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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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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Recent Episodes
May 21, 2026
Season 4 | Episode 18 – Dr. Jenny Bay-Williams, Productive Ways to Build Fluency with Basic Facts (Rerun)
<h1 dir="ltr">Dr. Jenny Bay-Williams, Productive Ways to Build Fluency with Basic Facts</h1> <p dir="ltr"><strong>ROUNDING UP: SEASON 4 | EPISODE 18</strong></p> <p dir="ltr">This summer we're replaying favorite listener episodes from the first four seasons of Rounding Up—like this one from Season 1. We'll return with all new episodes in early September.</p> <p dir="ltr">Ensuring students master their basic facts remains a shared goal among parents and educators. That said, many educators wonder what should replace the memorization drills that cause so much harm to their students' math identities.</p> <p dir="ltr">Today on the podcast, Jenny Bay-Williams talks about how to meet that goal and shares a set of productive practices that also support student reasoning and sensemaking. </p> <p dir="ltr"><strong>BIOGRAPHY</strong></p> <p dir="ltr">Jennifer Bay-Williams is a professor of mathematics education at the University of Louisville. She has authored over 40 books and 100 journal articles and book chapters that focus on making mathematics meaningful to all students. She is an international leader in the field of mathematics education, frequently speaking at state, national, and international conferences and serving on national boards. </p> <p dir="ltr"><strong>RESOURCES</strong></p> <p dir="ltr"><a href= "https://pubs.nctm.org/view/journals/mtlt/114/11/article-p830.xml?rskey=SdaRZv&result=1"> "Eight Unproductive Practices in Developing Fact Fluency"</a> article by Gina Kling and Jennifer M. Bay-Williams</p> <p dir="ltr"><a href= "https://www.ascd.org/books/math-fact-fluency?chapter=preface-math-fact-fluency"> Math Fact Fluency: 60+ Games and Assessment Tools to Support Learning and Retention</a> book by Jennifer M. Bay-Williams and Gina Kling</p> <p dir="ltr"><a href= "https://kcm.nku.edu/mathfactfluency/">Math Fact Fluency companion website</a> by Kentucky Center for Mathematics</p> <p dir="ltr"><strong>TRANSCRIPT</strong></p> <p dir="ltr"><strong>Mike Wallus:</strong> Welcome to the podcast, Jenny. We are excited to have you. </p> <p dir="ltr"><strong>Jennifer Bay-Williams:</strong> Well, thank you for inviting me. I'm thrilled to be here and excited to be talking about basic facts. </p> <p dir="ltr"><strong>Mike:</strong> Awesome. Let's jump in. So, your recommendations start with an emphasis on reasoning. I wonder if we could start by just having you talk about the why behind your recommendation and a little bit about what an emphasis on reasoning looks like in an elementary classroom when you're thinking about basic facts. </p> <p dir="ltr"><strong>Jenny:</strong> All right, well, I'm going to start with a little bit of a snarky response: that the non-reasoning approach doesn't work. </p> <p dir="ltr"><strong>Mike and Jenny:</strong> (laugh) </p> <p dir="ltr"><strong>Jenny:</strong> OK. So, one reason to move to reasoning is that memorization doesn't work. Drill doesn't work for most people. But the reason to focus on reasoning with basic facts beyond that fact, is that the reasoning strategies grow to strategies that can be used beyond basic facts. So, if you take something like the making 10 idea—that 9 plus 6, you can move one over and you have 10 plus 5—is a beautiful strategy for a 99 plus 35. So, you teach the reasoning upfront from the beginning, and it sets students up for success later on. </p> <p dir="ltr"><strong>Mike:</strong> That absolutely makes sense. So, you talk about the difference between telling a strategy and explicit instruction. And I raise this because I suspect that some people might struggle to think about how those are different. Could you describe what explicit instruction looks like and maybe share an example with listeners? </p> <p dir="ltr"><strong>Jenny:</strong> Absolutely. First of all, I like to use the whole phrase: "explicit strategy instruction." So, what you're trying to do is have that strategy be explicit, noticeable, visible. So, for example, if you're going to do the making 10 strategy we just talked about, you might have two 10-frames. One of them is filled with nine counters, and one of them is filled with six counters. And students can see that moving one counter over is the same quantity. So, they're seeing this flexibility that you can move numbers around, and you end up with the same sum. So, you're just making that idea explicit and then helping them generalize. You change the problems up and then they come back and they're like, "Oh, hey, we can always move some over to make a ten"—or a twenty, or a thirty, or whatever you're working on. And so, I feel like, in using the counters, or they could be stacking Unifix cubes or things like that. That's the explicit instruction. It's concrete. </p> <p dir="ltr">And then, if you need to be even more explicit, you ask students in the end to summarize the pattern that they noticed across the three or four problems that they solved. "Oh, that you take the bigger number, and then you go ahead and complete a ten to make it easier to add." And then, that's how you're really bringing those ideas out into the community to talk about. </p> <p dir="ltr">For multiplication, I'm just going to contrast. Let's say we're doing [the] add a group strategy with multiplication. If you were going to do direct instruction, and you're doing 6 times 8, you might say, "All right, so when you see a six," then a direct instruction would be like, "Take that first number and just assume it's a five." So then, "Five eights is how much? Write that down." That's direct instruction. You're like, "Here, do this step. Here, do this step. Here, do this step." The explicit strategy instruction would have, for example—I like, for eights, boxes of crayons because they oftentimes come in eights. So, but they'd have five boxes of crayons and then one more box of crayons. So, they could see you've got five boxes of crayons. They know that fact is 40, they—if they're working on their sixes, they should know their fives. And so, then what would one more group be about? </p> <p dir="ltr">So, just helping them see that with multiplication through visuals, you're adding on one group, not one more, but one group. So, they see that through the visuals that they're doing or through arrays or things like that. So, it's about them seeing the number of relationships and not being told what the steps are. </p> <p dir="ltr"><strong>Mike:</strong> And it strikes me, too, Jenny, that the role of the teacher in those two scenarios is pretty different. </p> <p dir="ltr"><strong>Jenny:</strong> Very different. Because the teacher is working very hard (chuckles) with the explicit strategy instruction to have the visuals that really highlight the strategy. Maybe it's the colors of the dots or the exact 10-frames they've picked and have they filled them or whether they choose to use the Unifix cubes and how they're going to color them and things like that. So, they're doing a lot of thinking to make that pattern noticeable, visible. As opposed to just saying, "Do this first, do that second, do that third."</p> <p dir="ltr"><strong>Mike:</strong> I love the way that you said that you're doing a lot of thinking and work as a teacher to make a pattern noticeable. That's powerful, and it really is a stark contrast to, "Let me just tell you what to do." </p> <p dir="ltr">I'd love to shift a little bit and ask you about another piece of your work. So, you advocate for teaching facts in an order that stresses relationships rather than simply teaching them in order. I'm wondering if you can tell me a little bit more about how relationships-based instruction has an impact on student thinking. </p> <p dir="ltr"><strong>Jenny:</strong> So, we want every student to enact the reasoning strategies. So, I'm going to go back to addition, for example. And I'm going to switch over to the strategy that I call "pretend-to-10", also called "use 10" or "compensation." But if you're going to set them up for using that strategy, there's a lot of steps to think through. So, if you're doing 9 plus 5, then in the pretend-to-10 strategy, you just pretend that 9 is a 10. So now you've got 10 plus 5 and then you've got to compensate in the end. You've got to fix your answer because it's 1 too much. And so, you've got to come back 1. That's some thinking. Those are some steps. So, what you want is to have the students automatic with certain things so that they're set up for that task. So, for that strategy, they need to be able to add a number onto 10 without much thought. </p> <p dir="ltr">Otherwise, the strategy is not useful. The strategy is useful when they already know 10 plus 5. So, you teach them this, you teach them that relationship—10 and some more—and then they know that 9's 1 less than 10. That relationship is hugely important, knowing 9 is 1 less than 10. And so then they know their answer has to be 1 less. 9's 1 less than 10. So, 9 plus a number is 1 less than 10 plus the number. Huge idea. </p> <p dir="ltr">And there's been a lot of research done in kindergarten on students understanding things like 7's 1 more than 6, 7's 1 less than 8. And they're predictive studies looking at student achievement in first grade, second grade, third grade. And students—it turns out that one of the biggest predictors of success is students understanding those number relationships. That 1 more, 1 less, 2 more, 2 less. Hugely important in doing the number sense. </p> <p dir="ltr">So that's what the relationship piece is, is sequencing facts so that what is going to be needed for the next thing they're going to do, the thinking that's going to be needed, is there for them. And then build on those relationships to learn the next strategy. </p> <p dir="ltr"><strong>Mike:</strong> I mean, it strikes me that there's a little bit of a twofer in that one. The first is this idea that what you're doing is purposely setting up a future idea, right? It's kind of like saying, "I'm going to build this prior knowledge about ten-ness, and then I'm going to have kids think about the relationship between 10 and 9." So, the care in this work is actually really understanding those relationships and how you're going to leverage them. </p> <p dir="ltr">The other thing that really jumps out from what you said [is] this has long term implications for students' thinking. It's not just fact acquisition; it's what you said: Research shows that this has implications for how kids are thinking further down the road. Am I understanding that right? </p> <p dir="ltr"><strong>Jenny:</strong> That's absolutely correct. So just that strategy alone. Let's say they're adding 29 plus 39. And they're like, "Oh hey, both of those numbers are right next to the next benchmark. So instead of 29 plus 39, I'm going to add 30 plus 40, [which equals] 70. And I got, I went up 2, so I'm going to come back down 2. And I know that 2 less than a benchmark's going to land on an 8." So that, again, it's coming back to this relationship of how far apart numbers are, what's right there within a set of 10, [which] helps then to generalize within tens or within hundreds. </p> <p dir="ltr">And by the way, how about fractions? </p> <p dir="ltr"><strong>Mike:</strong> Hmm. Talk about that. </p> <p dir="ltr"><strong>Jenny:</strong> (laughs) It generalizes to fractions. So, let's take that same idea of adding. Let's just say it's like, 2 and seven-eighths plus 2 and seven-eighths. So, if we just pretended those were both 3s because they're both super close to 3, then you'd have 6, and then you added on two-eighths too much. So, you come back two-eighths, or a fourth, and you have your answer. You don't have to do the regrouping with fractions and all the mess that really gets bogged down. And it's a much more efficient method that, again, you set students up for when they understand these number relationships. When you get into fractions, you're thinking about, "How close are you to the next whole number?" maybe, instead of to the next tens number. </p> <p dir="ltr"><strong>Mike:</strong> It strikes me that if you have a group of teachers who have a common understanding of this approach to facts, and everyone's kind of playing the long game and thinking about how what they're doing is going to support what's next, it just creates a system that's much more intentional in helping kids not only acquire the facts, but build a set of ways of thinking. </p> <p dir="ltr"><strong>Jenny:</strong> Mike, that's exactly