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

Descripción

DeAnn Huinker & Melissa Hedges, Math Trajectories for Young Learners, Part 1 ROUNDING UP: SEASON 4 | EPISODE 14 Research confirms that early mathematics experiences play a more significant role than we once imagined. Studies suggest that specific number competencies in 4-year-olds are strong predictors of fifth grade mathematics success. So what does it look like to provide meaningful mathematical experiences for our youngest learners? Today, we'll explore this question with DeAnn Huinker from UW-Milwaukee and Melissa Hedges from the Milwaukee Public Schools. BIOGRAPHY Dr. DeAnn Huinker is a professor of mathematics education in the Department of Teaching and Learning and directs the University of Wisconsin-Milwaukee Center for Mathematics and Science Education Research. Dr. Huinker teaches courses in mathematics education at the early childhood, elementary, and middle school levels. Dr. Melissa Hedges is a curriculum specialist who supports K–5 and K–8 schools for the Milwaukee Public Schools. RESOURCES Math Trajectories for Young Learners [https://www.nctm.org/Store/Products/Math-Trajectories-for-Young-Learners/] book by DeAnn Huinker and Melissa Hedges Learning Trajectories [https://www.learningtrajectories.org/] website, featuring the work of Doug Clements and Julie Sarama School Readiness and Later Achievement [https://pubmed.ncbi.nlm.nih.gov/18020822/] journal article by Greg Duncan and colleagues Early Math Trajectories: Low‐Income Children's Mathematics Knowledge From Ages 4 to 11 [https://onlinelibrary.wiley.com/doi/abs/10.1111/cdev.12662] journal article by Bethany Rittle-Johnson and colleagues TRANSCRIPT Mike Wallus: Welcome back to the podcast, DeAnn and Melissa. You have both been guests previously. It is a pleasure to have both of you back with us again to discuss your new book, Math Trajectories for Young Learners [https://www.nctm.org/Store/Products/Math-Trajectories-for-Young-Learners/]. Melissa Hedges: Thank you for having us. We're both very excited to be here. DeAnn Huinker: Yes, I concur. Good to see you and be here again. Mike: So DeAnn, I think what I'd like to do is just start with an important grounding question. What's a trajectory? DeAnn: That's exactly where we need to start, right? So as I think about, "What are learning trajectories?," I always envision them as these road maps of children's mathematical development. And what makes them so compelling is that these learning pathways are highly predictable. We can see where children are in their learning, and then we can be more intentional in our teaching when we know where they are currently at. But if I kind of think about the development of learning trajectories, they really are based on weaving together insights from research and practice to give us this clear picture of the typical development of children's learning. And as we always think about these learning trajectories, there are three main components. The first component is a mathematical goal. This is the big ideas of math that children are learning. For example, counting, subitizing, decomposing shapes. The second component of a learning trajectory are developmental progressions. This is really the heart of a trajectory. And the progression lays out a sequence of distinct levels of thinking and reasoning that grow in mathematical sophistication. And then the third component are activities and tasks that align to and support children's movement along that particular trajectory. Now, it's really important that we point out the learning trajectories that we use in our work with teachers and children were developed by Doug Clements and Julie Sarama. So we have taken their trajectories and worked to make them more usable and applicable for teachers in our area. So what Doug and Julie did is they mapped out children's learning starting at birth—when children are just-borns, 1-year-olds, 2-year-olds—and they mapped it out up till about age 8. And right now, last count, they have about 20 learning trajectories. And they're in different topics like number, operations, geometry, and measurement. And we have to put in a plug. They have a wonderful website. It's learningtrajectories.org [http://learningtrajectories.org]. We go there often to learn more about the trajectories and to get ideas for activities and tasks. Now, we're talking about this new book we have on math trajectories for young children. And in the book, we actually take a deep dive into just four of the trajectories. We look at counting, subitizing, composing numbers, and adding and subtracting. So back to your original question: What are they? Learning trajectories are highly predictable roadmaps of children's math learning that we can use to inform and support developmentally appropriate instruction. Mike: That's an incredibly helpful starting point. And I want to ask a follow-up just to get your thinking on the record. I wonder if you have thoughts about how you imagine educators could or should make use of the trajectories. Melissa: This is Melissa. I'll pick up with that question. So I'll piggyback on DeAnn's response and thinking around this highly predictable nature of a trajectory as a way to ground my first comment and that we want to always look at a trajectory as a tool. So it's really meant as an important tool to help us understand where a child is and their thinking right now, and then what those next steps might be to push for some deeper mathematical understanding. So the first thing that when we work with teachers that we like to keep in mind, and one of the things that actually draw teachers to the trajectories is that they're strength-based. So it's not what a child can't do. It's what a child can do right now based off of experience and opportunity that they've had. We also really caution against using our trajectories as a way to kind of pigeonhole kids or rank kids or label kids because what we know is that as children have more experience and opportunity, they grow and they learn and they advance along that trajectory. So really it's a tool that's incredibly powerful when in the hands of a teacher that understands how they work to be able to think about where are the children right now in their classroom and what can they do to advance them. And I think the other point that I would emphasize other than what moves children along is experience and opportunity. Children are going to be all over on the trajectory—that's been our experience—and they're in the same classroom. And it's not that some can't and some won't and some can; it's just some need more experience and some need more opportunity. So it's really opened up the door many ways to view a more equitable approach to mathematics instruction. The other thing that I would say is, and DeAnn and I had big conversations about this when we were first using the trajectories, is: Do we look at the ages? So the trajectories that Clements and Sarama develop do have age markers on them. And we were a bit back and forth on, "Do we use them?," "Do we not?," knowing that mathematical growth is meant to be viewed through a developmental lens. So we had them on and then we had them off and then we shared them with teachers and many of our projects and the teachers were like, "No, no, no, put the ages back on. Trust us. We'll use them well." (laughs) And so the ages are back onto the trajectories. And what we've noticed is that they really do help us understand how to take either intentional steps forward or intentional steps back, depending on what kids are showing us on that trajectory. The other spot that I would maybe put a plugin for on where we could use a trajectory and what would be an appropriate use for it would be for our special educators out there and to really start to use them to support clear, measurable IEP goals grounded in a developmental progress. So that's kind of what our rule of thumb would be around a "should" and "shouldn't" with the trajectories. Mike: That's really helpful. You mentioned the notion of experiences and opportunities being critical. So I wanted to take perhaps a bit of a detour and talk about what research tells us about the impact of early mathematics experiences, what impact that has on children. I wonder if you could share some of the research that you cite in the book with our listeners. DeAnn: Sure. This is DeAnn, and in the book we cite research throughout all of the chapters and aligned to all of the different trajectories. But as we think about our work, there really are a few studies that we anchor in, always, as we think about children's learning. And the research evidence