Rounding Up

ROUNDING UP: SEASON 4 | EPISODE 18

This summer we're replaying favorite listener episodes from the first four seasons of Rounding Up—like this one from Season 1. We'll return with all new episodes in early September.

Ensuring students master their basic facts remains a shared goal among parents and educators. That said, many educators wonder what should replace the memorization drills that cause so much harm to their students' math identities.

Today on the podcast, Jenny Bay-Williams talks about how to meet that goal and shares a set of productive practices that also support student reasoning and sensemaking.

BIOGRAPHY

Jennifer Bay-Williams is a professor of mathematics education at the University of Louisville. She has authored over 40 books and 100 journal articles and book chapters that focus on making mathematics meaningful to all students. She is an international leader in the field of mathematics education, frequently speaking at state, national, and international conferences and serving on national boards.

RESOURCES

"Eight Unproductive Practices in Developing Fact Fluency" article by Gina Kling and Jennifer M. Bay-Williams

Math Fact Fluency: 60+ Games and Assessment Tools to Support Learning and Retention book by Jennifer M. Bay-Williams and Gina Kling

Math Fact Fluency companion website by Kentucky Center for Mathematics

TRANSCRIPT

Click here for a full episode transcript.

What might it be like to engage in a number talk as a multilingual learner? How would you communicate your ideas, and what scaffolds might support your participation?

Today, we're talking with Jana Dean and Heather Byington about ways educators can support multilingual learners' engagement and participation during number talks.

Heather Byington has taught all grade levels over the span of her 27-year career as a bilingual public educator. She currently teaches middle school mathematics and English language support classes in Lacey, Washington. She is also a student at Washington State University pursuing a PhD in Mathematics Education.

Jana Dean currently serves as CEO of the Mathematics Education Collaborative and supports a fantastic team of middle school math teachers in North Thurston Public Schools. Her research focuses on the intersection of content learning and language learning.

Number Talks and Multilingual Learners article from Mathematics Education Collaborative by Jana Dean

Judit Moschkovich research

Math Between Us blog

"Number Talks: A Whole Class Routine for Learning Language for Learning Mathematics" article

Mathematics Education Collaborative website

[email protected] Jana Dean email

Mike Wallus: Welcome to the podcast, Jana and Heather. I am so excited to be talking with you both today.

Jana Dean: Good morning. Yeah, thanks for having us.

Heather Byington: Thanks so much for having us.

Mike: Absolutely.

Jana, before we begin talking about the ways that teachers can support multilingual learners during number talks, I wonder if you can offer a working definition that would help educators visualize what a number talk actually looks like.

Jana: Yeah, I'd be happy to do that. A number talk in terms of how we worked with the routine in this project consisted of the teacher providing some sort of visual prompt, starting either with a visual pattern of dots or a computation problem. And then the students get wait time, time to think about how they might solve that problem. And then as they share their strategies, the teacher records and asks them questions about their reasoning for why they approached the problem in the way that they approached it. The teacher creates what I like to think of as a visual mediator of student ideas. So the students' ideas become visible as they share them. So children who are listening can listen to the dialog or conversation between the person sharing and the teacher, but the ideas actually become visible as they're being shared. And the teacher always verifies with the student whether or not they've been understood. And the goal is not for the student to be right, but for the teacher and student to understand each other.

Mike: That's really helpful. Heather, is there anything else you'd add to that?

Heather: In terms of the way that we worked with it with multilingual learners and increasing their opportunities for engagement in the routine, we always gave them an option of talking to a partner and rehearsing their answer before they volunteered to share with the whole group. We prioritized calling on multilingual learners if they volunteered. And we also did a final reflection at the end. So those were some enhancements that we added onto the routine.

Mike: I think that's really helpful and I'm excited to talk a little bit more about the details of those, Heather.

One of the things that really struck me as we were preparing for this conversation was reading about the ways that some of the multilingual learners you worked with, how they described their experience during number talks. And it helped me to see the experience from their perspective and rethink some of the ways that I'd facilitated number talks in the past. And I'm wondering if you could share a bit about some of the feelings students told you that they were experiencing.

Jana: Yeah. One of the things we suspected before we started was that as a language learner myself, talking about ideas that you're just forming in a language you're in the process of learning can be really intimidating. It's very challenging. So they were nervous. And when I interviewed fourth graders about their experience in number talks, even facilitated with language acquisition in mind, they talked about how much courage it took them to share their ideas.

They also talked about and could very keenly remember moments when they had made a contribution that their teacher made use of or a time when they made a contribution that another student made use of later. So there was a lot of pride they felt in having shared their ideas once they found ways to do that.

They also talked about how much easier it was to share our ideas than it was to share my idea. And so if, for instance, we had given them the opportunity—and like Heather said, we almost always gave them the opportunity to talk with a partner—they would often share using the pronoun "we." "This is how we thought of it." And we picked up on that and began to ask them if it was OK to attribute a group of students with a unique idea rather than an individual. And that was also consistent with many of their home cultures. It's not every culture in which individual contributions are elevated, but rather when you dare to speak, you're definitely speaking for the group, for a collective. So that collective understanding was really important.

There was one child, and I'm really curious about how representative he was of many. He always talked to the same friend, and every time he shared, he, I'm going to say, nailed it. He really had it figured out what it was that he was going to say. And there was one particular day when he did a beautiful job sharing, and I asked him about that day and he said, "To be honest, that day I really didn't want to share, but I knew my teacher wanted to hear my idea, so I did anyway." And so there's that element of love and respect for their teacher that I think was also really motivating for them.

Heather: Yeah. Can I add something quickly to that?

So one aspect of that, I think that idea of a student sharing because it meant a lot to the teacher, we also tried to utilize individual conferring with students as much as possible and gave them opportunities to confer with us, whether it was just checking in briefly before the number talk started, encouraging them or maybe telling them, "Hey, you can share the idea with me after the number talk if that feels more comfortable to you." So it's giving them multiple opportunities to do that and encouraging them to share their thoughts.

Mike: What I appreciate about what you all are doing is even in this initial part of the conversation, really getting specific about the practices and the way that those practices played out for kids. And I think as an educator, one of the things that I've come to over all my years teaching is the need to have humility and also continue to be a learner. And that sometimes really leads me to questions about intent versus impact.

Heather, I wonder if you could talk about the parts of the number talk routine or facilitation practices that may have unintentionally provoked some of the anxiety that kids were experiencing.

Heather: So for multilingual learners, when I think about what they will need, the supports that they may need to be able to engage in a routine like a number talk, I think about first the processing time that they might need to understand and think about different ways of solving that prompt. And then I think about their understanding of the prompt. And then the other thing I think about is their ability to communicate their thoughts and ideas with others. So naturally, if it seems like there's a lot of pressure because of time, if they don't have much time, if they feel that pressure to do that processing and think of those ideas and share them quickly, that may provoke anxiety because this, of course, is still a language that they're still developing. So that ability to share with a partner and rehearse those ideas and process that with a partner, that really becomes, as Jana mentioned, more of a team effort.

And then being able to rehearse the words that they're going to use and the way they're going to convey that message and communicate it to others, that again reduces the anxiety because it's a lot less pressure to share my thoughts and ideas with one person than with a whole group. And if I share those thoughts with one person and they seem to understand what I mean, then now I might feel confident enough to share with more people. So I just think that naturally when it's a time constrained activity, that that naturally can provoke anxiety.

Mike: Yeah. I mean, that absolutely makes sense. I will say as a child who was not quick, even in my first language, the impact of that was profound, let alone trying to both process in a language that I was learning and feel like I was under pressure to produce an idea and describe it. That absolutely makes sense.

Jana: I want to back up a bit and quote something that you said, Heather, partway through our working together, which was that Heather had some familiarity with number talks before we started working together, but had a healthy skepticism as well. And at one point she said that she wondered if we might not actually be hurting students when we are facilitating a routine that they cannot find entry into. And so it became really like a guiding light or principle of our work together to work hard to help them find entry into the routine. And something that I didn't realize until a year after we began working together and I was really closely tracking the experiences of the multilingual learners themselves—and this is kind of back to your question about intent and impact—when we listen to children's mathematical ideas with the intent of not correcting them, trying to figure out what's right and what makes sense to them, we have to ask them questions about what their ideas are. And for many of the multilingual learners, engaging in that process itself was a huge lift language-wise. So I'm not just going to be able to say the answer or tell my teacher my strategy; I'm going to have to stick with my teacher until my teacher actually gets it. And a few of the multilingual learners that I followed over the course of a year actually said to me, "I don't like it when my teacher doesn't understand me." So while we absolutely, 100%, our intention is golden. It is about understanding them. But putting them in that position of that negotiating meaning with us until we do understand takes a great deal of trust on the part of the student. And so it's on us to develop that trust so that they're willing to do that with us.

Mike: I think that's a good segue because Jana, going into this, you mentioned three big ideas as starting points for supporting multilingual learners. One was negotiated meaning, one was the notion of voluntary sharing, and the last was the idea of using ambiguity as a resource. And I wonder if we can start this next part of the podcast with having you describe each of these for the listeners.

Jana: Yeah, absolutely.

Voluntary sharing means I've made a commitment to not ever put you on the spot as a student. And so any one of us who has learned a second language—which I've learned a couple, none of them to a super high level—but most people can relate to, say, standing in line in a grocery store and rehearsing what you're going to say so that you ask for the bag you want rather than the receipt that you don't want. There's a process in coming to speak, and I think there's a process in coming to speak publicly for just about every learner, especially about ideas that you're in the process of forming, but that pressure—and I've had many, many students over the year thank me for being the kind of teacher in a kind of classroom where they knew that I wasn't going to call on them unless they had volunteered to share. So the level of distraction, I think that that, again, well-intentioned pressure causes, is absolutely not worth it, and especially not for our multilingual learners.

Negotiated meaning really is the process of coming to understand each other, and we do it all the time. Unfortunately, often in classrooms, we end up in discourse routines that are actually not about teachers understanding students. They're about teachers asking questions for which students are supposed to have answers, which then the teacher evaluates. So what I would argue that the number talk routine turns that discourse pattern, which is often called I.R.E.—initiate, respond, evaluate—absolutely on its head. The child volunteers their idea, the teacher responds by trying to understand it as best they can, and then the student is the evaluator of whether or not the teacher actually understood them.

Mike: Heather, I was hoping we could go granular on a couple pieces that I heard you talk about too. You talk a lot about something very practical, the value of predictability, and I wonder if you can talk about how predictability impacted students and what does that mean for the teacher?

Heather: Absolutely.

When facilitating these number talks with this goal of engaging multilingual learners or helping them find those entry points, I found it helpful as a facilitator to utilize similar types of approaches to statements I would make during the routine, and then similar ways of asking students if I was seeing things the way that they were seeing them. It seemed to help the students that we were really hoping to engage to feel more comfortable with what was happening in the routine and to lean in more to that engagement. So I think that that is one thing as a facilitator to be aware of.

Jana, can you think of anything else that we haven't talked about yet?

Jana: There's the whole knowing the rules of the game aspect of really any classroom routine or instructional routine. So if the student knows how this thing goes, whatever "this thing" is, then that lifts off some of the cognitive load in terms of participation because they don't have to be figuring out how to participate. Judit Moschkovich writes about this a lot in her research, and I think she calls it the "sociocultural aspect of learning mathematics," and she uses the word "ecological". So the environment itself really matters. And in community, our social environment is made up of all kinds of routines. So I think that part of it is important. My favorite metaphor for it is learning a new card game. The first time you play the game, it is no fun because all you're doing is trying to figure out how the cards move, how the turns go, what the rules are, and how you can play. You can't do any strategy at all. But then as you learn the game, then you can really engage in it in a thoughtful way and have fun with it. So I really think that classroom routines are like that and not only for multilingual learners, but I have the privilege of being an instructional coach now in a middle school and have seen teachers engage in routines that I can tell are 100% soothing of trauma that students have as they come into the classroom, just because they know what to expect. So not only are those kinds of regular routines really helpful for multilingual learners, but they're also trauma-informed teaching. And when I say "routine," it can be easy to misunderstand and think it's boring. It has to be an open-ended routine so that something inside it that is engaging and fun can happen.

Heather: There are a couple of other things that occurred to me in terms of the students participating in the routine. I know that they started to see that we were elevating the status of gestures in terms of the communication to be another way to visualize the thinking in terms of the processing for themselves, but also a way to help others see what they were seeing and to understand their ideas. So that was one aspect of the routine that they could count on, that they could utilize gestures if needed, and that we would reinforce that. If they didn't have a mathematics label for the terminology that would typically be used in that conversation about those mathematics ideas, they could rely on describing what they understood, and then either I, the teacher, the facilitator, or another student, providing those words and the opportunity to practice that specific mathematics language within that routine. So those were some other things that were predictable and happened across all of the different number talks that happened, no matter what the prompt was.

Mike: You're making me think that part of what a teacher might do in response to this conversation is really to think about some of the things that they want to make normal, right? Like this notion of using gestures is both normal and accepted and valued. The idea that you are going to use rough draft, informal language, and that's OK, and that's a way that we get to more technical language of mathematics, and that's normal. And so thinking about what are the things that I want to become normal and predictable for kids, maybe homework recommendation number one for an educator that might be listening in.

Heather: So another thing that was predictable was the utilization of color-coding. And this is something that many teachers probably do already. But we did, when we were recording the students' ideas, we used different colors for each student, and that made it more accessible. Again, it was a support for our students to be able to distinguish between different chunks of information on the board as they were looking at each other's responses and reflecting on those responses. So really reading that.

Mike: Can I ask for a clarification on that, Heather?

Heather: Absolutely.

Mike: I think what you mean is that you use different [colors] to represent different students' contributions. So if a student shared something, you might write it in red, and if it was a different student, it might be in green. And then you can distinguish what contribution each student made.

Heather: Yes. Yes, that was a predictable aspect of the routine, as well as Jana had mentioned earlier, attributing the ideas to students using their initials. And if multiple students contributed to that idea and the original person who was sharing said that, yes, they would like to attribute more people, then we included all the people's initials who contributed to that idea that was shared in that number talk for that idea, that communication.

Mike: Speaking of contribution, I want to name something that we talked about in our preparation for this that seems incredibly simple but felt like it was really significant. You all talked about the importance of the teacher consistently—not just once, not just a handful of times—but consistently, on the regular stating to kids that they wanted to hear from all students. And I wonder if you can just talk about what did this sound like to make that happen and what was the impact on kids? Jana, I think this is one I'd love for you to start with.

Jana: Yeah, absolutely. It is simple. All you say is, "I'm so glad to be with you today. And let's remember that while we may not hear from everyone today, it's our goal to hear from almost everyone over the course of the week." And if you as a teacher have made a commitment to voluntary sharing, it's essential to say that, to really tell them that you do want to hear their voices. You need to tell them that. Otherwise, they're not going to know that you want to hear their voice. And like I shared a little while ago, there was one student who actually said to me, "I didn't want to share that day, and I knew my teacher wanted to hear from me, and so I did." And then in reflecting back on that share, to get at students' perspectives on what number talks have been like for them—they were fourth graders, only 10 years old. I showed them video of themselves participating in the number talk, and you should have seen the smile on that kid's face. The pride he had in having taken that risk because his teacher wanted him to. People rise to the expectations that we have for them, 100%, maybe not 100% of the time, but if we don't have that expectation, they don't get to choose to rise to the expectation. And you can't make anyone talk when they're not ready to talk yet.

Mike: Heather?

Heather: I also think that part of that goes back to something that we were talking about a little while ago, and that is establishing the norms in the community of learners. And in addition to communicating that to the whole group, our goal is to hear everyone's ideas over the course of the week. Something also as simple as when they were getting ready to do a pair-share and rehearse their thoughts with each other before launching into the whole-group discussion, also reminding them, "Hey, make sure that we're taking turns when we're sharing in that pair." So again, just to reinforce that we value everybody's contribution, we value everybody's voice and everybody needs to have a turn.