it. I mean, here we are, we're trying to make up for lost time. We never have enough time in the classroom. We want an efficient way to make sure our kids get the most learning in. And so, to me that is about investing early in the fact strategies. Because then actually when you get up to those other things that you're adding or subtracting or multiplying or whatever you're doing, you benefit from the fact that you took time early to learn those strategies. Because those strategies are now very useful for all this other math that you're doing. And then students are more successful in making good choices about how they're going to solve those problems that are, oftentimes—especially when, I like to mention fractions and decimals at least once in a basic facts talk because we get back, by the time we get into fractions and decimals—we're back to just sometimes only showing one way. The sort of standard algorithm way. When, in fact, those basic facts strategies absolutely apply to, almost always, more-efficient strategies for working with fractions and decimals. </p> <p dir="ltr"><strong>Mike:</strong> I want to shift a little bit. One of the things that was really helpful for me in growing my understanding is the way that you talk about a set of facts that you would describe as foundational facts and another set of facts that you would describe as derived facts. And I'm wondering if you can unpack what those two subsets are and how they're related to one another. </p> <p dir="ltr"><strong>Jenny:</strong> Yeah. So, the foundational facts are ones where automaticity is needed in order to enact a strategy. So, to me, the foundational fact strategies are their names. Like the doubling strategy—or double and double again, some people call it. Or add a group for multiplication. And the addition ones of making 10s and pretend-to-10 strategies. And in those strategies, you can solve lots of different facts. But there's too much going on (laughs) in your brain if you don't have automaticity with the facts you need. </p> <p dir="ltr">So, for example, if you have your 6 facts, and you're trying to get your 6 facts down. And you already know your 5s, like, automaticity with your 5s, then that becomes a useful way to get your 6s. So, if you have 6 times 8, and you know 5 times 8 is 40, then you're like, "I got one more 8, [which equals] 48." That's an added group strategy. </p> <p dir="ltr">But if you're not automatic with your 5s, this is how this sounds when you're interviewing a child. They're going to use add a group strategy, but they don't know their 5s. So, then they're like, "Let's see. 5 times 8 is 5, 10, 15, 20, 25, 30, 40. Now, what was I doing?" Like, they can't finish it because they were skip-counting with their 5s. They lose track of what they're doing, is my point. So, the key is that they just know those facts that they need in order to use a strategy. And that, going back to, like, the pretend-to-10, they got to know 10 and some more facts to be successful. They have to know 9's 1 less than 10 to be successful. So, that's the idea is, if they reach automaticity with the foundational fact sets, then their brain is freed up to go through those reasoning strategies. </p> <p dir="ltr"><strong>Mike:</strong> That totally makes sense. </p> <p dir="ltr">I want to shift a little bit now. One of the things that I really appreciated about the article [<a href= "https://pubs.nctm.org/view/journals/mtlt/114/11/article-p830.xml?rskey=SdaRZv&result=1">"Eight Unproductive Practices in Developing Fact Fluency"</a> by Gina Kling and Jennifer M. Bay-Williams] was that you made what I think is a very strong, unambiguous case for ending many of the past practices used for fact acquisition—worksheets and timed tests, in particular. This can be a tough sell because this is often what is associated with elementary mathematics, and families kind of expect this kind of practice. How would you help an educator explain the shift away from these practices to folks who are out in the larger community? What is it that we might help say to folks to help them understand this shift? </p> <p dir="ltr"><strong>Jenny:</strong> That's a great question, and the real answer is it depends, again, on [the] audience. So, who is your audience? Even if the audience is parents, what do those parents prioritize and want for their children? So, I feel like there's lots of reasons to do it, but to really speak to what matters to them. So, I'm going to give a very generic answer here. But for everyone, they want their child to be successful. So, I feel that that opportunity to show, to give a problem, like 29 plus 29, and ask how parents might add that problem. And if they think 30 plus 30 and subtract 2 to get to the answer or whatever, then that gives this case to say, "Well this is how we're going to work on basic facts. We're building up so that your child is ready to use these strategies. We're going to start right with the basic facts, learning these strategies. These really matter." And the example I gave could be whatever fits with the level of their kid. So, it could be like 302 minus 299. It's a classic one where you don't want your child to implement an algorithm there; you want them to notice those numbers are 3 apart. And so, there's this work that begins early. So, I think that's part of it. </p> <p dir="ltr">I think another part of it is helping people just reflect on their own learning experiences. What were your learning experiences with basic facts? And even if they liked the speed drills, they oftentimes recognize that it was not well-liked by most people. And also, then they really didn't learn strategies. So, I feel like we have to be showing that we're not taking something away; we're adding something in. They are going to become automatic with their facts. They're not going to forget them because we're not doing this memorizing that leads to a lot of forgetting. And, bonus, they're going to have these strategies that are super useful going forward. </p> <p dir="ltr">So, to me, those are some of the really strong speaking points. I like to play a game and then just stop and pause for a minute and just say, "Did you see how hard it was for me to get you quiet? Do you see how much fun you were having?" And then I just hold up a worksheet (laughs). I'm like, "And how about this?" You know, again, that emotional connection to the experience and the outcomes. </p> <p dir="ltr"><strong>Mike:</strong> That is wonderful. </p> <p dir="ltr">Since you brought it up, let's talk about replacements for worksheets and timed tests. </p> <p dir="ltr"><strong>Jenny:</strong> Mm-hmm. </p> <p dir="ltr"><strong>Mike:</strong> So, you advocate for games, as you said, and for an activity-based approach. I think that what I want to try to do is get really specific so that if I'm a classroom teacher, and I can't see a picture of that yet, can you help paint a picture? What might that look like? </p> <p dir="ltr"><strong>Jenny:</strong> I love that question because there's lots of good games and lots of places. But again, like I said earlier, this thinking really deeply about what game I'm choosing and for what—what do my students need to practice? And then being very intentional about game choice is really important. </p> <p dir="ltr">So, for example, if students are working on their 10 and some more facts, then you want to play a game where all the facts are 10 and some more facts. That's what they're working on. And then maybe you mix in some that aren't. Or you play a game with that and then they sort cards and find all the, solve the 10 and some more, or there's lots of things they can do. They can play Concentration, where the fact is hidden and the answer is hidden and things like that. So, you can be very focused. And then when you get to the strategies, you want to have a game that allows for students to say, allow their strategies. </p> <p dir="ltr">So, I'm a big fan of, like, sentence frames, for example. So, there's games that we have in our <a href= "https://www.ascd.org/books/math-fact-fluency?chapter=preface-math-fact-fluency"> Math Fact Fluency[: 60+ Games and Assessment Tools to Support Learning and Retention]</a> book [by Jennifer Bay-Williams and Gina Kling] that are in other places that specifically work on a strategy. So, for example, if I'm working on the pretend-to-10 strategy, I like to play the game Fixed-Addend War, which is the classic game of War, except there's an addend in the middle, and it's a 9, to start. And then each of the two players turns up a card. So, Mike, if you turn up a 7, then you're going to explain how you're going to use the pretend-to-10 strategy to add it. And I turned up a 6, so I'm going to, I'm going to do this then I'll, you can do it. So, I turned up a 6. So, I'm going to say, "Well, 10 and 6 is 16, so 9 and 6 is 1 less, [which equals] 15.' I've just explained the pretend-to-10 strategy. And then you get your turn. </p> <p dir="ltr"><strong>Mike:</strong> And I'd say, "Well, 7 and 10, I know 7 and 10 is 17, so 7 and 9 has to be 1 less, and that's 16."</p> <p dir="ltr"><strong>Jenny:</strong> Yeah. So, your total's higher than mine. You win those two cards, you put them in your deck, and we move on. So, that's a way to just practice thinking through that strategy. Notice there's no time factor in that. You have a different card than I have. You have as much time, and we're doing think-aloud. These are all high-leverage practices. Then we get to the games where it's like, you might turn up a 6 and a 5 where you're not going to use the pretend-to-10 strategy for that. You've got to think, "Oh, that doesn't really fit that strategy because neither one of those numbers is really close to 10. Oh, hey, it's near a double; I'm going to use my doubles." So, you sequence these games to—if you start with one of those open-ended games, it might be too big of a jump because students aren't ready to choose between their strategies. They have to first be adept at using their strategies. And once they're adept at using them, then they're ready to play games where they get to choose among the strategies. </p> <p dir="ltr"><strong>Mike:</strong> So, you're making me think a couple things, Jenny. One is, it's not just that we're shifting to using games as a venue to practice to get to automaticity. You're actually saying that when we think about the games, we really need to think about, "What are the strategies that we're after for kids?" And then make sure that the way that the game is structured, like, when you're talking about the pretend-to-10, with the fixed addend. That's designed to elicit that strategy and have kids work on developing their language and their thinking around that particularly. So, there's a level of intent around the game choice and the connection to the strategies that kids are thinking about. Am I understanding that right? </p> <p dir="ltr"><strong>Jenny:</strong> That's it. That's exactly right. That's exactly right. And a huge—a lot of intentionality so that they have that opportunity in a no-pressure, a low-stress, think-through-the-strategy [way]. If they make a mistake, their peer or themselves usually correct it in the moment, and they get so much practice in. I mean, imagine going through half a deck of cards playing that game. </p> <p dir="ltr"><strong>Mike:</strong> Yeah. </p> <p dir="ltr"><strong>Jenny:</strong> That's 26 facts. And then picture those 26 facts (laughs) on a page of paper. And then—and again, in the game that you've got the added benefit of think-aloud, and then you're hearing what your peer has said. </p> <p dir="ltr"><strong>Mike:</strong> You know, one of the things that strikes me is, if I'm a teacher, I might be thinking, like, "This is awesome, I'm super excited about it. Holy mackerel, do I have to figure these games out myself?" And I think the good news is, there's a lot of work that's been done on this. I know you've done some. Do you have any recommendations for folks? There's, of course, curriculum. But do you have recommendations for resources that you think, help a teacher think about this or help a teacher see some of the games that we're talking about? </p> <p dir="ltr"><strong>Jenny:</strong> Well, I'm going to start with my <a href= "https://www.ascd.org/books/math-fact-fluency?chapter=preface-math-fact-fluency">Math Fact Fluency</a> book because that is where we go through each of these strategies, each of the foundational facts sets and the strategies, and for each one supply a game. And then from those games they're easily adaptable to other settings. And some of the games are classic games. So, there's a game, for example, called Square Deal. And the idea is that you're covering a game board, and you're trying to make a square. So, you