is really clear that early mathematics matters. The math that children learn in these early years in prekindergarten, kindergarten, first grade—I mean, we're talking 4-, 5-, 6-year-olds, 7-year-olds—that their math learning is really more important than a lot of people think it is. OK? So as we think about these kind of anchor studies that we look at, one of the major studies in this area is from Greg Duncan and his colleagues [https://pubmed.ncbi.nlm.nih.gov/18020822/], and there was a study published in 2007. And what they did is they examined data from thousands of children drawing information from six large-scale studies, and they found that the math knowledge and abilities of 4- and 5-year-olds was the strongest predictor of later achievement. I mean, 4- and 5-year-olds, that's just as they're starting school. Mike: Wow. DeAnn: Yeah. One of the surprising findings was that they found early math knowledge and abilities was a stronger predictor than social emotional skills, stronger than family background, and stronger than family income. That it was the math knowledge that was predictive. Mike: That's incredible. DeAnn: Yes. A couple other surprising things from this study was that early math was a stronger predictor than early reading. Now, we know reading is really important, and we know reading gets a lot of emphasis in the early grades, but math is a stronger predictor than reading. And then one last thing I'll say about this study is that early math not only predicts later math achievement, it also predicts later reading achievement. So that is always a surprise as we share that information with teachers, that early math seems to matter as much and perhaps more than early reading abilities. There's a couple other studies I'll share with you as well. So there's this body of research that talks about [how] early math is very predictive of later learning, but we're teachers, we're educators. We like to know, "Well, what math seems to be most important?" So there was a study in 2016 [https://onlinelibrary.wiley.com/doi/abs/10.1111/cdev.12662] that looked at children's math learning in prekindergarten, 4-year-olds, and then looked at their learning again back in fifth grade. And what was unique about this study is they looked closely at what specific math topics seemed to matter the most. And what they found was that advanced number competencies were the strongest predictors of later achievement. Now, what are advanced number competencies? So these are the three that really stood out as being important. One was being able to count a set of objects with cardinality. So in other words, counting things, not just being able to recite a count sequence, no. So not verbal rote counting, but actually counting things, putting those numbers to objects. Another thing that they found [that] was really important was being able to count forward from any number. So if I said, "Start at 7 and keep counting," "Start at 23 and keep counting," that that was predictive of later learning. And the reason for that is when kids can count forward from a number, it helps them understand the structure of the number system, something we're always working on. And then the third thing that they found as part of advanced number competencies was conceptual subitizing. Now, what that is, is being able to see a number such as 5 as composed of subgroups, like 5 being composed of 4 and 1 or 3 and 2. So subitizing is being able to see the parts of a number, and that was really important for these 4-year-olds to begin working on for later learning. All right. One more, Mike, that I can share? Mike: Fire away! Yes. DeAnn: OK. So this last area of research that I want to share is actually really important as we think about the work of teachers in kindergarten and first grade in particular. So what these researchers did is they looked at children's learning at the beginning of kindergarten and then at the end of first grade. So, wow, think of the math kids learn from 5, 6 years old. And they found that these gains in what children can do was more predictive of later achievement than just what knowledge they had coming in. So learning gains, what children do and learn in math in kindergarten and first grade, is predictive of their mathematical success up through third grade. And then another study took it even further and said: Wait a minute, what they learn in kindergarten and first grade even predicts children's math achievement into high school. So there's just a growing body of research and evidence that early math is really important. The math learning of 4-year-olds, 5-year-olds, 6-year-olds, and 7-year-olds really builds this foundation that determines children's mathematical success many years later. Mike: This feels like a really great segue to a conversation about what it means to provide students opportunities for meaningful counting. That feels particularly significant when I heard all of the ideas that you were sharing in the research. I'm wondering if you could talk about the features of a meaningful counting experience. If we were to try to break that down and think about: What does that mean? What does that look like? What types of experiences count as meaningful when it comes to counting? Could you all talk about that a little bit? Melissa: Yeah, that's a great question, Mike. This is Melissa. So I think what's interesting about the idea of meaningful counting is, the more DeAnn and I studied the trajectory and spent time working with teachers and students, we came to the conclusion that the counting trajectory in particular is anchored, or a cornerstone of that counting trajectory is really meaningful counting. That once a skill is acquired—and we'll talk a little bit more about meaningful counting—but once that skill is acquired, it just builds and develops as kids grow and have more experience with number and quantity. So when we think about meaningful counting, the phrase that we like to use is that "Numbers represent quantity." And it's just not that kids are saying numbers out loud, it's that when they say "5," they know what 5 means. They know how many that is. They can connect it to a context that they can go grab five of something. They might know that 5 is bigger than 2 or that 10 is bigger than 5. So they start to really play with this idea of quantity. And specifically when we're talking about kids engaging in meaningful counting, there's really key skills and understandings that we're looking and watching for as children count. The first one DeAnn already alluded to, is this idea of cardinality. So when I count how many I have—1, 2, 3, 4, 5—if that's the size of my set, when someone asks me, "How many is it?," I can say "5" without needing to go back and count. So I can hold that quantity. Another one is stable count sequence. So we used to call it rote count sequence. And again, DeAnn referenced the idea that, really, when we're asking kids to count, we're asking more than just saying numbers. So we think about the stability and the confidence in their counting. One of the pieces that we've started to really watch very carefully and think carefully about with our children as we're watching many of them count is their ability to organize. So it's not the job of the teacher to organize the counter, to tell the child how to lay out the counters. It really is the work of the child because it brings to bear counting, saying the numbers, maintaining cardinality, as well as sets them up and sets us up to see where they at with that one-to-one correspondence. So can they organize a set of counters in such a way that allows them to say one number, one touch, one object? And then as they continue to coordinate those skills, are they able to say back and hold onto the idea of quantity? So the other ideas that we like to consider, mostly because they're embedded in the trajectory and we've seen them become incredibly important as we work with children, is the idea of producing a set. So when I ask a child, "Can you give me five?," they give me five, or are they able to stop when they get to five? Do they keep counting? Do they pick up a handful of counters and dump it in my hand? So all of those things are what we're looking for as we're thinking about the idea of producing a set. And then finally, even for our youngest ones, we really place a fair importance on the idea of representing a count. So can they demonstrate, can they show on paper what they did or how many they have? So we leave with a very rudimentary math sketch. So if they've counted a collection of five, how would they represent five on that paper? What that allows then the teacher to do is