Mike: Can you say more about why it's important to offer kids the option to talk with a classmate before they do any whole-group sharing? Why does that matter so much, particularly for multilingual learners? And either one of you, feel free to jump in and take this.

Heather: I'll start. My understanding is that when the originators of these number talks created this idea that they wanted, that idea of agency and giving students choice was really an important priority to them. And so I feel like part of the rationale for that is to give students choices as often as possible in this routine to elevate students to co-learners with the teacher. So I feel like that's kind of where it starts.

Mike: Jana, is there anything you want to add to that one?

Jana: Well, we've already mentioned the value of rehearsal before sharing with the whole group, but there's also another aspect of it that we may not have touched on yet, which is: As that person listens to us and we actually negotiate meeting and clear up ambiguity, we feel seen, heard, and understood. And if I feel seen, heard, and understood by Heather, it's going to be easier for me to share my idea with Mike, who I don't know quite as well as I know Heather. And so there's really a relational aspect of it that is about feeling understood.

Mike: I want to ask another question about something that feels eminently practical. You all talk about recommending that educators call on multilingual learners early in number talks. And I wonder if you could say more about the why behind that recommendation.

Heather: So as a learner of a new language, I may only have one way of explaining my thinking about that problem or the way that I'm seeing that. And if I have taken that risk and I've raised my hand, if somebody else answers first or maybe two other people answer first, maybe they've taken the only way that I knew to answer and share my thinking about this prompt. So for me, as a facilitator in that setting, that was really important for me to prioritize those volunteers if they raise their hand and call on them as one of the first contributors. I've also seen in some classes that I've been in, some math classes, if a student is not yet fluent in English, sometimes their classmates think that they don't know math, that they don't have ideas to share in math. So I also think that calling on those students first also, again, sets the norms in this community of learners that, again, we all have valid and valuable ideas to share.

And so Jana and I saw in particular with the pair-shares, we saw students starting to choose to work with students who still spoke primarily another language. And Jana captured on video where she had a student who didn't speak Spanish and a student who primarily spoke Spanish and they were sharing ideas with each other in that pair-share to get ready for the whole-group discussion. And honestly, I think that that worked more effectively because of that idea that everybody has valuable ideas to share. So I also think that that was another part of that idea of calling on those students first and making sure that they had a lot of opportunities to share their ideas.

Mike: Yeah. I'm really glad you mentioned that. You're making me think about this notion called positioning, meaning that the choices that we make—whether they're spoken or unspoken, like who we call on first or who gets called on more—they are sending a message to students. And often that message may not be the one we intended. So in this case, it really does show how the choices that you all were making in calling on multilingual learners early, it may have disrupted some narratives that people could have formed about how much those kids had to contribute to a mathematical conversation. I'm so glad you shared that.

Jana, I want to ask you this next question. It's something that, if I'm not mistaken, Heather brought up earlier, and I wanted to dig into it a little bit more if we could. You referenced the value of making gestures something that's a normal, accepted, valued practice, and I want to take a bit of time to clarify that. Perhaps for some folks who might not have a clear picture in their own mind of what we mean by that, can you say more about what we mean by gestures and maybe some examples of the ways that gestures either help students to communicate or even how they contributed to the conversation that was happening during the number talk where there might've been something that was lost if gestures weren't in play?

Jana: One thing I know for sure is that lately I've been learning from Heather about how some mathematical ideas are actually perhaps communicated better with gesture than verbally. And yet we have this traditional notion that there's some kind of language for expressing mathematics that's fancy and only occurs from the neck up, but that's not how we usually talk. So why would we tell people who are trying to explain their ideas that they can't use gesture as part of a person-to-person conversation? Gesture by no means keeps you from developing formal language. It actually helps you develop formal language. So one example of using gesture, it came up particularly during dot talks when we first started the routine, and the dot talks were a fabulous way to encourage and introduce that norm that gestures are welcome. But if a student is describing an array of dots and they say, "three on top," and then they use their hand to indicate it's horizontal, we would affirm, "Thank you so much for using your hands." I can tell that the three on top are in a horizontal line. And then, Heather is fabulous, and I've learned a lot about this from her at gesturing "horizontal" by bringing her hand across the space in front of her horizontally. And then everyone [says] "horizontal," and everyone gestures and says "horizontal" with them. And so we're pairing what's an academic word that is often very hard for students with any language background to remember with a physical gesture.

Mike: That's really helpful. As you all were talking about this, one of the things that I started thinking about is how there are ways that I use gestures to indicate a lot of mathematical ideas like partitioning into groups, indicating that I'm talking about a group and another group and another group, which is basically the seeds of multiplication or unitizing. How I'll gesture as a way to show that I'm combining or separating. How I gesture to show the way that I'm counting things. That all of those are ways that actually enhance what I might be saying and actually communicate that meaning more clearly both to my teacher and to the other students who are in the room.

Heather: Absolutely. Yeah. Another example of that, as you were talking about that, that I use all the time as a seventh grade mathematics teacher and we're working a lot with integers, is the idea of 0 in a horizontal hand as 0. And thinking about if that's 0 and I'm navigating between positive and negative numbers, what will that look [like] visually? And as you said, I just think that gestures are another tool for thinking and understanding and processing information and sometimes communicating that information.

Mike: Heather, I want to come back to you for something that, again, really struck me as important when we were preparing for this. You said that you recommend educators close their number talks with an opportunity for kids to make connections between strategies that emerged. And I wonder if you can just talk about: Why is it important to provide that opportunity for kids to make connections, particularly for our multilingual learners?

Heather: So first of all, I have a firm belief that development of conceptual understanding is really valuable in mathematics. And as we are engaging in this routine, in this whole-group discussion, and we're considering all these different possible ways of solving a prompt or seeing a prompt, then when we get to the end, it feels like that we should reflect on the different ideas that have been shared and draw some conclusions about what we can say across all of these different ideas as part of that development of conceptual understanding of what is happening there mathematically. In addition to that, in terms of student engagement, some of our students are multilingual learners. That was the time in the routine that they actually felt the most confident to contribute their thoughts and ideas. So maybe they didn't often raise their hand to speak in that whole-group discussion, but they did raise their hand to share something they noticed from the artifact, some kind of commonality or something that stood out to them.

So again, that was another opportunity for them to feel like they had a valid contribution, that their contribution needed to be heard. So those are a couple of good reasons why I feel like that final reflection is really important in particular for multilingual learners.

Mike: Well, Jana, before we close this conversation, I'm wondering if there are any resources that you'd recommend to a listener who wants to keep learning about the ideas and the practices that we've been discussing today. Is there anything that you could point them in the direction of, or perhaps even something that you'd invite them to try out as a first step?

Jana: Yes, absolutely. I have a couple of ideas. One would be to go to a blog I write that's called mathbetweenus.org. And I've published a short article there ["Number Talks: A Whole Class Routine for Learning Language for Learning Mathematics"] that is specifically about the adjustments we've made to the routine.

Also, I am now CEO of the Mathematics Education Collaborative, and we recently developed a grassroots workshop in making number talks meaningful. It only takes 2 hours. It's an introduction to the routine, ensuring that it's more than just something fun, but actually results in building number sense for students. It's a low-cost way for an individual teacher to get started. And then you can also go to our website at the Mathematics Education Collaborative, which is [www.mec-math.org] and reach out to us and see if you're interested in having us come to your district or your region. Or you can email me at [email protected]. So lots of ideas.

Mike: I think that's a great place to stop. I can't thank you both enough for joining me and being willing to have such an in-depth and detailed conversation. Jana and Heather, it's really been a pleasure talking with you both. Thank you.

Jana: You're welcome.

Heather: Thank you so much.

Jana: Thanks for your curiosity.

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

Research suggests that supporting students' fluency with addition and subtraction hinges on understanding how children's mathematical thinking develops. So what are the concepts and ideas that play a part in fluency with combinations to 10, 20, and beyond?

Today, we'll explore this question with Kristin Frang, director of instructional programs at Integrow Numeracy Solutions.

Kristin Frang is the director of instructional programs for Integrow Numeracy Solutions. She designs resources and services that support states, districts, schools, and individuals in transforming numeracy education.

"Understanding Units Coordination" Season 4, Episode 11 of the Rounding Up podcast

Integrow Numeracy Solutions

• website

• blog

• email address

On Track to Numeracy book by Lucinda "Petey" MacCarty, Kurt Kinsey, David Ellemor-Collins, and Robert J. Wright

Mike Wallus: Welcome to the podcast, Kristin. It is so great to be talking with you today.

Kristin Frang: It's great to be here. I feel so honored to be on this podcast.

Mike: Before we dive into a conversation about addition and subtraction, I'd like to do a bit of grounding. So you're currently the director of instructional programs for Integrow Numeracy Solutions. I wonder if briefly you could tell the listeners: What is Integrow Numeracy Solutions, and what's its mission?

Kristin: Yeah. Integrow Numeracy Solutions' mission is to transform numeracy education by connecting research with practice and empowering educators to advance student mathematical thinking and success. But I really want to bring that mission to life through a story, just a quick story, if I can.

Prior to my role with Integrow, I was a K–12 mathematics consultant. And one of the things that I did was, when the Common Core [State Standards] were released, I worked with teachers to transition to the then-new standards. We studied many documents together, including progression documents that were included in the standards, and teachers were honestly fascinated by this idea of a progression and that they were embedded into the standard. But I remember an instance where we had been studying these progressions and a teacher came up and said to me, "I know where my students are at; I can see them in these progressions. But how do I get them to the next stage?"

And I didn't have an answer (laughs) at that point. I was a former middle school and high school teacher. I was working with elementary teachers. I was studying, just like them, these progression documents, and I could only categorize the reasoning that was in front of us. And so that next step to say, "Oh, this is what I would do and bring into action in the classroom," I didn't have an answer for.

And so that's really where I was introduced to Integrow—formerly [the] US Math Recovery Council, but now Integrow Numeracy Solutions. And at the heart of our mission to empower educators is to bring research to the classroom in accessible and practical ways that advance student reasoning. We do this in professional learning, we do it in supplemental resources, and we also hire and train educators to deliver high-dosage tutoring for students to accelerate their learning.

Mike: I want to just linger on something you said, which was—and I really appreciate both the truth of the statement you made and also the vulnerability, which is to say—I think for many teachers, there's this experience of, "I can see my students in these progressions, but I'm not sure what to do when it comes to making moves to shift where they're at or help them move." And I think that's a profound truth for so many teachers. And I think it's really important that folks like you, who are doing this work, acknowledge that that's a place you were in once as well because that's so true for so many of us.

Kristin: Yeah. There's always a new thing where we're watching students, we're thinking about the next steps. And so often it boils down to categorizing the things that students are doing now, but not often figuring out: What are the true actions that we take with real children who are in front of us to get them to progress in their own reasoning? We can tell them the next step, but my belief system that is aligned with Integrow Numeracy Solutions is that the most powerful thing is to help students have those experiences and create that understanding themselves. And to do that, it's more complex than just knowing what the next benchmark is for them.

Mike: I think that's a helpful introduction. And I also find it to be a good segue for all the questions that I wanted to explore today. So let me start here: It feels important to acknowledge that supporting students' addition and subtraction fluency actually hinges on understanding how children's mathematical thinking develops. So I wonder if you can talk about some of the concepts and the ideas that play a part in fluency when it comes to combinations of 10, combinations to 20, and even beyond.

Kristin: Yeah. The words that we hear associated with fluency right now are "flexibility," "efficiency," "accuracy." So we've moved on from just speed, which I think is a really positive place for us to be in education. But at the heart of flexibility, efficiency and accuracy is a quantitative understanding of arithmetic. I'm really glad that you had Amy Hackenberg on [the podcast] recently who discussed this concept of units coordination because throughout what we'll talk about, you'll see units coordination come out, but she's definitely the expert to explain it. Just a nod. Just listen to that episode [Season 4, Episode 11]. It was amazing.

Thinking, though, specifically about fluency—fluency isn't just knowing all of these combinations. In the early stages of counting, students view a number simply as a count or result of a count of single items, and there's this critical shift in developing a unit as a fundamental tool of measurement. And that's the act of unitizing where a student conceives of a collection of items as one unit that's simultaneously made of smaller units.

It is a common progression that once a student counts on, that then we would shift to building strategies to solve addition and subtraction within 20, and then of course with 100, and beyond, and then in other domains. But this is all happening in first and second grade for that addition and subtraction to 20 fluency. So attending to this numerical composite—understanding that when a child says "7" and sees that that represents counting from 1 to 7 without having to count—is a really big cognitive shift in their mathematical understanding and can be undermined with, "Oh, now that they're counting on, we're going to tell them these strategies." And so we really do need to have some intentional instructional strategies to make sure that we're developing that first, that numerical composite, before we try to develop all these strategies for addition and subtraction to 20. Because that is the basis for children to move from a counting-based strategy to compose units.

So when they can use a quantity like, "Oh, 8 plus 5, I can break apart this 5 into smaller parts and I can give some of those parts to the 8." So children at that point have to simultaneously hold 5 as a single unit while recognizing the 2 and the 3 make up the 5, but they can be moved to the 8 as well. That's really sophisticated.

Mike: So I want to mark that because I think the notion that this is really sophisticated is important for folks to understand because I'll be vulnerable and honest: I didn't recognize the complexity of what children were grappling with when I started teaching, particularly as a person who was teaching kindergarten and first grade. I really saw my job as helping to build a set of rote procedures like counting and number sequence and memorizing combinations and the outcome of being able to count and the outcome of being able to quickly recall those. I think that's not in question, but understanding the mechanics and the evolution of kids' thinking that's going on, that's a big deal. This whole notion that you have a unit and the unit is composed of smaller units. And one of the things that you said that feels like a really big deal that could be lost is the idea that shifting from a counting-based strategy to a strategy that depends on this notion of units that have smaller units inside and that are also still a unit—that's such a big deal. In order to go from counting everything to counting on to being able to look at a number like 8 and say that it has a 5 and a 3 inside of it—all of that is connected to this notion of units inside of units. And I'm so glad you mentioned that.

Kristin: Yeah. The mental actions that students are doing, making those visible, when we see children do it developmentally, we just assume it's easy. But the shifts that they're making in their understanding of units to move from that pre-numerical stage of "Everything is a 1 and I have to repeat it" to "Now this word can stand in for the count" to "Now I can embed units inside of other units." There's so much happening, and they're so young at that age; we have to remember that too.

Mike: So let's talk about some other important components of developing fluency. What else is an important primer for how people are thinking about this?

Kristin: Yeah. Another important component is supporting students in developing the cognitive structures that allow students to anchor their understanding and quantitative meaning and develop that sophisticated reasoning. Many researchers, many authors have written in different ways and different names about these structures. So like a "mental structure," "mental residue," "mental tools," "patterns of thought." To name a few people, Zaretta Hammond, Betty [K.] Garner, Karen [S.] Karp are some people I've read and appreciate their thinking around that.

So it's more than just allowing students to use manipulatives to solve problems. There's an intentionality in how we use tools and an explicit process used by educators to bring their mathematical world to life. So first, identifying key settings that emphasize mathematical structures. So the tool in front of them has a big role to play in the "math"—I put that in quotations—in the "math" that they see. 10-frames that highlight a quantity of 10, but also can show other quantities within 10, such as, like, a five or a double. It has an added layer of boxes that contain a number. Some contain a number or a counter and others are empty. So there's ways that kids are coming to understand quantity with the structure.

Similarly, a bead rack can show a five structure, a double structure, depending on your representation. They can help kids think about exchanges and really kind of that movement of quantity in a real physical way. Using linking cubes, do you use them all in one color? Are you strategic about the color that you use to bring out mathematical structures for them?

