginning of kindergarten and then at the end of first grade. So, wow, think of the math kids learn from 5, 6 years old. And they found that these gains in what children can do was more predictive of later achievement than just what knowledge they had coming in. So learning gains, what children do and learn in math in kindergarten and first grade, is predictive of their mathematical success up through third grade. And then another study took it even further and said: Wait a minute, what they learn in kindergarten and first grade even predicts children's math achievement into high school. So there's just a growing body of research and evidence that early math is really important. The math learning of 4-year-olds, 5-year-olds, 6-year-olds, and 7-year-olds really builds this foundation that determines children's mathematical success many years later. Mike: This feels like a really great segue to a conversation about what it means to provide students opportunities for meaningful counting. That feels particularly significant when I heard all of the ideas that you were sharing in the research. I'm wondering if you could talk about the features of a meaningful counting experience. If we were to try to break that down and think about: What does that mean? What does that look like? What types of experiences count as meaningful when it comes to counting? Could you all talk about that a little bit? Melissa: Yeah, that's a great question, Mike. This is Melissa. So I think what's interesting about the idea of meaningful counting is, the more DeAnn and I studied the trajectory and spent time working with teachers and students, we came to the conclusion that the counting trajectory in particular is anchored, or a cornerstone of that counting trajectory is really meaningful counting. That once a skill is acquired—and we'll talk a little bit more about meaningful counting—but once that skill is acquired, it just builds and develops as kids grow and have more experience with number and quantity. So when we think about meaningful counting, the phrase that we like to use is that "Numbers represent quantity." And it's just not that kids are saying numbers out loud, it's that when they say "5," they know what 5 means. They know how many that is. They can connect it to a context that they can go grab five of something. They might know that 5 is bigger than 2 or that 10 is bigger than 5. So they start to really play with this idea of quantity. And specifically when we're talking about kids engaging in meaningful counting, there's really key skills and understandings that we're looking and watching for as children count. The first one DeAnn already alluded to, is this idea of cardinality. So when I count how many I have—1, 2, 3, 4, 5—if that's the size of my set, when someone asks me, "How many is it?," I can say "5" without needing to go back and count. So I can hold that quantity. Another one is stable count sequence. So we used to call it rote count sequence. And again, DeAnn referenced the idea that, really, when we're asking kids to count, we're asking more than just saying numbers. So we think about the stability and the confidence in their counting. One of the pieces that we've started to really watch very carefully and think carefully about with our children as we're watching many of them count is their ability to organize. So it's not the job of the teacher to organize the counter, to tell the child how to lay out the counters. It really is the work of the child because it brings to bear counting, saying the numbers, maintaining cardinality, as well as sets them up and sets us up to see where they at with that one-to-one correspondence. So can they organize a set of counters in such a way that allows them to say one number, one touch, one object? And then as they continue to coordinate those skills, are they able to say back and hold onto the idea of quantity? So the other ideas that we like to consider, mostly because they're embedded in the trajectory and we've seen them become incredibly important as we work with children, is the idea of producing a set. So when I ask a child, "Can you give me five?," they give me five, or are they able to stop when they get to five? Do they keep counting? Do they pick up a handful of counters and dump it in my hand? So all of those things are what we're looking for as we're thinking about the idea of producing a set. And then finally, even for our youngest ones, we really place a fair importance on the idea of representing a count. So can they demonstrate, can they show on paper what they did or how many they have? So we leave with a very rudimentary math sketch. So if they've counted a collection of five, how would they represent five on that paper? What that allows then the teacher to do is to continue to leverage where the trajectory goes as well as what they know about young children to bring in meaningful experiences tied to writing numbers, tied to having conversations about numbers. So the kids aren't doing worksheets, they're actually documenting something very important to them, which is this collection of whatever it is that they just counted in a way that makes sense to them. And so I think the other part that I like to talk about when we think about meaningful counting is this idea of hierarchical inclusion. It's that idea that children understand that numbers are nested one within each other and that each number in the count sequence is exactly 1 higher than what they said before. So, many times our reference with that is with our teachers are those little nesting dolls. So we think about 1 and then we wrap 2 around it and then we wrap 3 around it. So when we think about the number 3, we're thinking, "Well, it's actually the quantity of 2 and 1 more." And we see that as a really powerful understanding in particular as our children get older and we ask them not just what is 1 more or 1 less, but what is 10 more or 10 less, that they take that and they extend that in meaningful ways. So again, the idea of meaningful counting, regardless of where we are on the trajectory, it's the idea that numbers represent quantities. And the neat thing about the trajectory—the counting trajectory in particular—is that they give us really beautiful markers as to when to watch for these. So we tend to talk about the trajectories as levels. So we'll say at level 6 on our counting trajectory is where we see cardinality first start to kind of show up, where we're starting to look for it. And then we watch that idea of cardinality grow as children get older, as they have more experience and opportunity, and as they work with larger numbers. Mike: That's incredibly helpful. So I think one of the things that really jumped out, and I want to mark this and give you all an opportunity to be a little bit more explicit than you already were—this importance of linking numbers and quantities. And I wonder if you could say a bit more about what you mean, just to make sure that our listeners have a full understanding of why that is so significant. DeAnn: All right, this is DeAnn. I'll jump in and get started, and Melissa can add on. As we first started to study the learning trajectory, the one thing we noticed was the importance of connecting things to quantity. Even some of the original levels didn't necessarily say "quantity," but we anchor our work to developing meaning for our work. And we always think about, even when we're skip-counting, it should be done with objects that we should be able to see skip-counting as quantities, not just as words that I'm reciting. So across the trajectory, we put this huge emphasis on always connecting them to items, to things, or to actions and to movements so that it's not just a word, but that word has some meaning and significance for the child. Mike: I think that takes me to the other bit of language, Melissa, that you said that I want to come back to. You said at one point when you were describing meaningful counting experiences, you said, "One number, one touch, one object." And I wonder if you could unpack that, particularly "one touch," for young children and why that feels significant. Melissa: That's a great question. And I'll come at this through a lens of watching many, many children count and working with lots and lots of teachers. When children are counting a set, many times they'll look and they'll go, "1, 2, 3, 4, 5, 6, 7, 8, 9," and then however many are in the collection, they'll just say, "9" by just looking. And one of the things that we've noticed is that sometimes we need to explicitly give permission to children to do what they need to do with that collection to find out how many. Sometimes they're afraid to touch the items. Sometimes they don't know that they can. And we don't come right out and say, "Go ahead and touch them." But we just say, "Gosh, is there another way that you could find out how many?" And what we notice are some amazing and interesting ways kids organize their collections. So sometimes to be able to get to that "one touch, one, number one object," they'll lay them out in a row. Sometimes they'll lay them out in a circle and they'll mark the one that they started with. Sometimes, with our little guys in particular, we like to give them collections where they have to sit things up, so like, the little counting bears. So if the bears are lying down, the kids will be very intentional in, "I set it up and I count it. I set it up and I count it. " And they all, many times, have to be facing the same direction as well. So the kids are very particular about, "How does this fit into the counting experience?" And I would say that's one thing that's been really significant for us in understanding that it really is the work of the child to do that "one touch, one object, one count" in a way that matters to them. And that a teacher can very easily lay it out and say, "Find out how many. Remember to touch one and tell me the number." Then it's not coming from the child. Then we don't know what they know. So that's been a really, really interesting aspect for us to watch in kids is, "How are they choosing to go into and enter into counting that?" And we look at that as problem solving from our youngest, from our 3-year-olds, all