to continue to leverage where the trajectory goes as well as what they know about young children to bring in meaningful experiences tied to writing numbers, tied to having conversations about numbers. So the kids aren't doing worksheets, they're actually documenting something very important to them, which is this collection of whatever it is that they just counted in a way that makes sense to them. And so I think the other part that I like to talk about when we think about meaningful counting is this idea of hierarchical inclusion. It's that idea that children understand that numbers are nested one within each other and that each number in the count sequence is exactly 1 higher than what they said before. So, many times our reference with that is with our teachers are those little nesting dolls. So we think about 1 and then we wrap 2 around it and then we wrap 3 around it. So when we think about the number 3, we're thinking, "Well, it's actually the quantity of 2 and 1 more." And we see that as a really powerful understanding in particular as our children get older and we ask them not just what is 1 more or 1 less, but what is 10 more or 10 less, that they take that and they extend that in meaningful ways. So again, the idea of meaningful counting, regardless of where we are on the trajectory, it's the idea that numbers represent quantities. And the neat thing about the trajectory—the counting trajectory in particular—is that they give us really beautiful markers as to when to watch for these. So we tend to talk about the trajectories as levels. So we'll say at level 6 on our counting trajectory is where we see cardinality first start to kind of show up, where we're starting to look for it. And then we watch that idea of cardinality grow as children get older, as they have more experience and opportunity, and as they work with larger numbers. Mike: That's incredibly helpful. So I think one of the things that really jumped out, and I want to mark this and give you all an opportunity to be a little bit more explicit than you already were—this importance of linking numbers and quantities. And I wonder if you could say a bit more about what you mean, just to make sure that our listeners have a full understanding of why that is so significant. DeAnn: All right, this is DeAnn. I'll jump in and get started, and Melissa can add on. As we first started to study the learning trajectory, the one thing we noticed was the importance of connecting things to quantity. Even some of the original levels didn't necessarily say "quantity," but we anchor our work to developing meaning for our work. And we always think about, even when we're skip-counting, it should be done with objects that we should be able to see skip-counting as quantities, not just as words that I'm reciting. So across the trajectory, we put this huge emphasis on always connecting them to items, to things, or to actions and to movements so that it's not just a word, but that word has some meaning and significance for the child. Mike: I think that takes me to the other bit of language, Melissa, that you said that I want to come back to. You said at one point when you were describing meaningful counting experiences, you said, "One number, one touch, one object." And I wonder if you could unpack that, particularly "one touch," for young children and why that feels significant. Melissa: That's a great question. And I'll come at this through a lens of watching many, many children count and working with lots and lots of teachers. When children are counting a set, many times they'll look and they'll go, "1, 2, 3, 4, 5, 6, 7, 8, 9," and then however many are in the collection, they'll just say, "9" by just looking. And one of the things that we've noticed is that sometimes we need to explicitly give permission to children to do what they need to do with that collection to find out how many. Sometimes they're afraid to touch the items. Sometimes they don't know that they can. And we don't come right out and say, "Go ahead and touch them." But we just say, "Gosh, is there another way that you could find out how many?" And what we notice are some amazing and interesting ways kids organize their collections. So sometimes to be able to get to that "one touch, one, number one object," they'll lay them out in a row. Sometimes they'll lay them out in a circle and they'll mark the one that they started with. Sometimes, with our little guys in particular, we like to give them collections where they have to sit things up, so like, the little counting bears. So if the bears are lying down, the kids will be very intentional in, "I set it up and I count it. I set it up and I count it. " And they all, many times, have to be facing the same direction as well. So the kids are very particular about, "How does this fit into the counting experience?" And I would say that's one thing that's been really significant for us in understanding that it really is the work of the child to do that "one touch, one object, one count" in a way that matters to them. And that a teacher can very easily lay it out and say, "Find out how many. Remember to touch one and tell me the number." Then it's not coming from the child. Then we don't know what they know. So that's been a really, really interesting aspect for us to watch in kids is, "How are they choosing to go into and enter into counting that?" And we look at that as problem solving from our youngest, from our 3-year-olds, all the way up, is: "What are you going to do with that pile of stuff in front of you?" And that's an authentic problem for them, and it's meaningful. Mike: I think what jumps out about that from me is the structure of what you just described is actually an experience and it's an opportunity to make sense of counting versus what perhaps has typically happened, which is a procedure for counting that we're asking kids to replicate and show us again. And what strikes me is you're advocating for a sensemaking opportunity because that's the work of the child. As opposed to, "Let me show you how to do it; you do it again and show it back to me," but what might be missing is meaning or connection to something that's real and that sets up what we think might be a house of cards or at the very least it has significant implications as you described in the research. Melissa: One of the things, Mike, that I would add on that actually I just thought about is, when you were talking about the importance of us letting the children figure out how they want to approach that task of organizing their count, is: It's coming from the child. And Clements and Sarama talk about, the beautiful work about the trajectory is that we see that the mathematics comes from the child and we can nurture that along in developmentally appropriate ways. The other idea that popped into my mind is: It's kind of a parallel to when our children get older and we want to teach them a way to add and a way to subtract. And I'm going to show you how to do it and you follow my procedure. I'm going to show it; you follow my procedure. We know that that's not best practice either. And so we're really looking at: How do we grab onto that idea of number sense and move forward with it in a way that's meaningful with children from as young as 1 and 2 all the way up? Mike: I hope you've enjoyed the first half of our conversation with DeAnn and Melissa as much as I have. We'll release the second half of our conversation on April 9th. This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2026 The Math Learning Center | www.mathlearningcenter.org [http://www.mathlearningcenter.org]

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

