Rounding Up

The Math Learning Center

Welcome to "Rounding Up" with the Math Learning Center. These conversations focus on topics that are important to Bridges teachers, administrators and anyone interested in Bridges in Mathematics. Hosted by Mike Wallus, VP of Educator Support at MLC.

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

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

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

    12 мин.

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

    Amy Hackenberg, Understanding Units Coordination ROUNDING UP: SEASON 4 | EPISODE 11 Units coordination describes the ways students understand the organization of units (or a unit structure) when approaching problem-solving situations—and how students' understanding influences their problem-solving strategies. In this episode, we're talking with Amy Hackenberg from the University of Indiana about how educators can recognize and support students at different stages of units coordination. BIOGRAPHY Dr. Amy Hackenberg taught mathematics to middle and high school students for nine years in Los Angeles and Chicago, and is currently a professor of mathematics education at Indiana University-Bloomington. She conducts research on how students construct fractions knowledge and algebraic reasoning. She is the proud coauthor of the Math Recovery series book, Developing Fractions Knowledge. RESOURCES Integrow Numeracy Solutions Developing Fractions Knowledge by Amy J. Hackenberg, Anderson Norton, and Robert J. Wright TRANSCRIPT Mike Wallus: Welcome to the podcast, Amy. I'm excited to be chatting with you today about units coordination. Amy Hackenberg: Well, thank you for having me. I'm very excited to be here, Mike, and to talk with you. Mike: Fantastic. So we've had previous guests come on the podcast and they've talked about the importance of unitizing, but for guests who haven't heard those episodes, I'm wondering if we could start by offering a definition for unitizing, but then follow that up with an explanation of what units coordination is. Amy: Yeah, sure. So unitizing basically means to take a segment of experience as one thing, which we do all the time in order to even just relate to each other and tell stories about our day. I think of my morning as a segment of experience and can tell someone else about it. And we also do it mathematically when we construct number. And it's a very long process, but children began by compounding sensory experiences like sounds and rhythms as well as visual and tactical experiences of objects into experiential units—experiential segments of experience that they can think about, like hearing bells ringing could be an impetus to take a single bong as a unit. And later, people construct units from what they imagine and even later on, abstract units that aren't tied to any particular sensory material. It's again, a long process, but once we start to do that, we construct arithmetical units, which we can think of as discrete 1s. So, it all starts with unitizing segments of experience to create arithmetical items that we might count with whole numbers. Mike: What's really interesting about that is this notion of unitizing grows out of our lived experiences in a way that I think I hadn't thought about—this notion that a unit of experience might be something like a morning or lunchtime. That's a fascinating way to think about even before we get to, say, composing sets of 10 into a unit, that these notions of a unit [exist] in our daily lives. Amy: Yeah, and we make them out of our daily lives. That's how we make units. And what you said about a ten is also important because as we progress onward, we do take more than 1 one as a unit—like thinking of 4 flowers in a row in a garden as a single unit, as both 1 unit and as 4 little flowers—means it has a dual meaning, at least; we call it a composite unit at that point. That's a common term for that. So that's another example of unitizing that is of interest to teachers. Mike: Well, I'm excited to shift and talk about units coordination. How would you describe that? Amy: Yeah, so units coordination is a way for teachers and researchers to understand how children create units and organize units to interpret problem situations and to solve problems. So it originated in understanding how children construct whole number multiplication and division, but it has since expanded from just that to be thinking more broadly about units and structuring units and organizing and creating more units and how people do that in solving problems. Mike: Before we dig into the fine-grain details of students' thinking, I wonder if you can explain the role that units coordination plays in students' journey through elementary mathematics and maybe how that matters in middle school and beyond middle school. Amy: So that's where a lot of the research is right now, especially at the middle school level and starting to move into high school. But units coordination was originally about trying to understand how elementary school children construct whole number multiplication and division, but it's also found to greatly influence elementary school children's understanding of fractions, decimals, measurement and on into middle school students' understanding of those same ideas and topics: fractions ratios and proportional reasoning, rational numbers, writing and transforming algebraic equations, even combinatorial reasoning. So there's a lot of ways in which units coordination influences different aspects of children's thinking and is relevant in lots of different domains in the curriculum. Mike: Part of what's interesting for me is that I don't think I'm alone in saying that this big idea around units coordination sounds really new to me. It's not language that I learned in my preservice work[, nor] in my practice. So I think what's coming together for me is there's a larger set of ideas that flow through elementary school and into middle school and high school mathematics. And it's helpful to hear you talk about that, from the youngest children who are thinking about the notion of units in their daily lives to the way that this notion of units and units coordination continues to play through elementary school into middle school and high school. Amy: Yeah, it's nice that you're noticing that because I do think that's something that's a strength of units coordination in [that] it can be this unifying idea, although there's lots of variation and lots of variation in what you see with elementary students versus middle school students versus high school students versus even college students. Some of the research is on college students' unit coordination these days, but it is an interesting thread that can be helpful to think about in that way. Mike: OK. With that in mind, let's introduce a context for units coordination and talk a little bit about the stages of student thinking. Amy: Yeah. So, one way to understand some differences in how children up through, say, middle school students might coordinate units and engage in units coordination is to think about a problem and describe how solving it might happen.  Here's a garden problem: "Amaya is planting 4 pansies in a row. She plants 15 rows. How many pansies has she planted?" There are three stages of units coordination, broadly speaking—we've begun to understand more about the nuances there. But a stage refers to a set of ways of thinking that tend to fit together in how students understand and solve problems with whole numbers, fractions, quantities, and multiplicative relationships. It's sort of about a nexus of ideas, and—that we tend to see coming together and students don't usually think in a way that's characteristic of a different stage until they've made a significant change in their thinking, like a big reorganization happens for them to move from one stage to the next. So students at stage 1 of units coordination are primarily in a 1s world and their number sequence is not multiplicative. That's going to be hard to imagine. But they can take a group of 1s as one thing. So, they can make a composite unit and that means in the garden problem, they can take a row of pansies as 1 row as well as 4 little ones, and they can continue to do that over and over again. And so they can amass rows of 4 pansies and keep going. And what it usually looks like for them to solve the problem is they'll count by 1s after any known skip-counting patterns. So, in this case they might be like, "Oh, I know 4 and 8; that's two rows. 9, 10, 11, 12; that's three rows." Often using fingers or something to keep track, or in some way to keep track, and continuing to go up and get all the way, barring counting errors, to 60 pansies. And so for them the result, 60 pansies, is a composite unit. It's a unit of 60 units, but they don't maintain the structure that we see at all of the units of 60 as 15 fours. That's not something—even though they did track it in their thinking—they don't maintain that once they get to the 60, it's really just only a big composite unit of 60. So their view of the result is very different than an adult view might be.  So, the students at stage 1 can solve division problems, which means if they give some number of pansies and they're supposed to make rows of 4, they can definitely do it, they can solve that. But they don't think of multiplication and division as inverses. So let me say what I mean by that. If they had this problem next, so: "Amaya's mom gave her 28 pansies. How many rows of 4 can she make?" A student at stage 1 could solve that problem, and they would be able to track 4s over and over again and figure out that they got to 7 fours once they get to 28. But then if immediately afterwards a teacher said, "Well, so, how many pansies are there in 7 rows of 4?," the student at stage 1 would start over and solve the problem from the beginning. They wouldn't think that they had already solved it. And that's one telling sign of a student operating at stage 1. And the reason is that the mental actions they engage in to do the segmenting or the tracking off of the 4s and the 28 pansies are really different to them than what they use then the ways of thinking they use to create the 7 rows of 4 and make the 28 that way. And so they don't recognize them as similar, so they feel like they have to engage in new problem solving to solve that problem. So, to get back to the g

