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

    Season 4 | Episode 13 – Dr. Mike Steele, Pacing Discourse-Rich Lessons
    Mar 5 2026
    Mike Steele, Pacing Discourse-Rich Lessons ROUNDING UP: SEASON 4 | EPISODE 13 As a classroom teacher, pacing lessons was often my Achilles' heel. If my students were sharing their thinking or working on a task, I sometimes struggled to decide when to move on to the next phase of a lesson. Today we're talking with Mike Steele from Ball State University about several high-leverage practices that educators can use to plan and pace their lessons. BIOGRAPHY Mike Steele is a math education researcher focused on teacher knowledge and teacher learning. He is the past president of the Association of Mathematics Teacher Educators, editor in chief of the Mathematics Teacher Educator journal, and member of the NCTM board of directors. RESOURCES Journal Article "Pacing a Discourse-Rich Lesson: When to Move On" Books 5 Practices for Orchestrating Productive Mathematics Discussions The 5 Practices in Practice [Elementary] The 5 Practices in Practice [Middle School] The 5 Practices in Practice [High School] Coaching the 5 Practices TRANSCRIPT Mike Wallus: Well, hi, Mike. Welcome to the podcast. I'm excited to talk with you about discourse-rich lessons and what it looks like to pace them. Mike Steele: Well, I'm excited to talk with you too about this, Mike. This has been a real focus and interest, and I'm so excited that this article grabbed your attention. Mike Wallus: I suppose the first question I should ask for the audience is: What do you mean when you're talking about a discourse-rich lesson? What does that term mean about the lesson and perhaps also about the role of the teacher? Mike Steele: Yeah, I think that's a great question to start with. So when we're talking about a discourse-rich lesson, we're talking about one that has some mathematics that's worth talking about in it. So opportunities for thinking, reasoning, problem solving, in-progress thinking that leads to new mathematical understandings. And that kind of implicit in that discourse-rich lesson is student discourse-rich lesson. That we want not just teachers talking about sharing their own thinking about the mathematics, but opportunities for students to share their own thinking, to shape that thinking, to talk with each other, to see each other as intellectual resources in mathematics. And so to have a lesson like that, you've got to have a number of things in place. You've got to have a mathematical task that's worth talking about. So something that's not just a calculation and we end up at an answer and that the discourse isn't just, "Let me relay to you as a student the steps I took to do this." Because a lot of times when students are just starting to experience discourse-rich lessons, that's kind of mode one that they engage in is, "Let me recite for you the things that I did." But really opportunities to go beyond that and get into the reasoning and the why of the mathematics. And hopefully to explore some approaches or perspectives or representations that they may not have defaulted to in their first run-through or their first experience digging into a mathematical task. So the task has to have those opportunities and then we have to create learning environments that really foster those opportunities and students as the creators of mathematics and the teacher as the person who's shaping and guiding that discussion in a mathematically productive way. Mike Wallus: One of the things that struck me is there is likely a problem of practice that you're trying to solve in publishing this article, and I wonder if we could pull the curtain back and have you talk a bit about what was the genesis of this article for you? Mike Steele: Absolutely. So let me take us back about 20 or 25 years, and I'll take you back to some early work that went on around these sorts of rich tasks and discourse-rich lessons. So a lot of this legacy comes out of research or a project in the late nineties called the Quasar Project that helped identify: What is a rich task? What is a task, as the researchers described it, of high cognitive demand that has those opportunities for thinking and reasoning? The next question that that line of research brought forward is, "OK, so we know what a task looks like that gives these opportunities. How does this change what teachers do in the classroom? How they plan for lessons, how they make those moment-to-moment decisions as they're engaged in the teaching of that lesson?" Because it's very different than actually when I started teaching middle school in the nineties, where my preparation was: I looked at the content I had for that day, I wrote three example problems I wanted to write on the board that I very carefully got all the steps right and put those up and explained them and answered some questions. "Alright, everybody understand that? OK, great, moving on." And then the students went and reproduced that. That's fine for some procedural things, but if I really wanted them to engage in thinking and ...
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    35 min
  • Season 4 | Episode 12 – Kyndall Thomas, Building a Meaningful Understanding of Properties Through Fact Fluency Tasks

    Season 4 | Episode 12 – Kyndall Thomas, Building a Meaningful Understanding of Properties Through Fact Fluency Tasks
    Feb 19 2026
    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...
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    12 min
  • Season 4 | Episode 11 – Dr. Amy Hackenberg, Understanding Units Coordination

    Season 4 | Episode 11 – Dr. Amy Hackenberg, Understanding Units Coordination
    Feb 5 2026
    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 ...
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    31 min
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