Kristin Frang, Understanding the Roots of Fluency with Addition & Subtraction ROUNDING UP: SEASON 4 | EPISODE 16 Research suggests that supporting students' fluency with addition and subtraction hinges on understanding how children's mathematical thinking develops. So what are the concepts and ideas that play a part in fluency with combinations to 10, 20, and beyond? Today, we'll explore this question with Kristin Frang, director of instructional programs at Integrow Numeracy Solutions. BIOGRAPHY Kristin Frang is the director of instructional programs for Integrow Numeracy Solutions. She designs resources and services that support states, districts, schools, and individuals in transforming numeracy education. RESOURCES "Understanding Units Coordination" [https://www.mathlearningcenter.org/blog/understanding-units-coordination] Season 4, Episode 11 of the Rounding Up podcast Integrow Numeracy Solutions * website [http://www.integrowmath.org] * blog [https://www.integrowmath.org/blog] * email address [info@integrowmath.org] On Track to Numeracy [https://www.integrowmath.org/store/3978621] book by Lucinda "Petey" MacCarty, Kurt Kinsey, David Ellemor-Collins, and Robert J. Wright TRANSCRIPT Mike Wallus: Welcome to the podcast, Kristin. It is so great to be talking with you today. Kristin Frang: It's great to be here. I feel so honored to be on this podcast. Mike: Before we dive into a conversation about addition and subtraction, I'd like to do a bit of grounding. So you're currently the director of instructional programs for Integrow Numeracy Solutions. I wonder if briefly you could tell the listeners: What is Integrow Numeracy Solutions, and what's its mission? Kristin: Yeah. Integrow Numeracy Solutions' mission is to transform numeracy education by connecting research with practice and empowering educators to advance student mathematical thinking and success. But I really want to bring that mission to life through a story, just a quick story, if I can. Prior to my role with Integrow, I was a K–12 mathematics consultant. And one of the things that I did was, when the Common Core [State Standards] were released, I worked with teachers to transition to the then-new standards. We studied many documents together, including progression documents that were included in the standards, and teachers were honestly fascinated by this idea of a progression and that they were embedded into the standard. But I remember an instance where we had been studying these progressions and a teacher came up and said to me, "I know where my students are at; I can see them in these progressions. But how do I get them to the next stage?" And I didn't have an answer (laughs) at that point. I was a former middle school and high school teacher. I was working with elementary teachers. I was studying, just like them, these progression documents, and I could only categorize the reasoning that was in front of us. And so that next step to say, "Oh, this is what I would do and bring into action in the classroom," I didn't have an answer for. And so that's really where I was introduced to Integrow—formerly [the] US Math Recovery Council, but now Integrow Numeracy Solutions. And at the heart of our mission to empower educators is to bring research to the classroom in accessible and practical ways that advance student reasoning. We do this in professional learning, we do it in supplemental resources, and we also hire and train educators to deliver high-dosage tutoring for students to accelerate their learning. Mike: I want to just linger on something you said, which was—and I really appreciate both the truth of the statement you made and also the vulnerability, which is to say—I think for many teachers, there's this experience of, "I can see my students in these progressions, but I'm not sure what to do when it comes to making moves to shift where they're at or help them move." And I think that's a profound truth for so many teachers. And I think it's really important that folks like you, who are doing this work, acknowledge that that's a place you were in once as well because that's so true for so many of us. Kristin: Yeah. There's always a new thing where we're watching students, we're thinking about the next steps. And so often it boils down to categorizing the things that students are doing now, but not often figuring out: What are the true actions that we take with real children who are in front of us to get them to progress in their own reasoning? We can tell them the next step, but my belief system that is aligned with Integrow Numeracy Solutions is that the most powerful thing is to help students have those experiences and create that understanding themselves. And to do that, it's more complex than just knowing what the next benchmark is for them. Mike: I think that's a helpful introduction. And I also find it to be a good segue for all the questions that I wanted to explore today. So let me start here: It feels important to acknowledge that supporting students' addition and subtraction fluency actually hinges on understanding how children's mathematical thinking develops. So I wonder if you can talk about some of the concepts and the ideas that play a part in fluency when it comes to combinations of 10, combinations to 20, and even beyond. Kristin: Yeah. The words that we hear associated with fluency right now are "flexibility," "efficiency," "accuracy." So we've moved on from just speed, which I think is a really positive place for us to be in education. But at the heart of flexibility, efficiency and accuracy is a quantitative understanding of arithmetic. I'm really glad that you had Amy Hackenberg on [the podcast] recently who discussed this concept of units coordination because throughout what we'll talk about, you'll see units coordination come out, but she's definitely the expert to explain it. Just a nod. Just listen to that episode [https://www.mathlearningcenter.org/blog/understanding-units-coordination] [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 [https://www.corwin.com/author/zaretta-hammond], Betty [K.] Garner [https://www.corwin.com/author/betty-k-garner?srsltid=AfmBOopnpe66Guc5OO0mWjfPMO2JJh6sTBWkgXCKAxCbmA1iIkgTUyXs], Karen [S.] Karp [https://www.corwin.com/author/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 [https://www.corwin.com/author/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 [http://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 [http://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 info@integrowmath.org [info@integrowmath.org] 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] [https://www.integrowmath.org/store/3978621], 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 [http://www.mathlearningcenter.org]

23 de abr de 202633 min

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

Mike Steele, Pacing Discourse-Rich Lessons ROUNDING UP: SEASON 4 | EPISODE 13 As a classroom teacher, pacing lessons was often my Achilles' heel. If my students were sharing their thinking or working on a task, I sometimes struggled to decide when to move on to the next phase of a lesson. Today we're talking with Mike Steele from Ball State University about several high-leverage practices that educators can use to plan and pace their lessons. BIOGRAPHY Mike Steele is a math education researcher focused on teacher knowledge and teacher learning. He is the past president of the Association of Mathematics Teacher Educators, editor in chief of the Mathematics Teacher Educator journal, and member of the NCTM board of directors. RESOURCES Journal Article * "Pacing a Discourse-Rich Lesson: When to Move On" [https://pubs.nctm.org/view/journals/mtlt/118/11/article-p822.xml] Books * 5 Practices for Orchestrating Productive Mathematics Discussions [https://www.corwin.com/books/5-practices-262956] * The 5 Practices in Practice [Elementary] [https://www.corwin.com/books/5-practices-in-practice-260531] * The 5 Practices in Practice [Middle School] [https://www.corwin.com/books/the-5-practices-on-practice-260532] * The 5 Practices in Practice [High School] [https://www.corwin.com/books/mathematics-discussions-hs-260533] * Coaching the 5 Practices [https://www.corwin.com/books/coaching-the-5-practices-287127] TRANSCRIPT Mike Wallus: Well, hi, Mike. Welcome to the podcast. I'm excited to talk with you about discourse-rich lessons and what it looks like to pace them. Mike Steele: Well, I'm excited to talk with you too about this, Mike. This has been a real focus and interest, and I'm so excited that this article grabbed your attention. Mike Wallus: I suppose the first question I should ask for the audience is: What do you mean when you're talking about a discourse-rich lesson? What does that term mean about the lesson and perhaps also about the role of the teacher? Mike Steele: Yeah, I think that's a great question to start with. So when we're talking about a discourse-rich lesson, we're talking about one that has some mathematics that's worth talking about in it. So opportunities for thinking, reasoning, problem solving, in-progress thinking that leads to new mathematical understandings. And that kind of implicit in that discourse-rich lesson is student discourse-rich lesson. That we want not just teachers talking about sharing their own thinking about the mathematics, but opportunities for students to share their own thinking, to shape that thinking, to talk with each other, to see each other as intellectual resources in mathematics. And so to have a lesson like that, you've got to have a number of things in place. You've got to have a mathematical task that's worth talking about. So something that's not just a