So once we think about the key setting and the structure that we're trying to help kids reason about, we want to pose intentional questions that orient students to those structures. So how do they see that 5 inside? How are we going to bring that out? It's obvious to us, but are they seeing that or are they seeing something different in the tool? Are they reasoning about something different? And so the intentionality behind how we question students during those activities also aids to building their cognitive structures. So it's not the tool itself that is the 8. It's that the child is seeing the 8 and they're seeing the 5 and the 3 in some empty boxes.

And finally, I think the step that we miss a lot, especially in problem-based instruction or any kind of inquiry-based instruction, is this explicit time where we connect the symbols in formal mathematics directly to represent the child's thinking and the tool that they've been playing around with. So it's not just about knowing I can get an answer on the 10-frame, but it's [that] I'm abstracting that series of actions, and I'm then connecting it to this quantity that I've written in a symbol. And are there connections between those things? And if those things aren't happening—kids are doing all those parts and pieces, but really developing the cognitive structure that they can then themselves use and take with them, I think that's what's so powerful when we talk about fluency is they can take a cognitive structure with them and fill in the mathematics in the future [when] maybe they don't have an educator in front of them asking those questions. But if they've been through those processes, then they have that structure to fill in.

Mike: There's a lot that you just said that I think is important and we could probably linger on a lot of it. But on the front end of this conversation, you said it's one thing to be able to see students in a progression, and it's another thing to think about, "What's my role or what are the tools that I have to help them shift?"

What I heard in that last part, particularly is this notion of almost like a translation between the physical materials kids are engaging with and the meaning that they're making of that, and then helping them to abstract that in a way where we have symbols that are representing either actions or quantities and the relationships that are happening. That part of the teacher's job and part of the moves that teachers have in their toolbox is this notion of translation—taking what I'm seeing kids doing and how what I'm hearing them say or do to make meaning of it, and then helping them make that abstraction is kind of one of the tools that's really important in a teacher's toolbox when they're thinking about helping kids make moves.

In preparation for our interview, one of the things that stayed with me was you described how your own understanding of the meaning and the importance of fluency had shifted over time. And I'm wondering if you can talk about what you used to think and what is it that you think now about fluency. Could you talk about your own personal journey?

Kristin: For sure. I used to think that knowing facts, just knowing them in a very static way—like I know the answer to 5 plus 3, I keep coming back to that fact—reduces the cognitive load when they were getting into higher grade levels. Well, they don't need to think about that problem, and they can think about what we're doing in seventh grade math or in algebra.

But what I've come to understand is that the ways that students know their facts—more specifically how they're able to work with the units and the way they conceptualize the units that they are given, how they break them apart, how they put them back together—that's what matters as they go. So not just knowing the answer, but that these things can be taken apart and put back together.

Anderson Norton is a researcher that I really love to listen to. And I listened to him at an Integrow conference once. And he talked about developing mathematics through repeatable mental actions. So this kind of relates back to those cognitive structures. One example of a group of mental actions is this idea of composable, reversible, and associative. So when I think about 8 plus 5, 5 is composed of a 2 and a 3, and I can reverse that to focus on the unit of 2, and then I can associate that quantity with the 8 to make a new unit while keeping intact the unit of 5. That's really complex, but that idea transcends the domains of mathematics. Now, I'm not an expert in units coordination research, so I hope I represented that correctly, but I've certainly experienced students struggling to keep track of different units as they work. So thinking about exponent rules, and they break apart these powers and they're writing them and they're learning all these patterns, but they're struggling to keep track of the units that they're working with. Factoring functions in algebra. We're asking them to break apart something and put it back together in these different forms, and they're losing track of these units. So these actions of composable, reversible, and associative have implications in many domains of mathematics. So the bottom line is we want to develop not the fact itself, but the mental action behind that fact. Anderson Norton, I hope I did that justice.

Mike: I want to name something that I think is really important, particularly given the fact that your background is actually in secondary [education]. So what I take from this is this idea of working with units and the mental actions, that transcends arithmetic. It transcends whole numbers and even rational numbers. And it pays dividends and it keeps paying dividends in middle school and high school as kids are working in an algebra context. And I think that's worth saying out loud because it means that doing this work with elementary students to develop fluency is a bit of a twofer in the sense that you do get kids who end up with a bank of facts that they know, but they also have this underlying understanding of units and actions that pays dividends for them in the long run. Mathematics education, students' learning experience, is not a sprint or a series of handoffs. It's really a marathon. And those early experiences, they pay dividends and they keep paying dividends. I think that's really important because it reminds us, particularly as elementary educators, that we're part of a larger project.

Kristin: Not only part of a project, but part of building a lifelong interest in mathematics as an actual body of research that's dynamic and not a set of things to memorize and learn so that mathematics does become applicable in these different fields because the way that I approach a problem as an expert mathematician is that I take things apart, I put them back together. That transcends many careers. It's not just about being a math teacher or a math professor. It's about coming to understand that I have autonomy and how I see relationships of things, whether they're numbers or shapes or maybe parts that I'm working on in some sort of creative field that I'm in, but that I can do all of these things and that I can be curious and repeat those actions and see how they play out in that particular study.

Mike: That's well said.

Well, let's talk about the what, the why, the how of combinations to 10 and 20. To begin, I want to note that we use the term "combinations," and I'm wondering if you can say more about what you mean when you refer to combinations and why they matter.

Kristin: Yeah. I mean combinations not to literally mean "addition," but that combination is the idea of this relationship between parts and wholes. So that 2, 3, and 5 have this kind of additive relationship. I can put these parts together to make the whole; I can take a part out of the whole and be left with a part. I can have a part and wonder what part I need to make the whole. And so we sometimes talk about these in curriculums as "fact families," but the emphasis should be on the relationship of the parts to the whole and not filling out that kind of mimicking of like, "I know the four sentences because I know this thing." So, "If I know this, I also know this." It feels really nuanced, but in action really quite specific.

Mike: So I think that's really helpful and it really does lead me to my next question about how we help kids build their fluency with combinations to 10 and 20 and beyond. So given the why that you just articulated, it seems like the how is going to be substantially different from the ways that many, if not most, adults learn to build fluency. Can you talk about that, Kristin?

Kristin: We start from key combinations first. We consider a set of combinations that would be really useful in a lot of contexts. And I think many listeners will be familiar with those key combinations: doubles. Combinations of 10, of course. 5 plus because I have five fingers and then I can add some more on it, and I'm showing some finger patterns. So those are things we normally work on with students anyways. But starting again, going back to my original statement from a quantitative perspective—so not the memorization of those facts, but that I really come to understand them as quantities that are useful to me. And then building from those key combinations—I also want to name before I build onto that, is that some kids just have other facts that are interesting to them that they bring. So it might be their age, it might be the combination of their siblings' ages. And so we don't want to ignore that we introduce key combinations to students, but that students also have combinations that are useful to them naturally.

So once we have a set of those key combinations that we've come to think about and reason about, we can then build things that we don't know. We can transfer that. So 5 plus 3 can help me think about 4 plus 3. If I have a mental structure of a 10-frame or a bead rack that helps me think about, "Oh, there's just going to be one less counter on the top, and so I'm going to take that [counter] away." So that idea of taking the 1 out of the number is a really important mental action of them disembedding that quantity.

In addition, when we think about the 5 plus, the doubles, the partitions, we're thinking about combinations that will also transcend into multidigit combinations. So addition, subtraction—whether we're working with whole numbers or decimals, we can make tens, we can make hundreds, we can make wholes, we can make zeros. And those combinations of 10 are going to be really useful for us.

Mike: I'm struck by the fact that the combinations and also the mental actions that accompany them, as you said, they really do scale up quite nicely. And it seems like they scale up in the sense that they can get used to understand and solve problems with larger whole numbers, but they can also scale in the sense that ideas will help kids, but they can also scale in the sense that the ideas can really help kids when they encounter fractions and decimals. I wonder if you could talk about that idea just a little bit.

Kristin: Yeah. So thinking about a combination of 10 in this missing part. So 99 plus can help us when we're thinking about, that 99 is 1 away from 100. It can also help us think about 99 one-hundredths or 9 tenths as being one part or one unit away from a benchmark number that's really helpful for us. And so, it's just that the unit itself is different. So instead of just a whole, I'm one whole unit away from 100, I might be 1 tenth of a unit away from one whole, so the unit is just changing.

The view of mathematics this way, again, is very dynamic. We're creating a world where children are thinking about units and units away across domains, across number systems. And if we come to regard units as things that we can act on, whether it's a single object or a group of objects or a shape—we can put them together, take them apart and reassociate them—I can think of a lot of my mathematical knowledge in this way and not as a static set of information that I learned. And so then I'm able to transfer that because I've done that mental action or I've thought about something being a unit away.

Mike: That's fascinating because I'm going to go back to this whole notion of the relationship between 3 and 2 and 5. So 3 is 2 units away from a unit of 5 and three-fifths are 2 one-fifths away from a unit of five-fifths or one whole. This notion of units away from or units that combine to make other units, I really get now whether it's whole numbers or fractions, we're really talking about a unit that we've defined and then how many other units or how can we—how did you describe that? What was the language you used before about pulling a unit out? Was it "disembed"?

Kristin: "Disembed," yeah.

Mike: That really plays regardless of the type of unit we're talking about.

Kristin: Yeah. And remember back where we said this quantity had a meaning, so 7 stood for something. When we disembed, that unit still has meaning in the context of the original unit. So that's a really important point about disembedding is that it's not just that you take a part out, it's that part still has a relationship to the whole and you don't lose that relationship.

Mike: As I hear you talking, there seem to be some themes that are jumping out. One is the importance of key fact combinations and the mental actions. Another is the role visual models play in learning those combinations. And I think finally, I hear you indicating that it's important for students to make connections between different representations of the same combination. Tell me what I understood properly. Tell me what you'd revise or add to the summary that I just offered.

Kristin: Yes. I think we get a false sense that a student understands a concept when they're recognizing pattern, and that could be that they're recognizing pattern in a really intentional setting. Maybe they're using a 10-frame. But is that same relationship present in another setting? Success should not be measured by one instance of a child recognizing that pattern. And so one way of knowing that a child knows this is to see it in many contexts. And I think that's why it's so important for us to acknowledge the research around multiple representations in mathematics. And showing that knowledge in these multiple ways really does say that this is a connected set of knowledge that I can refer to as a child and not just be successful on this one day. That doesn't mean that that experience where they're recognizing the patterns is not important, but that can't be the measure of their success.

So this also becomes challenging in our system that values assessment events so heavily and measuring against a set benchmark. And I just want to name that because that's a real challenge for teachers. And of course we want to develop this rich set of knowledge, and sometimes we have to say that this is the system that we live in. But the true measure of that knowledge is being able to take that knowledge and transfer it into these multiple representations or in these multiple spaces and be able to use that. And that's why we talk so much about fluency being flexible and not just about accuracy.

Mike: You have me thinking more deeply than I have in a long time about the structure of some of the visual models and the physical materials that children use when they're engaged with the Bridges curriculum. I wonder if we could get specific and talk about a few of the visual models that support student learning. Are there features that make some models particularly valuable?

Kristin: One I want to mention that we might not have talked about is just a child's fingers. I think sometimes we think child's fingers are not models for them because they're counting by 1 and we tend to want students to move to more efficient strategies. But these fingers actually become really efficient tools. We can exchange fingers, we can move them very easily. We have control, and they're always with us. And so the finger use itself, I think, is a really powerful tool for us to encourage students to use in very sophisticated ways.

Mike: I mean, we literally have units of 1, units of 5, and a unit of 10 at our fingertips in front of us. I'm so glad you called that out because that's a tool that students can make use of, that teachers can make use of and that we can think of in a slightly different way than we had in the past when I just thought about fingers as a counting-by-1 resource, when actually fingers, [a hand], and hands, plural, are 1s, 5s, and 10s right there in front of you.

Kristin: And they can stand in for other units if we're really sophisticated with sequences. So a 1 can be a 7 if we wanted it to be, and we can think really creatively about that. I mean, I think that depends on some other skills. But yeah, we have 1s, 5s, and 10s built right into our hands.

Mike: That's exactly right. And you're making me think about the fact that when I skip-count or when I see students skip-count, oftentimes what's happening is I'm speaking the unit out loud and I'm holding up one finger to stand in for that unit on my hand to keep track of the number of units. So I totally hear what you're saying.

Kristin: Yeah, very sophisticated. And then there's even more complex content, right? So thinking about hours and elapsed time, and we're crossing different kinds of numerical systems where you go from a 12 to a 1 is very complex, and then we can have these fingers as units as well to help us keep track of things. So of course, frames are a really powerful tool. Frames—specifically, 10-frames, 5-frames, 20-frames—provide an extra structure for students, especially when they're really thinking hard about some quantity pieces. So they might not be completely solid in that unit, but we don't have to say, "Oh, you have to count on first before we're going to try to explore some other patterns." Those things can be developing simultaneously. So frames provide this box that contains the unit for them and it becomes this really obvious count for them. They can see those individual discrete items, but they can also see what's missing really clearly because they're empty.

Bead racks are a great support as well when you're thinking about that relational network that we want students to develop and not count by 1s. So we can exchange beads, and we can exchange quantities, and we don't have to exchange beads one by one. Sometimes frames, when we get to a space, it's inconvenient to have to move five counters at the same time where in a bead rack, you can just slide those five over or three over at the same time.

I also want to mention linear bead racks. So taking that stacked bead rack and making it align really helps students think about a continuous model, which transfers to a number line and the idea of units being measurement. So we were talking about, "It's one away," and so really conceptualizing that kind of next decade of numbers and one bead away. That's developing that idea of relative magnitude that's extremely helpful when we get to middle school and all of a sudden we're working in negative numbers.

Mike: We're reaching the end of our time together. And before we go, I'm wondering if you could share contact information for Integrow Numeracy Solutions with our listeners. I'd really love to be able to offer that because we've just touched the surface of some of the ideas that you help educators explore in some of the training and the support that you all offer.

Kristin: Yeah. If you'd like to find out more about us, a great place to go is our website, which is www.integrowmath.org, all one word. And we have a lot of different things you can explore from our events. There is actually, if you add a backslash "blog" to that [www.integrowmath.org/blog], you can go to our blog and read some of the ways that we think about our professional learning and some of the topics that I talked about today. If you want to reach out directly, feel free to email [email protected] and someone will get you to the right place based on your question.

Mike: And for listeners, we'll put a link to both of those in the show notes.

Before we leave, Kristin, I'll just ask one last question. Are there any recommendations that you have for folks interested in learning more about the ideas we've talked about today? It could be books, websites, articles, or even just a suggested practice for someone who wants to get started.

Kristin: Yeah. For sure, take a look at the blogs on our website. They're little snippets of pieces of our trainings that you can take right with you into the classroom. Some ideas that I've talked about—help with bead racks, ideas around multiplication and division, and supporting students to think about those units. Our new publication, On Track to Numeracy from [Lucinda] "Petey" MacCarty, Kurt Kinsey, [David Ellemor-Colons, and Robert J. Wright], is designed to be an accessible, relatable and practical tool focused on supporting classroom teachers. It not only has the progressions that I started this podcast off talking about, but it has those teaching tests and progressions that help us answer the question of, "What do I do next now that I can understand where my students are?"

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

Kristin: Thank you for having me. I've had a great time.

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

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.

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.

Learning Trajectories website, featuring the work of Doug Clements and Julie Sarama

Math Trajectories for Young Learners book by DeAnn Huinker and Melissa Hedges

Mike Wallus: A note to our listeners: This episode contains the second half of my conversation with DeAnn Huinker and Melissa Hedges about math trajectories for young learners. If you've not already listened to the first half of the conversation, I encourage you to go back and give it a listen. The second half of the conversation begins with DeAnn and Melissa discussing practices that educators can use to provide students a more meaningful experience with skip-counting.

