Rounding Up

Pam Harris, Exploring the Power & Purpose of Number Strings ROUNDING UP: SEASON 4 | EPISODE 4 I've struggled when I have a new strategy I want my students to consider and despite my best efforts, it just doesn't surface organically. While I didn't want to just tell my students what to do, I wasn't sure how to move forward. Then I discovered number strings.  Today, we're talking with Pam Harris about the ways number strings enable teachers to introduce new strategies while maintaining opportunities for students to discover important relationships.  BIOGRAPHY Pam Harris, founder and CEO of Math is Figure-out-able™, is a mom, a former high school math teacher, a university lecturer, an author, and a mathematics teacher educator. Pam believes real math is thinking mathematically, not just mimicking what a teacher does. Pam helps leaders and teachers to make the shift that supports students to learn real math. RESOURCES Young Mathematicians at Work [https://www.heinemann.com/products/e03385.aspx] by Catherine Fosnot and Maarten Dolk  Procedural fluency in mathematics: Reasoning and decision-making, not rote application of procedures position [https://www.nctm.org/Standards-and-Positions/Position-Statements/Procedural-Fluency-in-Mathematics/] by the National Council of Teachers of Mathematics Bridges number string example [https://educate.mathlearningcenter.org/pdf/12698/28#page=28] from Grade 5, Unit 3, Module 1, Session 1 (BES login required) Developing Mathematical Reasoning: Avoiding the Trap of Algorithms [https://www.corwin.com/books/dmr-289132?srsltid=AfmBOorJH-HVq1ibguASkTiA-ycE_iaO1cpFQE38SGF1k4tsJyMUNhF2] by Pamela Weber Harris and Cameron Harris Math is Figure-out-able!™ Problem Strings [http://mathisfigureoutable.com/ps] TRANSCRIPT Mike Wallus: Welcome to the podcast, Pam. I'm really excited to talk with you today. Pam Harris: Thanks, Mike. I'm super glad to be on. Thanks for having me. Mike: Absolutely.  So before we jump in, I want to offer a quick note to listeners. The routine we're going to talk about today goes by several different names in the field. Some folks, including Pam, refer to this routine as “problem strings,” and other folks, including some folks at The Math Learning Center, refer to them as “number strings.” For the sake of consistency, we'll use the term “strings” during our conversation today.  And Pam, with that said, I'm wondering if for listeners, without prior knowledge, could you briefly describe strings? How are they designed? How are they intended to work? Pam: Yeah, if I could tell you just a little of my history. When I was a secondary math teacher and I dove into research, I got really curious: How can we do the mental actions that I was seeing my son and other people use that weren't the remote memorizing and mimicking I'd gotten used to?  I ran into the work of Cathy Fosnot and Maarten Dolk, and [their book] Young Mathematicians at Work, and they had pulled from the Netherlands strings. They called them “strings.” And they were a series of problems that were in a certain order. The order mattered, the relationship between the problems mattered, and maybe the most important part that I saw was I saw students thinking about the problems and using what they learned and saw and heard from their classmates in one problem, starting to let that impact their work on the next problem. And then they would see that thinking made visible and the conversation between it and then it would impact how they thought about the next problem. And as I saw those students literally learn before my eyes, I was like, “This is unbelievable!” And honestly, at the very beginning, I didn't really even parse out what was different between maybe one of Fosnot's rich tasks versus her strings versus just a conversation with students. I was just so enthralled with the learning because what I was seeing were the kind of mental actions that I was intrigued with. I was seeing them not only happen live but grow live, develop, like they were getting stronger and more sophisticated because of the series of the order the problems were in, because of that sequence of problems. That was unbelievable. And I was so excited about that that I began to dive in and get more clear on: What is a string of problems?  The reason I call them “problem strings” is I'm K–12. So I will have data strings and geometry strings and—pick one—trig strings, like strings with functions in algebra. But for the purposes of this podcast, there's strings of problems with numbers in them. Mike: So I have a question, but I think I just want to make an observation first. The way you described that moment where students are taking advantage of the things that they made sense of in one problem and then the next part of the string offers them the opportunity to use that and to see a set of relationships. I vividly remember the first time I watched someone facilitate a string and feeling that same way, of this routine really offers kids an opportunity to take what they've made sense of and immediately apply it. And I think that is something that I cannot say about all the routines that I've seen, but it was really so clear. I just really resonate with that experience of, what will this do for children? Pam: Yeah, and if I can offer an additional word in there, it influences their work. We're taking the major relationships, the major mathematical strategies, and we're high-dosing kids with them. So we give them a problem, maybe a problem or two, that has a major relationship involved. And then, like you said, we give them the next one, and now they can notice the pattern, what they learned in the first one or the first couple, and they can let it influence. They have the opportunity for it to nudge them to go, “Hmm. Well, I saw what just happened there. I wonder if it could be useful here. I'm going to tinker with that. I'm going to play with that relationship a little bit.” And then we do it again. So in a way, we're taking the relationships that I think, for whatever reason, some of us can wander through life and we could run into the mathematical patterns that are all around us in the low dose that they are all around us, but many of us don't pick up on that low dose and connect them and make relationships and then let it influence when we do another problem.  