the way up, is: "What are you going to do with that pile of stuff in front of you?" And that's an authentic problem for them, and it's meaningful. Mike: I think what jumps out about that from me is the structure of what you just described is actually an experience and it's an opportunity to make sense of counting versus what perhaps has typically happened, which is a procedure for counting that we're asking kids to replicate and show us again. And what strikes me is you're advocating for a sensemaking opportunity because that's the work of the child. As opposed to, "Let me show you how to do it; you do it again and show it back to me," but what might be missing is meaning or connection to something that's real and that sets up what we think might be a house of cards or at the very least it has significant implications as you described in the research. Melissa: One of the things, Mike, that I would add on that actually I just thought about is, when you were talking about the importance of us letting the children figure out how they want to approach that task of organizing their count, is: It's coming from the child. And Clements and Sarama talk about, the beautiful work about the trajectory is that we see that the mathematics comes from the child and we can nurture that along in developmentally appropriate ways. The other idea that popped into my mind is: It's kind of a parallel to when our children get older and we want to teach them a way to add and a way to subtract. And I'm going to show you how to do it and you follow my procedure. I'm going to show it; you follow my procedure. We know that that's not best practice either. And so we're really looking at: How do we grab onto that idea of number sense and move forward with it in a way that's meaningful with children from as young as 1 and 2 all the way up? Mike: I hope you've enjoyed the first half of our conversation with DeAnn and Melissa as much as I have. We'll release the second half of our conversation on April 9th. 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 [http://www.mathlearningcenter.org]

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

Mike Steele, Pacing Discourse-Rich Lessons ROUNDING UP: SEASON 4 | EPISODE 13 As a classroom teacher, pacing lessons was often my Achilles' heel. If my students were sharing their thinking or working on a task, I sometimes struggled to decide when to move on to the next phase of a lesson. Today we're talking with Mike Steele from Ball State University about several high-leverage practices that educators can use to plan and pace their lessons. BIOGRAPHY Mike Steele is a math education researcher focused on teacher knowledge and teacher learning. He is the past president of the Association of Mathematics Teacher Educators, editor in chief of the Mathematics Teacher Educator journal, and member of the NCTM board of directors. RESOURCES Journal Article * "Pacing a Discourse-Rich Lesson: When to Move On" [https://pubs.nctm.org/view/journals/mtlt/118/11/article-p822.xml] Books * 5 Practices for Orchestrating Productive Mathematics Discussions [https://www.corwin.com/books/5-practices-262956] * The 5 Practices in Practice [Elementary] [https://www.corwin.com/books/5-practices-in-practice-260531] * The 5 Practices in Practice [Middle School] [https://www.corwin.com/books/the-5-practices-on-practice-260532] * The 5 Practices in Practice [High School] [https://www.corwin.com/books/mathematics-discussions-hs-260533] * Coaching the 5 Practices [https://www.corwin.com/books/coaching-the-5-practices-287127] 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 [https://www.corwin.com/books/5-practices-262956] 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 [https://www.corwin.com/books/5-practices-in-practice-260531], middle [https://www.corwin.com/books/the-5-practices-on-practice-260532], and high school [https://www.corwin.com/books/mathematics-discussions-hs-260533]. 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 [https://www.corwin.com/books/coaching-the-5-practices-287127] 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," [http://pubs.nctm.org/view/journals/mtlt/118/11/article-p822.xml] 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 [http://www.mathlearningcenter.org]

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

Kyndall Thomas, Building a Meaningful Understanding of Properties Through Fact Fluency Tasks ROUNDING UP: SEASON 4 | EPISODE 12 Building fluency with multiplication and division is essential for students in the upper elementary grades. This work also presents opportunities to build students' understanding of the algebraic properties that become increasingly important in secondary mathematics. In this episode, we're talking with Kyndall Thomas about practical ways educators can support fluency development and build students' understanding of algebraic properties. BIOGRAPHY Kyndall Thomas serves as a math interventionist and resource teacher with the Oregon Trail School District, focusing on data-driven support and empowering teachers to spark a love of numbers in their students. TRANSCRIPT Mike Wallus: Hi, Kyndall. Welcome to the podcast. I'm really excited to be talking with you today. Kyndall Thomas: Hi, Mike. Thanks for having me. I'm excited to dive into some math talk with you also. Mike: Kyndall, tell us a little bit about your background. What brought you to this work? Kyndall: Yeah. I started in the classroom. I was in upper elementary. I served fifth grade students, and I taught specifically math and science. And then I moved into a more interventionist role where I was a specialist that worked with teachers and also worked with small groups, intervention students. And through that I was able for the first time to really develop an understanding of that mathematical progression that happens at each grade level and the formative things that are introduced at the lower elementary [grades] and then kind of fade out, but still need to be brought back at the upper elementary. Mike: So I've heard other folks talk about the ways students can learn about the algebraic properties as they're building fluency, but I feel like you've taken this a step further. You have some ideas around how we can use visual models to make those properties visible. And I wonder if you could talk a little bit about what you mean by making properties visible and maybe why you think this is an opportunity that's too good to pass up? Kyndall: My thought is bringing visual models back into the classroom with our higher upper elementary students so that they can use those models to build a natural immersion of some of the algebraic properties so that they can emerge rather than just be rules that we are teaching. By supporting students' learning through building models with manipulatives, we're able to build a bridge in a student's mind between their experience with those models and then their mental capacity to visualize those models. This is where the opportunity to bring properties to life is too good to pass up. Mike: OK, so let's get specific. Where would you start? Which of the properties do you see as an opportunity to help students understand as they're building an understanding of fluency? Kyndall: So, when I begin laying the foundation for understanding of the operations and multiplication and division, I intentionally layer in two other major algebraic properties for discovery: the commutative property and the distributive property. We're not setting our students up for success when we simply introduce these properties as abstract rules to memorize. Strong visual models allow students to discover the why behind the rules. They're able to see these properties in action before I even spend any time naming them. For example, they get to witness or discover how factors can switch order without changing the product, how grouping affects computation, and how numbers can be broken apart and recombined for efficient counting and solving strategies. By teaching basic facts in this structured and intentional way through the behavior of numbers and the authentic discovery of properties, we're not only building fluency, but we're also developing deep conceptual understanding. Students begin to recognize patterns, understand rules, make connections, and rely on reasoning instead of rote memorization. That approach supports long-term mathematical flexibility, which is exactly what we want our students to be able to do. Mike: I want to ask you about two particular tools: the number rack and the 10-frame. Tell me a little bit about what's powerful about the way the [10-frame] is set up that helps students make sense of multiplication. What is it about the way it's designed that you love? Kyndall: The [10-frame] is so powerful because it's set up in our base ten system already. It introduces the tens in a way that is two rows of 5, which is going to lead into properties being identified. So, let me break that up into each individual thing that I love about it. First, the [10-frame] being broken up into the two rows of 5. That's going to allow students to be able to see that distributive property happening, where we're counting our 5s first and then adding some more into each group. So, when we're seeing a factor like 8 times 2, we're seeing that as two groups of 5 and two groups of 3. Mike: I think what you're making me remember is how it's difficult to help kids visualize that, right? It's a challenge. You can say "'4 times 4' is the same as '4 times 2 plus 4 times 2,'" but that's still an abstraction of what's happening, right? The visual really brings it to life in a way that—even if you're representing that with an equation and doing a true-false equation where it's 4 times 4 is the same as 4 times 2 plus 4 times 2—that's still at a level of abstraction that's not necessarily accessible for children. Kyndall: And as we're talking through this, if I see students and they're working on four groups of 3 and they're seeing those 3s as a double fact plus one more group, I'm on the board writing out the equation, and I'm using the parentheses as that introduction to what this looks like abstractly. They're building it, and they're building those visuals both with