Melissa Hedges: One of the things, Mike, that I would add on that actually I just thought about is when you were talking about the importance of us letting the children figure out how they want to approach that task of organizing their count is it's coming from the child. And Clements and Sarama talk about the beautiful work about the trajectory, [which] is that we see that the mathematics comes from the child and we can nurture that along in developmentally appropriate ways.

The other idea that popped into my mind is it's kind of a parallel to when our children get older and we want to teach them a way to add and a way to subtract, and I'm going to show you how to do it and you follow my procedure. I'm going to show it. You follow my procedure. We know that that's not best practice either. And so we're really looking at, how do we grab onto that idea of number sense and move forward with it in a way that's meaningful with children from as young as 1 and 2 all the way up?

Mike: DeAnn, I was going to ask a question to follow up on something that you said just now when you said even something like skip-counting should be done with quantities. And you, I think, anticipated the question I was going to ask, which is: What are the implications of this idea of connecting number and quantity for processes that we have used in the past, like rote counting or skip-counting? And I think what you're saying is we need to attend to those things that, like the counting sequence, we should not create an artificial barrier between speaking the words in sequence and quantity. Am I reading you right or is there more nuance than I'm describing?

DeAnn Huinker: I think you're right on target, Mike. (laughs) Connecting those things to quantity. And I mean, the one that's always salient for me is skip-counting. Skip-counting is such a rote skill for so many children that they don't realize when they go, "5, 10, 15" that they actually have seen, "Oh, there's five [items], there's five more items, there's five more items." So it's making that connection to quantity for something like skip-counting, but also on the counting trajectory, then we start thinking about, "What's a ten? And what makes a ten?" And, "What is 30? And how many tens are composing or embedded in that number 30?" And again, it's not just to rotely say, "3 tens." No. "Show me those objects. Can you make those tens?" Because sometimes we find disconnects. Kids will tell us things and then we say, "Can you show me?" And it doesn't match. (laughs) So we continually start thinking about quantities and putting [objects] with quantities.

Let me add one more thing. In the counting trajectory—and this was very intentional for Melissa—is when we have kids count, we'd like to give them like 31 or 32 counters to see whether [...] they can actually bridge that decade and to go beyond. The other thing that we did, so getting like beyond a ten, also we find when kids get to the number 100, they stop. They just think that's the end. I got to 100, I'm going to stop. And then we say, "Oh, what would be the next number?" And some will say 110, some will say 200, some will give us something else that we find bridging 100 is on the trajectory. And that's actually a really critical point. And again, we want it with quantities with objects.

Mike: I really appreciate this part of the conversation because I think for a teacher who's listening, it helps really get to the specific types of details that would allow them to create the kind of experiences that we think matter for children.

I do want to take a step back though and talk about what's going on for students under the hood, so to speak. So as they're engaging in meaningful counting, what are the cognitive processes that they're learning to coordinate?

Melissa: This is Melissa. So I'll start that one and then invite DeAnn to jump in as I work my way through my thinking.

One of the pieces that, in addition to everything we talked about with all of the skills and ideas and understanding that comes to bear when little ones count, one of the big pieces that we're starting to talk and learn about a whole lot more is this idea of executive functioning. And so executive functioning are those skills that help us manage our attention, help us manage our behavior. They help us stay focused. They help us complete tasks, keep track of things. So hopefully as I'm saying this, what you have in your mind is a little one counting and you're thinking, "Oh my gosh, how do they know where to start?" "How do they know when to stop?" "How do they know when this has been counted with that hasn't been counted?" "What am I going to say next?" All of that tends to be couched very strongly in this idea of executive functions. So when we watch kids count, we know that they're really drawing on those executive functions. And it's actually a really beautiful marriage. So again, we're looking for kids to—are they able to stay on task? Can they keep track? Do they monitor themselves as they go? If someone—this happens a lot—if someone bumps into their collection and their collection gets a little shaky because their desk got moved or someone kicked a counter across the floor, do they remember where that goes and what that stood for in quantity? And for us, that really kind of comes down to some of those higher order skills and in particular, those ideas of the executive functions.

So part of what we notice is that in particular with counting, though all of mathematics, much of what we do and ask kids to do, it takes planning, it takes self-monitoring, and it takes kind of a sense of control and agency over their work. We've talked a little bit about some of that other stuff in the way that it's the work of the child, and that's why we will always ask teachers to step back and just watch, just watch what they do, just watch what they do, because it gives us insight into so many skills, understandings, and kind of where they're at.

DeAnn: Yeah. This is DeAnn. I was thinking of that same thing, Melissa, about this is the work of the child, right? As adults, we're kind of prone sometimes to say, "Let me show you how to do it." But if we want to develop these executive function skills, these ideas and cognitive abilities under the hood, we have to give children opportunities. They need the time to think about how to organize that collection. That's always a great one to kind of think about. As adults, we're like, "Well, just line them up." And it's like, oh no, that's actually huge for a child to realize lining them up or organizing them in some way is a strategy, just like we do with larger numbers. It's a strategy for little kids. So again, that work needs to come from the child and they need to do some trial and error and adjustments in order to develop those things under the hood. And as adults, we can't take that opportunity away from children. We need to create the opportunities so they can explore more of their world and the quantitative world that we live in.

Mike: Everything that we're talking about has some pretty major implications for instructional practice, but what I find myself thinking about is my own time teaching kindergarten. And when I reflect on that, I sometimes found myself falling into something that I would call a readiness trap. And what I mean by that is I had this notion that kids had to have a certain set of skills in place before they were ready to do something like counting a collection. And I think what you're going to tell me is that perhaps I had it backwards. Am I right?

DeAnn: So this is DeAnn and I'm thinking, well, maybe it's not so much backwards, but it's a different perspective. So Melissa and I really struggle with this concept of readiness, and that's because we really frame our work from a developmental perspective. And as we think about learning trajectories, that's what they are. Learning trajectories is a developmental view of children's learning. So what really changes the question for us. We don't ask the question, "Are children ready?" What we ask is, "Oh, where are children currently in their learning?" And then we can start at that spot and then think about the experiences that would help support the next step in their learning. So from a learning trajectory perspective, we really view differences in children's understanding and abilities as just different starting points, OK? They're not deficits, nothing that needs to be remediated. Kids are ready to learn every single day.

It's really us as adults. We have to reframe our preconceptions and train ourselves to really look at what children can do, not what they can't do. And that's where learning trajectories are so powerful because they help us identify those starting points and then they help us as educators know more clearly what is the next developmental milestone that we should be working on with that child. So it's our responsibility to be ready for the children that come to us, not the other way around.

Mike: I really appreciate this idea of the progression as a series of starting points. I think that's a really helpful framing device, and it certainly puts the work that we do in the kind of light that you're advocating for.

One of the other things I wanted to talk about is in the book [Math Trajectories for Young Learners], you all make reference to how important it is to develop a playful pedagogy. And I wonder if we could just try to talk about, "Well, what does that mean? What might that look like in a classroom?"

Melissa: So this is Melissa. I think in any district or agency that's supporting young children, this is a very hot topic, the idea of play or playful pedagogy. And what I like to do is to think that we can use play as a teaching platform and not just as a break from learning. Play actually can kind of lay the foundation for a lot of those learning experiences. I think it's powerful because playful learning, it nurtures important habits of mind that we can develop in some ways, but for our little ones, they develop very naturally through the idea of play. So we think about curiosity, creativity, problem solving, flexibility, persistence, all of that comes up as kids are playing. And so I think that both DeAnn and I would agree that the idea around playful pedagogy and mathematics learning trajectories really partner well because the trajectories help us see that mathematics develops over time based on experience and opportunity. So the trajectories don't replace play so much as [...] strengthen educators in recognizing during times when kids are playing or during those playful moments that an educator can have a stronger perspective or a more keen eye, I guess, on what they're noticing with their children.

And when we think about playful pedagogy, where we're headed is not free play, but this idea of guided play. So in guided play, the teacher's going to set up the environment, they'll have a learning goal in mind. So for example, if I'm working and deepening my understanding as a classroom teacher around the counting trajectory, I'm going to have an idea of where my children are on the trajectory and what questions might I pose during play or ponderings might I provide to the children during play. So it's not me taking over that time or the teacher taking over that time, but it's really supporting or pushing the learning through some subtle prompts or some shared discoveries or maybe some purposeful questions. So, for example, if the kids are in the block area and they're building a tower or they just have blocks all over the floor, they're making a road, I might ask them, "How long is your road?" or "How tall is your tower?" and let them kind of ponder with that. And then, this is always a fun one, "What would happen if I put two more on?" or "What would happen if your tower grew by two more blocks? or "What would happen if three of them fell off?" And really just engaging in some of those playful conversations—not to take over the play, but to capitalize on the playful moment.

Mike: I love that, particularly the notion of, "What if three fell off?" or "What if I had four more blocks and I wanted to make it bigger or longer?" It's a lovely way of organically injecting or assessing kids' thinking within the context as opposed to imposing a task in a way that it just has an entirely different feel to it. And yet at the same time, it's really informed by the trajectory in a way that helps it be like, "This is the right point for me to ask that particular question."

Melissa: Exactly. So I can kind of give an example if, I'm thinking of maybe a 5-year-old and so, one of the levels of our counting trajectory is being able to do 1 more or 1 less, and it really sits around that idea of hierarchical inclusion. So if the kids are playing and I know that that's where this child might need or this group of children are ready to take that next step, those are questions I can pose in a very—you're right—in a very low-stress, not-high-stakes setting, and it's still very valuable information.

Mike: Actually, that's a great segue because I wanted to ask the two of you about some of the ways that teachers are using the learning trajectories and the assessment protocols that are found in the book to monitor their students' growth. So I wonder if you could say a little bit more about that.

DeAnn: This is DeAnn. I'll start and then I'll pass it back to Melissa.

So, you ask us about the assessment protocols. So maybe we should explain what an assessment protocol is. One thing that we've done with the trajectories that were developed by Doug Clements and Julie Sarama, we've taken those trajectories, but as we're thinking about making them useful for teachers, we actually have developed some structured assessment protocols that are aligned to the trajectory with [tasks] and prompts that we can use with children to help find those starting points. As I mention in the book, we have five assessment protocols in there, like one for counting, one for subitizing, one for adding and subtracting and so on. And then teachers can take these and use them to [say], "Let me ask this question. Oh, they did great there. Let me jump up a couple levels. Let me ask a question there." Or maybe I want to back up to a previous level and ask so that we can kind of get a sense of those starting points for then building instruction.

All right, and then Melissa, you can share how else teachers are using them in and out in the district.

Melissa: I think one of the important aspects that I firmly believe in when a teacher approaches their teaching of mathematics through the lens of a learning trajectory, a mathematics learning trajectory, is that it really does lay the foundation for equitable teaching and learning opportunities. So not only does it lay the path for a developmental approach, it's also incredibly equitable in that we've looked at trajectories as identifying children's strengths. And in that way, it's not what they don't know, it's, "Where are they, and what are those [experiences] that they need?" So it's not that somebody is never going to learn it. Again, they need more experience and opportunity. And that's, I think, probably been one of the biggest takeaways as we've looked at how we are using trajectories here in the Milwaukee Public Schools, in particular the counting trajectory. So to get a really nice handle on where children are developmentally, if we have, for example, in a first grade classroom where they're moving into composing that unit of 10, and we know that we've got kids that are struggling with cardinality, even counting collections of one, two, three, four, five [objects], we know that that's going to be a struggle. So what is it that we can do to accelerate some of those learning opportunities and give more learning opportunities for children so when they get to those big key milestones, we have an idea of why they may be struggling? And it's not that they can't; it's not that they won't; it's not that they don't understand. They just need more experience and more opportunity and more guidance with that work.

So that's one of the ways I think that has really allowed us to support our teachers and have our teachers feel a great sense of autonomy in making instructional decisions for their students. That it's not, "The book is telling me to do this or this is telling me to do that." It's, "Here is something that's really honored a developmental approach to what kids know, and how can I take that then and apply that in my classroom with my students?"

The other thing that it really has helped us do on a big broad level is think about, "Where do we want children to work towards by the end of 3-year-old kindergarten or 4-year-old kindergarten or 5-year-old kindergarten or first grade or second grade in a way that, again, matches the developmental nature of children's mathematical growth?"

Mike: What I really appreciate about what you shared is there's certainly the systems level way of thinking about using this as a tool, but I appreciate the fact that as an educator who might be reading the book, I can also see directly into my own classroom practice and think about moves that I can make to support students and also to understand where they are and what comes next for them. That's super helpful.

Melissa: Yeah. It's those small little moments. It's really as, just staying keyed in and tuned to those small moments.

Mike: I'm going to ask a question at this point in the interview that I suspect is difficult to narrow down an answer, but I want to give it a try just because there's so much from my reading of the book that was powerful. And at the same time, I'm hoping that we can give people a chance to think about how they might start to take action.

So here's the question: If you were to, say, recommend one or two small-scale practices for listeners who want to take the ideas we're talking about and put them into action in their classrooms, what might you recommend?

DeAnn: This is DeAnn. I'll get us started.

First of all, [...] developing this book really came out of our own work with teachers. We have spent many, many hours with teachers of the young grades and helping them to improve their practice. And then with them, we started learning about the trajectories and learning with them as they started thinking about and applying these to the classrooms. So a place to start for one's own professional learning and to deepen your understanding is just to pick a trajectory and just read through it. "Ooh, what's happening with children that are 1-, 2-years-old, all the way up through children who are 6, 7, and 8." And just reading through this progression of levels and then starting to deepen your knowledge of what are these kind of steps that we're taking them through.

And I'll use an example. I think one of the biggest surprises I had for myself in this work is I never really understood before studying the trajectories that counting then leads to unitizing, which then leads to looking at groups, which then takes us to place value. And we talk about counting as being the on-ramp to place value. And I didn't really think about that connection until I just started reading and studying the counting learning trajectory myself and thinking about, "How do children go across all these levels?"

Mike: I want to just jump in and say thank you for saying that because that's something that I've been pondering as I've been listening to you all the way back, DeAnn, to when you talked about connecting skip-counting to physical quantities. What struck me about that is that it allows me to start to imagine a unit that's not just 1, right? If I'm skip-counting by 2s, and I have 2, that's kind of the starting point for unitizing, which—I think the other thing that jumps out is, that's actually eventually going to lead to a deeper understanding of, say, multiplication. So there's a lot in this that really when you understand what's going on across that trajectory, it really helps you understand what's critical about what kids are learning and what also is critical about the kind of experience that I as an educator want to make sure that I'm offering to students.

DeAnn: I'll just build on that a little bit. (laughs) Melissa might too, is, "Wow, counting is amazing." And I think Melissa would say, "Counting is the foundation of everything." But that counting is much more than I think most of us as adults realized. That counting does take us through this idea of making these groups and then thinking about units and units of 10, understanding the place value or our base ten system and understanding place value. It's just amazing when you really start to dig into a little deeper about the math, and the math learning, but how it goes across so many years.

Mike: Melissa, how about you? Do you have a recommendation or do you want to build on something that DeAnn shared?