get a 2-by-2 grid taken, and you score a point or 5 points or whatever you want to score. Well, we have that game housed under the 10 and some more facts. So, all the answers are, like, 19, 16, 15, and the students turn over a 10 card and another card, and if it's a 10 and a 5, they get to claim a 15 spot on the game board. </p> <p dir="ltr">Well, that game board can be easily adapted to any multiplication fact sets, any other addition [sets]. I like to do a Square Deal with 10 and some more, and then I like to do Square Deal with 9 and some more. There's my (laughs) effort, again, to come back to either pretend-to-10 or making 10. Where they're like, "Oh, I just played 10 and some more. Now we're doing the same game, but it's 9 and some more." So, I feel like there's a lot of games there. And there is a free companion website that has about half of the games ready to download in English and in Spanish. </p> <p dir="ltr"><strong>Mike:</strong> Any chance you'd be willing to share it? </p> <p dir="ltr"><strong>Jenny:</strong> Yeah, absolutely. So, you can just google it. The Kentucky Center for Mathematics created it during COVID-19, actually, as a gift to the math community. And so, if you type in "Kentucky Center for Math" or "<a href= "https://kcm.nku.edu/mathfactfluency/">KCM Math Fact Fluency companion website</a>," it will pop up. </p> <p dir="ltr"><strong>Mike:</strong> That's awesome. </p> <p dir="ltr">I want to ask you about one more thing before we close because we've really talked about the replacement for worksheets, the replacements for timed tests. But there is a piece of this where people think about "How do I know?" right? "How can I tell that kids have started to build this automaticity?" And you make a pretty strong case for interviewing students to understand their thinking. I'm wondering if you could just talk again about the why behind it, and a little bit about what it might look like. </p> <p dir="ltr"><strong>Jenny:</strong> So, first of all, timed tests are definitely a mistake for many reasons. And one of the reasons—beyond the anxiety they cause—they're just very poor assessment tools. So, you can't see if the student is skip-counting or not, for example, for multiplication facts. You can't see if they're counting by 1s for the addition facts. You can't see that when they're doing the test, and you can't assume that they're working at a constant rate; that they're just solving one every, you know, couple of seconds, which is the way those tests are designed. Because I can spend a lot of time on one and less time on the other. So, they're just not, they're just not effective as an assessment tool. </p> <p dir="ltr">So, if you flip that. Let's say they're playing the game we were talking about earlier, and you just want to know, "Can they use the pretend-to-10 strategy?" That's your assessment question of the day. Well, you can just wander around with a little checklist (chuckles), you know? "Yes, they can." "No, they can't." And so, a checklist can get at the strategies, and a checklist can also get at the facts, like, "How well are they doing with their facts?" So, once they do some of those games that are more open-ended, you can just observe and listen to them and get a feel for that. If they're playing Square Deal with whatever fact, you know. So, what happens is, you're like, "I wonder how they're doing with their 4s. We've really been working with their 4s a lot.' Well, you can play Square Deal or a number of other games where that day you're working on 4s. The [game] Fixed-Addend War can become Fixed-Factor War, and you put a 4 in the middle. So adaptable games and then you're just listening and watching. </p> <p dir="ltr">And if you're not comfortable with that approach, then they can be playing those games, and you can have students channeling through where you do a little mini-interview. It only takes a few questions to get a feel for whether a student knows their facts. And you can really see who's automatic and who's still thinking. So, for example, a student who's working on their 4s, if you give them 4 times 7, they might say, "28." I call that automatic. Or they might, they might do 4 times 7, and they pause, and they're like, "28." Then I'm like, "How did you think about that?" And they're like, "Well, I doubled and doubled again." "Great." So, I can mark off that they are using a strategy, but they're not automatic yet. So that to me is a check, not a star. And if I ask, "How did you do it?" And they say, "Well, I skip-counted." Well then, I'm marking down they skip-counted. Because that means they need a strategy to help them move toward automaticity.</p> <p dir="ltr"><strong>Mike:</strong> I think what strikes me about that, too, is, when you understand where they're at on their journey to automaticity, you can actually do something about it as opposed to just looking at the quantity that you might see on a timed test. What's actionable about that? I'm not sure, but I think what you're suggesting really makes the case that I can do something with data that I observe or data that I hear in an interview or see in an interview. </p> <p dir="ltr"><strong>Jenny:</strong> Absolutely. I mean this whole different positioning of the teacher as coaching the student toward their growth; helping them grow in their math proficiency, their math fluency. You see where they're at and then you're monitoring that in order to move them forward instead of just marking them right or wrong on a timed test. I think that's a great way to synthesize that. </p> <p dir="ltr"><strong>Mike:</strong> Well, I have to say, it has been a pleasure talking with you. Thank you so much for joining us today.</p> <p dir="ltr"><strong>Jenny:</strong> Thank you so much. I am, again, thrilled to be invited and always happy to talk about this topic. </p> <p dir="ltr"><strong>Mike:</strong> This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.</p>
May 7, 2026
Season 4 | Episode 17 – Jana Dean & Heather Byington, Supporting Multilingual Learners During Number Talks
<h1 dir="ltr">Jana Dean & Heather Byington, Supporting Multilingual Learners During Number Talks</h1> <h3 dir="ltr">ROUNDING UP: SEASON 4 | EPISODE 17</h3> <p dir="ltr">What might it be like to engage in a number talk as a multilingual learner? How would you communicate your ideas, and what scaffolds might support your participation? </p> <p dir="ltr">Today, we're talking with Jana Dean and Heather Byington about ways educators can support multilingual learners' engagement and participation during number talks. </p> <h3 dir="ltr">BIOGRAPHIES</h3> <p dir="ltr">Heather Byington has taught all grade levels over the span of her 27-year career as a bilingual public educator. She currently teaches middle school mathematics and English language support classes in Lacey, Washington. She is also a student at Washington State University pursuing a PhD in Mathematics Education. </p> <p dir="ltr">Jana Dean currently serves as CEO of the Mathematics Education Collaborative and supports a fantastic team of middle school math teachers in North Thurston Public Schools. Her research focuses on the intersection of content learning and language learning. </p> <h3 dir="ltr">RESOURCES</h3> <p dir="ltr"><a href= "https://campusdirectory.ucsc.edu/cd_detail?uid=jmoschko">Judit Moschkovich</a> research </p> <p dir="ltr"><a href="http://mathbetweenus.org">Math Between Us</a> blog</p> <p dir="ltr"><a href= "https://www.mathbetweenus.org/2025/12/01/number-talks-a-whole-class-routine-for-learning-language-for-learning-mathematics/"> "Number Talks: A Whole Class Routine for Learning Language for Learning Mathematics"</a> article </p> <p dir="ltr"><a href="https://www.mec-math.org/">Mathematics Education Collaborative</a> website </p> <p dir="ltr"><a href= "mailto:jdean@mec-math.org">jdean@mec-math.org</a> Jana Dean email</p> <h3 dir="ltr">TRANSCRIPT</h3> <p dir="ltr"><strong>Mike Wallus:</strong> Welcome to the podcast, Jana and Heather. I am so excited to be talking with you both today.</p> <p dir="ltr"><strong>Jana Dean:</strong> Good morning. Yeah, thanks for having us. </p> <p dir="ltr"><strong>Heather Byington:</strong> Thanks so much for having us. </p> <p dir="ltr"><strong>Mike:</strong> Absolutely. </p> <p dir="ltr">Jana, before we begin talking about the ways that teachers can support multilingual learners during number talks, I wonder if you can offer a working definition that would help educators visualize what a number talk actually looks like.</p> <p dir="ltr"><strong>Jana:</strong> Yeah, I'd be happy to do that. A number talk in terms of how we worked with the routine in this project consisted of the teacher providing some sort of visual prompt, starting either with a visual pattern of dots or a computation problem. And then the students get wait time, time to think about how they might solve that problem. And then as they share their strategies, the teacher records and asks them questions about their reasoning for why they approached the problem in the way that they approached it. The teacher creates what I like to think of as a visual mediator of student ideas. So the students' ideas become visible as they share them. So children who are listening can listen to the dialog or conversation between the person sharing and the teacher, but the ideas actually become visible as they're being shared. And the teacher always verifies with the student whether or not they've been understood. And the goal is not for the student to be right, but for the teacher and student to understand each other. </p> <p dir="ltr"><strong>Mike:</strong> That's really helpful. Heather, is there anything else you'd add to that? </p> <p dir="ltr"><strong>Heather:</strong> In terms of the way that we worked with it with multilingual learners and increasing their opportunities for engagement in the routine, we always gave them an option of talking to a partner and rehearsing their answer before they volunteered to share with the whole group. We prioritized calling on multilingual learners if they volunteered. And we also did a final reflection at the end. So those were some enhancements that we added onto the routine. </p> <p dir="ltr"><strong>Mike:</strong> I think that's really helpful and I'm excited to talk a little bit more about the details of those, Heather. </p> <p dir="ltr">One of the things that really struck me as we were preparing for this conversation was reading about the ways that some of the multilingual learners you worked with, how they described their experience during number talks. And it helped me to see the experience from their perspective and rethink some of the ways that I'd facilitated number talks in the past. And I'm wondering if you could share a bit about some of the feelings students told you that they were experiencing. </p> <p dir="ltr"><strong>Jana:</strong> Yeah. One of the things we suspected before we started was that as a language learner myself, talking about ideas that you're just forming in a language you're in the process of learning can be really intimidating. It's very challenging. So they were nervous. And when I interviewed fourth graders about their experience in number talks, even facilitated with language acquisition in mind, they talked about how much courage it took them to share their ideas. </p> <p dir="ltr">They also talked about and could very keenly remember moments when they had made a contribution that their teacher made use of or a time when they made a contribution that another student made use of later. So there was a lot of pride they felt in having shared their ideas once they found ways to do that. </p> <p dir="ltr">They also talked about how much easier it was to share our ideas than it was to share my idea. And so if, for instance, we had given them the opportunity—and like Heather said, we almost always gave them the opportunity to talk with a partner—they would often share using the pronoun "we." "This is how we thought of it." And we picked up on that and began to ask them if it was OK to attribute a group of students with a unique idea rather than an individual. And that was also consistent with many of their home cultures. It's not every culture in which individual contributions are elevated, but rather when you dare to speak, you're definitely speaking for the group, for a collective. So that collective understanding was really important. </p> <p dir="ltr">There was one child, and I'm really curious about how representative he was of many. He always talked to the same friend, and every time he shared, he, I'm going to say, nailed it. He really had it figured out what it was that he was going to say. And there was one particular day when he did a beautiful job sharing, and I asked him about that