calculation and we end up at an answer and that the discourse isn't just, "Let me relay to you as a student the steps I took to do this." Because a lot of times when students are just starting to experience discourse-rich lessons, that's kind of mode one that they engage in is, "Let me recite for you the things that I did." But really opportunities to go beyond that and get into the reasoning and the why of the mathematics. And hopefully to explore some approaches or perspectives or representations that they may not have defaulted to in their first run-through or their first experience digging into a mathematical task. So the task has to have those opportunities and then we have to create learning environments that really foster those opportunities and students as the creators of mathematics and the teacher as the person who's shaping and guiding that discussion in a mathematically productive way. Mike Wallus: One of the things that struck me is there is likely a problem of practice that you're trying to solve in publishing this article, and I wonder if we could pull the curtain back and have you talk a bit about what was the genesis of this article for you? Mike Steele: Absolutely. So let me take us back about 20 or 25 years, and I'll take you back to some early work that went on around these sorts of rich tasks and discourse-rich lessons. So a lot of this legacy comes out of research or a project in the late nineties called the Quasar Project that helped identify: What is a rich task? What is a task, as the researchers described it, of high cognitive demand that has those opportunities for thinking and reasoning? The next question that that line of research brought forward is, "OK, so we know what a task looks like that gives these opportunities. How does this change what teachers do in the classroom? How they plan for lessons, how they make those moment-to-moment decisions as they're engaged in the teaching of that lesson?" Because it's very different than actually when I started teaching middle school in the nineties, where my preparation was: I looked at the content I had for that day, I wrote three example problems I wanted to write on the board that I very carefully got all the steps right and put those up and explained them and answered some questions. "Alright, everybody understand that? OK, great, moving on." And then the students went and reproduced that. That's fine for some procedural things, but if I really wanted them to engage in thinking and reasoning, I had to start changing my whole practice. So this bubbles up out of the original work of the 5 Practices for Orchestrating Productive Discussions [book] from Peg Smith and Mary Kay Stein. I had the opportunity actually to work with them both in the early two thousands at the University of Pittsburgh. And so as we were working on this five-practices framework that was supposed to help teachers think about, "What does a different conceptualization of planning and teaching look like that really gets us to this discourse-rich classroom environment where students are making sense of and grappling with mathematics and talking to each other in a meaningful way about it?" We worked with teachers around that and the five-practices [framework] is certainly helpful, but then as teachers were working with the five practices and they were anticipating student thinking, they were writing questions that assess and advance student thinking, some of the things that came up were, "OK, what are the moment-to-moment decisions and challenges related to that as we start planning and teaching in this way?" And a number of common challenges came up. A lot of times when we were using a five-practice lesson, we were doing kind of a launch, explore, share, and discuss sort of format where we've got the teacher who's getting us started on a task, but we're not giving the farm away on that task. We're not saying too much and guiding their thinking. And then we let students have some time individually and in small groups to start messing around with the mathematics, working, talking. And then at some point we're going to call everybody together and we're going to share what the different ways of thinking were. We're going to try to draw that together. Peg Smith likes to talk about this as being more than a show-and-tell. So it's not just, "We stand up, we give our answer, we do that. Great." Next group, doing the same thing, and oftentimes they start to look alike. But there's some really meaningful thinking that goes on in that whole-class discussion. So one of the really pragmatic concerns here is, "How do I know when to move?" So I've got students working individually, and maybe I gave them 3 minutes to get started. Was that enough? What can I see in the work they're doing? What questions am I going to hear to tell me, "OK, now it's a good moment to move to small groups." And then similarly, when you've got small groups working, they're cranking away on a task. There might be multiple subquestions in that task. What's my cue that we're ready to go on to that whole-class discussion? We were in so many classrooms where teachers were really working hard to do this work, and this happens to me all the time. I have somehow miscalculated what students are going to be able to do—either how quickly they're going to be able to do it, or I expected them to draw on this piece of prior knowledge and it took us a while to get there, or they've flown through something that I didn't expect them to fly through. So I'm having to make some choice in a moment, saying, "This isn't exactly how I imagined it, so what do I do here?" And frequently with teachers that get caught in that dilemma, the first response is to take control back, [to] say, "OK, you're all struggling with this. Let's come back together and let me show you what you should have figured out here." And it's done with the best of intentions. We need to get some closure on the mathematical ideas. But then it takes us right away from what we were trying to do, which was have our students grapple with the mathematics. And so we do this lovely polished job of putting that together and maybe students took the important things away from that, that they wanted to, maybe they didn't, but they didn't get all the way they were on their own. So that's really the problem of practice that this helps us to solve is, when we get in those positions of, "OK, I've got to make a call. I've got this much time left. I've got this sort of work that I see going on in the classroom. Am I ready? What can I do next?" That really keeps that ownership of the mathematics with our students but still gives me some ability to orchestrate, to shape that discussion in a way that's mathematically meaningful and that gets at the goals I had for the lesson. Mike Wallus: Yeah, I appreciated that part of the article and even just hearing you describe that so much, Mike, because you gave words to I think what sat behind the dilemma that I found myself in so often, which was: I was either trying to gauge whether there was enough—and I think the challenge is we're going to get into, what "enough" actually might mean—but given enough time, whether I was confident that there was understanding, how much understanding was necessary. And what that translates into is a lack of clarity around "How do I use my time? How do I gauge when it's worth expending some of the time that I maybe hadn't thought about and when it's worth recognizing that perhaps I didn't need all of that and I'm ready to do something?" So I think the next question probably should be: Let's talk about "enough." When you talk about knowing if you have enough, say a little bit more about what you mean and perhaps what a teacher might be looking and listening for. Mike Steele: Absolutely. And I think this is a hidden thread in that five-practices model because we say: "OK, we want that whole-class discussion to still be a site for learning where there are some new ideas that are coming together." So that then backs me up to thinking about the small-group work. I'm putting myself in that mode where I've got six groups working around the classroom. I'm circulating around; I'm asking questions. I of course don't see every single thing at any given moment that the small groups are doing. So I'm getting these little excerpts, these little 2- to 3-minute excerpts, when you stop into a group. So I think when we think about "enough," I want to think about, with that task that I'm doing, with what my mathematical goals are and knowing that we're going to have time on the backend of this whole-class discussion to pull some ideas together, to sharpen some