their hands and with their minds, and then I'm bringing it to life in the equation on the board. Mike: So, I think what I see in my mind as I hear you describe that is, you have kids with a set of materials. You're doing, for lack of a better word, a translation into a more abstract version of that, and you're helping kids connect the physical materials that they have in front of them to that abstraction and really kind of drawing the connection between the two. Am I getting that right? Kyndall: Yeah. As the students are doing the physical work of math, I'm translating it into its own language up on the board. Absolutely. Mike: I think what's clear to me from this conversation is the way that the tools can illuminate the property, and I think this also helps me think about what my role is as a teacher in terms of building a bridge to an abstraction. Do you actually feel like there's a point where you do introduce the formal language of it? And if you do, how do you decide when? Kyndall: So, the vocabulary kind of comes after the concept has been discovered. But I don't like to introduce the vocabulary first as a rote memorization tool because that has no meaning to it. Mike: I think if I were to summarize this, you're giving them a physical experience with the properties. You're translating that into an abstraction. And then once they've got an experience that they can hang those ideas on top of, then you might decide to introduce the formal language to them at some point. Kyndall: Yeah, absolutely. Mike: So, just as a refresher, for folks who might teach upper elementary and don't have a lot of lived experiences with the number rack—be it the ten or twenty or the hundred—can you describe a little bit about the structure, and maybe what about the structure in particular is important? Kyndall: The structure of a number rack has rows, and each row has 10 beads in it. And typically those beads are divided into two sets of 5: five red beads and five white beads. Then we typically move into a number rack that has two rows so that we're working within 20. Now, my thought is to take that [to] our third, fourth, and fifth grade, our upper elementary students, and use the hundreds rekenrek [i.e., number rack], where now we have 10 rows and we have 10 beads in each row—still split up into five red [beads] and five white—so that we can use that to teach things. If we're looking at the zero property, students are starting to notice that the rows represent the groups—the rows with the beads on it, that's one group. And so, if we're building zero groups of 3, we don't have a group that we can access to put three beads in. If we're looking at it with the commutative property, students are able to say, "One group of 3. We have one row and we're putting three beads in it." But what happens when we switch those factors? Now we're utilizing three of our rows, but we're only sliding over one bead. The number rack is also so important when we get to the distributive property because of the way that they have separated those colors. So when we're looking at a factor like 7 times 6—seven groups of 6—then we're gonna be accessing seven rows with six beads in each. That is already set up in the structure of the tool to have five red beads and one white bead showing seven groups of 5 and seven groups of 1 put together. Mike: That is super powerful. One of the things that really jumped out that I want to mark is: If I treat the rows like the groups and then I treat the beads like the number of things in each group, I can model one group with three inside of it, or I can model three groups with one inside of it, and I can really make the difference between those things clear, but also [I can make] the way that the product is still the same clear, right? So, I've got an actual physical model that helps kids understand what was often a rule that was just like 1 times 3 is the same as 3 times 1, because it is. But you're actually saying this is a tool that helps us make meaning of that. The other thing that jumps out from what you said is: If I'm doing 6 times 5 or 6 times 7 and I push over six [beads], and six looks like five red, one white, I'm automatically set up to make sense of the distributive property because the visual helps me see it. Am I getting that right? Kyndall: Yes, except let me correct you on that last one. You said "6 times 5," and you said, "If I slide over six," Now, six is our group number. We have to be deliberate; that's six groups of 5. So, we're grabbing our groups first, but absolutely, yes. That is the key structure there. [laughs] That's the idea. Mike: This is why this would've been very helpful for a young Mike Wallus. Kyndall: [laughs] Mike: Well, before we go, are there any resources that you'd recommend to a listener that have either informed your thinking or that might help someone take what you've been talking about and put these ideas into action? Kyndall: Yeah. I've been putting this practice into play here at my own district and tracking its progress for a while now. After seeing the success in my own halls here in Sandy, [Oregon,] I've started to reach out and work with other educators on purposeful tool use and mathematical progression. If it resonates with you, whether you're in the classroom or in a leadership role, I would genuinely love to connect and learn alongside you. You're always welcome to reach out to me directly at KyndallThomas56@yahoo.com [KyndallThomas56@yahoo.com]. 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 [http://www.mathlearningcenter.org]

Season 4 | Episode 11 – Dr. Amy Hackenberg, Understanding Units Coordination

Amy Hackenberg, Understanding Units Coordination ROUNDING UP: SEASON 4 | EPISODE 11 Units coordination describes the ways students understand the organization of units (or a unit structure) when approaching problem-solving situations—and how students' understanding influences their problem-solving strategies. In this episode, we're talking with Amy Hackenberg from the University of Indiana about how educators can recognize and support students at different stages of units coordination. BIOGRAPHY Dr. Amy Hackenberg taught mathematics to middle and high school students for nine years in Los Angeles and Chicago, and is currently a professor of mathematics education at Indiana University-Bloomington. She conducts research on how students construct fractions knowledge and algebraic reasoning. She is the proud coauthor of the Math Recovery series book, Developing Fractions Knowledge. RESOURCES Integrow Numeracy Solutions [https://www.integrowmath.org/] Developing Fractions Knowledge [https://www.integrowmath.org/store/67] by Amy J. Hackenberg, Anderson Norton, and Robert J. Wright TRANSCRIPT Mike Wallus: Welcome to the podcast, Amy. I'm excited to be chatting with you today about units coordination. Amy Hackenberg: Well, thank you for having me. I'm very excited to be here, Mike, and to talk with you. Mike: Fantastic. So we've had previous guests come on the podcast and they've talked about the importance of unitizing, but for guests who haven't heard those episodes, I'm wondering if we could start by offering a definition for unitizing, but then follow that up with an explanation of what units coordination is. Amy: Yeah, sure. So unitizing basically means to take a segment of experience as one thing, which we do all the time in order to even just relate to each other and tell stories about our day. I think of my morning as a segment of experience and can tell someone else about it. And we also do it mathematically when we construct number. And it's a very long process, but children began by compounding sensory experiences like sounds and rhythms as well as visual and tactical experiences of objects into experiential units—experiential segments of experience that they can think about, like hearing bells ringing could be an impetus to take a single bong as a unit. And later, people construct units from what they imagine and even later on, abstract units that aren't tied to any particular sensory material. It's again, a long process, but once we start to do that, we construct arithmetical units, which we can think of as discrete 1s. So, it all starts with unitizing segments of experience to create arithmetical items that we might count with whole numbers. Mike: What's really interesting about that is this notion of unitizing grows out of our lived experiences in a way that I think I hadn't thought about—this notion that a unit of experience might be something like a morning or lunchtime. That's a fascinating way to think about even before we get to, say, composing sets of 10 into a unit, that these notions of a unit [exist] in our daily lives. Amy: Yeah, and we make them out of our daily lives. That's how we make units. And what you said about a ten is also important because as we progress onward, we do take more than 1 one as a unit—like thinking of 4 flowers in a row in a garden as a single unit, as both 1 unit and as 4 little flowers—means it has a dual meaning, at least; we call it a composite unit at that point. That's a common term for that. So that's another example of unitizing that is of interest to teachers. Mike: Well, I'm excited to shift and talk about units coordination. How would you describe that? Amy: Yeah, so units coordination is a way for teachers and researchers to understand how children create units and organize units to interpret problem situations and to solve problems. So it originated in understanding how children construct whole number multiplication and division, but it has since expanded from just that to be thinking more broadly about units and structuring units and organizing and creating more units and how people do that in solving problems. Mike: Before we dig into the fine-grain details of students' thinking, I wonder if you can explain the role that units coordination plays in students' journey through elementary mathematics and maybe how that matters in middle school and beyond middle school. Amy: So that's where a lot of the research is right now, especially at the middle school level and starting to move into high school. But units coordination was originally about trying to understand how elementary school