Melissa: I think what I'll recommend might be a build on. One of the best ways that I would encourage folks to get into understanding how a trajectory could be a really powerful tool in the classroom is pick a child, or one or two children, and give that trajectory a try. Just do it as written, don't stray from it. Just kind of give it a feel, and see what it is that you're learning about your children or child as they work through that trajectory. Because again, it's those small moments when we're looking for those small transitions. Like, if a child—one of the tasks in the counting trajectory assessment is counting a collection of 31. And what do we notice? Do they try to count by 2s? Do they just count by 1s? Do they begin to do some of that grouping of 10 and 10 and 10 and 1 more? One of the most fascinating things we found out as we've watched kids work through the trajectory is when they get a collection, a little bit of a larger collection, let's say 43, they might begin to do some of that grouping and they'll go, "10, 20, 30, 40." And then they hit what we would say as "41" and they say "50, 60, 70". (laughs)

So I would encourage folks just to probably start with where DeAnn started us, which is understand the mathematics a little bit that you're looking for, read through that trajectory, get a feel for what that progression is looking like. Maybe you start to very naturally think of a child that you know, maybe they're kind of sitting here, maybe they're kind of sitting there, and then give that trajectory a try, and see what you learn about your kids.

The other thing that I would say is that we've got a lovely set of videos in the book. There's over 50 videos of many of us doing little tasks with children that will help illustrate what some of those assessment tasks look like or what kids' thinking sounds like.

The other lovely part is that we've provided some activities as well. So if you're thinking, "Oh, somebody is at this level or I'm looking to expand my teaching of number and quantity in my classroom," there's lots of really lovely tested, tried-and-true tasks in there that a teacher could pick up and use tomorrow.

Mike: I think one of the things I'd like to do before we close is just give you all an opportunity to share with listeners where they could go if they wanted to buy the book and learn a little bit more. And then I'll also offer, is there anything else that you think might be a reference point for teachers who want to continue learning about the things you've shared today?

DeAnn: So the book we're talking about, Math Trajectories for Young Learners is published by NCTM, so the National Council of Teachers of Mathematics. It can be purchased from them. They even have a nice little website, nctm.org/trajectories, and that will take you right to a website that can give you access for ordering the book. I will also point out that it's available certainly in paperback, but it can also be purchased in digital formats. So you can download a PDF or something to read on your Kindle or some other reading device.

Mike: I think that's a great place to stop. DeAnn and Melissa, thank you so much for your time. It's really been a pleasure talking with you today.

DeAnn: Always a pleasure talking with you and thinking with you, Mike, about children's learning.

Melissa: Completely agree. Thank you again for having us, Mike.

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

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

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.

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.

Math Trajectories for Young Learners book by DeAnn Huinker and Melissa Hedges

Learning Trajectories website, featuring the work of Doug Clements and Julie Sarama

School Readiness and Later Achievement journal article by Greg Duncan and colleagues

Early Math Trajectories: Low‐Income Children's Mathematics Knowledge From Ages 4 to 11 journal article by Bethany Rittle-Johnson and colleagues

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.

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. 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, 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 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

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.

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.

Journal Article

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

Books

• 5 Practices for Orchestrating Productive Mathematics Discussions

• The 5 Practices in Practice [Elementary]

• The 5 Practices in Practice [Middle School]

• The 5 Practices in Practice [High School]

• Coaching the 5 Practices

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Kyndall: Yeah, absolutely.

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

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

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

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

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

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

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

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

Kyndall: [laughs]

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

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

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

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

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

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

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

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.

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.

Integrow Numeracy Solutions

Developing Fractions Knowledge by Amy J. Hackenberg, Anderson Norton, and Robert J. Wright

Mike Wallus: Welcome to the podcast, Amy. I'm excited to be chatting with you today about units coordination.

Amy Hackenberg: Well, thank you for having me. I'm very excited to be here, Mike, and to talk with you.

Mike: Fantastic. So we've had previous guests come on the podcast and they've talked about the importance of unitizing, but for guests who haven't heard those episodes, I'm wondering if we could start by offering a definition for unitizing, but then follow that up with an explanation of what units coordination is.

Amy: Yeah, sure. So unitizing basically means to take a segment of experience as one thing, which we do all the time in order to even just relate to each other and tell stories about our day. I think of my morning as a segment of experience and can tell someone else about it. And we also do it mathematically when we construct number. And it's a very long process, but children began by compounding sensory experiences like sounds and rhythms as well as visual and tactical experiences of objects into experiential units—experiential segments of experience that they can think about, like hearing bells ringing could be an impetus to take a single bong as a unit. And later, people construct units from what they imagine and even later on, abstract units that aren't tied to any particular sensory material. It's again, a long process, but once we start to do that, we construct arithmetical units, which we can think of as discrete 1s. So, it all starts with unitizing segments of experience to create arithmetical items that we might count with whole numbers.

Mike: What's really interesting about that is this notion of unitizing grows out of our lived experiences in a way that I think I hadn't thought about—this notion that a unit of experience might be something like a morning or lunchtime. That's a fascinating way to think about even before we get to, say, composing sets of 10 into a unit, that these notions of a unit [exist] in our daily lives.

Amy: Yeah, and we make them out of our daily lives. That's how we make units. And what you said about a ten is also important because as we progress onward, we do take more than 1 one as a unit—like thinking of 4 flowers in a row in a garden as a single unit, as both 1 unit and as 4 little flowers—means it has a dual meaning, at least; we call it a composite unit at that point. That's a common term for that. So that's another example of unitizing that is of interest to teachers.

Mike: Well, I'm excited to shift and talk about units coordination. How would you describe that?

Amy: Yeah, so units coordination is a way for teachers and researchers to understand how children create units and organize units to interpret problem situations and to solve problems. So it originated in understanding how children construct whole number multiplication and division, but it has since expanded from just that to be thinking more broadly about units and structuring units and organizing and creating more units and how people do that in solving problems.

Mike: Before we dig into the fine-grain details of students' thinking, I wonder if you can explain the role that units coordination plays in students' journey through elementary mathematics and maybe how that matters in middle school and beyond middle school.

Amy: So that's where a lot of the research is right now, especially at the middle school level and starting to move into high school. But units coordination was originally about trying to understand how elementary school children construct whole number multiplication and division, but it's also found to greatly influence elementary school children's understanding of fractions, decimals, measurement and on into middle school students' understanding of those same ideas and topics: fractions ratios and proportional reasoning, rational numbers, writing and transforming algebraic equations, even combinatorial reasoning. So there's a lot of ways in which units coordination influences different aspects of children's thinking and is relevant in lots of different domains in the curriculum.

Mike: Part of what's interesting for me is that I don't think I'm alone in saying that this big idea around units coordination sounds really new to me. It's not language that I learned in my preservice work[, nor] in my practice. So I think what's coming together for me is there's a larger set of ideas that flow through elementary school and into middle school and high school mathematics. And it's helpful to hear you talk about that, from the youngest children who are thinking about the notion of units in their daily lives to the way that this notion of units and units coordination continues to play through elementary school into middle school and high school.

Amy: Yeah, it's nice that you're noticing that because I do think that's something that's a strength of units coordination in [that] it can be this unifying idea, although there's lots of variation and lots of variation in what you see with elementary students versus middle school students versus high school students versus even college students. Some of the research is on college students' unit coordination these days, but it is an interesting thread that can be helpful to think about in that way.

Mike: OK. With that in mind, let's introduce a context for units coordination and talk a little bit about the stages of student thinking.

Amy: Yeah. So, one way to understand some differences in how children up through, say, middle school students might coordinate units and engage in units coordination is to think about a problem and describe how solving it might happen.

Here's a garden problem: "Amaya is planting 4 pansies in a row. She plants 15 rows. How many pansies has she planted?" There are three stages of units coordination, broadly speaking—we've begun to understand more about the nuances there. But a stage refers to a set of ways of thinking that tend to fit together in how students understand and solve problems with whole numbers, fractions, quantities, and multiplicative relationships. It's sort of about a nexus of ideas, and—that we tend to see coming together and students don't usually think in a way that's characteristic of a different stage until they've made a significant change in their thinking, like a big reorganization happens for them to move from one stage to the next.

So students at stage 1 of units coordination are primarily in a 1s world and their number sequence is not multiplicative. That's going to be hard to imagine. But they can take a group of 1s as one thing. So, they can make a composite unit and that means in the garden problem, they can take a row of pansies as 1 row as well as 4 little ones, and they can continue to do that over and over again. And so they can amass rows of 4 pansies and keep going. And what it usually looks like for them to solve the problem is they'll count by 1s after any known skip-counting patterns. So, in this case they might be like, "Oh, I know 4 and 8; that's two rows. 9, 10, 11, 12; that's three rows." Often using fingers or something to keep track, or in some way to keep track, and continuing to go up and get all the way, barring counting errors, to 60 pansies. And so for them the result, 60 pansies, is a composite unit. It's a unit of 60 units, but they don't maintain the structure that we see at all of the units of 60 as 15 fours. That's not something—even though they did track it in their thinking—they don't maintain that once they get to the 60, it's really just only a big composite unit of 60. So their view of the result is very different than an adult view might be.

So, the students at stage 1 can solve division problems, which means if they give some number of pansies and they're supposed to make rows of 4, they can definitely do it, they can solve that. But they don't think of multiplication and division as inverses. So let me say what I mean by that. If they had this problem next, so: "Amaya's mom gave her 28 pansies. How many rows of 4 can she make?" A student at stage 1 could solve that problem, and they would be able to track 4s over and over again and figure out that they got to 7 fours once they get to 28. But then if immediately afterwards a teacher said, "Well, so, how many pansies are there in 7 rows of 4?," the student at stage 1 would start over and solve the problem from the beginning. They wouldn't think that they had already solved it. And that's one telling sign of a student operating at stage 1. And the reason is that the mental actions they engage in to do the segmenting or the tracking off of the 4s and the 28 pansies are really different to them than what they use then the ways of thinking they use to create the 7 rows of 4 and make the 28 that way. And so they don't recognize them as similar, so they feel like they have to engage in new problem solving to solve that problem.

So, to get back to the garden problem, students at stage 2 have a multiplicative number sequence, so they think of 60 as a one that they could repeat. Iterating is a term we often use. They could imagine it just being repeated over and over again. And this is a contrast to students at stage 1 who think of 60 as like, "Oh, I got to have all 60 pansies there if I'm going to think about a number like 60." Whereas students at stage 2 do have a multiplicative number sequence and so they think, "Oh, I don't have to have all my 60 pansies. I can just think about one pansy and I just repeat it however many times I need, to have however many pansies I want to imagine in my problem solving." So they anticipate 60 as 1 sixty times. And that's obviously a great relief for kids who are dealing with big numbers. You can imagine it feels really onerous to think about 1,000 if you feel like you have to have 1,000 items in your mind, "Oh, how could I possibly do that?" But, "Oh, I don't have to have 1,000; I can just have 1 and I can repeat it." That's a great economy, efficiency in thinking that happens.

So in terms of the garden problem, students at stage 2 also have constructed a row as a thing to count, so a composite unit's one item as well, so 4 little items. And they can amass 4s just like I was talking about with students at stage 1. But what they are also able to do is break apart 4s as they go along. They might say, "Well, I've got 4 and 4 is 8 and one more [4] is 12 and one more is 16 and one more is 20 and one more is 24 and one more is 28." Maybe at that point they say, "Oh, let's see. I don't know what one more 4 is, but two more [4s] is 30 and then two more is 32." So they can take the row apart. They don't all do this, but they can; they have the mental capabilities to do that because they're not right in the midst of making the coordination happen. They're sort of a little bit able to stand above the coordination and take their rows apart if they need to.

Mike: It sounds like part of what happens at stage 1 is you might have a kid who potentially could count by 4s for lack of a better way of saying it. And they might say, "Well, 4 and 4, so 2 sets of 4s, [is] 8." And then at some point it kind of breaks down where that memorized list of what happens when you count by 4. And then kids are back to saying, "OK, 12, 13, 14, 15, 16." And if you were watching this, listeners, you would see that I stuck out four fingers and then I'm like, "OK, so that's 3 fours, and so on." And so I would see a student who might appear to be thinking about units, but tell me if I'm correct in thinking that it's more a function of that they know a set of numbers in accounting sequence for counting by 4s.

Amy: So students at any stage may vary in the skip-counting patterns they know. I call it knowing a skip-counting pattern, to know automatically, like, 4, 8, 12, 16, or whatever it is. So you could have a student at stage 2 who doesn't know their skip-counting patterns very well, and you also could have a student at stage 2 who counts by 1s. So that's the issue there, is you can't always tell just from what you see if you have to do more than the test of what I'm saying. It's just to give a sense of the stages. But the main thing is the outer boundary of what they can do at stage 2 is they don't have to count by 1s. They can do other things because of the fact that their composite units have this special feature where they're multiplicative in nature. I mean the fancy term for it is they have iterable units of 1.

But let me say a little bit more about what happens when they get to 60. So, let's say a student at stage 2, they've gotten up to 60, there are 60 pansies and there are 15 rows of 4. They will think of the 60 as 15 fours as they make it. So we call it a three levels of unit structure. 60 is a unit of 15 units, each containing 4 little ones. They'll think about [it] that way as they solve the problem, but as they continue to work further and add more pansies on or do a further extension of the problem, they wouldn't maintain that three levels of units structure that we see. So that's important because it has implications for how they can build from what they've done.

Mike: How would you know that they hadn't maintained it? What might they say or do that would give you that cue?

Amy: Well, so you see it most if, let's say I say, "Oh, guess what? We got 12 more pansies and you're going to put 'em in rows of 4. Can you put those on?" And then they put 'em on. OK, they find out it's 72 now. "OK, so how many rows are we talking here?" It would be a new problem for them to figure that out. It wouldn't be like they would be able to maintain that, "Oh, I had 15 rows and then I now have the 3 more added on."

Mike: Got you. OK.

Amy: So, you see they're having to remake stuff as adult learners. We would think, "Oh, you should already know that that's 15 fours, right?" But they'll have to redo that in solving an extension of the problem like I was talking about there.

So students at stage 3, they also can definitely take 4 as a row of 1 and also 4 pansies. They can arrive at 60 and view it as a unit of units, but they also can view it as a unit of 15 units, each containing 4, and they maintain that. So, if they were asked a further problem, like, "Hey, we're going to rearrange this garden; we're going to actually 3 rows together at a time. Can you do that, and how many rows would you have and how many pansies in each row? And what would be the total?" They'd be able to say, "Oh, yeah, I can, let's see, put my 3 rows together, that's going to be 12, and then I'm going to end up with 5 of them." And now they've created 60 as a unit of 5 rows, each containing 12, and they can still think of 60 as a unit of 15 units, each containing 4, or 15 rows, each containing 4. So they can switch between different unit structures.

It doesn't mean they automatically know it without thinking it through, but they can do it and they can go back and forth. And that has great implications for anticipating and for solving division problems and seeing them as inverses of multiplication and a whole lot of stuff: proportional reasoning, fractions, lots of things. [laughs]

Mike: I think what's really interesting about this is I really appreciate you walking through the mental processes or maybe even the mental scripts that the kids might engage in to help see behind the curtain, for lack of a better word. Because what strikes me is that there is a point, probably early in my teaching career, where I would've attended and focused mostly on, "Did they get the answer?" And I think what you're helping remind me of is that it's the "how," but there are particular ideas. And now I think I understand why the notion of units—plural—units coordination matters so much because a lot of what's happening is their ability to coordinate a unit made of units and then to be flexible with the units within that unit of units. Am I making proper sense of that, Amy?

Amy: Yeah, for sure. That's great; that's exactly it. So the process and what units get created and how they get thought about and used is actually really, really important in trying to support kids' multiplicative thinking among other kinds of thinking too.

Mike: I think this is a great segue because I suspected a lot of teachers are wondering about the kinds of tasks or practices or questions that they might use that could nudge students' thinking regarding units coordination. And I'm wondering: What are some ideas you'd recommend for teachers as they're trying to think about how they assess but also advance their students' thinking when it comes to units coordination?