things to clarify some of the mathematics. Do I have enough mathematical grist for the mill here in what the small groups are doing to be able to then take that and make progress with students' thinking at the center—again, not taking over the thinking myself—to be able to do that work. So, for any given mathematical idea, as I've started thinking about this when I plan lessons using the five-practices model, I am really taking that apart. What's the mathematical nugget that I'm listening for here, that I'm looking for in students' work that tells me: "OK, we've gotten to a point where, if I were to call people together right now and get them thinking about it, that there's more to think about, but we're well on our way." And also when I'm looking for that, knowing that I'm also not looking at those six groups all at exactly the same time. So, I want to look for those mile markers along the way that tell me we're getting close, but we're not all the way there. Because if I pick one that's, we're pretty much all the way there, that's the first group I come to and I'm going to circulate around to five more. They're going to have run out of interesting things to do, and they're off talking about, thinking about something else. So, that really becomes the fine line: "What are those little mathematical ideas along the way that are far enough that get us towards our goals, but still we've got a little bit of the journey to go that we're going to go on together?" Mike Wallus: This is so fascinating. The analogy that's coming together in my mind is almost like you're listening for the ingredients for a conversation that you want to have as a group. So it's not necessarily "Has everyone finished?" And that's your threshold. It's actually "Did I hear this idea starting to bubble up? Did I hear elements of this idea or this strategy start to bubble up? Is there an insight that's percolating in different groups?" And it's the combination of those things that the teacher is listening for, and that's kind of the gauge of enoughness. Is that an accurate analogy? Mike Steele: It is, and I love that analogy because it reminds me of a favorite in our household as we're relaxing. We love to watch The Great British Baking Show. So, you're watching people take something from ingredients to a finished product. Now as you're watching that 20-minute segment, they're working on their technical challenge and they're all baking the same thing. I don't have to wait until the end of that, where they've presented their finished product, to have a good idea of what's going to happen. As I'm going through, as I'm watching 'em through that baking process, we're at the middle, my wife and I are talking, like, "Ooh, I've got concerns about that one. That one's looking good though." We get an idea of where it's going. So I think the ingredient analogy really lands with me. We don't have to wait. We're looking for those pieces to be able to pull that together, those ingredients. We're not waiting until there's a final product and saying—because then, what is there to say about it? "Oh, look, that looks great. Oh, that one, maybe not exactly what we'd intended." So, it's giving us those ingredients for that whole-class discussion. Mike Wallus: The other thing that struck me as I was listening to you is: We're not teaching a task; we're teaching a set of ideas or relationships. The task is the vehicle. So, it's perfectly reasonable, it seems, to say, "We're going to pause at this point in the task, or at a place where students might not be entirely finished with the task. And we might have a conversation at that point because we have enough that we can have part of the conversation." And that doesn't mean that they don't go back to the task. But you're really helping me recognize that one of the places where I sometimes get stuck, or got stuck, when I was teaching, is task completion was part of my time marking. And I think really what you're challenging me and other educators to do is to say, "The task is just the vehicle. What's going on? What's percolating around that task as it's happening?" How does that strike you? Mike Steele: Yeah, absolutely. And it was the same challenge with me and sometimes still is the same challenge with me. (laughs) Yeah, you give this task, and we think about that task as our unit of analysis as a teacher when we're planning. And so we want our students as we're using it to get to the end of it. It's a very natural thing to do. And let me make this really concrete. If I'm doing a visual pattern task with third graders, we have, I think there's one of the elementary [5 Practices in Practice] book called "Tables & Chairs." So you've got these square tables that have four seats around them, and you're putting a string of tables together and asking kids to get at the generalization. "If you have any number of tables, how many people can you seat?" And so I think early when I started giving those tasks, I was looking for, "OK, has everybody gotten to the rule? Have they gotten to that generalization? OK, now we can talk about it." And we can talk about the different ways people made sense of that geometrically and those connections, and that's what I want to get out of the whole-class discussion. But we don't even have to get there if groups have a sense of how that pattern is growing, even if they haven't gotten to the formal description of the rule. Because if they've gotten to that point, they've made some sense of the visual. They've made some of those connections. They've parsed that in different ways. That's plenty for me to have a good conversation, that we can come to that rule as a group and we can even come to it in different ways as a group. But it frees me up from being like, "OK, everybody got the rule? Everybody got the rule? Everybody got the rule?" Because that often resulted in, I'd have a couple of groups that maybe had been a little slower getting started and they're still getting there. And then I'm sitting there and I'm talking to them, I'm giving them these terribly leading questions. "Can we just get to the rule? Come on, let's go. You're almost there. We got it. We got it." And that then is, again, me taking over that thinking and not giving them the space for those ideas to breathe. Mike Wallus: What else is jumping out for me is the ramifications for how thinking this way actually might shift the way that I would plan for teaching, but also how it might shift the way that I'm looking for evidence to assess students' progress during the task. So I wonder if you have situations or maybe some recommendations for: How might a person plan in ways that help them recognize the ways that the task can be a vehicle but also plan for the kind of evidence that they might be looking for along the way? Could you talk a little bit about that? Mike Steele: Absolutely. So I'll give kind of a multi-layered description of this. When we're using a task that's got multiple solution paths that has these opportunities for diverse thinking, the five-practices framework tells us anticipating student thinking is a critical part of it. So, what are the different solution paths that students can take through it? So, if it's a visual pattern task, they may look at it this way with a visual. They may think about those tables like the tops and the bottoms and then the sides. They may think about the two ends of the tables having different numbers of chairs and the ones in between having a different number of chairs and parsing it that way. And we can develop those. It's actually, for me, quite a lot of fun to develop those fully formed solutions that students can do. And early on when I was enacting lessons like this, I would do that. I'd have those that I was looking for. I'd also think about questions I'd want to ask students who are struggling to get started or maybe were going down a path that may not be mathematically productive and the questions I might ask them to get them on a more mathematically productive path. And I'd go around and I'd look for those solutions, and I'd use that to think about my selecting, my sequencing, my connecting my whole-class discussion. So, great, check. That's layer one. I think responding to the challenge of what's enough requires us to then take those solution paths apart—both the fully formed ones, maybe the incomplete thinking—and say, "OK, within that solution, what are the things that I want to see and hear that gives me some confidence that we're on this path, even if we're not at the end of this path, and that give me enough to think about?" So, if I think about, I'll go back again to this visual pattern task