children construct whole number multiplication and division, but it's also found to greatly influence elementary school children's understanding of fractions, decimals, measurement and on into middle school students' understanding of those same ideas and topics: fractions ratios and proportional reasoning, rational numbers, writing and transforming algebraic equations, even combinatorial reasoning. So there's a lot of ways in which units coordination influences different aspects of children's thinking and is relevant in lots of different domains in the curriculum. Mike: Part of what's interesting for me is that I don't think I'm alone in saying that this big idea around units coordination sounds really new to me. It's not language that I learned in my preservice work[, nor] in my practice. So I think what's coming together for me is there's a larger set of ideas that flow through elementary school and into middle school and high school mathematics. And it's helpful to hear you talk about that, from the youngest children who are thinking about the notion of units in their daily lives to the way that this notion of units and units coordination continues to play through elementary school into middle school and high school. Amy: Yeah, it's nice that you're noticing that because I do think that's something that's a strength of units coordination in [that] it can be this unifying idea, although there's lots of variation and lots of variation in what you see with elementary students versus middle school students versus high school students versus even college students. Some of the research is on college students' unit coordination these days, but it is an interesting thread that can be helpful to think about in that way. Mike: OK. With that in mind, let's introduce a context for units coordination and talk a little bit about the stages of student thinking. Amy: Yeah. So, one way to understand some differences in how children up through, say, middle school students might coordinate units and engage in units coordination is to think about a problem and describe how solving it might happen. Here's a garden problem: "Amaya is planting 4 pansies in a row. She plants 15 rows. How many pansies has she planted?" There are three stages of units coordination, broadly speaking—we've begun to understand more about the nuances there. But a stage refers to a set of ways of thinking that tend to fit together in how students understand and solve problems with whole numbers, fractions, quantities, and multiplicative relationships. It's sort of about a nexus of ideas, and—that we tend to see coming together and students don't usually think in a way that's characteristic of a different stage until they've made a significant change in their thinking, like a big reorganization happens for them to move from one stage to the next. So students at stage 1 of units coordination are primarily in a 1s world and their number sequence is not multiplicative. That's going to be hard to imagine. But they can take a group of 1s as one thing. So, they can make a composite unit and that means in the garden problem, they can take a row of pansies as 1 row as well as 4 little ones, and they can continue to do that over and over again. And so they can amass rows of 4 pansies and keep going. And what it usually looks like for them to solve the problem is they'll count by 1s after any known skip-counting patterns. So, in this case they might be like, "Oh, I know 4 and 8; that's two rows. 9, 10, 11, 12; that's three rows." Often using fingers or something to keep track, or in some way to keep track, and continuing to go up and get all the way, barring counting errors, to 60 pansies. And so for them the result, 60 pansies, is a composite unit. It's a unit of 60 units, but they don't maintain the structure that we see at all of the units of 60 as 15 fours. That's not something—even though they did track it in their thinking—they don't maintain that once they get to the 60, it's really just only a big composite unit of 60. So their view of the result is very different than an adult view might be. So, the students at stage 1 can solve division problems, which means if they give some number of pansies and they're supposed to make rows of 4, they can definitely do it, they can solve that. But they don't think of multiplication and division as inverses. So let me say what I mean by that. If they had this problem next, so: "Amaya's mom gave her 28 pansies. How many rows of 4 can she make?" A student at stage 1 could solve that problem, and they would be able to track 4s over and over again and figure out that they got to 7 fours once they get to 28. But then if immediately afterwards a teacher said, "Well, so, how many pansies are there in 7 rows of 4?," the student at stage 1 would start over and solve the problem from the beginning. They wouldn't think that they had already solved it. And that's one telling sign of a student operating at stage 1. And the reason is that the mental actions they engage in to do the segmenting or the tracking off of the 4s and the 28 pansies are really different to them than what they use then the ways of thinking they use to create the 7 rows of 4 and make the 28 that way. And so they don't recognize them as similar, so they feel like they have to engage in new problem solving to solve that problem. So, to get back to the garden problem, students at stage 2 have a multiplicative number sequence, so they think of 60 as a one that they could repeat. Iterating is a term we often use. They could imagine it just being repeated over and over again. And this is a contrast to students at stage 1 who think of 60 as like, "Oh, I got to have all 60 pansies there if I'm going to think about a number like 60." Whereas students at stage 2 do have a multiplicative number sequence and so they think, "Oh, I don't have to have all my 60 pansies. I can just think about one pansy and I just repeat it however many times I need, to have however many pansies I want to imagine in my problem solving." So they anticipate 60 as 1 sixty times. And that's obviously a great relief for kids who are dealing with big numbers. You can imagine it feels really onerous to think about 1,000 if you feel like you have to have 1,000 items in your mind, "Oh, how could I possibly do that?" But, "Oh, I don't have to have 1,000; I can just have 1 and I can repeat it." That's a great economy, efficiency in thinking that happens. So in terms of the garden problem, students at stage 2 also have constructed a row as a thing to count, so a composite unit's one item as well, so 4 little items. And they can amass 4s just like I was talking about with students at stage 1. But what they are also able to do is break apart 4s as they go along. They might say, "Well, I've got 4 and 4 is 8 and one more [4] is 12 and one more is 16 and one more is 20 and one more is 24 and one more is 28." Maybe at that point they say, "Oh, let's see. I don't know what one more 4 is, but two more [4s] is 30 and then two more is 32." So they can take the row apart. They don't all do this, but they can; they have the mental capabilities to do that because they're not right in the midst of making the coordination happen. They're sort of a little bit able to stand above the coordination and take their rows apart if they need to. Mike: It sounds like part of what happens at stage 1 is you might have a kid who potentially could count by 4s for lack of a better way of saying it. And they might say, "Well, 4 and 4, so 2 sets of 4s, [is] 8." And then at some point it kind of breaks down where that memorized list of what happens when you count by 4. And then kids are back to saying, "OK, 12, 13, 14, 15, 16." And if you were watching this, listeners, you would see that I stuck out four fingers and then I'm like, "OK, so that's 3 fours, and so on." And so I would see a student who might appear to be thinking about units, but tell me if I'm correct in thinking that it's more a function of that they know a set of numbers in accounting sequence for counting by 4s. Amy: So students at any stage may vary in the skip-counting patterns they know. I call it knowing a skip-counting pattern, to know automatically, like, 4, 8, 12, 16, or whatever it is. So you could have a student at stage 2 who doesn't know their skip-counting patterns very well, and you also could have a student at stage 2 who counts by 1s. So that's the issue there, is you can't always tell just from what you see if you have to do more than the test of what I'm saying. It's just to give a sense of the stages. But the main thing is the outer boundary of what they can do at stage 2 is they don't have to count by 1s. They can do other things because of the fact that their composite units have this special feature where they're multiplicative in nature. I mean the fancy term for it is they have iterable units of 1. But let me say a little bit more about what happens when they get to 60. So, let's say a student at stage 2, they've gotten up to 60, there are 60 pansies and there are 15 rows of 4. They will think of the 60 as 15 fours as they make it. So we call it a three levels of unit structure. 