Amy: That's a great question. And, I mean, the big response is: Have students engage in lots of reasoning with units—composite units, breaking apart numbers strategically, thinking about different solution pathways. So not just one solution pathway, but can you come up with multiple solutions for the problem? Really sharing student solutions that involve breaking apart units. So if you're doing something like 5 sevens and finding out that kids are thinking of it as 5 fives and 5 twos, let's share that. How else could we break apart the 5 sevens? 5 fives and 5 twos? Why is that maybe helpful compared to other ways we might think about it? We might know 5 fives and 5 twos more easily than other ways of breaking it apart. And then even how are kids thinking about the 5 twos and the 5 fives and evaluating each of those. So basic things like that are super important.

How many rows can we make with 36 flowers with 4 per row? Thinking strategically about that, like: I know that 5 fours is 20 and I need 16 more flowers, so that's 4 fours because it's double 2 fours, so 8, so that means 9 rows total. So I'm just kind of really briefly talking through, but posing these kinds of tasks and then asking for how students can break them up and think about them and presenting and making public that kind of thinking and reasoning. So valuing it in that way and sharing it.

Same thing with lots of even more advanced multiplication problems. So for example, my daughter's in fourth grade right now, and so we've been working with her on, like, 30 times 20 and doing something other than knowing 3 times 2 and then putting 0s on because she doesn't remember that. So to do 30 times 20, we asked her about 10 twenties. Oh, she can figure that out; that's 200. And then can I iterate? Oh yeah, another 10 twenties, another 10 twenties. And then we did like 40 thirties, which was definitely harder. And so as part of the process of that, after she figured out 10 thirties, when she was iterating her thirties, that was harder than iterating the twenties. She had to break apart numbers. When she got to 90 plus 30, she had to think about 90 plus 10 plus 20. So doing embedded, breaking apart of units with the prospect of trying to figure out a larger multiplication problem, is super important. And interestingly, she could do 900 plus 300 and figure out that that was 900 and 100 to get 1,000 and then 200 more. So that's additive reasoning, but it's the breaking apart of units and reconstituting them. That's what's really important in the process of solving multiplication and division problems.

So that's my big thought about [laughs] that. And the other thing is to not go to patterns too soon. I mean, this is related to what I just said about not thinking that I can just do 3 times 2 and then add 0s and count the 0s because that really doesn't develop. It misses so much in what you can do with units. And so even if some kids do remember that and get the answer right, they're really robbed of the experience that we're trying to give to my daughter of really thinking about, "Well, how can I figure out 40 thirties or 30 forties or 30 twenties?" [laughs] Right now I'm a big advocate of actually doing lots of counting by decade numbers because I feel like it's a way of really enhancing kids' work with larger multiplication.

Mike: I've been sitting listening to you talk about this, Amy, and there are multiple things where I'm like, I need to ask her about this. I need to ask her about that. I need to ask about this other thing. So I'm going to ask you a couple of follow-ups.

One of the things that is just an observation is the language you used when you were talking about your work with your daughter. When the original task was "30 times 20" and you shifted the language to say "30 twenties," and then you step back even a little bit from there and you said, "Well, what's 30 tens?" This language that you were using, I wonder if you could be explicit about what you think that shift in language accomplishes.

Amy: Yeah, I've been also thinking a lot about this, so it's great. Yeah, one of the problems with multiplication notation is that it doesn't make clear anything about what the group is and what the number of groups you have are. And so just saying "30 times 20," I mean, you can think of that as "30 twenties" or I can think of that as "20 thirties," but the language doesn't contain it, so it doesn't refer to the action I might do in thinking about how to actually figure it out. And kids have to bring a lot to the table, then, to really read that into that multiplication notation. It's even more so with fractions. I can say more about that in a second. So I really am advocating with my preservice teachers is that we speak in iterative language with the multiplication. So we try to always say, "I'm talking about 5 sevens," or "I'm talking about 7 fives, 30 forties, 40 thirties." And then of course with the decade numbers, knowing that we can go down to 10 of something and that that's easier to figure out, and then we can build on that. So like 10 twenties and then, "Oh, I'm going to need 3 of those 10 twenties to get to 30 twenties."

Mike: Which really to some degree is helping them make meaningful sense of the associative property as well.

Amy: Right! Yeah, exactly. It's very mathematically rich. Unfortunately, it's not necessarily worked on [laughs] a lot, I am finding, and I think it's a real missed opportunity. Because I think there's a lot that kids could do with that that would really build strong meanings for multiplication and strong ideas of base ten as well.

Mike: Yeah, absolutely. I think one of the things that I've been obsessed with lately is this notion of "nudge" or small-sized shifts in my practice that I can make. Part of what I'd like to mark for the audience is the shift in the language, as you described—30 twenties or 5 sevens—those are moves that a teacher could make to help clarify the fact that units are involved and help students visualize with a bit more clarity what's going on. That feels like something that a teacher could take up and really have an impact on students' understanding.

Amy: Yeah, I think so. I think it is something that is reasonable, and what's nice is it also can flow right into fractions because then instead of saying just, "three-fifths," we say, "3 one-fifths, 4 one-fifths, 5 one-fifths, 6 one-fifths, 7 one-fifths." It allows for fractions larger than 1 to have maybe more of an iterative meaning. Not that that's a simple thing at all; that's a whole nother podcast we could do, but [laughs] I've done a lot of research on that.

Mike: Well, I think you're hitting on something important, though, Amy, because this notion of, "What is a unit fraction?," it's really, "Four-fifths is a group of 4 one-fifths," right? And that's a critical understanding that I think often floats underneath students' understanding in ways that, if we could make that clearer or help build that understanding, that also has huge ramifications for what comes later in their mathematics learning experience.

Amy: Yeah, so I'm a big proponent of iterative language there as well.

Mike: You have me thinking about something else too, which is the importance of context and having students deal with measurement division problems specifically as a way to build their understanding. And I know I'm using language right now for the audience that might not be super clear, but I'm wondering if you could talk a little bit about what measurement division means in context and maybe why that would be valuable for students.

Amy: Yeah. Right. So, in multiplication and division structures, if we're talking about equal groups, there's always some number of equal groups, some number in the equal group, so a size of the group, and then a total number of items. And so, with measurement division, we know the total number of items, and we know the number of items in a group, but we don't know the number of groups. So my example of, "You've got 36 flowers, and you want to put them in rows of 4" would be a measurement division problem because we know that there are 4 in each row, and we know we have 36, but we don't know how many rows we're going to make. And so those are really nice to pair with work on equal groups multiplication problems because they are very closely related. And for kids, they can become closely related as they solve them and realize, like, "Oh, I can use my multiplication strategies to build up my 4s and find out when I get to 36," and, "Oh, then I do, I know how many rows I've made." So it's highly linked to what we're talking about here.

Mike: What I found myself thinking about is that in solving that problem, one of the ways that a kid could do that is they're iterating a set, right? So, potentially, they're iterating a set of 4s multiple times, and then they're finding out how many of those sets of 4 they have, right? So I think part of what you're helping me think about is the way that the structure of a measurement division problem maybe shines a flashlight on this notion of groups and the number in each group, and also some of the ideas you were talking about earlier with units coordination.

Amy: Yeah, for sure. And in terms of continuing the theme of using iterative language, then when you get the result of that problem, 9 rows, "Oh, what does that 9 mean?" "Oh, it means 9 fours make 36." So that's a meaning both for 4 times 9 equals 36, as well as 36 divided by 4 equals 9. So it's nice to emphasize that. And yeah, as students build those meanings and have repeated work with that kind of thing, they usually, often—[laughs] we don't know all the mechanisms here—but they usually come to be able to at least make that coordination in their problem-solving activity, and ultimately make it so they can anticipate it, like we're talking about with stage 3.

Mike: One of the things that is really helpful is, in the course of this interview, we've talked a lot about what might the behavior of a student at stage 1 or stage 2 or stage 3 not only look like, but what might it mean for how they're thinking. And I think what I'm really appreciating about this, Amy, is there are a few practical things that an educator could do to support students. One is iterative language as we've been talking about. And the other is measurement division, using a particular problem structure like measurement division to shine a light on these parts that we think are really important for kids to attend to if they're in fact going to make some of the shifts that we're hoping for.

Amy: Yeah, for sure. And then also exploring the boundaries of what the kids' strategies are and asking for multiple solutions. Because you might see kids, even students at stage 3, that might be counting by 1s, and so you want to [prompt], "Oh, can you solve that another way? Is there another way you can do it?" And so seeing what they see as possible, what they're able to think about is also really important to support units coordination.

Mike: Absolutely. Before we close, I typically ask a question about resources or training or learning experiences that would help someone who's listening continue learning or continue to think about how they could take up these ideas in their practice. You, particularly, I know have written some work around this and I also suspect that you might have some recommendations in terms of organizations that can help educators really dig into these ideas if they saw that as something that was important for their growth. Would you be willing to talk a little bit about resources, organizations, or even the types of experience you think support teachers as they're making sense of all of this?

Amy: Yeah. Well, yes. I was planning to talk about Integrow at this point because Integrow Numeracy Solutions has a lot of great supportive materials for all this kind of work. And everything that I'm talking about is something that is sort of built into much of what they do. For people who are unfamiliar, it's a bit—council, used to be called a council, of people who got together and have really developed materials that are supportive of teachers working one-on-one to support students who might be struggling as well as whole-group instruction all around developing strong number sense. And it's a very well developed set of materials, both for classroom use as well as for teacher development.

And we—meaning me and my two coauthors, Andy Norton and Bob Wright—wrote a book in the series for teachers on fractions called Developing Fractions Knowledge. And that was published—oh my gosh—nine years ago now. So Andy and I are working on a second edition right now, and in that book we address units coordination and talk about its usefulness for teachers. It's mostly, though, a book about fractions and about how units coordination is relevant in trying to support students' fractions knowledge and to help assess students' thinking and also promote their learning. So that is one resource I can recommend on units coordination with a revision coming in the next year [2026].

Mike: That's fantastic. So I'll say for listeners, we'll include a link to Integrow Numeracy Solutions if you want to check out the organization. And Amy will also add a link directly to the book so that if someone wanted to dig in and explore that way they had the option.

I think that's probably a great place to stop, although I certainly would love to continue. I want to thank you so much for joining us. It's really been a pleasure talking with you.

Amy: Yeah, likewise, Mike. I've really enjoyed it, and I look forward to further conversations.

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

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

What counts as counting? 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.

Christopher Danielson started teaching in 1994 in the Saint Paul (MN) Public Schools. He earned his PhD in mathematics education from Michigan State University in 2005 and taught at the college level for 10 years after that.

Christopher is the author of Which One Doesn't Belong?, How Many?, and How Did You Count? Christopher also founded Math On-A-Stick, a large-scale family math playspace at the Minnesota State Fair.

How Did You Count? A Picture Book by Christopher Danielson

How Many?: A Counting Book by Christopher Danielson

Following Learning blog by Simon Gregg

Connecting Mathematical Ideas by Jo Boaler and Cathleen Humphreys

Mike Wallus: Before we start today's episode, I'd like to offer a bit of context to our listeners. This is the second half of a conversation that we originally had with Christopher Danielson back in the fall of 2025. At that time, we were talking about [the instructional routine] Which one doesn't belong? This second half of the conversation focuses deeply on the question "What counts as counting?" I hope you'll enjoy the conversation as much as I did.

Well, welcome to the podcast, Christopher. I'm excited to be talking with you today.

Christopher Danielson: Thank you for the invitation. Delightful to be invited.

Mike: So I'd like to talk a little bit about your recent work, the book How Did You Count?[: A Picture Book] In it, you touch on what seems like a really important question, which is: "What is counting?" Would you care to share how your definition of counting has evolved over time?

Christopher: Yeah. So the previous book to How Did You Count? was called How Many?[: A Counting Book], and it was about units. So the conversation that the book encourages would come from children and adults all looking at the same picture, but maybe counting different things. So "how many?" was sort of an ill-formed question; you can't answer that until you've decided what to count.

So for example, on the first page, the first photograph is a pair of shoes, Doc Marten shoes, sitting in a shoebox on a floor. And children will count the shoes. They'll count the number of pairs of shoes. They'll count the shoelaces. They'll count the number of little silver holes that the shoelaces go through, which are called eyelets. And so the conversation there came from there being lots of different things to count. If you look at it, if I look at it, if we have a sufficiently large group of learners together having a conversation, there's almost always going to be somebody who notices some new thing that they could count, some new way of describing the thing that they're counting.

One of the things that I noticed in those conversations with children—I noticed it again and again and again—was a particular kind of interaction. And so we're going to get now to "What does it mean to count?" and how my view of that has changed. The eyelets, there are five eyelets on each side of each shoe. Two little flaps that come over, each has five of those little silver rings. Super compelling for kids to count them. Most of the things on that page, there's not really an interesting answer to "How did you count them?" Shoelaces, they're either two or four; it's obvious how you counted them. But the eyelets, there's often an interesting conversation to be had there. So if a kid would say, "I counted 20 of those little silver holes," I would say, "Fabulous. How do you know there are 20?" And they would say, "I counted." In my mind, that was like an evasion. They felt like what they had been called on to do by this strange man who's just come into our classroom and seems friendly enough, what they had been called on to do was say a number and a unit. And they said they had 20 silver things. We're done now. And so by my asking them, "How do you know? " And they say, "I counted." It felt to me like an evasion because I counted as being 1, 2, 3, 4, 5, all the way up to 20. And they didn't really want to tell me about anything more complicated than that. It was just sort of an obvious "I counted." So in order to counter what I felt like was an evasion, I would say, "Oh, so you said to yourself, 1, 2, 3, and then blah, blah, blah, 18, 19, 20." And they'd be like, "No, there were 10 on each shoe." Or, "No, there's 5 on each side." Or rarely there would be the kid who would see there were 4 bottom eyelets across the 4 flaps on the 2 shoes and then another row and another row. Some kids would say there's 5 rows of 4 of them, which are all fabulous answers. But I thought, initially, that that didn't count as counting. After hearing it enough times, I started to wonder, "Is it possible that kids think 5 rows of 4, 4 groups of 5, 2 groups of 10, counted by 2s and 1, 2, 3, 4, all the way up to 19 and 20—is it possible that kids conceive of all of those things as ways of counting, that all of those are encapsulated under counting?" And so I began because of the ways children were responding to me to think differently about what it means to count.

So when I first started working on this next book, How Did You Count?, I wanted it to be focused on that. The focus was deliberately going to be on the ways that you count. We're all going to agree that we're counting tangerines; we're all going to agree that we're counting eggs, but the conversation is going to come because there are rich ways that these things are arranged, rich relationships that are embedded inside of the photographs.

And what I found was, when I would go on Twitter and throw out a picture of some tangerines and ask how people counted, and I would get back the kind of thing that was how I had previously seen counting. So I would get back from some people, "There are 12." I'd ask, "How did you count?" And they'd say, "I didn't. I multiplied 3 times 4." "I didn't. I multiplied 2 times 6."

But then, on reflection through my own mathematical training, I know that there's a whole field of mathematics called combinatorics. Which if you asked a mathematician, "What is combinatorics?," 9 times out of 10, the answer is going to be, "It's the mathematics of counting." And it's not mathematicians sitting around going "1, 2, 3, 4" or "2, 4, 6, 8." It's looking for structures and ways to count the number of possibilities there are, the number of—if we're thinking about calculating probabilities of winning the lottery, somebody's got to know what the probabilities are of choosing winning numbers, of choosing five out of six winning numbers. And the field of combinatorics is what does that. It counts possibilities.

So I know that mathematicians and kindergartners—this is what I've learned in both my graduate education and in my postgraduate education working with kindergartners—is that they both think about counting in this rich way. It's any work that you do to know how many there are. And that might be one by one; it might be skip-counting; it might be multiplication; it might be using some other kind of structure.