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

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

    What Counts as Counting? with Dr. Christopher Danielson ROUNDING UP: SEASON 4 | EPISODE 10 What counts as counting? The question may sound simple, but take a moment to think about how you would answer. After all, we count all kinds of things: physical quantities, increments of time, lengths, money, as well as fractions and decimals.  In this episode, we'll talk with Christopher Danielson about what counts as counting and how our definition might shape the way we engage with our students. BIOGRAPHY Christopher Danielson started teaching in 1994 in the Saint Paul (MN) Public Schools. He  earned his PhD in mathematics education from Michigan State University in 2005 and taught at the college level for 10 years after that. Christopher is the author of Which One Doesn't Belong?, How Many?, and How Did You Count? Christopher also founded Math On-A-Stick, a large-scale family math playspace at the Minnesota State Fair. RESOURCES How Did You Count? A Picture Book by Christopher Danielson How Many?: A Counting Book by Christopher Danielson Following Learning blog by Simon Gregg Connecting Mathematical Ideas by Jo Boaler and Cathleen Humphreys  TRANSCRIPT Mike Wallus: Before we start today's episode, I'd like to offer a bit of context to our listeners. This is the second half of a conversation that we originally had with Christopher Danielson back in the fall of 2025. At that time, we were talking about [the instructional routine] Which one doesn't belong? This second half of the conversation focuses deeply on the question "What counts as counting?" I hope you'll enjoy the conversation as much as I did.  Well, welcome to the podcast, Christopher. I'm excited to be talking with you today. Christopher Danielson: Thank you for the invitation. Delightful to be invited. Mike: So I'd like to talk a little bit about your recent work, the book How Did You Count?[: A Picture Book] In it, you touch on what seems like a really important question, which is: "What is counting?" Would you care to share how your definition of counting has evolved over time? Christopher: Yeah. So the previous book to How Did You Count? was called How Many?[: A Counting Book], and it was about units. So the conversation that the book encourages would come from children and adults all looking at the same picture, but maybe counting different things. So "how many?" was sort of an ill-formed question; you can't answer that until you've decided what to count.  So for example, on the first page, the first photograph is a pair of shoes, Doc Marten shoes, sitting in a shoebox on a floor. And children will count the shoes. They'll count the number of pairs of shoes. They'll count the shoelaces. They'll count the number of little silver holes that the shoelaces go through, which are called eyelets. And so the conversation there came from there being lots of different things to count. If you look at it, if I look at it, if we have a sufficiently large group of learners together having a conversation, there's almost always going to be somebody who notices some new thing that they could count, some new way of describing the thing that they're counting. One of the things that I noticed in those conversations with children—I noticed it again and again and again—was a particular kind of interaction. And so we're going to get now to "What does it mean to count?" and how my view of that has changed. The eyelets, there are five eyelets on each side of each shoe. Two little flaps that come over, each has five of those little silver rings. Super compelling for kids to count them. Most of the things on that page, there's not really an interesting answer to "How did you count them?" Shoelaces, they're either two or four; it's obvious how you counted them. But the eyelets, there's often an interesting conversation to be had there. So if a kid would say, "I counted 20 of those little silver holes," I would say, "Fabulous. How do you know there are 20?" And they would say, "I counted." In my mind, that was like an evasion. They felt like what they had been called on to do by this strange man who's just come into our classroom and seems friendly enough, what they had been called on to do was say a number and a unit. And they said they had 20 silver things. We're done now. And so by my asking them, "How do you know? " And they say, "I counted." It felt to me like an evasion because I counted as being 1, 2, 3, 4, 5, all the way up to 20. And they didn't really want to tell me about anything more complicated than that. It was just sort of an obvious "I counted." So in order to counter what I felt like was an evasion, I would say, "Oh, so you said to yourself, 1, 2, 3, and then blah, blah, blah, 18, 19, 20." And they'd be like, "No, there were 10 on each shoe." Or, "No, there's 5 on each side." Or rarely there would be the kid who would see there were 4 bottom eyelets across the 4 flaps on the 2 shoes and then another row and another row. Some kids would say there's 5 rows of 4 of them, which are all fabulous answers. But I thought, initially, that that didn't count as counting. After hearing it enough times, I started to wonder, "Is it possible that kids think 5 rows of 4, 4 groups of 5, 2 groups of 10, counted by 2s and 1, 2, 3, 4, all the way up to 19 and 20—is it possible that kids conceive of all of those things as ways of counting, that all of those are encapsulated under counting?" And so I began because of the ways children were responding to me to think differently about what it means to count.  So when I first started working on this next book, How Did You Count?, I wanted it to be focused on that. The focus was deliberately going to be on the ways that you count. We're all going to agree that we're counting tangerines; we're all going to agree that we're counting eggs, but the conversation is going to come because there are rich ways that these things are arranged, rich relationships that are embedded inside of the photographs. And what I found was, when I would go on Twitter and throw out a picture of some tangerines and ask how people counted, and I would get back the kind of thing that was how I had previously seen counting. So I would get back from some people, "There are 12." I'd ask, "How did you count?" And they'd say, "I didn't. I multiplied 3 times 4." "I didn't. I multiplied 2 times 6."  But then, on reflection through my own mathematical training, I know that there's a whole field of mathematics called combinatorics. Which if you asked a mathematician, "What is combinatorics?," 9 times out of 10, the answer is going to be, "It's the mathematics of counting." And it's not mathematicians sitting around going "1, 2, 3, 4" or "2, 4, 6, 8." It's looking for structures and ways to count the number of possibilities there are, the number of—if we're thinking about calculating probabilities of winning the lottery, somebody's got to know what the probabilities are of choosing winning numbers, of choosing five out of six winning numbers. And the field of combinatorics is what does that. It counts possibilities.  So I know that mathematicians and kindergartners—this is what I've learned in both my graduate education and in my postgraduate education working with kindergartners—is that they both think about counting in this rich way. It's any work that you do to know how many there are. And that might be one by one; it might be skip-counting; it might be multiplication; it might be using some other kind of structure. Mike: I think that's really interesting because there was a point in time where I saw counting as a fairly rote process, right? Where I didn't understand that there were all of these elements of counting, meaning one-to-one correspondence and quantity versus being able to just say the rote count out loud. And so one way that I think counting and its meaning have expanded for me is to kind of understand some of those pieces. But the thing that occurs to me as I hear you talk is that I think one of the things that I've done at different points, and I wonder if people do, is say, "That's all fine and good, but counting is counting." And then we've suddenly shifted and we're doing something called addition or multiplication. And this is really interesting because it feels like you're drawing a much clearer connection between those critical, emergent ideas around counting and these other things we do to try to figure out the answer to how many or how did you count. Tell me what you think about that. Christopher: Yeah. So this for me is the project, right? This book is an instantiation of this larger project, a way of viewing the world of mathematics through the lens of what it means to learn it. And I would describe that larger project through some imagery and appealing to teachers' ideas about what it means to have a classroom conversation.  For me, learning is characterized by increasing sophistication, increasing expertise with whatever it is that I'm studying. And so when I put several different triangular arrangements of things—in the book, there's a triangular arrangement of bowling pins, which lots of kids know from having bowled in their lives and other kids don't have any experiences with them, but the image is rich and vivid and they're able to do that counting. And then later on, there's a triangular arrangement of what turned out to be very bland, gooey, and nasty, but beautiful to photograph: pink pudding cups. Later on, there are two triangles of eggs. And so what I'm asking of kids—I'm always imagining a child and a parent sitting on a couch reading these books together, but also building them for classrooms. Any of this could be like a thing that happens at home, a thing that happens for a kid individually or a classroom full of children led by a teacher. Thinking about the second picture of the pudding cups, my hope and expectation is that at least some children will say, "OK, there are 6 rows