60 is a unit of 15 units, each containing 4 little ones. They'll think about [it] that way as they solve the problem, but as they continue to work further and add more pansies on or do a further extension of the problem, they wouldn't maintain that three levels of units structure that we see. So that's important because it has implications for how they can build from what they've done. Mike: How would you know that they hadn't maintained it? What might they say or do that would give you that cue? Amy: Well, so you see it most if, let's say I say, "Oh, guess what? We got 12 more pansies and you're going to put 'em in rows of 4. Can you put those on?" And then they put 'em on. OK, they find out it's 72 now. "OK, so how many rows are we talking here?" It would be a new problem for them to figure that out. It wouldn't be like they would be able to maintain that, "Oh, I had 15 rows and then I now have the 3 more added on." Mike: Got you. OK. Amy: So, you see they're having to remake stuff as adult learners. We would think, "Oh, you should already know that that's 15 fours, right?" But they'll have to redo that in solving an extension of the problem like I was talking about there. So students at stage 3, they also can definitely take 4 as a row of 1 and also 4 pansies. They can arrive at 60 and view it as a unit of units, but they also can view it as a unit of 15 units, each containing 4, and they maintain that. So, if they were asked a further problem, like, "Hey, we're going to rearrange this garden; we're going to actually 3 rows together at a time. Can you do that, and how many rows would you have and how many pansies in each row? And what would be the total?" They'd be able to say, "Oh, yeah, I can, let's see, put my 3 rows together, that's going to be 12, and then I'm going to end up with 5 of them." And now they've created 60 as a unit of 5 rows, each containing 12, and they can still think of 60 as a unit of 15 units, each containing 4, or 15 rows, each containing 4. So they can switch between different unit structures. It doesn't mean they automatically know it without thinking it through, but they can do it and they can go back and forth. And that has great implications for anticipating and for solving division problems and seeing them as inverses of multiplication and a whole lot of stuff: proportional reasoning, fractions, lots of things. [laughs] Mike: I think what's really interesting about this is I really appreciate you walking through the mental processes or maybe even the mental scripts that the kids might engage in to help see behind the curtain, for lack of a better word. Because what strikes me is that there is a point, probably early in my teaching career, where I would've attended and focused mostly on, "Did they get the answer?" And I think what you're helping remind me of is that it's the "how," but there are particular ideas. And now I think I understand why the notion of units—plural—units coordination matters so much because a lot of what's happening is their ability to coordinate a unit made of units and then to be flexible with the units within that unit of units. Am I making proper sense of that, Amy? Amy: Yeah, for sure. That's great; that's exactly it. So the process and what units get created and how they get thought about and used is actually really, really important in trying to support kids' multiplicative thinking among other kinds of thinking too. Mike: I think this is a great segue because I suspected a lot of teachers are wondering about the kinds of tasks or practices or questions that they might use that could nudge students' thinking regarding units coordination. And I'm wondering: What are some ideas you'd recommend for teachers as they're trying to think about how they assess but also advance their students' thinking when it comes to units coordination? Amy: That's a great question. And, I mean, the big response is: Have students engage in lots of reasoning with units—composite units, breaking apart numbers strategically, thinking about different solution pathways. So not just one solution pathway, but can you come up with multiple solutions for the problem? Really sharing student solutions that involve breaking apart units. So if you're doing something like 5 sevens and finding out that kids are thinking of it as 5 fives and 5 twos, let's share that. How else could we break apart the 5 sevens? 5 fives and 5 twos? Why is that maybe helpful compared to other ways we might think about it? We might know 5 fives and 5 twos more easily than other ways of breaking it apart. And then even how are kids thinking about the 5 twos and the 5 fives and evaluating each of those. So basic things like that are super important. How many rows can we make with 36 flowers with 4 per row? Thinking strategically about that, like: I know that 5 fours is 20 and I need 16 more flowers, so that's 4 fours because it's double 2 fours, so 8, so that means 9 rows total. So I'm just kind of really briefly talking through, but posing these kinds of tasks and then asking for how students can break them up and think about them and presenting and making public that kind of thinking and reasoning. So valuing it in that way and sharing it. Same thing with lots of even more advanced multiplication problems. So for example, my daughter's in fourth grade right now, and so we've been working with her on, like, 30 times 20 and doing something other than knowing 3 times 2 and then putting 0s on because she doesn't remember that. So to do 30 times 20, we asked her about 10 twenties. Oh, she can figure that out; that's 200. And then can I iterate? Oh yeah, another 10 twenties, another 10 twenties. And then we did like 40 thirties, which was definitely harder. And so as part of the process of that, after she figured out 10 thirties, when she was iterating her thirties, that was harder than iterating the twenties. She had to break apart numbers. When she got to 90 plus 30, she had to think about 90 plus 10 plus 20. So doing embedded, breaking apart of units with the prospect of trying to figure out a larger multiplication problem, is super important. And interestingly, she could do 900 plus 300 and figure out that that was 900 and 100 to get 1,000 and then 200 more. So that's additive reasoning, but it's the breaking apart of units and reconstituting them. That's what's really important in the process of solving multiplication and division problems. So that's my big thought about [laughs] that. And the other thing is to not go to patterns too soon. I mean, this is related to what I just said about not thinking that I can just do 3 times 2 and then add 0s and count the 0s because that really doesn't develop. It misses so much in what you can do with units. And so even if some kids do remember that and get the answer right, they're really robbed of the experience that we're trying to give to my daughter of really thinking about, "Well, how can I figure out 40 thirties or 30 forties or 30 twenties?" [laughs] Right now I'm a big advocate of actually doing lots of counting by decade numbers because I feel like it's a way of really enhancing kids' work with larger multiplication. Mike: I've been sitting listening to you talk about this, Amy, and there are multiple things where I'm like, I need to ask her about this. I need to ask her about that. I need to ask about this other thing. So I'm going to ask you a couple of follow-ups. One of the things that is just an observation is the language you used when you were talking about your work with your daughter. When the original task was "30 times 20" and you shifted the language to say "30 twenties," and then you step back even a little bit from there and you said, "Well, what's 30 tens?" This language that you were using, I wonder if you could be explicit about what you think that shift in language accomplishes. Amy: Yeah, I've been also thinking a lot about this, so it's great. Yeah, one of the problems with multiplication notation is that it doesn't make clear anything about what the group is and what the number of groups you have are. And so just saying "30 times 20," I mean, you can think of that as "30 twenties" or I can think of that as "20 thirties," but the language doesn't contain it, so it doesn't refer to the action I might do in thinking about how to actually figure it out. And kids have to bring a lot to the table, then, to really read that into that multiplication notation. It's even more so with fractions. I can say more about that in a second. So I really am advocating with my preservice teachers is that we speak in iterative language with the multiplication. So we try to always say, "I'm talking about 5 sevens," or "I'm talking about 7 fives, 30 forties, 40 thirties." And then of course with the decade numbers, knowing that we can go down to 10 of something and that that's easier to figure out, and then we can build on that. So like 10 twenties and then, "Oh, I'm going to need 3 of those 10 twenties to get to 30 twenties." Mike: Which really to some degree is helping them make meaningful sense of the associative property as well. Amy: Right! Yeah, exactly. It's very mathematically rich. Unfortunately, it's not necessarily worked on [laughs] a lot, I am finding, and I think it's a real missed opportunity. Because I think there's a lot that kids could do with that that would really build strong meanings for multiplication and strong ideas of base ten as well. Mike: Yeah, absolutely. I think one of the things that I've been obsessed with lately is this notion of "nudge" or small-sized shifts in my practice that I can make. Part of what I'd like to mark for the audience is the shift in the language, as you described—30 twenties or 5 sevens—those are moves that a teacher could make to help clarify the fact that units are involved and help students visualize with a bit more clarity what's going on. That feels like something that a teacher could take up and really have an impact on students' understanding. Amy: Yeah, I think so. I think it is something that is reasonable, and what's nice is it also can flow right into fractions because then instead of saying just, "three-fifths," we say, "3 one-fifths, 4 one-fifths, 5 one-fifths, 6 one-fifths, 7 one-fifths." It allows for fractions larger than 1 to have maybe more of an iterative meaning. Not that that's a simple thing at all; that's a whole nother podcast we could do, but [laughs] I've done a lot of research on that. Mike: Well, I think you're hitting on something important, though, Amy, because this notion of, "What is a unit fraction?," it's really, "Four-fifths is a group of 4 one-fifths," right? And that's a critical understanding that I think often floats underneath students' understanding in ways that, if we could make that clearer or help build that understanding, that also has huge ramifications for what comes later in their mathematics learning experience. Amy: Yeah, so I'm a big proponent of iterative language there as well. Mike: You have me thinking about something else too, which is the importance of context and having students deal with measurement division problems specifically as a way to build their understanding. And I know I'm using language right now for the audience that might not be super clear, but I'm wondering if