day and he said, "To be honest, that day I really didn't want to share, but I knew my teacher wanted to hear my idea, so I did anyway." And so there's that element of love and respect for their teacher that I think was also really motivating for them. </p> <p dir="ltr"><strong>Heather:</strong> Yeah. Can I add something quickly to that? </p> <p dir="ltr">So one aspect of that, I think that idea of a student sharing because it meant a lot to the teacher, we also tried to utilize individual conferring with students as much as possible and gave them opportunities to confer with us, whether it was just checking in briefly before the number talk started, encouraging them or maybe telling them, "Hey, you can share the idea with me after the number talk if that feels more comfortable to you." So it's giving them multiple opportunities to do that and encouraging them to share their thoughts. </p> <p dir="ltr"><strong>Mike:</strong> What I appreciate about what you all are doing is even in this initial part of the conversation, really getting specific about the practices and the way that those practices played out for kids. And I think as an educator, one of the things that I've come to over all my years teaching is the need to have humility and also continue to be a learner. And that sometimes really leads me to questions about intent versus impact. </p> <p dir="ltr">Heather, I wonder if you could talk about the parts of the number talk routine or facilitation practices that may have unintentionally provoked some of the anxiety that kids were experiencing. </p> <p dir="ltr"><strong>Heather:</strong> So for multilingual learners, when I think about what they will need, the supports that they may need to be able to engage in a routine like a number talk, I think about first the processing time that they might need to understand and think about different ways of solving that prompt. And then I think about their understanding of the prompt. And then the other thing I think about is their ability to communicate their thoughts and ideas with others. So naturally, if it seems like there's a lot of pressure because of time, if they don't have much time, if they feel that pressure to do that processing and think of those ideas and share them quickly, that may provoke anxiety because this, of course, is still a language that they're still developing. So that ability to share with a partner and rehearse those ideas and process that with a partner, that really becomes, as Jana mentioned, more of a team effort. </p> <p dir="ltr">And then being able to rehearse the words that they're going to use and the way they're going to convey that message and communicate it to others, that again reduces the anxiety because it's a lot less pressure to share my thoughts and ideas with one person than with a whole group. And if I share those thoughts with one person and they seem to understand what I mean, then now I might feel confident enough to share with more people. So I just think that naturally when it's a time constrained activity, that that naturally can provoke anxiety. </p> <p dir="ltr"><strong>Mike:</strong> Yeah. I mean, that absolutely makes sense. I will say as a child who was not quick, even in my first language, the impact of that was profound, let alone trying to both process in a language that I was learning and feel like I was under pressure to produce an idea and describe it. That absolutely makes sense. </p> <p dir="ltr"><strong>Jana:</strong> I want to back up a bit and quote something that you said, Heather, partway through our working together, which was that Heather had some familiarity with number talks before we started working together, but had a healthy skepticism as well. And at one point she said that she wondered if we might not actually be hurting students when we are facilitating a routine that they cannot find entry into. And so it became really like a guiding light or principle of our work together to work hard to help them find entry into the routine. And something that I didn't realize until a year after we began working together and I was really closely tracking the experiences of the multilingual learners themselves—and this is kind of back to your question about intent and impact—when we listen to children's mathematical ideas with the intent of not correcting them, trying to figure out what's right and what makes sense to them, we have to ask them questions about what their ideas are. And for many of the multilingual learners, engaging in that process itself was a huge lift language-wise. So I'm not just going to be able to say the answer or tell my teacher my strategy; I'm going to have to stick with my teacher until my teacher actually gets it. And a few of the multilingual learners that I followed over the course of a year actually said to me, "I don't like it when my teacher doesn't understand me." So while we absolutely, 100%, our intention is golden. It is about understanding them. But putting them in that position of that negotiating meaning with us until we do understand takes a great deal of trust on the part of the student. And so it's on us to develop that trust so that they're willing to do that with us. </p> <p dir="ltr"><strong>Mike:</strong> I think that's a good segue because Jana, going into this, you mentioned three big ideas as starting points for supporting multilingual learners. One was negotiated meaning, one was the notion of voluntary sharing, and the last was the idea of using ambiguity as a resource. And I wonder if we can start this next part of the podcast with having you describe each of these for the listeners. </p> <p dir="ltr"><strong>Jana:</strong> Yeah, absolutely. </p> <p dir="ltr">Voluntary sharing means I've made a commitment to not ever put you on the spot as a student. And so any one of us who has learned a second language—which I've learned a couple, none of them to a super high level—but most people can relate to, say, standing in line in a grocery store and rehearsing what you're going to say so that you ask for the bag you want rather than the receipt that you don't want. There's a process in coming to speak, and I think there's a process in coming to speak publicly for just about every learner, especially about ideas that you're in the process of forming, but that pressure—and I've had many, many students over the year thank me for being the kind of teacher in a kind of classroom where they knew that I wasn't going to call on them unless they had volunteered to share. So the level of distraction, I think that that, again, well-intentioned pressure causes, is absolutely not worth it, and especially not for our multilingual learners. </p> <p dir="ltr">Negotiated meaning really is the process of coming to understand each other, and we do it all the time. Unfortunately, often in classrooms, we end up in discourse routines that are actually not about teachers understanding students. They're about teachers asking questions for which students are supposed to have answers, which then the teacher evaluates. So what I would argue that the number talk routine turns that discourse pattern, which is often called I.R.E.—initiate, respond, evaluate—absolutely on its head. The child volunteers their idea, the teacher responds by trying to understand it as best they can, and then the student is the evaluator of whether or not the teacher actually understood them. </p> <p dir="ltr"><strong>Mike:</strong> Heather, I was hoping we could go granular on a couple pieces that I heard you talk about too. You talk a lot about something very practical, the value of predictability, and I wonder if you can talk about how predictability impacted students and what does that mean for the teacher? </p> <p dir="ltr"><strong>Heather:</strong> Absolutely. </p> <p dir="ltr">When facilitating these number talks with this goal of engaging multilingual learners or helping them find those entry points, I found it helpful as a facilitator to utilize similar types of approaches to statements I would make during the routine, and then similar ways of asking students if I was seeing things the way that they were seeing them. It seemed to help the students that we were really hoping to engage to feel more comfortable with what was happening in the routine and to lean in more to that engagement. So I think that that is one thing as a facilitator to be aware of. </p> <p dir="ltr">Jana, can you think of anything else that we haven't talked about yet? </p> <p dir="ltr"><strong>Jana:</strong> There's the whole knowing the rules of the game aspect of really any classroom routine or instructional routine. So if the student knows how this thing goes, whatever "this thing" is, then that lifts off some of the cognitive load in terms of participation because they don't have to be figuring out how to participate. <a href= "https://campusdirectory.ucsc.edu/cd_detail?uid=jmoschko">Judit Moschkovich</a> writes about this a lot in her research, and I think she calls it the "sociocultural aspect of learning mathematics," and she uses the word "ecological". So the environment itself really matters. And in community, our social environment is made up of all kinds of routines. So I think that part of it is important. My favorite metaphor for it is learning a new card game. The first time you play the game, it is no fun because all you're doing is trying to figure out how the cards move, how the turns go, what the rules are, and how you can play. You can't do any strategy at all. But then as you learn the game, then you can really engage in it in a thoughtful way and have fun with it. So I really think that classroom routines are like that and not only for multilingual learners, but I have the privilege of being an instructional coach now in a middle school and have seen teachers engage in routines that I can tell are 100% soothing of trauma that students have as they come into the classroom, just because they know what to expect. So not only are those kinds of regular routines really helpful for multilingual learners, but they're also trauma-informed teaching. And when I say "routine," it can be easy to misunderstand and think it's boring. It has to be an open-ended routine so that something inside it that is engaging and fun can happen. </p> <p dir="ltr"><strong>Heather:</strong> There are a couple of other things that occurred to me in terms of the students participating in the routine. I know that they started to see that we were elevating the status of gestures in terms of the communication to be another way to visualize the thinking in terms of the processing for themselves, but also a way to help others see what they were seeing and to understand their ideas. So that was one aspect of the routine that they could count on, that they could utilize gestures if needed, and that we would reinforce that. If they didn't have a mathematics label for the terminology that would typically be used in that conversation about those mathematics ideas, they could rely on describing what they understood, and then either I, the teacher, the facilitator, or another student, providing those words and the opportunity to practice that specific mathematics language within that routine. So those were some other things that were predictable and happened across all of the different number talks that happened, no matter what the prompt was. </p> <p dir="ltr"><strong>Mike:</strong> You're making me think that part of what a teacher might do in response to this conversation is really to think about some of the things that they want to make normal, right? Like this notion of using gestures is both normal and accepted and valued. The idea that you are going to use rough draft, informal language, and that's OK, and that's a way that we get to more technical language of mathematics, and that's normal. And so thinking about what are the things that I want to become normal and predictable for kids, maybe homework recommendation number one for an educator that might be listening in. </p> <p dir="ltr">Heather: So another thing that was predictable was the utilization of color-coding. And this is something that many teachers probably do already. But we did, when we were recording the students' ideas, we used different colors for each student, and that made it more accessible. Again, it was a support for our students to be able to distinguish between different chunks of information on the board as they were looking at each other's responses and reflecting on those responses. So really reading that. </p> <p dir="ltr"><strong>Mike:</strong> Can I ask for a clarification on that, Heather? </p> <p dir="ltr"><strong>Heather:</strong> Absolutely. </p> <p dir="ltr"><strong>Mike:</strong> I think what you mean is that you use different [colors] to