    22 мин.
  4. Season 4 | Episode 9 - Dr. Todd Hinnenkamp, Enacting Talk Moves with Intention

    8 ЯНВ.

    Season 4 | Episode 9 - Dr. Todd Hinnenkamp, Enacting Talk Moves with Intention

    Dr. Todd Hinnenkamp, Enacting Talk Moves with Intention ROUNDING UP: SEASON 4 | EPISODE 9 All students deserve a classroom rich in meaningful mathematical discourse. But what are the talk moves educators can use to bring this goal to life in their classrooms?  Today, we're talking about this question with Todd Hinnenkamp from the North Kansas City Schools. Whether talk moves are new to you or already a part of your practice, this episode will deepen your understanding of the ways they impact your classroom community.  BIOGRAPHY Dr. Todd Hinnenkamp is the instructional coordinator for mathematics for the North Kansas City Schools.  RESOURCES Talk Moves with Intention for Math Learning Center Standards for Mathematical Practice by William McCallum  5 Practices for Orchestrating Productive Mathematics Discussions by Margaret (Peg) Smith and Mary Kay Stein  TRANSCRIPT Mike Wallus: Before we begin, I'd like to offer a quick note to listeners. During this episode, we'll be referencing a series of talk moves throughout the conversation. You can find a link to these talk moves included in the show notes for this episode. Welcome to the podcast, Todd. I'm really excited to be chatting with you today. Todd Hinnenkamp: I'm excited to be here with you, Mike. Talk through some things. Mike: Great. So I've heard you present on using talk moves with intention, and one of the things that you shared at the start was the idea that talk moves advance three aspects of teaching and learning: a productive classroom community, student agency, and students' mathematical practice. So as a starting point, can you unpack that statement for listeners? Todd: Sure. I think all talk moves with intention contribute to advancing all three of those, maybe some more than others. But all can be impactful in this endeavor, and I really think that identifying them or understanding them well upfront is super important.  So if you unpack "productive community" first, I think about the word "productive" as an individual word. In different situations, it means a quality or a power of producing, bringing about results, benefits, those types of things. And then if you pair that word "community" alongside, I think about the word "community" as a unified body of individuals, an interacting population. I even like to think about it as joint ownership or participation. When that's present, that's a pretty big deal. So I like to think about those two concepts individually and then also together. So when you think about the "productivity" word and the "community" word and then pairing them well together, is super important. And I think about student agency. Specifically the word "agency" means something pretty powerful that I think we need to have in mind. When you think about it in a way of, like, having the capacity or the condition or state of acting or even exerting some power in your life. I think about students being active in the learning process. I think about engagement and motivation and them owning the learning. I think oftentimes we see that because they feel like they have the capacity to do that and have that agency. So I think about that, that being a thing that we would want in every single classroom so they can be productive contributors later in life as well. So I feel like sometimes there's too many students in classrooms today with underdeveloped agencies. So I think if we can go after agency, that's pretty powerful as well. And when you think about students' math practice, super important habits of what we want to develop in students. I mean, we're fortunate to have some clarity around those things, those practices, thanks to the work of Dr. [William] McCallum and his team more than a decade ago when they provided us the standards for mathematical practice. But if you think about the word "practice" alone, it's interesting. I've done some research on this. I think the transitive verb meaning is to do or perform often, customarily or maybe habitually. The transitive verb meaning is to pursue something actively. Or if you think about it with a noun, it's just a usual way of doing something or condition of being proficient through a systematic exercise. So I think all those things are, if we can get kids to develop their math practice in a way it becomes habitual and is really strong within them, it's pretty powerful. So I do think it's important that we start with that. We can't glaze over these three concepts because I think that right now, if you can tie some intentional talk moves to them, I think that it can be a pretty powerful lever to student understanding. Mike: Yeah. You have me thinking about a couple things. One of the first things that jumped out as I was listening to you talk is there's the "what," which are the talk moves, but you're really exciting the stage with the "why." Why do we want to do these things? And what I'd like to do is take each one of them in turn. So can we first talk about some of the moves that set up productive community for learners? Todd: Yeah. I think all the moves that are on my mind contribute, but there's probably a couple that I think go after productive community even more so than others. And I would say the "student restates" move, that first move where you're expecting students to repeat or restate in their own words what another student shared, promotes some really special things. I think first it communicates to everyone in the room that "We're going to talk about math in here. We're going to listen to and respectfully consider what others say and think." It really upholds my expectation as an educator that we're going to interact with and understand the mathematical thinking that's present so that student restates is a great one to get going.  And I would also offer the "think, turn, and learn" move is a highly impactful one as well. The general premise here is that you're offering time upfront. Always starting with "think," you're offering time upfront. And what that should be communicating to students is that "You have something to offer. I'm providing you time to think about it, to organize it, so then you're more apt to share it with either your partner or the community." It really increases the likelihood that kids have something to contribute. And as you literally turn your body and learn from each other—and those words are intentional, "turn" and "learn"—it opens the door to share, to expand your thinking, to then refine what you're thinking and build to develop both speaking and listening skills that help the community bond become stronger. So in the end it says, "I have something to offer here. I'm valued through my interactions." And I feel like that there's something that comes out of that process for kids. Mike: You talked about the practice of "think, turn, learn." And one of the things that jumps out is "think." Because we've often used language like "turn and talk," and that's in there with "turn and learn," but "think" feels really important. I wonder if you could say more about why "think"? Let's just make it explicit. Why "think"? Todd: Sure. No, and I'm not trying to throw shade at "turn and talks" or anything like that, but I do think when we have intention with our moves, they're super impactful relative to other opportunities where maybe we're just not getting the most out of it. So that idea of offering time or providing or ensuring time for kids to think upfront—and depending on the situation, that can be 10 seconds, that can be 30 seconds—where you feel like students have had a chance to internalize what's going on [and] think about what they would say, it puts them in an entirely different mode to build a share with somebody else. I'm often in classrooms, and if we don't provide that think time, you see kids turn and talk to each other, and the first part is them still trying to figure out what should be said. And it just doesn't seem like it's as impactful or as productive during that time as it could be without that "think" first. Mike: Yeah, absolutely.  I want to go back to something you said earlier too, when you were describing the value that comes out of restating or rephrasing, having a student do that with another student's thinking. One of the things that struck me is there were points in time when you were talking about that and you were talking about the value for an individual student who's in that spot.  Todd: Mm. Mike: But I also heard you come back to it and say, "There's something in this for the group, for the community as well." And I wonder if you could unpack a little bit: What's in it for the kid when they go through that restating another student's [idea], or having their [own] idea restated, and then what's in it for the community? Todd: Sure. Well, let's start with the individual, Mike. And I think that with what we know about learning and how much more deeply we learn when we internalize something and reflect on it and actually link it to our past learning and think about what it means to us, is probably the most important thing that comes out of that. So the student that's restating what another student says, they really have to think about what that student said and then internalize it and make sense of it in a way where they can actually say it out to the community again. That's a big deal! So to talk about the impact on the community in that mode, Mike, when you get one or two [ideas], and maybe you ask for a couple more, you now have student thinking in four different forms out in the community rather than, say, one student sharing something and a teacher restating it and moving on. And I just love how those moves together can cause the thinking to linger in the classroom longer for kids. Often when I'm in classrooms, the kids actually learn it more when somebody else says it rather than me. And it kind of ties to that where, like, they just need to hear other kids thinking and start to