Mike: I think that's really interesting because there was a point in time where I saw counting as a fairly rote process, right? Where I didn't understand that there were all of these elements of counting, meaning one-to-one correspondence and quantity versus being able to just say the rote count out loud. And so one way that I think counting and its meaning have expanded for me is to kind of understand some of those pieces. But the thing that occurs to me as I hear you talk is that I think one of the things that I've done at different points, and I wonder if people do, is say, "That's all fine and good, but counting is counting." And then we've suddenly shifted and we're doing something called addition or multiplication. And this is really interesting because it feels like you're drawing a much clearer connection between those critical, emergent ideas around counting and these other things we do to try to figure out the answer to how many or how did you count. Tell me what you think about that.

Christopher: Yeah. So this for me is the project, right? This book is an instantiation of this larger project, a way of viewing the world of mathematics through the lens of what it means to learn it. And I would describe that larger project through some imagery and appealing to teachers' ideas about what it means to have a classroom conversation.

For me, learning is characterized by increasing sophistication, increasing expertise with whatever it is that I'm studying. And so when I put several different triangular arrangements of things—in the book, there's a triangular arrangement of bowling pins, which lots of kids know from having bowled in their lives and other kids don't have any experiences with them, but the image is rich and vivid and they're able to do that counting. And then later on, there's a triangular arrangement of what turned out to be very bland, gooey, and nasty, but beautiful to photograph: pink pudding cups. Later on, there are two triangles of eggs. And so what I'm asking of kids—I'm always imagining a child and a parent sitting on a couch reading these books together, but also building them for classrooms. Any of this could be like a thing that happens at home, a thing that happens for a kid individually or a classroom full of children led by a teacher. Thinking about the second picture of the pudding cups, my hope and expectation is that at least some children will say, "OK, there are 6 rows in this triangle and there were 4 rows previously. So I already know these first four are 10. I don't have to do any more work, and then 5 plus 6, right?" And then that demonstrates some learning. They're more expert with this triangle than they would have been previously.

I'm also expecting that there's going to be some kid who's counting them 1 by 1, and I'm expecting that there are going to be some kids who are like, "You know what? That 6 up top and the 1 makes 7 and the 5 and the 2 make 7, and the 4 and the 3. So it's 3 sevens. There's 21." I'm expecting that we're going to have—in a reasonably large population of third, fourth, fifth graders, sort of the target audience for this book—we're going to have some kids who are doing each of these. And for me, getting back to this larger project, that is a rich task, which can be approached in a bunch of different ways, and all of those children are doing the same sort of task. They're all counting at various levels of sophistication representing various opportunities to learn previously, various ways of applying their new learning as they're having conversations, looking at new images, hearing other people's ideas, but that larger project of building something that is rich enough for everybody to be able to find something new in, but simple enough for everybody to have access to—yeah, that's the larger project.

Mike: So one of the things that I found myself thinking about when I was thinking about my own experiences with dot talks or some of the subitizing images that I've used and the book that you have, is: There's something about the way that a set of items can be arranged. And I think what's interesting about that is I've heard you say that that arrangement can both reveal structure, in terms of number, but it can also make connections to ideas in geometry. And I wonder if you could talk a little bit about that.

Christopher: Yeah. I'll draw a quick distinction that I think will be helpful. If you've ever seen bowling pins, right? It's four, three, two, one. The one [pin] is at the front; the [row of] four is at the back. Arranged so that the three fit into the spaces between the four as you're looking at it from the front. Very iconic arrangement. And you can quickly tell that it's a symmetric triangle and the longest row is four. You might just know that that's 10. But if you take those same bowling pins and just toss them around inside of a classroom or inside of a closet and they're just lying on the floor, so they're all in your field of vision, you don't know that there's 10 right away. You have to do a different kind of work in order to know that there are 10 of them. In that sense, the structure of the triangle with the longest row of four is a thing that you can start to recognize as you learn about triangles and ultimately what mathematicians refer to as triangular numbers.

That's a thing you can learn to recognize, but learning to recognize 10 in that arrangement doesn't afford you anything when it's 10 [pins] scattered around on the floor. Unless you do a little abstraction. There's a story in the book about a lovely sixth grader who proceeded to tell me about how the bowling pin arrangement matches a way that she thinks about things. Because if she's ever going about her life, I don't know, making a bracelet or buying groceries, collecting pencils for the first day of school or whatever. If she wants to count them, and it looks like there's probably fewer than 100 but more than 5, she will grab a set of 4, a set of 3, a set of 2, a set of 1, and she'll know that's 10. Unprompted by me, except that we had this bowling pin arrangement.

So there are ways to abstract from that. You can use these structures that you've noticed in order to do something that isn't structured that way, but the 4, 3, 2, 1 thing probably came from recognizing that 4, 3, 2, 1 made this nice little geometric arrangement. So our eyes, our brains, are tuned to symmetry and to beauty and elegance, and there is something much more lovely about a nice arrangement of 4, 3, 2, 1 than there is about a bunch of scattered things. And so a lot of those things are things that have been captured by mathematicians. So we have words for square numbers—3 times 3 is 9 because you can make 3 rows of 3 and you make something that looks nice that way. Triangular numbers, there are other figurate numbers like hexagonal numbers, but yet innate in our minds, there is an appeal to symmetry. And so if we start arranging things in symmetric patterned ways that will be appealing to our brains and to our eyes and to our mathematical minds, and my goal is to try to tap into that in order to help kids become more powerful mathematicians.

Mike: So I want to go back to something you said earlier, and I think it's an important distinction before I ask this next question. One of the things that's fascinating is that a child could engage with this kind of image, and there doesn't necessarily have to be an adult in the room or a teacher who's guiding them. But what I was thinking about is: If there is a student or a pair of students or a classroom of students, and you're an educator and you're engaging them with one of these images, how do you think about the educator's role in that space? What are they trying to do? How should they think about their purpose? And then I'm going to ask a sub-question: To what extent do you feel like annotation is a part of what an educator might do?

Christopher: Yes. One thing that teachers are generally more expert at than young children is being able to state something simply, clearly, concisely in a way that lots of other people can understand. If you listen to children thinking aloud, it is often hesitant and halting and it goes in different directions and units get left off. So they'll say, "3 and then 4 more is 8" and they've left off the fact that the 4 were—I mean, you could just easily get lost. And so one of the roles that a teacher plays can certainly be to help make clear to other students the ideas that a particular student is expressing and at the same time, often helping make it more clear for that student, right? Often a restating or a question or an introduction of a vocabulary word that seems like it's going to be helpful right now will not just be helpful to other people to understand it for the whole class, but will be helpful for the student in clarifying their own ideas and their own thinking, solidifying it in some kind of way.

So that's one of the roles. I know that there are also roles that involve—and I think about this a lot whenever I'm working with learners—status, right? Making sure that children that have different perceived status in the classroom are able to be lifted up. That we're not just hearing from the kid who's been identified as "the math kid." So I think intellectual status, social status, those are going to be balances, right?

I also understand that teachers have a role in making sure that children are listening to each other. If I'm working with learners, I can't always be the one to do the restating. I've got to make sure there are times where kids are required to try to understand each other's thinking and not just the teacher's restatement of that thinking. There are just so many balances. But I would say that some top ones for me, if I'm thinking about how to make choices, thinking about raising up the status of all learners as intellectual resources, making good on a promise that I make to children, which is that any way of counting these things is valid and not telling a kid, "Oh no, no, no, we're not counting 1 by 1 today" or, "Oh no, no, no, that's too sophisticated. That's too advanced of a—We can't share that because nobody will understand it."

So making good on that promise that I make at the beginning, which is, "I really want to know how you counted." Making sure that learners are able to get better at expressing the ideas that are in their heads using language and gesture and making sure that learners are communicating with each other and not just with me as a teacher. Those seem like four important tensions, and a talented and experienced elementary teacher could probably name like 10 other tensions that they're keeping in mind all at the same time: behavior, classroom management, but also some ideas around multilingual learners. Yeah, a lot of respect for the kind of balances that teachers have to maintain and the kinds of tensions that they have to choose when to use and when to gloss over or not worry about for right now.

So you ask about annotation and, absolutely, I think about multiple representations of mathematical ideas. And so far I've only focused on the role of the teacher in a classroom discussion and thinking about gesture, thinking about words and other language forms, but I haven't focused on writing and annotation is absolutely a role that teachers can play. For me, the thing that I want to have happen is I want children to see their ideas represented in multiple ways. So if they've described for the class something in words and gestures, then there are sort of two natural easy annotations for a teacher to do or a teacher to have students do, which is, one, make those gestures and words explicit in the image. And that's where something like a smartboard or projecting onto a whiteboard—lots of technologies that teachers use for this kind of stuff—but where we can write directly on the image. So if you said you put the 1 and the 4 together in the bowling pins and then the 3 and the 2, then I might make a loopy thing that goes around the 4 and the 1, and I might circle the 3 and the 2, right? And so that adds both some clarity for students looking, but also is a model for: Here's how we can start to annotate our images.

But then I'm also probably going to want to write 4 plus 1, maybe in parentheses, plus 3 plus 2 in parentheses, so that we can connect the 4 to the four [items] that are circled, the 1 to the one that is circled, the 4 plus 1 in parentheses, identifying that as a group, like a thing that has a mathematical purpose. It's communicating part of an idea and that that connects back. Teachers are super skilled at using color to do that, right? So 4 plus 1 might be written in red to match the red circle that goes around here, using not green because of color blindness. They're using blue to do 3 plus 2 in parentheses over here. And teachers might make other choices, right? We might sometimes use color to annotate in the image, but then just black here so that we aren't doing all of that work of corresponding for kids and are asking kids to try to do some of that corresponding work. And we might do it the other way around as well.

So annotation as a way of adding, I think, a couple of dimensions to the conversation. And I have to shout out a fabulous teacher who I know through math Twitter. Simon Gregg is a teacher in an international school in Toulouse, France. And he has done amazing work with using and producing his own Which one doesn't belong?s, and annotating them and having kids do them; how many?; and then there are a few examples of his work with kids in the teacher guide for How Did You Count? Yeah, he's just a true master at annotation. So go find Simon Gregg on social media if you want to learn some beautiful things about representing kids' ideas in writing.

Mike: Love it. So the question that I typically will ask any guest before the close of the interview is: What are some resources that educators might grab onto, be they yours or other work in the field that you think is really powerful that supports the kind of work that we've been talking about? What would you offer to someone who's interested in continuing to learn and maybe to try this out?

Christopher: In the teacher guide of How Did You Count?, I make mention of which of the number talks books was most powerful for me. But if you want to take a look at that page in the teacher book and then throw a link in and a shout out to the folks who wrote it. Jo Boaler and Cathleen Humphreys wrote a book called Connecting Mathematical Ideas. It's old enough that there are some CD-ROMs in it. I don't know if there's a new edition; I'm sure used ones are available on all the places you buy used books. But the expert work that the teacher Cathy Humphreys does, as described in the book—even if you can't use the CD-ROMS in your computer—expert work at drawing out students' ideas, and then the two collaborating to reflect on that lesson, the connections they were drawing. It's been a while since I read it, but I imagine the annotations have got to come up. Fabulous resources for thinking about how these ideas pertain to middle school classrooms, but absolutely stuff that we can learn as college teachers or as elementary teachers on either side of that bridge from arithmetic to algebra.

Mike: So for listeners, just so you know, we're going to add links to the resources that Christopher referred to in all of our show notes for folks' convenience.

Christopher, I think this is probably a good place to stop. Thank you so much for joining us. It's absolutely been a pleasure chatting with you.

Christopher: Yeah. Thank you for the invitation, for your thoughtful prep work and support of both the small and the larger projects along the way. I appreciate that. I appreciate all of you at Bridges and The Math Learning Center. You do fabulous work.

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

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

All students deserve a classroom rich in meaningful mathematical discourse. But what are the talk moves educators can use to bring this goal to life in their classrooms?

Today, we're talking about this question with Todd Hinnenkamp from the North Kansas City Schools. Whether talk moves are new to you or already a part of your practice, this episode will deepen your understanding of the ways they impact your classroom community.

Dr. Todd Hinnenkamp is the instructional coordinator for mathematics for the North Kansas City Schools.

Talk Moves with Intention for Math Learning Center

Standards for Mathematical Practice by William McCallum

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

Mike Wallus: Before we begin, I'd like to offer a quick note to listeners. During this episode, we'll be referencing a series of talk moves throughout the conversation. You can find a link to these talk moves included in the show notes for this episode.

Welcome to the podcast, Todd. I'm really excited to be chatting with you today.

Todd Hinnenkamp: I'm excited to be here with you, Mike. Talk through some things.

Mike: Great. So I've heard you present on using talk moves with intention, and one of the things that you shared at the start was the idea that talk moves advance three aspects of teaching and learning: a productive classroom community, student agency, and students' mathematical practice. So as a starting point, can you unpack that statement for listeners?

Todd: Sure. I think all talk moves with intention contribute to advancing all three of those, maybe some more than others. But all can be impactful in this endeavor, and I really think that identifying them or understanding them well upfront is super important.

So if you unpack "productive community" first, I think about the word "productive" as an individual word. In different situations, it means a quality or a power of producing, bringing about results, benefits, those types of things. And then if you pair that word "community" alongside, I think about the word "community" as a unified body of individuals, an interacting population. I even like to think about it as joint ownership or participation. When that's present, that's a pretty big deal. So I like to think about those two concepts individually and then also together. So when you think about the "productivity" word and the "community" word and then pairing them well together, is super important.

And I think about student agency. Specifically the word "agency" means something pretty powerful that I think we need to have in mind. When you think about it in a way of, like, having the capacity or the condition or state of acting or even exerting some power in your life. I think about students being active in the learning process. I think about engagement and motivation and them owning the learning. I think oftentimes we see that because they feel like they have the capacity to do that and have that agency. So I think about that, that being a thing that we would want in every single classroom so they can be productive contributors later in life as well. So I feel like sometimes there's too many students in classrooms today with underdeveloped agencies. So I think if we can go after agency, that's pretty powerful as well.

And when you think about students' math practice, super important habits of what we want to develop in students. I mean, we're fortunate to have some clarity around those things, those practices, thanks to the work of Dr. [William] McCallum and his team more than a decade ago when they provided us the standards for mathematical practice. But if you think about the word "practice" alone, it's interesting. I've done some research on this. I think the transitive verb meaning is to do or perform often, customarily or maybe habitually. The transitive verb meaning is to pursue something actively. Or if you think about it with a noun, it's just a usual way of doing something or condition of being proficient through a systematic exercise. So I think all those things are, if we can get kids to develop their math practice in a way it becomes habitual and is really strong within them, it's pretty powerful.

So I do think it's important that we start with that. We can't glaze over these three concepts because I think that right now, if you can tie some intentional talk moves to them, I think that it can be a pretty powerful lever to student understanding.

Mike: Yeah. You have me thinking about a couple things. One of the first things that jumped out as I was listening to you talk is there's the "what," which are the talk moves, but you're really exciting the stage with the "why." Why do we want to do these things? And what I'd like to do is take each one of them in turn. So can we first talk about some of the moves that set up productive community for learners?

Todd: Yeah. I think all the moves that are on my mind contribute, but there's probably a couple that I think go after productive community even more so than others. And I would say the "student restates" move, that first move where you're expecting students to repeat or restate in their own words what another student shared, promotes some really special things. I think first it communicates to everyone in the room that "We're going to talk about math in here. We're going to listen to and respectfully consider what others say and think." It really upholds my expectation as an educator that we're going to interact with and understand the mathematical thinking that's present so that student restates is a great one to get going.