represent different students' contributions. So if a student shared something, you might write it in red, and if it was a different student, it might be in green. And then you can distinguish what contribution each student made. </p> <p dir="ltr"><strong>Heather:</strong> Yes. Yes, that was a predictable aspect of the routine, as well as Jana had mentioned earlier, attributing the ideas to students using their initials. And if multiple students contributed to that idea and the original person who was sharing said that, yes, they would like to attribute more people, then we included all the people's initials who contributed to that idea that was shared in that number talk for that idea, that communication. </p> <p dir="ltr"><strong>Mike:</strong> Speaking of contribution, I want to name something that we talked about in our preparation for this that seems incredibly simple but felt like it was really significant. You all talked about the importance of the teacher consistently—not just once, not just a handful of times—but consistently, on the regular stating to kids that they wanted to hear from all students. And I wonder if you can just talk about what did this sound like to make that happen and what was the impact on kids? Jana, I think this is one I'd love for you to start with. </p> <p dir="ltr"><strong>Jana:</strong> Yeah, absolutely. It is simple. All you say is, "I'm so glad to be with you today. And let's remember that while we may not hear from everyone today, it's our goal to hear from almost everyone over the course of the week." And if you as a teacher have made a commitment to voluntary sharing, it's essential to say that, to really tell them that you do want to hear their voices. You need to tell them that. Otherwise, they're not going to know that you want to hear their voice. And like I shared a little while ago, there was one student who actually said to me, "I didn't want to share that day, and I knew my teacher wanted to hear from me, and so I did." And then in reflecting back on that share, to get at students' perspectives on what number talks have been like for them—they were fourth graders, only 10 years old. I showed them video of themselves participating in the number talk, and you should have seen the smile on that kid's face. The pride he had in having taken that risk because his teacher wanted him to. People rise to the expectations that we have for them, 100%, maybe not 100% of the time, but if we don't have that expectation, they don't get to choose to rise to the expectation. And you can't make anyone talk when they're not ready to talk yet. </p> <p dir="ltr"><strong>Mike:</strong> Heather? </p> <p dir="ltr"><strong>Heather:</strong> I also think that part of that goes back to something that we were talking about a little while ago, and that is establishing the norms in the community of learners. And in addition to communicating that to the whole group, our goal is to hear everyone's ideas over the course of the week. Something also as simple as when they were getting ready to do a pair-share and rehearse their thoughts with each other before launching into the whole-group discussion, also reminding them, "Hey, make sure that we're taking turns when we're sharing in that pair." So again, just to reinforce that we value everybody's contribution, we value everybody's voice and everybody needs to have a turn. </p> <p dir="ltr"><strong>Mike:</strong> Can you say more about why it's important to offer kids the option to talk with a classmate before they do any whole-group sharing? Why does that matter so much, particularly for multilingual learners? And either one of you, feel free to jump in and take this. </p> <p dir="ltr"><strong>Heather:</strong> I'll start. My understanding is that when the originators of these number talks created this idea that they wanted, that idea of agency and giving students choice was really an important priority to them. And so I feel like part of the rationale for that is to give students choices as often as possible in this routine to elevate students to co-learners with the teacher. So I feel like that's kind of where it starts. </p> <p dir="ltr"><strong>Mike:</strong> Jana, is there anything you want to add to that one? </p> <p dir="ltr"><strong>Jana:</strong> Well, we've already mentioned the value of rehearsal before sharing with the whole group, but there's also another aspect of it that we may not have touched on yet, which is: As that person listens to us and we actually negotiate meeting and clear up ambiguity, we feel seen, heard, and understood. And if I feel seen, heard, and understood by Heather, it's going to be easier for me to share my idea with Mike, who I don't know quite as well as I know Heather. And so there's really a relational aspect of it that is about feeling understood. </p> <p dir="ltr"><strong>Mike:</strong> I want to ask another question about something that feels eminently practical. You all talk about recommending that educators call on multilingual learners early in number talks. And I wonder if you could say more about the why behind that recommendation. </p> <p dir="ltr"><strong>Heather:</strong> So as a learner of a new language, I may only have one way of explaining my thinking about that problem or the way that I'm seeing that. And if I have taken that risk and I've raised my hand, if somebody else answers first or maybe two other people answer first, maybe they've taken the only way that I knew to answer and share my thinking about this prompt. So for me, as a facilitator in that setting, that was really important for me to prioritize those volunteers if they raise their hand and call on them as one of the first contributors. I've also seen in some classes that I've been in, some math classes, if a student is not yet fluent in English, sometimes their classmates think that they don't know math, that they don't have ideas to share in math. So I also think that calling on those students first also, again, sets the norms in this community of learners that, again, we all have valid and valuable ideas to share. </p> <p dir="ltr">And so Jana and I saw in particular with the pair-shares, we saw students starting to choose to work with students who still spoke primarily another language. And Jana captured on video where she had a student who didn't speak Spanish and a student who primarily spoke Spanish and they were sharing ideas with each other in that pair-share to get ready for the whole-group discussion. And honestly, I think that that worked more effectively because of that idea that everybody has valuable ideas to share. So I also think that that was another part of that idea of calling on those students first and making sure that they had a lot of opportunities to share their ideas. </p> <p dir="ltr"><strong>Mike:</strong> Yeah. I'm really glad you mentioned that. You're making me think about this notion called positioning, meaning that the choices that we make—whether they're spoken or unspoken, like who we call on first or who gets called on more—they are sending a message to students. And often that message may not be the one we intended. So in this case, it really does show how the choices that you all were making in calling on multilingual learners early, it may have disrupted some narratives that people could have formed about how much those kids had to contribute to a mathematical conversation. I'm so glad you shared that. </p> <p dir="ltr">Jana, I want to ask you this next question. It's something that, if I'm not mistaken, Heather brought up earlier, and I wanted to dig into it a little bit more if we could. You referenced the value of making gestures something that's a normal, accepted, valued practice, and I want to take a bit of time to clarify that. Perhaps for some folks who might not have a clear picture in their own mind of what we mean by that, can you say more about what we mean by gestures and maybe some examples of the ways that gestures either help students to communicate or even how they contributed to the conversation that was happening during the number talk where there might've been something that was lost if gestures weren't in play? </p> <p dir="ltr"><strong>Jana:</strong> One thing I know for sure is that lately I've been learning from Heather about how some mathematical ideas are actually perhaps communicated better with gesture than verbally. And yet we have this traditional notion that there's some kind of language for expressing mathematics that's fancy and only occurs from the neck up, but that's not how we usually talk. So why would we tell people who are trying to explain their ideas that they can't use gesture as part of a person-to-person conversation? Gesture by no means keeps you from developing formal language. It actually helps you develop formal language. So one example of using gesture, it came up particularly during dot talks when we first started the routine, and the dot talks were a fabulous way to encourage and introduce that norm that gestures are welcome. But if a student is describing an array of dots and they say, "three on top," and then they use their hand to indicate it's horizontal, we would affirm, "Thank you so much for using your hands." I can tell that the three on top are in a horizontal line. And then, Heather is fabulous, and I've learned a lot about this from her at gesturing "horizontal" by bringing her hand across the space in front of her horizontally. And then everyone [says] "horizontal," and everyone gestures and says "horizontal" with them. And so we're pairing what's an academic word that is often very hard for students with any language background to remember with a physical gesture. </p> <p dir="ltr"><strong>Mike:</strong> That's really helpful. As you all were talking about this, one of the things that I started thinking about is how there are ways that I use gestures to indicate a lot of mathematical ideas like partitioning into groups, indicating that I'm talking about a group and another group and another group, which is basically the seeds of multiplication or unitizing. How I'll gesture as a way to show that I'm combining or separating. How I gesture to show the way that I'm counting things. That all of those are ways that actually enhance what I might be saying and actually communicate that meaning more clearly both to my teacher and to the other students who are in the room. </p> <p dir="ltr"><strong>Heather:</strong> Absolutely. Yeah. Another example of that, as you were talking about that, that I use all the time as a seventh grade mathematics teacher and we're working a lot with integers, is the idea of 0 in a horizontal hand as 0. And thinking about if that's 0 and I'm navigating between positive and negative numbers, what will that look [like] visually? And as you said, I just think that gestures are another tool for thinking and understanding and processing information and sometimes communicating that information. </p> <p dir="ltr"><strong>Mike:</strong> Heather, I want to come back to you for something that, again, really struck me as important when we were preparing for this. You said that you recommend educators close their number talks with an opportunity for kids to make connections between strategies that emerged. And I wonder if you can just talk about: Why is it important to provide that opportunity for kids to make connections, particularly for our multilingual learners?