    27 мин.
  5. Season 4 | Episode 8 – Janet Walkoe & Margaret Walton, Exploring the Seeds of Algebraic Thinking

    18.12.2025

    Season 4 | Episode 8 – Janet Walkoe & Margaret Walton, Exploring the Seeds of Algebraic Thinking

    Janet Walkoe & Margaret Walton, Exploring the Seeds of Algebraic Thinking ROUNDING UP: SEASON 4 | EPISODE 8 Algebraic thinking is defined as the ability to use symbols, variables, and mathematical operations to represent and solve problems. This type of reasoning is crucial for a range of disciplines.  In this episode, we're talking with Janet Walkoe and Margaret Walton about the seeds of algebraic thinking found in our students' lived experiences and the ways we can draw on them to support student learning.  BIOGRAPHIES Margaret Walton joined Towson University's Department of Mathematics in 2024. She teaches mathematics methods courses to undergraduate preservice teachers and courses about teacher professional development to education graduate students. Her research interests include teacher educator learning and professional development, teacher learning and professional development, and facilitator and teacher noticing. Janet Walkoe is an associate professor in the College of Education at the University of Maryland. Janet's research interests include teacher noticing and teacher responsiveness in the mathematics classroom. She is interested in how teachers attend to and make sense of student thinking and other student resources, including but not limited to student dispositions and students' ways of communicating mathematics. RESOURCES "Seeds of Algebraic Thinking: a Knowledge in Pieces Perspective on the Development of Algebraic Thinking" "Seeds of Algebraic Thinking: Towards a Research Agenda" NOTICE Lab  "Leveraging Early Algebraic Experiences"  TRANSCRIPT Mike Wallus: Hello, Janet and Margaret, thank you so much for joining us. I'm really excited to talk with you both about the seeds of algebraic thinking. Janet Walkoe: Thanks for having us. We're excited to be here.  Margaret Walton: Yeah, thanks so much. Mike: So for listeners, without prayer knowledge, I'm wondering how you would describe the seeds of algebraic thinking. Janet: OK. For a little context, more than a decade ago, my good friend and colleague, [Mariana] Levin—she's at Western Michigan University—she and I used to talk about all of the algebraic thinking we saw our children doing when they were toddlers—this is maybe 10 or more years ago—in their play, and just watching them act in the world. And we started keeping a list of these things we saw. And it grew and grew, and finally we decided to write about this in our 2020 FLM article ["Seeds of Algebraic Thinking: Towards a Research Agenda" in For the Learning of Mathematics] that introduced the seeds of algebraic thinking idea. Since they were still toddlers, they weren't actually expressing full algebraic conceptions, but they were displaying bits of algebraic thinking that we called "seeds." And so this idea, these small conceptual resources, grows out of the knowledge and pieces perspective on learning that came out of Berkeley in the nineties, led by Andy diSessa. And generally that's the perspective that knowledge is made up of small cognitive bits rather than larger concepts. So if we're thinking of addition, rather than thinking of it as leveled, maybe at the first level there's knowing how to count and add two groups of numbers. And then maybe at another level we add two negative numbers, and then at another level we could add positives and negatives. So that might be a stage-based way of thinking about it.  And instead, if we think about this in terms of little bits of resources that students bring, the idea of combining bunches of things—the idea of like entities or nonlike entities, opposites, positives and negatives, the idea of opposites canceling—all those kinds of things and other such resources to think about addition. It's that perspective that we're going with. And it's not like we master one level and move on to the next. It's more that these pieces are here, available to us. We come to a situation with these resources and call upon them and connect them as it comes up in the context. Mike: I think that feels really intuitive, particularly for anyone who's taught young children. That really brings me back to the days when I was teaching kindergartners and first graders.  I want to ask you about something else. You all mentioned several things like this notion of "do, undo" or "closing in" or the idea of "in-betweenness" while we were preparing for this interview. And I'm wondering if you could describe what these things mean in some detail for our audience, and then maybe connect them back with this notion of the seeds of algebraic thinking. Margaret: Yeah, sure. So we would say that these are different seeds of algebraic thinking that kids might activate as they learn math and then also learn more formal algebra. So the first seed, the doing and undoing that you mentioned, is really completing some sort of action or process and then reversing it.  So an example might be when a toddler stacks blocks or cups. I have lots of nieces and nephews or friends' kids who I've seen do this often—all the time, really—when they'll maybe make towers of blocks, stack them up one by one and then sort of unstack them, right? So later this experience might apply to learning about functions, for example, as students plug in values as inputs, that's kind of the doing part, but also solve functions at certain outputs to find the input. So that's kind of one example there.  And then you also talked about closing in and in-betweenness, which might both be related to intervals. So closing in is a seed where it's sort of related to getting closer and closer to a desired value. And then in formal algebra, and maybe math leading up to formal algebra, the seed might be activated when students work with inequalities maybe, or maybe ordering fractions.  And then the last seed that you mentioned there, in-betweenness, is the idea of being between two things. For example, kids might have experiences with the story of Goldilocks and the Three Bears, and the porridge being too hot, too cold, or just right. So that "just right" is in-between. So these seats might relate to inequalities and the idea that solutions of math problems might be a range of values and not just one. Mike: So part of what's so exciting about this conversation is that the seeds of algebraic thinking really can emerge from children's lived experience, meaning kids are coming with informal prior knowledge that we can access. And I'm wondering if you can describe some examples of children's play, or even everyday tasks, that cultivate these seeds of algebraic thinking. Janet: That's great. So when I think back to the early days when we were thinking about these ideas, one example stands out in my head. I was going to the grocery store with my daughter who was about three at the time, and she just did not like the grocery store at all. And when we were in the car, I told her, "Oh, don't worry, we're just going in for a short bit of time, just a second." And she sat in the back and said, "Oh, like the capital letter A." I remember being blown away thinking about all that came together for her to think about that image, just the relationship between time and distance, the amount of time highlighting the instantaneous nature of the time we'd actually be in the store, all kinds of things.  And I think in terms of play examples, there were so many. When she was little, she was gifted a play doctor kit. So it was a plastic kit that had a stethoscope and a blood pressure monitor, all these old-school tools. And she would play doctor with her stuffed animals. And she knew that any one of her stuffed animals could be the patient, but it probably wouldn't be a cup. So she had this idea that these could be candidates for patients, and it was this—but only certain things. We refer to this concept as "replacement," and it's this idea that you can replace whatever this blank box is with any number of things, but maybe those things are limited and maybe that idea comes into play when thinking about variables in formal algebra. Margaret: A couple of other examples just from the seeds that you asked about in the previous question. One might be if you're talking about closing in, games like when kids play things like "you're getting warmer" or "you're getting colder" when they're trying to find a hidden object or you're closing in when tuning an instrument, maybe like a guitar or a violin.  And then for in-betweeness, we talked about Goldilocks, but it could be something as simple as, "I'm sitting in between my two parents" or measuring different heights and there's someone who's very tall and someone who's very short, but then there are a bunch of people who also fall in between. So those are some other examples. Mike: You're making me wonder about some of these ideas, these concepts, these habits of mind that these seeds grow into during children's elementary learning experiences. Can we talk about that a bit? Janet: Sure. Thank you for that question.  So we think of seeds as a little more general. So rather than a particular seed growing into something or being destined for something, it's more that a seed becomes activated more in a particular context and connections with other seeds get strengthened. So for example, the idea of like or nonlike terms with the positive and negative numbers. Like or nonlike or opposites can come up in so many different contexts. And that's one seed that gets evoked when thinking potentially when thinking about addition. So rather than a seed being planted and growing into things, it's more like there are these seeds, these resources that children collect as they act on the world and experience things. And in particular contexts, certain seeds are evoked and then connected. And then in other contexts, as the context becomes more familiar, maybe they're evoked more often and connected more strongly. And then that becomes something that's connected with tha