you could talk a little bit about what measurement division means in context and maybe why that would be valuable for students. Amy: Yeah. Right. So, in multiplication and division structures, if we're talking about equal groups, there's always some number of equal groups, some number in the equal group, so a size of the group, and then a total number of items. And so, with measurement division, we know the total number of items, and we know the number of items in a group, but we don't know the number of groups. So my example of, "You've got 36 flowers, and you want to put them in rows of 4" would be a measurement division problem because we know that there are 4 in each row, and we know we have 36, but we don't know how many rows we're going to make. And so those are really nice to pair with work on equal groups multiplication problems because they are very closely related. And for kids, they can become closely related as they solve them and realize, like, "Oh, I can use my multiplication strategies to build up my 4s and find out when I get to 36," and, "Oh, then I do, I know how many rows I've made." So it's highly linked to what we're talking about here. Mike: What I found myself thinking about is that in solving that problem, one of the ways that a kid could do that is they're iterating a set, right? So, potentially, they're iterating a set of 4s multiple times, and then they're finding out how many of those sets of 4 they have, right? So I think part of what you're helping me think about is the way that the structure of a measurement division problem maybe shines a flashlight on this notion of groups and the number in each group, and also some of the ideas you were talking about earlier with units coordination. Amy: Yeah, for sure. And in terms of continuing the theme of using iterative language, then when you get the result of that problem, 9 rows, "Oh, what does that 9 mean?" "Oh, it means 9 fours make 36." So that's a meaning both for 4 times 9 equals 36, as well as 36 divided by 4 equals 9. So it's nice to emphasize that. And yeah, as students build those meanings and have repeated work with that kind of thing, they usually, often—[laughs] we don't know all the mechanisms here—but they usually come to be able to at least make that coordination in their problem-solving activity, and ultimately make it so they can anticipate it, like we're talking about with stage 3. Mike: One of the things that is really helpful is, in the course of this interview, we've talked a lot about what might the behavior of a student at stage 1 or stage 2 or stage 3 not only look like, but what might it mean for how they're thinking. And I think what I'm really appreciating about this, Amy, is there are a few practical things that an educator could do to support students. One is iterative language as we've been talking about. And the other is measurement division, using a particular problem structure like measurement division to shine a light on these parts that we think are really important for kids to attend to if they're in fact going to make some of the shifts that we're hoping for. Amy: Yeah, for sure. And then also exploring the boundaries of what the kids' strategies are and asking for multiple solutions. Because you might see kids, even students at stage 3, that might be counting by 1s, and so you want to [prompt], "Oh, can you solve that another way? Is there another way you can do it?" And so seeing what they see as possible, what they're able to think about is also really important to support units coordination. Mike: Absolutely. Before we close, I typically ask a question about resources or training or learning experiences that would help someone who's listening continue learning or continue to think about how they could take up these ideas in their practice. You, particularly, I know have written some work around this and I also suspect that you might have some recommendations in terms of organizations that can help educators really dig into these ideas if they saw that as something that was important for their growth. Would you be willing to talk a little bit about resources, organizations, or even the types of experience you think support teachers as they're making sense of all of this? Amy: Yeah. Well, yes. I was planning to talk about Integrow at this point because Integrow Numeracy Solutions [https://www.integrowmath.org/] has a lot of great supportive materials for all this kind of work. And everything that I'm talking about is something that is sort of built into much of what they do. For people who are unfamiliar, it's a bit—council, used to be called a council, of people who got together and have really developed materials that are supportive of teachers working one-on-one to support students who might be struggling as well as whole-group instruction all around developing strong number sense. And it's a very well developed set of materials, both for classroom use as well as for teacher development. And we—meaning me and my two coauthors, Andy Norton and Bob Wright—wrote a book in the series for teachers on fractions called Developing Fractions Knowledge [https://www.corwin.com/books/developing-fractions-knowledge-245818]. And that was published—oh my gosh—nine years ago now. So Andy and I are working on a second edition right now, and in that book we address units coordination and talk about its usefulness for teachers. It's mostly, though, a book about fractions and about how units coordination is relevant in trying to support students' fractions knowledge and to help assess students' thinking and also promote their learning. So that is one resource I can recommend on units coordination with a revision coming in the next year [2026]. Mike: That's fantastic. So I'll say for listeners, we'll include a link to Integrow Numeracy Solutions if you want to check out the organization. And Amy will also add a link directly to the book so that if someone wanted to dig in and explore that way they had the option. I think that's probably a great place to stop, although I certainly would love to continue. I want to thank you so much for joining us. It's really been a pleasure talking with you. Amy: Yeah, likewise, Mike. I've really enjoyed it, and I look forward to further conversations. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2026 The Math Learning Center | www.mathlearningcenter.org

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

What Counts as Counting? with Dr. Christopher Danielson ROUNDING UP: SEASON 4 | EPISODE 10 What counts as counting? The question may sound simple, but take a moment to think about how you would answer. After all, we count all kinds of things: physical quantities, increments of time, lengths, money, as well as fractions and decimals. In this episode, we'll talk with Christopher Danielson about what counts as counting and how our definition might shape the way we engage with our students. BIOGRAPHY Christopher Danielson started teaching in 1994 in the Saint Paul (MN) Public Schools. He earned his PhD in mathematics education from Michigan State University in 2005 and taught at the college level for 10 years after that. Christopher is the author of Which One Doesn't Belong?, How Many?, and How Did You Count? Christopher also founded Math On-A-Stick [https://www.mnstatefair.org/location/math-on-a-stick/], a large-scale family math playspace at the Minnesota State Fair. RESOURCES How Did You Count? A Picture Book [https://www.routledge.com/How-Did-You-Count-Picture-Book/Danielson/p/book/9781032898353] by Christopher Danielson How Many?: A Counting Book [https://www.routledge.com/How-Many-A-Counting-Book/Danielson/p/book/9781625311825?srsltid=AfmBOorIFjAgrjwlQe3nrOiyU5hFKbatwWeYQfXRXn6KBT3xWB1J6L-I] by Christopher Danielson Following Learning blog [https://followinglearning.blogspot.com/] by Simon Gregg Connecting Mathematical Ideas [https://www.heinemann.com/products/connecting-mathematical-ideas-e07818.aspx] by Jo Boaler and Cathleen Humphreys TRANSCRIPT Mike Wallus: Before we start today's episode, I'd like to offer a bit of context to our listeners. This is the second half of a conversation that we originally had with Christopher Danielson back in the fall of 2025. At that time, we were talking about [the instructional routine] Which one doesn't belong? [https://www.mathlearningcenter.org/blog/christopher-danielson-which-one-doesnt-belong-fostering-flexible-reasoning] This second half of the conversation focuses deeply on the question "What counts as counting?" I hope you'll enjoy the conversation as much as I did. Well, welcome to the podcast, Christopher. I'm excited to be talking with you today. Christopher Danielson: Thank you for the invitation. Delightful to be invited. Mike: So I'd like to talk a little bit about your recent work, the book How Did You Count?[: A Picture Book] [https://www.routledge.com/How-Did-You-Count-Teachers-Guide-with-Picture-Book/Danielson/p/book/9781625312938] In it, you touch on what seems like a really important question, which is: "What is counting?" Would you care to share how your definition of counting has evolved over time? Christopher: Yeah. So the previous book to How Did You Count? was called How Many?