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Rounding Up

Rounding Up

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Welcome to Rounding Up, the professional learning podcast brought to you by The Math Learning Center. Two things have always been true in education: Ongoing professional learning is essential, and teachers are extremely busy people. Rounding Up is a podcast designed to provide meaningful, bite-sized professional learning for busy educators and instructional leaders. I'm Mike Wallus, vice president for educator support at The Math Learning Center and host of the show. In each episode, we'll explore topics important to teachers, instructional leaders, and anyone interested in elementary mathematics education. Topics such as posing purposeful questions, effectively recording student thinking, cultivating students' math identity, and designing asset-based instruction from multilingual learners. Don't miss out! Subscribe now wherever you get your podcasts. Each episode will also be published on the Bridges Educator Site. We hope you'll give Rounding Up a try, and that the ideas we discuss have a positive impact on your teaching and your students' learning.2022 The Math Learning Center | www.mathlearningcenter.org Ciencia Matemáticas
Episodios
  • Season 4 | Episode 10 – What Counts as Counting? Guest: Dr. Christopher Danielson, Part 2

    Season 4 | Episode 10 – What Counts as Counting? Guest: Dr. Christopher Danielson, Part 2
    Jan 22 2026
    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 ...
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    22 m
  • Season 4 | Episode 9 - Dr. Todd Hinnenkamp, Enacting Talk Moves with Intention

    Season 4 | Episode 9 - Dr. Todd Hinnenkamp, Enacting Talk Moves with Intention
    Jan 8 2026
    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 ...
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    27 m
  • Season 4 | Episode 8 – Janet Walkoe & Margaret Walton, Exploring the Seeds of Algebraic Thinking

    Season 4 | Episode 8 – Janet Walkoe & Margaret Walton, Exploring the Seeds of Algebraic Thinking
    Dec 18 2025
    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...
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    18 m
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