    18 мин.
  6. Season 4 | Episode 7 - Tutita Casa, Anna Strauss, Jenna Waggoner & Mhret Wondmagegne, Developing Student Agency: The Strategy Showcase

    04.12.2025

    Season 4 | Episode 7 - Tutita Casa, Anna Strauss, Jenna Waggoner & Mhret Wondmagegne, Developing Student Agency: The Strategy Showcase

    Tutita Casa, Anna Strauss, Jenna Waggoner & Mhret Wondmagegne, Developing Student Agency: The Strategy Showcase ROUNDING UP: SEASON 4 | EPISODE 7 When students aren't sure how to approach a problem, many of them default to asking the teacher for help. This tendency is one of the central challenges of teaching: walking the fine line between offering support and inadvertently cultivating dependence.  In this episode, we're talking with a team of educators about a practice called the strategy showcase, designed to foster collaboration and help students engage with their peers' ideas.  BIOGRAPHIES Tutita Casa is an associate professor of elementary mathematics education at the Neag School of Education at the University of Connecticut. Mhret Wondmagegne, Anna Strauss, and Jenna Waggoner are all recent graduates of the University of Connecticut School of Education and early career elementary educators who recently completed their first years of teaching. RESOURCE National Council of Teachers of Mathematics  TRANSCRIPT Mike Wallus: Well, we have a full show today and I want to welcome all of our guests. So Anna, Mhret, Jenna, Tutita, welcome to the podcast. I'm really excited to be talking with you all about the strategy showcase. Jenna Waggoner: Thank you.  Tutita Casa: It's our pleasure.  Anna Strauss: Thanks.  Mhret Wondmagegne: Thank you. Mike: So for listeners who've not read your article, Anna, could you briefly describe a strategy showcase? So what is it and what could it look like in an elementary classroom? Anna: So the main idea of the strategy showcase is to have students' work displayed either on a bulletin board—I know Mhret and Jenna, some of them use posters or whiteboards. It's a place where students can display work that they've either started or that they've completed, and to become a resource for other students to use. It has different strategies that either students identified or you identified that serves as a place for students to go and reference if they need help on a problem or they're stuck, and it's just a good way to have student work up in the classroom and give students confidence to have their work be used as a resource for others. Mike: That was really helpful. I have a picture in my mind of what you're talking about, and I think for a lot of educators that's a really important starting point.  Something that really stood out for me in what you said just now, but even in our preparation for the interview, is the idea that this strategy showcase grew out of a common problem of practice that you all and many teachers face. And I'm wondering if we can explore that a little bit. So Tutita, I'm wondering if you could talk about what Anna and Jenna and Mhret were seeing and maybe set the stage for the problem of practice that they were working on and the things that may have led into the design of the strategy showcase. Tutita: Yeah. I had the pleasure of teaching my coauthors when they were master's students, and a lot of what we talk about in our teacher prep program is how can we get our students to express their own reasoning? And that's been a problem of practice for decades now. The National Council of Teachers of Mathematics has led that work. And to me, [what] I see is that idea of letting go and really being curious about where students are coming from. So that reasoning is really theirs. So the question is what can teachers do? And I think at the core of that is really trying to find out what might be limiting students in that work. And so Anna, Jenna, and Mhret, one of the issues that they kept bringing back to our university classroom is just being bothered by the fact that their students across the elementary grades were just lacking the confidence, and they knew that their students were more than capable. Mike: Jenna, I wonder if you could talk a little bit about, what did that actually look like? I'm trying to imagine what that lack of confidence translated into. What you were seeing potentially or what you and Anna and Mhret were seeing in classrooms that led you to this work. Jenna: Yeah, I know definitely we were reflecting, we were all in upper elementary, but we were also across grade levels anywhere from fourth to fifth grade all the way to sixth and seventh. And across all of those places, when we would give students especially a word problem or something that didn't feel like it had one definite answer or one way to solve it or something that could be more open-ended, we a lot of times saw students either looking to teachers. "I'm not sure what to do. Can you help me?" Or just sitting there looking at the problem and not even approaching it or putting something on their paper, or trying to think, "What do I know?" A lot of times if they didn't feel like there was one concrete approach to start the problem, they would shut down and feel like they weren't doing what they were supposed to or they didn't know what the right way to solve it was. And then that felt like kind of a halting thing to them. So we would see a lot of hesitancy and not that courage to just kind of be productively struggling. They wanted to either feel like there was something to do or they would kind of wait for teacher guidance on what to do. Mike: So we're doing this interview and I can see Jenna and the audience who's listening, obviously Jenna, they can't see you, but when you said "the right way," you used a set of air quotes around that. And I'm wondering if you or Anna or Mhret would like to talk about this notion of the right way and how when students imagined there was a right way, that had an effect on what you saw in the classroom. Jenna: I think it can be definitely, even if you're working on a concept like multiplication or division, whatever they've been currently learning, depending on how they're presented instruction, if they're shown one way how to do something but they don't understand it, they feel like that's how they're supposed to understand to solve the problem. But if it doesn't make sense for them or they can't see how it connects to the problem and the overall concept, if they don't understand the concept for multiplication, but they've been taught one strategy that they don't understand, they feel like they don't know how to approach it. So I think a lot of it comes down to they're not being taught how to understand the concept, but they're more just being given one direct way to do something. And if that doesn't make sense to them or they don't understand the concepts through that, then they have a really difficult time of being able to approach something independently. Mike: Mhret, I think Jenna offered a really nice segue here because you all were dealing with this question of confidence and with kids who, when they didn't see a clear path or they didn't see something that they could replicate, just got stuck, or for lack of a better word, they kind of turned to the teacher or imagined that that was the next step. And I was really excited about the fact that you all had designed some really specific features into the strategy showcase that addressed that problem of practice. So I'm wondering if you could just talk about the particular features or the practices that you all thought were important in setting up the strategy showcase and trying to take up this practice of a strategy showcase. Mhret: Yeah, so we had three components in this strategy showcase. The first one, we saw it being really important, being open-ended tasks, and that combats what Jenna was saying of "the right way." The questions that we asked didn't ask them to use a specific strategy. It was open-ended in a way that it asked them if they agreed or disagreed with a way that someone found an answer, and it just was open to see whatever came to their mind and how they wanted to start the task. So that was very important as being the first component.  And the second one was the student work displayed, which Anna was talking about earlier. The root of this being we want students' confidence to grow and have their voices heard. And so their work being displayed was very important—not teacher work or not an example being given to them, but what they had in their mind. And so we did that intentionally with having their names covered up in the beginning because we didn't want the focus to be on who did it, but just seeing their work displayed—being worth it to be displayed and to learn from—and so their names were covered up in the beginning and it was on one side of the board.  And then the third component was the students' co-identified strategies. So that's when after they have displayed their individual work, we would come up as a group and talk about what similarities did we see, what differences in what the students have used. And they start naming strategies out of that. They start giving names to the strategies that they see their peers using, and we co-identify and create this strategy that they are owning. So those are the three important components. Mike: OK. Wow. There's a lot there. And I want to spend a little bit of time digging into each one of these and I'm going to invite all four of you to feel free to jump in and just let us know who's talking so that everybody has a sense of that.  I wonder if you could talk about this whole idea that, when you say open-ended tasks, I think that's really important because it's important that we build a common definition. So when you all describe open-ended tasks, let's make sure that we're talking the same language. What does that mean? And Tutita, I wonder if you want to just jump in on that one. Tutita: Sure. Yeah. An open-ended task, as it suggests, it's not a direct line where, for example, you can prompt students to say, "You must use 'blank' strategy to solve this particular problem." To me, it's just mathematical. That's what a really good rich problem is, is that it really allows for that problem s

    34 мин.

  7. 20.11.2025

    Season 4 | Episode 6 - Christy Pettis & Terry Wyberg, The Case for Choral Counting with Fractions