</p> <p dir="ltr"><strong>Heather:</strong> So first of all, I have a firm belief that development of conceptual understanding is really valuable in mathematics. And as we are engaging in this routine, in this whole-group discussion, and we're considering all these different possible ways of solving a prompt or seeing a prompt, then when we get to the end, it feels like that we should reflect on the different ideas that have been shared and draw some conclusions about what we can say across all of these different ideas as part of that development of conceptual understanding of what is happening there mathematically. In addition to that, in terms of student engagement, some of our students are multilingual learners. That was the time in the routine that they actually felt the most confident to contribute their thoughts and ideas. So maybe they didn't often raise their hand to speak in that whole-group discussion, but they did raise their hand to share something they noticed from the artifact, some kind of commonality or something that stood out to them. </p> <p dir="ltr">So again, that was another opportunity for them to feel like they had a valid contribution, that their contribution needed to be heard. So those are a couple of good reasons why I feel like that final reflection is really important in particular for multilingual learners. </p> <p dir="ltr"><strong>Mike:</strong> Well, Jana, before we close this conversation, I'm wondering if there are any resources that you'd recommend to a listener who wants to keep learning about the ideas and the practices that we've been discussing today. Is there anything that you could point them in the direction of, or perhaps even something that you'd invite them to try out as a first step? </p> <p dir="ltr"><strong>Jana:</strong> Yes, absolutely. I have a couple of ideas. One would be to go to a blog I write that's called <a href= "http://mathbetweenus.org">mathbetweenus.org</a>. And I've published a short article there [<a href= "https://www.mathbetweenus.org/2025/12/01/number-talks-a-whole-class-routine-for-learning-language-for-learning-mathematics/">"Number Talks: A Whole Class Routine for Learning Language for Learning Mathematics"</a>] that is specifically about the adjustments we've made to the routine. </p> <p dir="ltr">Also, I am now CEO of the Mathematics Education Collaborative, and we recently developed a grassroots workshop in making number talks meaningful. It only takes 2 hours. It's an introduction to the routine, ensuring that it's more than just something fun, but actually results in building number sense for students. It's a low-cost way for an individual teacher to get started. And then you can also go to our website at the Mathematics Education Collaborative, which is [<a href= "http://www.mec-math.org">www.mec-math.org</a>] and reach out to us and see if you're interested in having us come to your district or your region. Or you can email me at <a href= "mailto:jdean@mec-math.org">jdean@mec-math.org</a>. So lots of ideas. </p> <p dir="ltr"><strong>Mike:</strong> I think that's a great place to stop. I can't thank you both enough for joining me and being willing to have such an in-depth and detailed conversation. Jana and Heather, it's really been a pleasure talking with you both. Thank you. </p> <p dir="ltr"><strong>Jana:</strong> You're welcome. </p> <p dir="ltr"><strong>Heather:</strong> Thank you so much. </p> <p dir="ltr"><strong>Jana:</strong> Thanks for your curiosity. </p> <p dir="ltr"><strong>Mike:</strong> This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. </p> <p dir="ltr">© 2026 The Math Learning Center | <a href= "http://www.mathlearningcenter.org">www.mathlearningcenter.org</a></p>
April 23, 2026
Season 4 | Episode 16 – Kristin Frang, Understanding the Roots of Fluency with Addition & Subtraction
<h1 dir="ltr">Kristin Frang, Understanding the Roots of Fluency with Addition & Subtraction</h1> <h3 dir="ltr">ROUNDING UP: SEASON 4 | EPISODE 16</h3> <p dir="ltr">Research suggests that supporting students' fluency with addition and subtraction hinges on understanding how children's mathematical thinking develops. So what are the concepts and ideas that play a part in fluency with combinations to 10, 20, and beyond? </p> <p dir="ltr">Today, we'll explore this question with Kristin Frang, director of instructional programs at Integrow Numeracy Solutions. </p> <h3 dir="ltr">BIOGRAPHY</h3> <p dir="ltr">Kristin Frang is the director of instructional programs for Integrow Numeracy Solutions. She designs resources and services that support states, districts, schools, and individuals in transforming numeracy education.</p> <h3 dir="ltr">RESOURCES</h3> <p dir="ltr"><a href= "https://www.mathlearningcenter.org/blog/understanding-units-coordination"> "Understanding Units Coordination"</a> Season 4, Episode 11 of the Rounding Up podcast</p> <p dir="ltr">Integrow Numeracy Solutions</p> <ul> <li dir="ltr" aria-level="1"> <p dir="ltr" role="presentation"><a href= "http://www.integrowmath.org">website</a></p> </li> <li dir="ltr" aria-level="1"> <p dir="ltr" role="presentation"><a href= "https://www.integrowmath.org/blog">blog</a> </p> </li> <li dir="ltr" aria-level="1"> <p dir="ltr" role="presentation"><a href= "mailto:info@integrowmath.org">email address</a></p> </li> </ul> <p dir="ltr"><a href= "https://www.integrowmath.org/store/3978621">On Track to Numeracy</a> book by Lucinda "Petey" MacCarty, Kurt Kinsey, David Ellemor-Collins, and Robert J. Wright</p> <h3 dir="ltr">TRANSCRIPT</h3> <p dir="ltr"><strong>Mike Wallus:</strong> Welcome to the podcast, Kristin. It is so great to be talking with you today. </p> <p dir="ltr"><strong>Kristin Frang:</strong> It's great to be here. I feel so honored to be on this podcast. </p> <p dir="ltr"><strong>Mike:</strong> Before we dive into a conversation about addition and subtraction, I'd like to do a bit of grounding. So you're currently the director of instructional programs for Integrow Numeracy Solutions. I wonder if briefly you could tell the listeners: What is Integrow Numeracy Solutions, and what's its mission? </p> <p dir="ltr"><strong>Kristin:</strong> Yeah. Integrow Numeracy Solutions' mission is to transform numeracy education by connecting research with practice and empowering educators to advance student mathematical thinking and success. But I really want to bring that mission to life through a story, just a quick story, if I can. </p> <p dir="ltr">Prior to my role with Integrow, I was a K–12 mathematics consultant. And one of the things that I did was, when the Common Core [State Standards] were released, I worked with teachers to transition to the then-new standards. We studied many documents together, including progression documents that were included in the standards, and teachers were honestly fascinated by this idea of a progression and that they were embedded into the standard. But I remember an instance where we had been studying these progressions and a teacher came up and said to me, "I know where my students are at; I can see them in these progressions. But how do I get them to the next stage?" </p> <p dir="ltr">And I didn't have an answer (laughs) at that point. I was a former middle school and high school teacher. I was working with elementary teachers. I was studying, just like them, these progression documents, and I could only categorize the reasoning that was in front of us. And so that next step to say, "Oh, this is what I would do and bring into action in the classroom," I didn't have an answer for.</p> <p dir="ltr">And so that's really where I was introduced to Integrow—formerly [the] US Math Recovery Council, but now Integrow Numeracy Solutions. And at the heart of our mission to empower educators is to bring research to the classroom in accessible and practical ways that advance student reasoning. We do this in professional learning, we do it in supplemental resources, and we also hire and train educators to deliver high-dosage tutoring for students to accelerate their learning. </p> <p dir="ltr"><strong>Mike:</strong> I want to just linger on something you said, which was—and I really appreciate both the truth of the statement you made and also the vulnerability, which is to say—I think for many teachers, there's this experience of, "I can see my students in these progressions, but I'm not sure what to do when it comes to making moves to shift where they're at or help them move." And I think that's a profound truth for so many teachers. And I think it's really important that folks like you, who are doing this work, acknowledge that that's a place you were in once as well because that's so true for so many of us. </p> <p dir="ltr"><strong>Kristin:</strong> Yeah. There's always a new thing where we're watching students, we're thinking about the next steps. And so often it boils down to categorizing the things that students are doing now, but not often figuring out: What are the true actions that we take with real children who are in front of us to get them to progress in their own reasoning? We can tell them the next step, but my belief system that is aligned with Integrow Numeracy Solutions is that the most powerful thing is to help students have those experiences and create that understanding themselves. And to do that, it's more complex than just knowing what the next benchmark is for them. </p> <p dir="ltr"><strong>Mike:</strong> I think that's a helpful introduction. And I also find it to be a good segue for all the questions that I wanted to explore today. So let me start here: It feels important to acknowledge that supporting students' addition and subtraction fluency actually hinges on understanding how children's mathematical thinking develops. So I wonder if you can talk about some of the concepts and the ideas that play a part in fluency when it comes to combinations of 10, combinations to 20, and even beyond. </p> <p dir="ltr"><strong>Kristin:</strong> Yeah. The words that we hear associated with fluency right now are "flexibility," "efficiency," "accuracy." So we've moved on from just speed, which I think is a really positive place for us to be in education. But at the heart of flexibility, efficiency and accuracy is a quantitative understanding of arithmetic. I'm really glad that you had Amy Hackenberg on [the podcast] recently who discussed this concept of units coordination because throughout what we'll talk about, you'll see units coordination come out, but she's definitely the expert to explain it. Just a nod. Just <a href= "https://www.mathlearningcenter.org/blog/understanding-units-coordination">listen to that episode</a> [Season 4, Episode 11]. It was amazing. </p> <p dir="ltr">Thinking, though, specifically about fluency—fluency isn't just knowing all of these combinations. In the early stages of counting, students view a number simply as a count or result of a count of single items, and there's this critical shift in developing a unit as a fundamental tool of measurement. And that's the act of unitizing where a student conceives of a collection of items as one unit that's simultaneously made of smaller units. </p> <p dir="ltr">It is a common progression that once a student counts on, that then we would shift to building strategies to solve addition and subtraction within 20, and then of course with 100, and beyond, and then in other domains. But this is all happening in first and second grade for that addition and subtraction to 20 fluency. So attending to this numerical composite—understanding that when a child says "7" and sees that that represents counting from 1 to 7 without having to count—is a really big cognitive shift in their mathematical understanding and can be undermined with, "Oh, now that they're counting on, we're going to tell them these strategies." And so we really do need to have some intentional instructional strategies to make sure that we're developing that first, that numerical composite, before we try to develop all these strategies for addition and subtraction to 20. Because that is the basis for children to move from a counting-based strategy to compose units. </p> <p dir="ltr">So when they can use a quantity like, "Oh, 8 plus 5, I can break apart this 5 into smaller parts and I can give some of those parts to the 8." So children at that point have to simultaneously hold 5 as a single unit while recognizing the 2 and the 3 make up the 5, but they can be moved to the 8 as well. That's really sophisticated. </p> <p dir="ltr"><strong>Mike:</strong> So I want to mark that because I think the notion that this is really sophisticated is important for folks to understand because I'll be vulnerable and honest: I didn't recognize the complexity of what children were grappling with when I started teaching, particularly as a person who was teaching kindergarten and first grade. I really saw my job as helping to build a set of rote procedures like counting and number sequence and memorizing combinations and the outcome of being able to count and the outcome of being able to quickly recall those. I think that's not in question, but understanding the mechanics and the evolution of kids' thinking that's going on, that's a big deal. This whole notion that you have a unit and the unit is composed of smaller units. And one of the things that you said that feels like a really big deal that could be lost is the idea that shifting from a counting-based strategy to a strategy that depends on this notion of units that have smaller units inside and that are also still a unit—that's such a big deal. In order to go from counting everything to counting on to being able to look at a number like 8 and say that it has a 5 and a 3 inside of it—all of that is connected to this notion of units inside of units. And I'm so glad you mentioned that. </p> <p dir="ltr"><strong>Kristin:</strong> Yeah. The mental actions that students are doing, making those visible, when we see children do it developmentally, we just assume it's easy. But the shifts that they're making in their understanding of units to move from that pre-numerical stage of "Everything is a 1 and I have to