analogy. If I see groups that are talking about increases, so when we add a table, we're adding two chairs or they're making that distinction between those end tables and the center tables. And I've asked them a couple of questions like: OK, they've done that for 4, they've done that for 5. We may not have done that for 10 or 100 or a generalization, but that might be enough. So, I'm trying to take apart the mathematics and look for those little ideas within it. We've got this idea of a constant rate of change. We've got an idea that the number of tables and the number of chairs have a direct relationship here. So we're setting the stage for that functional thinking, even if, at a third grade level, we're not going to talk about that word. And those might be the important goals that I have for the lesson. So that's the next phase of what I'm doing. In addition to those fully formed solutions, I'm figuring out: What are the little mathematical ideas in each that I would want to see or hear in my classroom that tell me, "OK, I have a good sense of where they are. I know where this bake's going to turn out 5 minutes from now on the show when they've taken it out of the oven." So, that's I think the next layer of that planning, of trying to figure out how to plan. And then as we're in the moment in the classroom, being able to know what we're looking for and listening for. And the listening for me is really, really important. I think when I started doing this and I had a sense of, "What are the mathematical ideas I need to draw on?" I made the mistake of overly looking for those on paper. And if we think about how students make sense of writing things down, and sometimes despite our best efforts, the finality that comes with it: "If I've written it down, I have made it real." And if our thinking is still kind of this in-progress thinking, we may not be ready to write it down. So if I wait for it to be written on the page, I may have waited too long, or longer than I needed to, for everybody to get that idea. So again I want to make sure I listen for key words and phrases. And I might have a couple of questions teed up to help me hear those. And once I've heard those, I'm like, "OK, I am ready to go." And then for me—at least in my early fifties and not having the memory that I did when I was a 22-year-old, fresh-out-of-the-box classroom teacher—I need to have a way of keeping track of that and writing that down. So be it physical, be it digital, I want to say, "OK, I know what I'm listening for, what I'm looking for." And sometimes those may be interchangeable. If it's written on the page, great. If not, if I hear it, that's great too. And then if I've got a pretty good roster of that as I've moved through and say, "OK, I feel like all of my groups or most of my groups are at this point, there we go." I feel confident that when I pull us back together, it's not going to be me asking a question and then that terribly awkward sea of crickets out there. I'm like, "I know you were thinking about stuff; just give it to me. I know you've got this." But it gives me much more confidence that we're going to have that nice transition into a good whole-class discussion. Mike Wallus: OK. There's a ton of powerful stuff that you just said. So I want to try to mark two things that really jump out for me. One is an observation that I think is important, and then one is a thought that I want to pick your brain around a little bit further. I think the biggest piece that I heard you say, which as you were talking about, is this notion that I'm waiting for something to appear in written form. And it feels really freeing and it gives me a lot more space to say, "This is something I could hear or I could even see in the way that kids were manipulating materials. That that counts as evidence, and I don't have to literally see it written on a paper in order for me to count that that idea is in the room." I just want to name that for the audience because that feels tremendously important. Because from a practical standpoint, if we're waiting for it to be written, that takes more time. And it doesn't necessarily mean that suddenly it appeared and before when it was just in a child's mind or in the way that they were manipulating something, that it wasn't there. It was there. So I just want to mark that. The other thing that you had me thinking about is, I know for myself, I've gone through and done some of the anticipation work in the five practices, but what struck me is when my colleagues and I would do that, we often would generate quite a few alternative strategies or ideas. But I feel like what we were looking at is the final outcome, like, "This counting by 1 strategy is what we might see. This decomposing numbers more flexibly is something we might see. This counting on strategy is something we might see." But what we didn't talk about that I think you're advocating for is: What are the moments within that that matter? It's almost like: What in the process of getting to this anticipated strategy is something that is useful or important that counts as one of those ingredients? So I want to run that past you and say, does that follow or am I missing something? Mike Steele: It does. And I think those two things go together in a really important way because as you're talking about that pivotal moment in student thinking, as they're coming to this new understanding, as they're grappling with that mathematical idea, and thinking about, "What are the implications if we leverage that moment right there to then ask more questions to connect different ways of student thinking as compared to waiting till it's written down?" Because when it's written down, that exciting moment of the new discovery has passed. And so then when we want them to come revisit—"Tell us what you were thinking when you did that."—they're having to rewind and go back and reenact that. If we have the ability to capture those neurons firing at full throttle in that moment of a new mathematical insight and then use that to build on as a teacher and to really get where we want to go with the lesson, I feel like we're doing the right thing by kids by trying to seize that moment, to leverage it. We always have time to write down what we think we learned later on at the end of the lesson. It's a great task for homework. And that's another thing I love about leaving some things unfinished with a task is, that's just a delightful homework assignment. And the kids love it because they don't feel like I've asked them to do anything new. (laughs) Just write down what you understood about this, and now we're codifying it kind of at a different place in the process. Mike Wallus: Well, OK, and that makes me think about something else. Because you've helped me recognize that I don't have to wait for a final solution in writing that's fleshed out in order to start a whole-group conversation. But I think what you're saying is, it changes the tone and maybe also the purpose and the impact of that conversation on students. Because if I have a task that I'm midway through and suddenly there's a conversation that helps create some understanding, some aha moments, if my task is unfinished and I had an aha, I probably really want to go back to that and see if I can apply that aha. And that's kind of cool to imagine like a classroom where you have a bunch of kids dying to go back and see if they can figure out how they can put that to use. Now you wouldn't always have to do that, but that strikes me as different than a consolidation conversation where it's kind of like, "Well, everything's finished. What have we learned?" Those are valuable. But I'm just really, I think in love with the possibility that a conversation that doesn't always wait until final solutions creates for learning. Mike Steele: And when I've seen this done effectively, there are these moments that happen. Mike, they're exactly what you're describing, is that there's an insight that comes up in the whole-class conversation, and you will see people going back to their paper or their tablet that they were doing their original work on and start writing. And we know oftentimes with kids, I remember so many times in my classroom where we're having this discussion, this important point comes up, and everybody's kind of frozen. And I'm like, "No, you should write that down. That's the important thing. Write that down." And when you see it happen organically, it's because something really catalyzed in insight that was important enough that they went back to that work and said, "Oh, I want to capture this." Mike Wallus: So, I'm wondering if there are habits of mind, habits in planning, or habits in practice that we could distill down. So, how would you unpack the things that a person might do if they're listening and they're like, "I want to do this today," or "I want to do this at my next planning."