    Christy Pettis & Terry Wyberg, The Case for Choral Counting with Fractions ROUNDING UP: SEASON 4 | EPISODE 6 How can educators help students recognize similarities in the way whole numbers and fractions behave? And are there ways educators can build on students' understanding of whole numbers to support their understanding of fractions?  The answer from today's guests is an emphatic yes. Today we're talking with Terry Wyberg and Christy Pettis about the ways choral counting can support students' understanding of fractions.  BIOGRAPHIES Terry Wyberg is a senior lecturer in the Department of Curriculum and Instruction at the University of Minnesota. His interests include teacher education and development, exploring how teachers' content knowledge is related to their teaching approaches. Christy Pettis is an assistant professor of teacher education at the University of Wisconsin-River Falls. RESOURCES Choral Counting & Counting Collections: Transforming the PreK-5 Math Classroom by Megan L. Franke, Elham Kazemi, and Angela Chan Turrou  Teacher Education by Design Number Chart app by The Math Learning Center TRANSCRIPT Mike Wallus: Welcome to the podcast, Terry and Christy. I'm excited to talk with you both today. Christy Pettis: Thanks for having us. Terry Wyberg: Thank you. Mike: So, for listeners who don't have prior knowledge, I'm wondering if we could just offer them some background. I'm wondering if one of you could briefly describe the choral counting routine. So, how does it work? How would you describe the roles of the teacher and the students when they're engaging with this routine? Christy: Yeah, so I can describe it. The way that we usually would say is that it's a whole-class routine for, often done in kind of the middle grades. The teachers and the students are going to count aloud by a particular number. So maybe you're going to start at 5 and skip-count by 10s or start at 24 and skip-count by 100 or start at two-thirds and skip-count by two-thirds.  So you're going to start at some number, and you're going to skip-count by some number. And the students are all saying those numbers aloud. And while the students are saying them, the teacher is writing those numbers on the board, creating essentially what looks like an array of numbers. And then at certain points along with that talk, the teacher will stop and ask students to look at the numbers and talk about things they're noticing. And they'll kind of unpack some of that. Often they'll make predictions about things. They'll come next, continue the count to see where those go. Mike: So you already pivoted to my next question, which was to ask if you could share an example of a choral count with the audience. And I'm happy to play the part of a student if you'd like me to. Christy: So I think it helps a little bit to hear what it would sound like. So let's start at 3 and skip-count by 3s. The way that I would usually tell my teachers to start this out is I like to call it the runway. So usually I would write the first three numbers. So I would write "3, 6, 9" on the board, and then I would say, "OK, so today we're going to start at 3 and we're going to skip-count by 3s. Give me a thumbs-up or give me the number 2 when you know the next two numbers in that count." So I'm just giving students a little time to kind of think about what those next two things are before we start the count together. And then when I see most people kind of have those next two numbers, then we're going to start at that 3 and we're going to skip-count together.  Are you ready? Mike: I am. Christy: OK. So we're going to go 3…  Mike & Christy: 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36.  Christy: Keep going.  Mike & Christy: 39, 42, 45, 48, 51. Christy: Let's stop there.  So we would go for a while like that until we have an array of numbers on the board. In this case, I might've been recording them, like where there were five in each row. So it would be 3, 6, 9, 12, 15 would be the first row, and the second row would say 18, 21, 24, 27, 30, and so on. So we would go that far and then I would stop and I would say to the class, "OK, take a minute, let your brains take it in. Give me a number 1 when your brain notices one thing. Show me 2 if your brain notices two things, 3 if your brain notices three things." And just let students have a moment to just take it in and think about what they notice.  