repeat it" to "Now this word can stand in for the count" to "Now I can embed units inside of other units." There's so much happening, and they're so young at that age; we have to remember that too. </p> <p dir="ltr"><strong>Mike:</strong> So let's talk about some other important components of developing fluency. What else is an important primer for how people are thinking about this? </p> <p dir="ltr"><strong>Kristin:</strong> Yeah. Another important component is supporting students in developing the cognitive structures that allow students to anchor their understanding and quantitative meaning and develop that sophisticated reasoning. Many researchers, many authors have written in different ways and different names about these structures. So like a "mental structure," "mental residue," "mental tools," "patterns of thought." To name a few people, <a href= "https://www.corwin.com/author/zaretta-hammond">Zaretta Hammond</a>, <a href= "https://www.corwin.com/author/betty-k-garner?srsltid=AfmBOopnpe66Guc5OO0mWjfPMO2JJh6sTBWkgXCKAxCbmA1iIkgTUyXs"> Betty [K.] Garner</a>, <a href= "https://www.corwin.com/author/karen-s-karp">Karen [S.] Karp</a> are some people I've read and appreciate their thinking around that. </p> <p dir="ltr">So it's more than just allowing students to use manipulatives to solve problems. There's an intentionality in how we use tools and an explicit process used by educators to bring their mathematical world to life. So first, identifying key settings that emphasize mathematical structures. So the tool in front of them has a big role to play in the "math"—I put that in quotations—in the "math" that they see. 10-frames that highlight a quantity of 10, but also can show other quantities within 10, such as, like, a five or a double. It has an added layer of boxes that contain a number. Some contain a number or a counter and others are empty. So there's ways that kids are coming to understand quantity with the structure. </p> <p dir="ltr">Similarly, a bead rack can show a five structure, a double structure, depending on your representation. They can help kids think about exchanges and really kind of that movement of quantity in a real physical way. Using linking cubes, do you use them all in one color? Are you strategic about the color that you use to bring out mathematical structures for them? </p> <p dir="ltr">So once we think about the key setting and the structure that we're trying to help kids reason about, we want to pose intentional questions that orient students to those structures. So how do they see that 5 inside? How are we going to bring that out? It's obvious to us, but are they seeing that or are they seeing something different in the tool? Are they reasoning about something different? And so the intentionality behind how we question students during those activities also aids to building their cognitive structures. So it's not the tool itself that is the 8. It's that the child is seeing the 8 and they're seeing the 5 and the 3 in some empty boxes. </p> <p dir="ltr">And finally, I think the step that we miss a lot, especially in problem-based instruction or any kind of inquiry-based instruction, is this explicit time where we connect the symbols in formal mathematics directly to represent the child's thinking and the tool that they've been playing around with. So it's not just about knowing I can get an answer on the 10-frame, but it's [that] I'm abstracting that series of actions, and I'm then connecting it to this quantity that I've written in a symbol. And are there connections between those things? And if those things aren't happening—kids are doing all those parts and pieces, but really developing the cognitive structure that they can then themselves use and take with them, I think that's what's so powerful when we talk about fluency is they can take a cognitive structure with them and fill in the mathematics in the future [when] maybe they don't have an educator in front of them asking those questions. But if they've been through those processes, then they have that structure to fill in. </p> <p dir="ltr"><strong>Mike:</strong> There's a lot that you just said that I think is important and we could probably linger on a lot of it. But on the front end of this conversation, you said it's one thing to be able to see students in a progression, and it's another thing to think about, "What's my role or what are the tools that I have to help them shift?"</p> <p dir="ltr">What I heard in that last part, particularly is this notion of almost like a translation between the physical materials kids are engaging with and the meaning that they're making of that, and then helping them to abstract that in a way where we have symbols that are representing either actions or quantities and the relationships that are happening. That part of the teacher's job and part of the moves that teachers have in their toolbox is this notion of translation—taking what I'm seeing kids doing and how what I'm hearing them say or do to make meaning of it, and then helping them make that abstraction is kind of one of the tools that's really important in a teacher's toolbox when they're thinking about helping kids make moves. </p> <p dir="ltr">In preparation for our interview, one of the things that stayed with me was you described how your own understanding of the meaning and the importance of fluency had shifted over time. And I'm wondering if you can talk about what you used to think and what is it that you think now about fluency. Could you talk about your own personal journey? </p> <p dir="ltr"><strong>Kristin:</strong> For sure. I used to think that knowing facts, just knowing them in a very static way—like I know the answer to 5 plus 3, I keep coming back to that fact—reduces the cognitive load when they were getting into higher grade levels. Well, they don't need to think about that problem, and they can think about what we're doing in seventh grade math or in algebra. </p> <p dir="ltr">But what I've come to understand is that the ways that students know their facts—more specifically how they're able to work with the units and the way they conceptualize the units that they are given, how they break them apart, how they put them back together—that's what matters as they go. So not just knowing the answer, but that these things can be taken apart and put back together. </p> <p dir="ltr"><a href= "https://www.corwin.com/author/anderson-norton">Anderson Norton</a> is a researcher that I really love to listen to. And I listened to him at an Integrow conference once. And he talked about developing mathematics through repeatable mental actions. So this kind of relates back to those cognitive structures. One example of a group of mental actions is this idea of composable, reversible, and associative. So when I think about 8 plus 5, 5 is composed of a 2 and a 3, and I can reverse that to focus on the unit of 2, and then I can associate that quantity with the 8 to make a new unit while keeping intact the unit of 5. That's really complex, but that idea transcends the domains of mathematics. Now, I'm not an expert in units coordination research, so I hope I represented that correctly, but I've certainly experienced students struggling to keep track of different units as they work. So thinking about exponent rules, and they break apart these powers and they're writing them and they're learning all these patterns, but they're struggling to keep track of the units that they're working with. Factoring functions in algebra. We're asking them to break apart something and put it back together in these different forms, and they're losing track of these units. So these actions of composable, reversible, and associative have implications in many domains of mathematics. So the bottom line is we want to develop not the fact itself, but the mental action behind that fact. Anderson Norton, I hope I did that justice. </p> <p dir="ltr"><strong>Mike:</strong> I want to name something that I think is really important, particularly given the fact that your background is actually in secondary [education]. So what I take from this is this idea of working with units and the mental actions, that transcends arithmetic. It transcends whole numbers and even rational numbers. And it pays dividends and it keeps paying dividends in middle school and high school as kids are working in an algebra context. And I think that's worth saying out loud because it means that doing this work with elementary students to develop fluency is a bit of a twofer in the sense that you do get kids who end up with a bank of facts that they know, but they also have this underlying understanding of units and actions that pays dividends for them in the long run. Mathematics education, students' learning experience, is not a sprint or a series of handoffs. It's really a marathon. And those early experiences, they pay dividends and they keep paying dividends. I think that's really important because it reminds us, particularly as elementary educators, that we're part of a larger project. </p> <p dir="ltr"><strong>Kristin:</strong> Not only part of a project, but part of building a lifelong interest in mathematics as an actual body of research that's dynamic and not a set of things to memorize and learn so that mathematics does become applicable in these different fields because the way that I approach a problem as an expert mathematician is that I take things apart, I put them back together. That transcends many careers. It's not just about being a math teacher or a math professor. It's about coming to understand that I have autonomy and how I see relationships of things, whether they're numbers or shapes or maybe parts that I'm working on in some sort of creative field that I'm in, but that I can do all of these things and that I can be curious and repeat those actions and see how they play out in that particular study. </p> <p dir="ltr"><strong>Mike:</strong> That's well said. </p> <p dir="ltr">Well, let's talk about the what, the why, the how of combinations to 10 and 20. To begin, I want to note that we use the term "combinations," and I'm wondering if you can say more about what you mean when you refer to combinations and why they matter. </p> <p dir="ltr"><strong>Kristin:</strong> Yeah. I mean combinations not to literally mean "addition," but that combination is the idea of this relationship between parts and wholes. So that 2, 3, and 5 have this kind of additive relationship. I can put these parts together to make the whole; I can take a part out of the whole and be left with a part. I can have a part and wonder what part I need to make the whole. And so we sometimes talk about these in curriculums as "fact families," but the emphasis should be on the relationship of the parts to the whole and not filling out that kind of mimicking of like, "I know the four sentences because I know this thing." So, "If I know this, I also know this." It feels really nuanced, but in action really quite specific. </p> <p dir="ltr"><strong>Mike:</strong> So I think that's really helpful and it really does lead me to my next question about how we help kids build their fluency with combinations to 10 and 20 and beyond. So given the why that you just articulated, it seems like the how is going to be substantially different from the ways that many, if not most, adults learn to build fluency. Can you talk about that, Kristin? </p> <p dir="ltr"><strong>Kristin:</strong> We start from key combinations first. We consider a set of combinations that would be really useful in a lot of contexts. And I think many listeners will be familiar with those key combinations: doubles. Combinations of 10, of course. 