[: A Counting Book] [https://www.routledge.com/How-Many-A-Counting-Book-Teachers-Guide-with-Student-Book/Danielson/p/book/9781625312181], and it was about units. So the conversation that the book encourages would come from children and adults all looking at the same picture, but maybe counting different things. So "how many?" was sort of an ill-formed question; you can't answer that until you've decided what to count. So for example, on the first page, the first photograph is a pair of shoes, Doc Marten shoes, sitting in a shoebox on a floor. And children will count the shoes. They'll count the number of pairs of shoes. They'll count the shoelaces. They'll count the number of little silver holes that the shoelaces go through, which are called eyelets. And so the conversation there came from there being lots of different things to count. If you look at it, if I look at it, if we have a sufficiently large group of learners together having a conversation, there's almost always going to be somebody who notices some new thing that they could count, some new way of describing the thing that they're counting. One of the things that I noticed in those conversations with children—I noticed it again and again and again—was a particular kind of interaction. And so we're going to get now to "What does it mean to count?" and how my view of that has changed. The eyelets, there are five eyelets on each side of each shoe. Two little flaps that come over, each has five of those little silver rings. Super compelling for kids to count them. Most of the things on that page, there's not really an interesting answer to "How did you count them?" Shoelaces, they're either two or four; it's obvious how you counted them. But the eyelets, there's often an interesting conversation to be had there. So if a kid would say, "I counted 20 of those little silver holes," I would say, "Fabulous. How do you know there are 20?" And they would say, "I counted." In my mind, that was like an evasion. They felt like what they had been called on to do by this strange man who's just come into our classroom and seems friendly enough, what they had been called on to do was say a number and a unit. And they said they had 20 silver things. We're done now. And so by my asking them, "How do you know? " And they say, "I counted." It felt to me like an evasion because I counted as being 1, 2, 3, 4, 5, all the way up to 20. And they didn't really want to tell me about anything more complicated than that. It was just sort of an obvious "I counted." So in order to counter what I felt like was an evasion, I would say, "Oh, so you said to yourself, 1, 2, 3, and then blah, blah, blah, 18, 19, 20." And they'd be like, "No, there were 10 on each shoe." Or, "No, there's 5 on each side." Or rarely there would be the kid who would see there were 4 bottom eyelets across the 4 flaps on the 2 shoes and then another row and another row. Some kids would say there's 5 rows of 4 of them, which are all fabulous answers. But I thought, initially, that that didn't count as counting. After hearing it enough times, I started to wonder, "Is it possible that kids think 5 rows of 4, 4 groups of 5, 2 groups of 10, counted by 2s and 1, 2, 3, 4, all the way up to 19 and 20—is it possible that kids conceive of all of those things as ways of counting, that all of those are encapsulated under counting?" And so I began because of the ways children were responding to me to think differently about what it means to count. So when I first started working on this next book, How Did You Count?, I wanted it to be focused on that. The focus was deliberately going to be on the ways that you count. We're all going to agree that we're counting tangerines; we're all going to agree that we're counting eggs, but the conversation is going to come because there are rich ways that these things are arranged, rich relationships that are embedded inside of the photographs. And what I found was, when I would go on Twitter and throw out a picture of some tangerines and ask how people counted, and I would get back the kind of thing that was how I had previously seen counting. So I would get back from some people, "There are 12." I'd ask, "How did you count?" And they'd say, "I didn't. I multiplied 3 times 4." "I didn't. I multiplied 2 times 6." But then, on reflection through my own mathematical training, I know that there's a whole field of mathematics called combinatorics. Which if you asked a mathematician, "What is combinatorics?," 9 times out of 10, the answer is going to be, "It's the mathematics of counting." And it's not mathematicians sitting around going "1, 2, 3, 4" or "2, 4, 6, 8." It's looking for structures and ways to count the number of possibilities there are, the number of—if we're thinking about calculating probabilities of winning the lottery, somebody's got to know what the probabilities are of choosing winning numbers, of choosing five out of six winning numbers. And the field of combinatorics is what does that. It counts possibilities. So I know that mathematicians and kindergartners—this is what I've learned in both my graduate education and in my postgraduate education working with kindergartners—is that they both think about counting in this rich way. It's any work that you do to know how many there are. And that might be one by one; it might be skip-counting; it might be multiplication; it might be using some other kind of structure. Mike: I think that's really interesting because there was a point in time where I saw counting as a fairly rote process, right? Where I didn't understand that there were all of these elements of counting, meaning one-to-one correspondence and quantity versus being able to just say the rote count out loud. And so one way that I think counting and its meaning have expanded for me is to kind of understand some of those pieces. But the thing that occurs to me as I hear you talk is that I think one of the things that I've done at different points, and I wonder if people do, is say, "That's all fine and good, but counting is counting." And then we've suddenly shifted and we're doing something called addition or multiplication. And this is really interesting because it feels like you're drawing a much clearer connection between those critical, emergent ideas around counting and these other things we do to try to figure out the answer to how many or how did you count. Tell me what you think about that. Christopher: Yeah. So this for me is the project, right? This book is an instantiation of this larger project, a way of viewing the world of mathematics through the lens of what it means to learn it. And I would describe that larger project through some imagery and appealing to teachers' ideas about what it means to have a classroom conversation. For me, learning is characterized by increasing sophistication, increasing expertise with whatever it is that I'm studying. And so when I put several different triangular arrangements of things—in the book, there's a triangular arrangement of bowling pins, which lots of kids know from having bowled in their lives and other kids don't have any experiences with them, but the image is rich and vivid and they're able to do that counting. And then later on, there's a triangular arrangement of what turned out to be very bland, gooey, and nasty, but beautiful to photograph: pink pudding cups. Later on, there are two triangles of eggs. And so what I'm asking of kids—I'm always imagining a child and a parent sitting on a couch reading these books together, but also building them for classrooms. Any of this could be like a thing that happens at home, a thing that happens for a kid individually or a classroom full of children led by a teacher. Thinking about the second picture of the pudding cups, my hope and expectation is that at least some children will say, "OK, there are 6 rows in this triangle and there were 4 rows previously. So I already know these first four are 10. I don't have to do any more work, and then 5 plus 6, right?" And then that demonstrates some learning. They're more expert with this triangle than they would have been previously. I'm also expecting that there's going to be some kid who's counting them 1 by 1, and I'm expecting that there are going to be some kids who are like, "You know what? That 6 up top and the 1 makes 7 and the 5 and the 2 make 7, and the 4 and the 3. So it's 3 sevens. There's 21." I'm expecting that we're going to have—in a reasonably large population of third, fourth, fifth graders, sort of the target audience for this book—we're going to have some kids who are doing each of these. And for me, getting back to this larger project, that is a rich task, which can be approached in a bunch of different ways, and all of those children are doing the same sort of task. They're all counting at various levels of sophistication representing various opportunities to learn previously, various ways of applying their new learning as they're having conversations, looking at new images, hearing other people's ideas, but that larger project of building something that is rich enough for everybody to be able to find something new in, but simple enough for everybody to have access to—yeah, that's the larger project. Mike: So one of the things that I found myself thinking about when I was thinking about my own experiences with dot talks or some of the subitizing images that I've used and the book that you have, is: There's something about the way that a set of items can be arranged. And I think what's interesting about that is I've heard you say that that arrangement can both reveal structure, in terms of number, but it can also make connections to ideas in geometry. And I wonder if you could talk a little bit about that. Christopher: Yeah. I'll draw a quick distinction that I think will be helpful. If you've ever seen bowling pins, right? It's four, three, two, one. The one [pin] is at the front; the [row of] four is at the back. Arranged so that the three fit into the spaces between the four as you're looking at it from the front. Very iconic arrangement. And you can quickly tell that it's a symmetric triangle and the longest row is four. You might just know that that's 10. But if you take those same bowling pins and just toss them around inside of a classroom or inside of a closet and they're just lying on the floor, so they're all in your field of vision, you don't know that there's 10 right away. You have to do a different kind of work in order to know that there are 10 of them. In that sense, the structure of the triangle with the longest row of four is a thing that you can start to recognize as you learn about triangles and ultimately what mathematicians refer to as triangular numbers. That's a thing you can learn to recognize, but learning to recognize 10 in that arrangement doesn't afford you anything when it's 10 [pins] scattered around on the floor. Unless you do a little abstraction. There's a story in the book about a lovely sixth grader who proceeded to tell me about how the bowling pin arrangement matches a way that she thinks about things. Because if she's ever going about her life, I don't know, making a bracelet or buying groceries, collecting pencils for the first day of school or