And once we've seen them have some time, then I would say, "Turn and talk to your neighbor, and tell them some things that you notice." So they would do that. They would talk back and forth. And then I would usually warm-call someone from that and say something like, "Terry, why don't you tell me what you and Mike talked about?" So Terry, do you have something that you would notice? Terry: Yeah, I noticed that the last column goes up by 15, Christy: The last column goes up by 15. OK, so you're saying that you see this 15, 30, 45? Terry: Yes. Christy: In that last column. And you're thinking that 15 plus 15 is 30 and 30 plus 15 is 45. Is that right? Terry: Yes. Christy: Yeah. And so then usually what I would say to the students is say, "OK, so if you also noticed that last column is increasing by 15, give me a 'me too' sign. And if you didn't notice it, show an 'open mind' sign." So I like to give everybody something they can do. And then we'd say, "Let's hear from somebody else. So how about you, Mike? What's something that you would notice?" Mike: So one of the things that I was noticing is that there's patterns in the digits that are in the ones place. And I can definitely see that because the first number 3 [is] in the first row. In the next row, the first number is 18 and the 8 is in the ones place. And then when I look at the next row, 33 is the first number in that row, and there's a 3 again. So I see this column pattern of 3 in the ones place, 8 in the ones place, 3 in the ones place, 8 in the ones place. And it looks like that same kind of a number, a different number. The same number is repeating again, where there's kind of like a number and then another number. And then it repeats in that kind of double, like two numbers and then it repeats the same two numbers. Christy: So, what I would say in that one is try to revoice it, and I'd probably be gesturing, where I'd do this. But I'd say, "OK, so Mike's noticing in this ones place, in this first column, he's saying he notices it's '3, 8, 3, 8.' And then in other columns he's noticing that they do something similar. So the next column, or whatever, is like '6, 1, 6, 1' in the ones place. Why don't you give, again, give me a 'me too' [sign] if you also noticed that pattern or an 'open mind' [sign] if you didn't."  So, that's what we would do. So, we would let people share some things. We would get a bunch of noticings while students are noticing those things. I would be, like I said, revoicing and annotating on the board. So typically I would revoice it and point it out with gestures, and then I would annotate that to take a record of this thing that they've noticed on the board. Once we've gotten several students' noticings on the board, then we're going to stop and we're going to unpack some of those. So I might do something like, "Oh, so Terry noticed this really interesting thing where he said that the last column increases by 15 because he saw 15, 30, 45, and he recognized that. I'm wondering if the other columns do something like that too. Do they also increase by the same kind of number? Hmm, why don't you take a minute and look at it and then turn and talk to your neighbor and see what you notice." And we're going to get them to notice then that these other ones also increase by 15. So if that hadn't already come out, I could use it as a press move to go in and unpack that one further.  And then we would ask the question, in this case, "Why do they always increase by 15?" And we might then use that question and that conversation to go and talk about Mike's observation, and to say, like, "Huh, I wonder if we could use what we just noticed here to figure out about why this idea that [the numbers in the] ones places are going back and forth between 3, 8, 3, 8. I wonder if that has something to do with this." Right? So we might use them to unpack it. They'll notice these patterns. And while the students were talking about these things, I'd be taking opportunities to both orient them to each other with linking moves to say, "Hey, what do you notice? What can you add on to what Mike said, or could you revoice it?" And also to annotate those things to make them available for conversation. Mike: There was a lot in your description, Christy, and I think that provides a useful way to understand what's happening because there's the choice of numbers, there's the choice of how big the array is when you're recording initially, there are the moves that the teacher's making. What you've set up is a really cool conversation that comes forward. We did this with whole numbers just now, and I'm wondering if we could take a step forward and think about, OK, if we're imagining a choral count with fractions, what would that look and sound like? Christy: Yeah, so one of the ones I really like to do is to do these ones that are just straight multiples, like start at 3 and skip-count by 3s. And then to either that same day or the very next day—so very, very close in time in proximity—do one where we're going to do something similar but with fractions. So one of my favorites is for the parallel of the whole number of skip-counting by 3s is we'll start at 3 fourths and we'll skip-count by 3 fourths. And when we write those numbers, we're not going to put them in simplest form; we're just going to write 3 fourths, 6 fourths, 9 fourths. So in this case, I would probably set it up in the exact same very parallel structure to that other one that we just did with the whole numbers. And I would pu

    37 мин.
  8. Season 4 | Episode 5 - Ramsey Merritt, Improving Students' Turn & Talk Experience

    06.11.2025

    Season 4 | Episode 5 - Ramsey Merritt, Improving Students' Turn & Talk Experience