5 plus because I have five fingers and then I can add some more on it, and I'm showing some finger patterns. So those are things we normally work on with students anyways. But starting again, going back to my original statement from a quantitative perspective—so not the memorization of those facts, but that I really come to understand them as quantities that are useful to me. And then building from those key combinations—I also want to name before I build onto that, is that some kids just have other facts that are interesting to them that they bring. So it might be their age, it might be the combination of their siblings' ages. And so we don't want to ignore that we introduce key combinations to students, but that students also have combinations that are useful to them naturally. </p> <p dir="ltr">So once we have a set of those key combinations that we've come to think about and reason about, we can then build things that we don't know. We can transfer that. So 5 plus 3 can help me think about 4 plus 3. If I have a mental structure of a 10-frame or a bead rack that helps me think about, "Oh, there's just going to be one less counter on the top, and so I'm going to take that [counter] away." So that idea of taking the 1 out of the number is a really important mental action of them disembedding that quantity. </p> <p dir="ltr">In addition, when we think about the 5 plus, the doubles, the partitions, we're thinking about combinations that will also transcend into multidigit combinations. So addition, subtraction—whether we're working with whole numbers or decimals, we can make tens, we can make hundreds, we can make wholes, we can make zeros. And those combinations of 10 are going to be really useful for us. </p> <p dir="ltr"><strong>Mike:</strong> I'm struck by the fact that the combinations and also the mental actions that accompany them, as you said, they really do scale up quite nicely. And it seems like they scale up in the sense that they can get used to understand and solve problems with larger whole numbers, but they can also scale in the sense that ideas will help kids, but they can also scale in the sense that the ideas can really help kids when they encounter fractions and decimals. I wonder if you could talk about that idea just a little bit. </p> <p dir="ltr"><strong>Kristin:</strong> Yeah. So thinking about a combination of 10 in this missing part. So 99 plus can help us when we're thinking about, that 99 is 1 away from 100. It can also help us think about 99 one-hundredths or 9 tenths as being one part or one unit away from a benchmark number that's really helpful for us. And so, it's just that the unit itself is different. So instead of just a whole, I'm one whole unit away from 100, I might be 1 tenth of a unit away from one whole, so the unit is just changing. </p> <p dir="ltr">The view of mathematics this way, again, is very dynamic. We're creating a world where children are thinking about units and units away across domains, across number systems. And if we come to regard units as things that we can act on, whether it's a single object or a group of objects or a shape—we can put them together, take them apart and reassociate them—I can think of a lot of my mathematical knowledge in this way and not as a static set of information that I learned. And so then I'm able to transfer that because I've done that mental action or I've thought about something being a unit away. </p> <p dir="ltr"><strong>Mike:</strong> That's fascinating because I'm going to go back to this whole notion of the relationship between 3 and 2 and 5. So 3 is 2 units away from a unit of 5 and three-fifths are 2 one-fifths away from a unit of five-fifths or one whole. This notion of units away from or units that combine to make other units, I really get now whether it's whole numbers or fractions, we're really talking about a unit that we've defined and then how many other units or how can we—how did you describe that? What was the language you used before about pulling a unit out? Was it "disembed"? </p> <p dir="ltr"><strong>Kristin:</strong> "Disembed," yeah. </p> <p dir="ltr"><strong>Mike:</strong> That really plays regardless of the type of unit we're talking about. </p> <p dir="ltr"><strong>Kristin:</strong> Yeah. And remember back where we said this quantity had a meaning, so 7 stood for something. When we disembed, that unit still has meaning in the context of the original unit. So that's a really important point about disembedding is that it's not just that you take a part out, it's that part still has a relationship to the whole and you don't lose that relationship. </p> <p dir="ltr"><strong>Mike:</strong> As I hear you talking, there seem to be some themes that are jumping out. One is the importance of key fact combinations and the mental actions. Another is the role visual models play in learning those combinations. And I think finally, I hear you indicating that it's important for students to make connections between different representations of the same combination. Tell me what I understood properly. Tell me what you'd revise or add to the summary that I just offered. </p> <p dir="ltr"><strong>Kristin:</strong> Yes. I think we get a false sense that a student understands a concept when they're recognizing pattern, and that could be that they're recognizing pattern in a really intentional setting. Maybe they're using a 10-frame. But is that same relationship present in another setting? Success should not be measured by one instance of a child recognizing that pattern. And so one way of knowing that a child knows this is to see it in many contexts. And I think that's why it's so important for us to acknowledge the research around multiple representations in mathematics. And showing that knowledge in these multiple ways really does say that this is a connected set of knowledge that I can refer to as a child and not just be successful on this one day. That doesn't mean that that experience where they're recognizing the patterns is not important, but that can't be the measure of their success. </p> <p dir="ltr">So this also becomes challenging in our system that values assessment events so heavily and measuring against a set benchmark. And I just want to name that because that's a real challenge for teachers. And of course we want to develop this rich set of knowledge, and sometimes we have to say that this is the system that we live in. But the true measure of that knowledge is being able to take that knowledge and transfer it into these multiple representations or in these multiple spaces and be able to use that. And that's why we talk so much about fluency being flexible and not just about accuracy. </p> <p dir="ltr"><strong>Mike:</strong> You have me thinking more deeply than I have in a long time about the structure of some of the visual models and the physical materials that children use when they're engaged with the Bridges curriculum. I wonder if we could get specific and talk about a few of the visual models that support student learning. Are there features that make some models particularly valuable? </p> <p dir="ltr"><strong>Kristin:</strong> One I want to mention that we might not have talked about is just a child's fingers. I think sometimes we think child's fingers are not models for them because they're counting by 1 and we tend to want students to move to more efficient strategies. But these fingers actually become really efficient tools. We can exchange fingers, we can move them very easily. We have control, and they're always with us. And so the finger use itself, I think, is a really powerful tool for us to encourage students to use in very sophisticated ways. </p> <p dir="ltr"><strong>Mike:</strong> I mean, we literally have units of 1, units of 5, and a unit of 10 at our fingertips in front of us. I'm so glad you called that out because that's a tool that students can make use of, that teachers can make use of and that we can think of in a slightly different way than we had in the past when I just thought about fingers as a counting-by-1 resource, when actually fingers, [a hand], and hands, plural, are 1s, 5s, and 10s right there in front of you. </p> <p dir="ltr"><strong>Kristin:</strong> And they can stand in for other units if we're really sophisticated with sequences. So a 1 can be a 7 if we wanted it to be, and we can think really creatively about that. I mean, I think that depends on some other skills. But yeah, we have 1s, 5s, and 10s built right into our hands. </p> <p dir="ltr"><strong>Mike:</strong> That's exactly right. And you're making me think about the fact that when I skip-count or when I see students skip-count, oftentimes what's happening is I'm speaking the unit out loud and I'm holding up one finger to stand in for that unit on my hand to keep track of the number of units. So I totally hear what you're saying. </p> <p dir="ltr"><strong>Kristin:</strong> Yeah, very sophisticated. And then there's even more complex content, right? So thinking about hours and elapsed time, and we're crossing different kinds of numerical systems where you go from a 12 to a 1 is very complex, and then we can have these fingers as units as well to help us keep track of things. So of course, frames are a really powerful tool. Frames—specifically, 10-frames, 5-frames, 20-frames—provide an extra structure for students, especially when they're really thinking hard about some quantity pieces. So they might not be completely solid in that unit, but we don't have to say, "Oh, you have to count on first before we're going to try to explore some other patterns." Those things can be developing simultaneously. So frames provide this box that contains the unit for them and it becomes this really obvious count for them. They can see those individual discrete items, but they can also see what's missing really clearly because they're empty. </p> <p dir="ltr">Bead racks are a great support as well when you're thinking about that relational network that we want students to develop and not count by 1s. So we can exchange beads, and we can exchange quantities, and we don't have to exchange beads one by one. Sometimes frames, when we get to a space, it's inconvenient to have to move five counters at the same time where in a bead rack, you can just slide those five over or three over at the same time. </p> <p dir="ltr">I also want to mention linear bead racks. So taking that stacked bead rack and making it align really helps students think about a continuous model, which transfers to a number line and the idea of units being measurement. So we were talking about, "It's one away," and so really conceptualizing that kind of next decade of numbers and one bead away. That's developing that idea of relative magnitude that's extremely helpful when we get to middle school and all of a sudden we're working in negative numbers. </p> <p dir="ltr"><strong>Mike:</strong> We're reaching the end of our time together. And before we go, I'm wondering if you could share contact information for Integrow Numeracy Solutions with our listeners. I'd really love to be able to offer that because we've just touched the surface of some of the ideas that you help educators explore in some of the training and the support that you all offer. </p> <p dir="ltr"><strong>Kristin:</strong> Yeah. If you'd like to find out more about us, a great place to go is our website, which is <a href= "http://www.integrowmath.org">www.integrowmath.org</a>, all one word. And we have a lot of different things you can explore from our events. There is actually, if you add a backslash "blog" to that [<a href= "http://www.integrowmath.org/blog">www.integrowmath.org/blog</a>], you can go to our blog and read some of the ways that we think about our professional learning and some of the topics that I talked about today. If you want to reach out directly, feel free to email <a href= "mailto:info@integrowmath.org">info@integrowmath.org</a> and someone will get you to the right place based on your question. </p> <p dir="ltr"><strong>Mike:</strong> And for listeners, we'll put a link to both of those in the show notes.</p> <p dir="ltr">Before we leave, Kristin, I'll just ask one last question. Are there any recommendations that you have for folks interested in learning more about the ideas we've talked about today? It could be books, websites, articles, or even just a suggested practice for someone who wants to get started. </p> <p dir="ltr"><strong>Kristin:</strong> Yeah. For sure, take a look at the blogs on our website. They're little snippets of pieces of our trainings that you can take right with you into the classroom. Some ideas that I've talked about—help with bead racks, ideas around multiplication and division, and supporting students to think about those units. Our new publication, <a href= "https://www.integrowmath.org/store/3978621">On Track to Numeracy from [Lucinda] "Petey" MacCarty, Kurt Kinsey, [David Ellemor-Colons, and Robert J. Wright]</a>, is designed to be an accessible, relatable and practical tool focused on supporting classroom teachers. It not only has the progressions that I started this podcast off talking about, but it has those teaching tests and progressions that help us answer the question of, "What do I do next now that I can understand where my students are?"</p> <p dir="ltr"><strong>Mike:</strong> I think it's a great place to stop, Kristin. I want to thank you so much for joining us. It's really been a pleasure talking with you. </p> <p dir="ltr"><strong>Kristin:</strong> Thank you for having me. I've had a great time. </p> <p dir="ltr"><strong>Mike:</strong> This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. </p> <p dir="ltr">© 2026 The Math Learning Center | <a href= "http://www.mathlearningcenter.org">www.mathlearningcenter.org</a></p>
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