whatever. If she wants to count them, and it looks like there's probably fewer than 100 but more than 5, she will grab a set of 4, a set of 3, a set of 2, a set of 1, and she'll know that's 10. Unprompted by me, except that we had this bowling pin arrangement. So there are ways to abstract from that. You can use these structures that you've noticed in order to do something that isn't structured that way, but the 4, 3, 2, 1 thing probably came from recognizing that 4, 3, 2, 1 made this nice little geometric arrangement. So our eyes, our brains, are tuned to symmetry and to beauty and elegance, and there is something much more lovely about a nice arrangement of 4, 3, 2, 1 than there is about a bunch of scattered things. And so a lot of those things are things that have been captured by mathematicians. So we have words for square numbers—3 times 3 is 9 because you can make 3 rows of 3 and you make something that looks nice that way. Triangular numbers, there are other figurate numbers like hexagonal numbers, but yet innate in our minds, there is an appeal to symmetry. And so if we start arranging things in symmetric patterned ways that will be appealing to our brains and to our eyes and to our mathematical minds, and my goal is to try to tap into that in order to help kids become more powerful mathematicians. Mike: So I want to go back to something you said earlier, and I think it's an important distinction before I ask this next question. One of the things that's fascinating is that a child could engage with this kind of image, and there doesn't necessarily have to be an adult in the room or a teacher who's guiding them. But what I was thinking about is: If there is a student or a pair of students or a classroom of students, and you're an educator and you're engaging them with one of these images, how do you think about the educator's role in that space? What are they trying to do? How should they think about their purpose? And then I'm going to ask a sub-question: To what extent do you feel like annotation is a part of what an educator might do? Christopher: Yes. One thing that teachers are generally more expert at than young children is being able to state something simply, clearly, concisely in a way that lots of other people can understand. If you listen to children thinking aloud, it is often hesitant and halting and it goes in different directions and units get left off. So they'll say, "3 and then 4 more is 8" and they've left off the fact that the 4 were—I mean, you could just easily get lost. And so one of the roles that a teacher plays can certainly be to help make clear to other students the ideas that a particular student is expressing and at the same time, often helping make it more clear for that student, right? Often a restating or a question or an introduction of a vocabulary word that seems like it's going to be helpful right now will not just be helpful to other people to understand it for the whole class, but will be helpful for the student in clarifying their own ideas and their own thinking, solidifying it in some kind of way. So that's one of the roles. I know that there are also roles that involve—and I think about this a lot whenever I'm working with learners—status, right? Making sure that children that have different perceived status in the classroom are able to be lifted up. That we're not just hearing from the kid who's been identified as "the math kid." So I think intellectual status, social status, those are going to be balances, right? I also understand that teachers have a role in making sure that children are listening to each other. If I'm working with learners, I can't always be the one to do the restating. I've got to make sure there are times where kids are required to try to understand each other's thinking and not just the teacher's restatement of that thinking. There are just so many balances. But I would say that some top ones for me, if I'm thinking about how to make choices, thinking about raising up the status of all learners as intellectual resources, making good on a promise that I make to children, which is that any way of counting these things is valid and not telling a kid, "Oh no, no, no, we're not counting 1 by 1 today" or, "Oh no, no, no, that's too sophisticated. That's too advanced of a—We can't share that because nobody will understand it." So making good on that promise that I make at the beginning, which is, "I really want to know how you counted." Making sure that learners are able to get better at expressing the ideas that are in their heads using language and gesture and making sure that learners are communicating with each other and not just with me as a teacher. Those seem like four important tensions, and a talented and experienced elementary teacher could probably name like 10 other tensions that they're keeping in mind all at the same time: behavior, classroom management, but also some ideas around multilingual learners. Yeah, a lot of respect for the kind of balances that teachers have to maintain and the kinds of tensions that they have to choose when to use and when to gloss over or not worry about for right now. So you ask about annotation and, absolutely, I think about multiple representations of mathematical ideas. And so far I've only focused on the role of the teacher in a classroom discussion and thinking about gesture, thinking about words and other language forms, but I haven't focused on writing and annotation is absolutely a role that teachers can play. For me, the thing that I want to have happen is I want children to see their ideas represented in multiple ways. So if they've described for the class something in words and gestures, then there are sort of two natural easy annotations for a teacher to do or a teacher to have students do, which is, one, make those gestures and words explicit in the image. And that's where something like a smartboard or projecting onto a whiteboard—lots of technologies that teachers use for this kind of stuff—but where we can write directly on the image. So if you said you put the 1 and the 4 together in the bowling pins and then the 3 and the 2, then I might make a loopy thing that goes around the 4 and the 1, and I might circle the 3 and the 2, right? And so that adds both some clarity for students looking, but also is a model for: Here's how we can start to annotate our images. But then I'm also probably going to want to write 4 plus 1, maybe in parentheses, plus 3 plus 2 in parentheses, so that we can connect the 4 to the four [items] that are circled, the 1 to the one that is circled, the 4 plus 1 in parentheses, identifying that as a group, like a thing that has a mathematical purpose. It's communicating part of an idea and that that connects back. Teachers are super skilled at using color to do that, right? So 4 plus 1 might be written in red to match the red circle that goes around here, using not green because of color blindness. They're using blue to do 3 plus 2 in parentheses over here. And teachers might make other choices, right? We might sometimes use color to annotate in the image, but then just black here so that we aren't doing all of that work of corresponding for kids and are asking kids to try to do some of that corresponding work. And we might do it the other way around as well. So annotation as a way of adding, I think, a couple of dimensions to the conversation. And I have to shout out a fabulous teacher who I know through math Twitter. Simon Gregg [https://followinglearning.blogspot.com/] is a teacher in an international school in Toulouse, France. And he has done amazing work with using and producing his own Which one doesn't belong?s, and annotating them and having kids do them; how many?; and then there are a few examples of his work with kids in the teacher guide for How Did You Count? Yeah, he's just a true master at annotation. So go find Simon Gregg on social media if you want to learn some beautiful things about representing kids' ideas in writing. Mike: Love it. So the question that I typically will ask any guest before the close of the interview is: What are some resources that educators might grab onto, be they yours or other work in the field that you think is really powerful that supports the kind of work that we've been talking about? What would you offer to someone who's interested in continuing to learn and maybe to try this out? Christopher: In the teacher guide of How Did You Count?, I make mention of which of the number talks books was most powerful for me. But if you want to take a look at that page in the teacher book and then throw a link in and a shout out to the folks who wrote it. Jo Boaler and Cathleen Humphreys wrote a book called Connecting Mathematical Ideas [https://www.heinemann.com/products/connecting-mathematical-ideas-e07818.aspx]. It's old enough that there are some CD-ROMs in it. I don't know if there's a new edition; I'm sure used ones are available on all the places you buy used books. But the expert work that the teacher Cathy Humphreys does, as described in the book—even if you can't use the CD-ROMS in your computer—expert work at drawing out students' ideas, and then the two collaborating to reflect on that lesson, the connections they were drawing. It's been a while since I read it, but I imagine the annotations have got to come up. Fabulous resources for thinking about how these ideas pertain to middle school classrooms, but absolutely stuff that we can learn as college teachers or as elementary teachers on either side of that bridge from arithmetic to algebra. Mike: So for listeners, just so you know, we're going to add links to the resources that Christopher referred to in all of our show notes for folks' convenience. Christopher, I think this is probably a good place to stop. Thank you so much for joining us. It's absolutely been a pleasure chatting with you. Christopher: Yeah. Thank you for the invitation, for your thoughtful prep work and support of both the small and the larger projects along the way. I appreciate that. I appreciate all of you at Bridges and The Math Learning Center. You do fabulous work. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2026 The Math Learning Center | www.mathlearningcenter.org