    Ramsey Merritt, Improving Students' Turn & Talk Experience ROUNDING UP: SEASON 4 | EPISODE 5 Most educators know what a turn and talk is—but are your students excited to do them?  In this episode, we put turn and talks under a microscope. We'll talk with Ramsey Merritt from the Harvard Graduate School of Education about ways to revamp and better scaffold turn and talks to ensure your students are having productive mathematical discussions.  BIOGRAPHY Ramsey Merritt is a lecturer in education at Brandeis University and the director of leadership development for Reading (MA) Public Schools. He has taught and coached at every level of the U.S. school system in both public and independent schools from New York to California. Ramsey also runs an instructional leadership consulting firm, Instructional Success Partners, LLC. Prior to his career in education, he worked in a variety of roles at the New York Times. He is currently completing his doctorate in education leadership at Harvard Graduate School of Education. Ramsey's book, Diving Deeper with Upper Elementary Math, will be released in spring 2026. TRANSCRIPT Mike Wallus: Welcome to the podcast, Ramsey. So great to have you on. Ramsey Merritt: It is my pleasure. Thank you so much for having me. Mike: So turn and talk's been around for a while now, and I guess I'd call it ubiquitous at this point. When I visit classrooms, I see turn and talks happen often with quite mixed results. And I wanted to start with this question: At the broadest level, what's the promise of a turn and talk? When strategically done well, what's it good for? Ramsey: I think at the broadest level, we want students talking about their thinking and we also want them listening to other students' thinking and ideally being open to reflect, ask questions, and maybe even change their minds on their own thinking or add a new strategy to their thinking. That's at the broadest level.  I think if we were to zoom in a little bit, I think turn and talks are great for idea generation. When you are entering a new concept or a new lesson or a new unit, I think they're great for comparing strategies. They're obviously great for building listening skills with the caveat that you put structures in place for them, which I'm sure we'll talk about later. And building critical-thinking and questioning skills as well.  I think I've also seen turn and talks broadly categorized into engagement, and it's interesting when I read that because to me I think about engagement as the teacher's responsibility and what the teacher needs to do no matter what the pedagogical tool is. So no matter whether it's a turn and talk or something else, engagement is what the teacher needs to craft and create a moment. And I think a lot of what we'll probably talk about today is about crafting moments for the turn and talk. In other words, how to engage students in a turn and talk, but not that a turn and talk is automatically engagement. Mike: I love that, and I think the language that you've used around crafting is really important. And it gets to the heart of what I was excited about in this conversation because a turn and talk is a tool, but there is an art and a craft to designing its implementation that really can make or break the tool itself. Ramsey: Yeah. If we look back a little bit as to where turn and talk came from, I sort of tried to dig into the papers on this. And what I found was that it seems as if turn and talks may have been a sort of spinoff of the think-pair-share, which has been around a little bit longer. And what's interesting in looking into this is, I think that turn and talks were originally positioned as a sort of cousin of think-pair-share that can be more spontaneous and more in the moment. And I think what has happened is we've lost the "think" part. So we've run with it, and we've said, "This is great," but we forgot that students still need time to think before they turn and talk. And so what I see a lot is, it gets to be somewhat too spontaneous, and certain students are not prepared to just jump into conversations. And we have to take a step back and sort of think about that. Mike: That really leads into my next question quite well because I have to confess that when I've attended presentations, there are points in time when I've been asked to turn and talk when I can tell you I had not a lot of interest nor a lot of clarity about what I should do. And then there were other points where I couldn't wait to start that conversation. And I think this is the craft and it's also the place where we should probably think about, "What are the pitfalls that can derail or have a turn and talk kind of lose the value that's possible?" How would you talk about that? Ramsey: Yeah, it is funny that we as adults have that reaction when people say, "Turn and talk."  The three big ones that I see the most, and I should sort of say here, I've probably been in 75 to 100 buildings and triple or quadruple that for classrooms. So I've seen a lot of turn and talks, just like you said. And the three big ones for me, I'll start with the one that I see less frequently but still see it enough to cringe and want to tell you about it. And it's what I call the "stall" turn and talk. So it's where teachers will sometimes use it to buy themselves a little time. I have literally heard teachers say something along the lines of, "OK, turn and talk to your neighbor while I go grab something off the printer."  But the two biggest ones I think lead to turn and talk failure are a lack of specificity. And in that same vein too, what are you actually asking them to discuss? So there's a bit of vagueness in the prompting, so that's one of the big ones.  The other big one for me is, and it seems so simple, and I think most elementary teachers are very good at using an engaging voice. They've learned what tone does for students and what signals tone sends to them about, "Is now the time to engage? Should I be excited?" But I so often see the turn and talk launched unenthusiastically, and that leads to an engagement deficit. And that's what you're starting out with if you don't have a good launch: Students are already sort of against you because you haven't made them excited to talk. Mike: I mean those things resonate. And I have to say there are some of them that I cringe because I've been guilty of doing, definitely the first thing when I've been unprepared. But I think these two that you just shared, they really go to this question of how intentionally I am thinking about building that sense of engagement and also digging into the features that make a turn and talk effective and engaging.  So let's talk about the features that make turn and talks effective and engaging for students. I've heard you talk about the importance of picking the right moment for a turn and talk. So what's that mean? Ramsey: So for me, I break it down into three key elements. And one of them, as you say, is the timing. And this might actually be the most important element, and it goes back to the origin story, is: If you ask a question, and say you haven't planned a turn and talk, but you ask a question to a whole group and you see 12 hands shoot up, that is an ideal moment for a turn and talk. You automatically know that students are interested in this topic. So I think that's the sort of origin story, is: Instead of whipping around the room and asking all 12 students—because especially at the elementary level, if students don't get their chance to share, they are very disappointed. So I've also seen these moments drag out far too long. So it's kind of a good way to get everyone's voice heard. Maybe they're not saying it out to the whole group, but they get to have everyone's voice heard. And also you're buying into the engagement that's already there. So that would be the more spontaneous version, but you can plan in your lesson planning to time a turn and talk at a specific moment if you know your students well enough that you know can get them engaged in.  And so that leads to one of the other points is the launch itself. So then you're really thinking about, "OK, I think this could be an interesting moment for students. Let me think a little bit deeper about what the hook is." Almost every teacher knows what a hook is, but they typically think about the hook at the very top of their lesson. And they don't necessarily think about, "How do I hook students in to every part of my lesson?" And maybe it's not a full 1-minute launch, maybe it's not a full hook, but you've got to reengage students, especially now in this day and time, we're seeing students with increasingly smaller attention spans. So it's important to think about how you're launching every single piece of your lesson.  And then the third one, which goes against that origin story that I may or may not even be right about, but it goes against that sort of spontaneous nature of turn and talks, is: I think the best turn and talks are usually planned out in advance.  So for me it's planning, timing, and launching. Those are my elements to success when I'm coaching teachers on doing a turn and talk. Mike: Another question that I wanted to unpack is: Talk about what. The turn and talk is a vehicle, but there's also content, right? So I'm wondering about that. And then I'm also wondering are there prompts or particular types of questions that educators can use that are more interesting and engaging, and they help draw students in and build that engagement experience you were talking about? Ramsey: Yeah, and it's funny you say, "Talk about what" because that's actually feedback that I've given to teachers, when I say, "How did that go for you?" And they go, "Well, it went OK." And I say, "Well, what did you ask them to talk about? Talk about what is important to think about in that planning process." So I hate to throw something big out there, but I would actually argue

    28 мин.
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Welcome to "Rounding Up" with the Math Learning Center. These conversations focus on topics that are important to Bridges teachers, administrators and anyone interested in Bridges in Mathematics. Hosted by Mike Wallus, VP of Educator Support at MLC.

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