? Could you talk a little bit about what are the baby steps, so to speak, for a person? Mike Steele: Yeah, and I think the first one is really about getting into the mathematics and going deep with the mathematics in the task that you're hoping to teach. As somebody who is trained as a secondary math teacher, and early in my career, I was like, "Oh, I know what the math is. I don't need to spend the time on the math." I can't tell you how wrong I was about that. So anticipating those ways of thinking, thinking about where those challenges are, that sort of thing, is absolutely critically important to doing that work. And giving the time and space for that to happen. I mean, it was almost without fail. Every time I shorted myself on the time to think about the mathematics and just popped open my instructional resource and said, "Here we go. Class starts in 5 minutes. Let's get going on this," I'd bump into things that I was like, "Oh, I wish I had thought about that mathematical idea first." Or there'd be a question that would come up that I'd be totally unprepared to answer and I could have been prepared to answer. Now, we're not going to anticipate every way of thinking that students have or every question that they'll have, but I always find that if I've thought through it, I'm probably in a better position to give a meaningful answer to it or ask a good question back in response. And it also frees up my cognitive load to actually spend some time on those questions that I didn't expect rather than trying to make sense of everything as if it's the first time I'm seeing it. And then along with that, doing this as a group, we used to sit in our PLC sessions and start to solve tasks together and share our thinking about, "OK, what are the mathematical ideas that we're really trying to take apart here?" And there were always insights that didn't occur to me that would occur to somebody else that added to my own thinking. And now in an increasingly digitally connected age, we don't necessarily have to be in the same room with people to do that. We can do that at a distance and still be very effective. And then the last thing I'll talk about here in terms of getting started is: We are so good as teachers at sharing an interesting task that we found or that we used with our students with our colleagues. "Here's this thing I use in my class. It was great. You're a couple days behind me in the pacing. Maybe you can use this next Tuesday." What we I think are less good at is bringing back the outcomes of that and talking about that. "Here's what students did." I loved it when we had opportunities to gather a group of teachers in the PLC with student work from a task they did and talk about it and see: What did students make sense of? What were the questions that I asked that were helpful, or that maybe weren't helpful, in teaching that lesson. Because we'll share the task, but my goodness, the questions that we came up with to ask students in the moment, those are just as portable from one classroom to another. And we should be thinking about, just like we think about digital archives to share those tasks and those lesson plans—like sharing those questions, sharing that student work—those are the other legs of that stool that are important for really helping us do this work in a meaningful and collaborative way. Because if we don't talk about the outcomes of what students learned, the task could be great, it could be interesting, but so what? What's the important mathematical insights that kids took away from it? Mike Wallus: Yeah, I'm kind of in love with this notion that in addition to sharing tasks, sharing questions that really generated an impact in the classroom space or sharing moments of insight that led to something that jumped out. It's fascinating to think about taking those ideas and building them into a regular PLC process. It just has so much potential. Before we close the conversation, I wanted to ask you a question that I ask almost every guest: If someone wanted to learn more about the ideas that you've shared today, what are some of the resources you'd recommend? Mike Steele: Well, I've talked quite a bit about the work of the 5 Practices for Orchestrating Productive Discussions [https://www.corwin.com/books/5-practices-262956] and that series of books that have been written over the past 15 years on that—the resources that are available online for that, I think, would be a great place to start. I've only scratched the surface at taking you through those five practices—which are actually six practices, because early on we realized that attention to the task we select and the goals for that task is the important "practice zero." In fact, it was a teacher that pointed that out to Peg Smith. And that's the lovely thing. So the reason I've stayed in touch with and helped to develop this work over the years is because when we see teachers taking it up, not only is it meaningful, but the feedback we get from teachers then shapes the next things that we do with it. So there's the original 5 practices book that kind of presents the model, shows some examples of tasks and how you go through the model. But then in 2019 and 2020, we published a series called The 5 Practices in Practice that, there's a book for each grade band—elementary [https://www.corwin.com/books/5-practices-in-practice-260531], middle [https://www.corwin.com/books/the-5-practices-on-practice-260532], and high school [https://www.corwin.com/books/mathematics-discussions-hs-260533]. But those were the ones that really aggregated the challenges that we heard from teachers over 10 years of doing this work and started to address those challenges. How do you overcome those things? We also, for each of those books, there's brand-new original video that we took in urban classrooms that illustrated teachers working really effectively with the five practices. I was able to be in the room when we filmed all of the high school classrooms in Milwaukee, Wisconsin, and it was just amazing to see that work. And then the last piece that I'll suggest to that, which is a book that came out relatively recently in that series. There is a Coaching the 5 Practices [https://www.corwin.com/books/coaching-the-5-practices-287127] book. So if you are a coach, instructional leader who's looking to support a team and a PLC in doing exactly this sort of work that we've been talking about, the Coaching the 5 Practices book is an incredible resource for thinking about how you can structure that work. Mike Wallus: OK. I have to also ask you, can you give a shout out to the article that you recently wrote and published as well, the title and where people could find it? Mike Steele: Absolutely. Yes. The article is called "Pacing a Discourse-Rich Lesson: When to Move On," [http://pubs.nctm.org/view/journals/mtlt/118/11/article-p822.xml] and I authored it alongside an elementary and middle school teacher who provided a reflection on it. It comes from the classroom of a high school teacher, Michael Moore, in Milwaukee, who we filmed for the [5 Practices in Practice] high school book. So I drew from his classroom. And then Kara Benson in Zionsville Community Schools right here in Zionsville, Indiana. And Kelly Agnew who teaches in Muncie Community Schools, which is where Ball State [University] is located. Each provided a reflection from an elementary and middle school standpoint about the ideas in the article. It was published in NCTM'S practitioner journal, Mathematics Teacher: Learning and Teaching PK-12, in the Volume 118, Issue 11, from November of 2025. Mike Wallus: That's fantastic. And for listeners, just so you know, we're going to put a link to all of the resources that Mike shared. I think this is probably a good place to stop, Mike. I suspect we could talk for much longer. I just want to thank you, though, for taking the time to join the podcast. It has been an absolute pleasure chatting with you. Mike Steele: The pleasure has been all mine. As you can tell, I love talking about these ideas, and I was so glad to have the opportunity to share a little bit of this with the audience. Mike Wallus: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2026 The Math Learning Center | www.mathlearningcenter.org [http://www.mathlearningcenter.org]

5 de mar de 202634 min

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

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