And I would also offer the "think, turn, and learn" move is a highly impactful one as well. The general premise here is that you're offering time upfront. Always starting with "think," you're offering time upfront. And what that should be communicating to students is that "You have something to offer. I'm providing you time to think about it, to organize it, so then you're more apt to share it with either your partner or the community." It really increases the likelihood that kids have something to contribute. And as you literally turn your body and learn from each other—and those words are intentional, "turn" and "learn"—it opens the door to share, to expand your thinking, to then refine what you're thinking and build to develop both speaking and listening skills that help the community bond become stronger. So in the end it says, "I have something to offer here. I'm valued through my interactions." And I feel like that there's something that comes out of that process for kids.

Mike: You talked about the practice of "think, turn, learn." And one of the things that jumps out is "think." Because we've often used language like "turn and talk," and that's in there with "turn and learn," but "think" feels really important. I wonder if you could say more about why "think"? Let's just make it explicit. Why "think"?

Todd: Sure. No, and I'm not trying to throw shade at "turn and talks" or anything like that, but I do think when we have intention with our moves, they're super impactful relative to other opportunities where maybe we're just not getting the most out of it. So that idea of offering time or providing or ensuring time for kids to think upfront—and depending on the situation, that can be 10 seconds, that can be 30 seconds—where you feel like students have had a chance to internalize what's going on [and] think about what they would say, it puts them in an entirely different mode to build a share with somebody else. I'm often in classrooms, and if we don't provide that think time, you see kids turn and talk to each other, and the first part is them still trying to figure out what should be said. And it just doesn't seem like it's as impactful or as productive during that time as it could be without that "think" first.

Mike: Yeah, absolutely.

I want to go back to something you said earlier too, when you were describing the value that comes out of restating or rephrasing, having a student do that with another student's thinking. One of the things that struck me is there were points in time when you were talking about that and you were talking about the value for an individual student who's in that spot.

Todd: Mm.

Mike: But I also heard you come back to it and say, "There's something in this for the group, for the community as well." And I wonder if you could unpack a little bit: What's in it for the kid when they go through that restating another student's [idea], or having their [own] idea restated, and then what's in it for the community?

Todd: Sure. Well, let's start with the individual, Mike. And I think that with what we know about learning and how much more deeply we learn when we internalize something and reflect on it and actually link it to our past learning and think about what it means to us, is probably the most important thing that comes out of that. So the student that's restating what another student says, they really have to think about what that student said and then internalize it and make sense of it in a way where they can actually say it out to the community again. That's a big deal! So to talk about the impact on the community in that mode, Mike, when you get one or two [ideas], and maybe you ask for a couple more, you now have student thinking in four different forms out in the community rather than, say, one student sharing something and a teacher restating it and moving on. And I just love how those moves together can cause the thinking to linger in the classroom longer for kids. Often when I'm in classrooms, the kids actually learn it more when somebody else says it rather than me. And it kind of ties to that where, like, they just need to hear other kids thinking and start to process that a little bit more on their level. And we get to shore that up too as teachers. We can shore up whatever's missing if we need to later. But I think the depth that comes from thinking about it, putting it out in the community, having more kids think about [it] is pretty powerful.

Mike: I think what's cool about that is the idea that there's four or five ideas floating around and how different that is than [when] a kid says something, the teacher restates it and moves on. I might not have made sense of it on the first kid's description or the teacher's description, but when those things linger around, there's a much better chance that I'm going to make sense of it.

Todd: Yeah. And I agree, Mike.

And what's really important in that process as well is the first move I always talk about is "wait." You literally have to wait. When the student restates something, we've got to let that sit for a little bit for it to really be something that other kids can grasp onto and then give them time to process what they heard and then ask if someone could restate. At that point, it's causing all this cognition in the brain, and it's making me think about what I understand and what I don't understand about what was said. And it just starts to build and make a huge difference over time.

Mike: Yeah. I'm glad you said that because I'm a person who talks to think, but that is not true of a lot of folks.

Todd: (laughs)

Mike: A lot of people need time to think…

Todd: Sure.

Mike: …before they talk. And so I think it's really important to recognize that that wait time is really an opportunity for mental space. And if we don't do that, it actually might fall flat.

Todd: Totally agree. I'd see it day in and day out in classrooms I'm in, where if we can offer that time to let that concept or thinking permeate across the room for a little bit longer, it's a whole different outcome.

Mike: Nice.

I'm wondering if we can pivot and talk a little bit about moves that support student agency and their mathematical practice. They really do feel like they're kind of interconnected.

Todd: Yeah, I think they are somewhat interconnected as I think about them. And I see agency as like a broader concept, like really that development of capacity to act or have power in a situation. But when you think about math practices—thinking about the standards for mathematical practices—it's a little more specific. So when you think about the math practice of perseverance, I think we have to think about the move [called] wait time that I just talked about. When used with intention, I think it can communicate to kids, "I've got confidence in you. You have something to offer. I believe in you and that you're capable of contributing here." I just think that we have to think about our use of wait time and the messages that kids get from that and be careful not to squelch their opportunity to grow in those situations.

Mike: OK. I have a follow-up. You're making me think about ways to do wait time well and ways to do wait time that might have an unintended consequence. So walk me through a really productive use of wait time—what the language is that the teacher uses or how they manage what can feel uncomfortable for most of us.

Todd: Sure. And I will be very upfront that anytime you start to use wait time, if you haven't before, there's going to be some discomfort. (laughs) You think about, if you're a person that always wants to fill that space or feel like you need to because students aren't quite contributing, then you start to shift your practice to cause there to be a little more extended wait time, there's going to be some discomfort that plays out in that situation.

So I think honestly, Mike, part of it is having the right question or the right prompt, and setting up the expectation and upholding it over time. I talk a lot with teachers about establishing and maintaining productive community. I think that we have to establish it over time and then maintain it. And what I mean by that is if you start to use wait time, you're establishing that norm in your classroom, is that I'm always going to give you time to think, and that's super important in here because we want to make sure that we get the most out of the experience.

The maintaining part of it, I believe, is where we uphold that over time. We don't start to back off if kids don't then share their thinking. We can't always fill that space. And I think sometimes an inappropriate use of wait time is if we do it pretty well, but then we rescue when there's a time that kids aren't sharing something. So I do believe that no matter what classroom you're in, there is always one kid that can give you at least a nugget that you can go with. So I think as much as you can wait and try to draw that out before interjecting is super important.

Mike: Yeah. You make me think about a scenario that I encountered a fair amount when I was teaching elementary, which was: I'd ask a question and there were two or three kids who immediately put their hand up. There were quite a few that were still thinking, and it was really uncomfortable for me, but I think also for some of the kids who had their hands up, that I didn't immediately call on them, that I actually waited and let the question marinate…

Todd: Yes.

Mike: …and the end product was great. I had more kids who had something to say because they had that space. But it was a little uncomfortable, especially for those kids who were like, "Wait, I know it immediately. Why aren't you calling on me?

Todd: (laughs) Yes. And I think it's super important what you just shared, Mike, because in our practice, we have to be aware that the day-to-day practices or actions that we enact in our lessons, they're impacting everything from community, agency, practice. All the things that we're talking about today are sometimes just suddenly being impacted either positively or negatively. And I think the scenario you described about your practice is, like, you were intentional about it. You became aware, you realized that there's a handful of kids that I'm probably letting drive the discourse maybe more than I need to. And you're right: You've got other kids in classrooms that I'm in that are really waiting to talk and never have the chance. And I do feel like those are the kids that are going to have a hard time staying caught up with everybody because they're not getting that opportunity to develop some of those habits.

Mike: Yeah. It makes me think about when I was a kid as well. I was not the fast kid, right? I was thinking about it, but I was not the first kid with my hand up.

You've really got me thinking about how wait time is a real subtle way of saying, "You're not necessarily the most competent person just because you have your hand up first." There's no added bonus that says, like, "You're the best just because your hand's up first. Everybody can contribute. You might need a little bit of time to process. That's super normal in a math class."

Todd: Yes, it is. And you go back to what we discussed earlier about being a valued contributor in the community, and you think about what those kids feel once they experience that wait time and then their ideas being the ones that drive the discourse or that are highlighted or presented. That's where you draw that in, and if you can have 50% of your kids be the ones that are feeling that, then you got to shoot for 60, and then 70, and so on. But you gotta start to expand the number of kids that talk and share and restate and do all the things around discourse, but wait time is a super powerful tool to do that.

Mike: Another thing that you shared when I saw your presentation was the idea that you can pair talk moves in a sequence and that those sequence talk moves can have a powerful impact on kids. And I'm wondering if you can talk a bit about some of the ways that educators can sequence talk moves to have maximum impact.

Todd: Sure. Yeah. And I'm not necessarily suggesting that there is an always or a 100% correct way to line them up and sequence them, but I do think there's some [that], if you can go after them in particular instances in your professional practice, I think it's going to change your practice, I think more quickly and more deeply. And the same goes with a lesson. I think right off the bat, we first must wait. We have to start to build that into our practice where we wait. So if we offer a prompt [or] pose a question, let it sit for a second. I always talk about 4 to 6 seconds would be about how long you'd want to just let it sit for a little bit. Then, if you've got the right question and the right prompt, I think you could just say, "OK, now I'm going to give you some time to think and then I'm going to have you turn and actually learn with a partner. So I want you to think about the prompt that's on the board. What would you share with your partner?" And literally you give them time to think and then you can turn and learn.

So at that point, I think it's important that you're walking about the community, listening in, getting a feel for what's being discussed, because I think at that point you can have a feel for maybe what you might want to go to next, what insight you want to make sure is surfaced that is aligned to the learning goal of the day. That's how you get all that headed in the right direction. So you gotta lean in and figure that out. And I think at that point you could ask someone to share. "OK, who can share what you and your partner talked about?" See what happens; see what you get. You can be strategic if nobody offers. You can just say, "Hey, would you end up sharing? I listened to what you had. Would you mind sharing?"

And then I think at that point you could use a "Do you agree or disagree and why?" So here's their thinking on this situation. So I want you to really think about it. Do you agree with what they're sharing or not? And then I'm going to ask you why. Let that sit. Give them some time to think. Let that play out. I think at that point you could offer the floor to whoever wants to argue about that and try to convince the community that they agree or disagree and why.

And then I think, even, (laughs) I guess to keep going, Mike, I think you could at that point use the "tell us more," when that student's offering the reasoning on why they agree or disagree, and you don't feel like it's enough or maybe there's other kids in the room not quite understanding where they're going. "OK. So tell us a little bit more. Keep going." And offer that space and time for them to do that.

So yeah, there's several ways that you can sequence them, but I really think you have to figure out the learning goal, be intentional about the discourse and how you can get it headed in the right direction and also slow it down enough that there's some depth to it as well.

Mike: We had a guest on [Rounding Up] earlier this season, and he was talking about the importance of "agree or disagree." He called it "pick a side," but I think the idea is the same.

Todd: Oh yeah. Yeah, same concept.

Mike: And I wonder if you could talk about, what is it about agree [or] disagree that you think is particularly powerful for kids?

Todd: Sure, Mike. Do you agree [or] disagree? It does make you take a stand. Like you have to understand the situation well enough to be able to say, "Hey, I agree with this thinking because..." fill in the blank. I think it puts you in a position where you've got to weigh everything that's playing out in the discourse and then actually understand it well enough to be able to then communicate about it. Your approach may be different than the thinking that was shared, but if you can understand it well enough and then state whether you agree or disagree and why, that's some pretty deep understanding. I mean, there's some high value in that if you can get to that point.

Mike: Absolutely. I get the sense that a fair number of these talk moves might start to feel pretty organic. They might happen almost like muscle memory when an educator starts to use them, but you really have me thinking about planning for talk moves. Do you have any guidance for an educator who might be trying to think about, "Hey, I want to purposefully integrate some of these moves into my practice." What would it look like to plan for that?

Todd: Sure. I think first of all, when you talk about muscle memory, that's a great way to put it. Some of these moves may not be strong in a professional practice for a person right now, but the more you get to using them and trying them out and implementing them and seeing what they'll do for productive community agency math practice, you're going to start to develop a level of growth in your practice that I think is going to be tremendous. But as you think about being intentional with them as you plan a lesson and go after a particular learning goal for a lesson, the one thing that comes to mind for me is really Dr. Peg Smith's work around the five practices and orchestrating discussions, right? You think about anticipating and then selecting and sequencing and connecting for sure all come to mind in that. So I guess I would go as far as saying, as we prepare for what students may say or do, we can intentionally think about the moves that might be most impactful in different scenarios.

For instance, let's just say [there's a] a third grade student; you're working on a model to represent multiplication. The student draws a model to represent a multiplication scenario. You can plan to project or show the model and then simply use the think, turn, and learn move. You can show the student thinking, ask students to not talk upfront. We need to give people individual time to think. And then I want you to think about what you see and then I want you to turn and learn with your partner about that. So I think at that point, with that move being used, you're going to get a lot of discourse around whether the model is an appropriate representation or not. I think there's going to be depth in what kids take away from the experience. And you can go back to that as like, OK, so if I know that a kid says this and just says, "Well, it looks like there's this many rows and this many arrays," [then] the tell us more move is a great one there. "Tell us more. What do you mean by that?" Then they have to extend to give more depth to their thinking and then refine it a little bit more.

So I think as you think about the learning goal, there's certainly ways that you can think about any of these talk moves, and in a way where you want to make sure that the right move is being used to get you closer to the goal of sense being made and such. So yeah.

Mike: I want to come back to something that you touched on in the beginning, but it feels like a through line, which is: These talk moves are about building engagement and math, but really they're about so much more. What do you see as the long-term payoff for kids who experience this type of a learning experience?

Todd: Well, (laughs) it often feels counterintuitive when I'm in schools and talking about these things because I think we've shifted into a mode where professional learning communities are so honed in on that exact math content standard, what do we want kids to know, be able to do? How will we know? What are we going to do when they don't? And I really believe that the more I'm in these situations, Mike, I'm understanding that we can't shift that learning like we want to until we deal with some of these that—I call them more general pedagogy practices, like discourse and talk moves with intention. Those are more general practices, not a content-specific practice teaching kids how to find a common denominator so they can add fractions and such. But I really think if we can get at some of these general pedagogy things to build up community, agency, math practice—all those things that I think will transcend time—I often talk about it, we're going after something bigger than just the priority standards or the most important standards within our state. We're going after things that are deeper, bigger, pay off more later in life than we may even realize that we're experiencing in the moment.

Mike: Yeah. I mean, things like flexibility, problem solving, citizenship are all pieces that really jump out when I listen to you talk about that, Todd.

Todd: Sure. No, you think about—and I mean, as we were talking about productive community earlier, I always offer them: Is there anything different you would want in your classroom or your school? You think about the words "productive" and "community," can we all come together and think about things in a way where we're contributing, we're all valued, we're producing together. And that's not something that I think we spend a lot of time talking about in schools now that it's so specific to content and how kids do on state assessments and such, but these things transcend all that.

Mike: Absolutely.

We're at that point in time where I could probably keep chatting with you about this for hours, but you are a busy school educator and you gotta get out of here. I'm wondering though, if you could leave listeners with a thought or a question or maybe a nudge related to their practice, what would you share?

Todd: I first would say I just think it's important to always be reflecting on whose talk is driving the experience. As you think about everything we've talked about today, Mike, is the student talk driving it? Is their reasoning driving it or is it ours? And I think understanding that these talk moves with intention and what they go after and using them consistently with intention, it just starts to shift the balance to favor more student-to-student discourse. And I think it presents as, in turn, more developed community, agency and math practice. And I just think that you get more out of that than [a] high quantity of teacher-to-student discourse or student-to-teacher discourse. So I always offer [to] just pick out a move, try it for a week, find a wing person, collaborate around that, share ideas. How'd it go? What were your barriers? What did you see happening? Just these small shifts I think can create some big opportunities for people down the road.

Mike: I think that's a great place to stop. Thank you so much for joining us, Todd. It's really been a pleasure chatting.

Todd: Mine as well. Thank you, Mike.

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

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