We need a higher dose. I needed a higher dose of those major patterns. I think most kids do. Problem strings or number strings are so brilliant because of that sequence and the way that the problems are purposely one after the other. Give students the opportunity to, like you said, apply what they've been learning instantly [snaps]. And then not just then, but on the next problem and then sometimes in a particular structure we might then say, “Mm, based on what you've been seeing, what could you do on this last problem?” And we might make that last problem even a little bit further away from the pattern, a little bit more sophisticated, a little more difficult, a little less lockstep, a little bit more where they have to think outside the box but still could apply that important relationship. Mike: So I have two thoughts, Pam, as I listen to you talk.  One is that for both of us, there's a really clear payoff for children that we've seen in the way that strings are designed and the way that teachers can use them to influence students' thinking and also help kids build a recognition or high-dose a set of relationships that are really important.  The interesting thing is, I taught kindergarten through second grade for most of my teaching career, and you've run the gamut. You've done this in middle school and high school. So I think one of the things that might be helpful is to share a few examples of what a string could look like at a couple different grade levels. Are you OK to share a few? Pam: You bet. Can I tack on one quick thing before I do? Mike: Absolutely. Pam: You mentioned that the payoff is huge for children. I'm going to also suggest that one of the things that makes strings really unique and powerful in teaching is the payoff for adults. Because let's just be clear, most of us—now, not all, but most of us, I think—had a similar experience to me that we were in classrooms where the teacher said, “Do this thing.” That's the definition of math is for you to rote memorize these disconnected facts and mimic these procedures. And for whatever reason, many of us just believed that and we did it. Some people didn't. Some of us played with relationships and everything. Regardless, we all kind of had the same learning experience where we may have taken at different places, but we still saw the teacher say, “Do these things. Rote memorize. Mimic.”  And so as we now say to ourselves, “Whoa, I've just seen how cool this can be for students, and we want to affect our practice.” We want to take what we do, do something—we now believe this could be really helpful, like you said, for children, but doing that's not trivial. But strings make it easier. Strings are, I think, a fantastic differentiated kind of task for teachers because a teacher who's very new to thinking and using relationships and teaching math a different way than they were taught can dive in and do a problem string. Learn right along with your students. A veteran teacher, an expert teacher who's really working on their teacher moves and really owns the landscape of learning and all the things still uses problem strings because they're so powerful. Like, anybody across the gamut can use strings—I just said problem strings, sorry—number strengths—[laughs] strings, all of us no matter where we are in our teaching journey can get a lot out of strings. Mike: So with all that said, let's jump in. Let's talk about some examples across the elementary span. Pam: Nice. So I'm going to take a young learner, not our youngest, but a young learner. I might ask a question like, “What is 8 plus 10?” And then if they're super young learners, I expect some students might know that 10 plus a single digit is a teen, but I might expect many of the students to actually say “8, 9, 10, 11, 12,” or “10, 11,” and they might count by ones given—maybe from the larger, maybe from the whatever. But anyway, we're going to kind of do that. I'm going to get that answer from them. I'm going to write on the board, “8 plus 10 is 18,” and then I would have done some number line work before this, but then I'm going to represent on the board: 8 plus 10, jump of 10, that's 18. And then the next problem's going to be something like 8 plus 9. And I'm going to say, “Go ahead and solve it any way you want, but I wonder—maybe you could use the first problem, maybe not.” I'm just going to lightly suggest that you consider what's on the board. Let them do whatever they do. I'm going to expect some students to still be counting. Some students are going to be like, “Oh, well I can think about 9 plus 8 counting by ones.” I think by 8—”maybe I can think about 8 plus 8. Maybe I can think about 9 plus 9.” Some students are going to be using relationships, some are counting. Kids are over the map.  When I get an answer, they're all saying, like, 17. Then I'm going to say, “Did anybody use the first problem to help? You didn't have to, but did anybody?” Then I'm going to grab that kid. And if no one did, I'm going to say, “Could you?” and pause.  Now, if no one sparks at that moment, then I'm not going to make a big deal of it. I'll just go, “Hmm, OK, alright,” and I'll do the next problem. And the next problem might be something like, “What's 5 plus 10?” Again, same thing, we're going to get 15. I'm going to draw it on the board.  Oh, I should have mentioned: When we got to the 8 plus 9, right underneath that 8, jump, 10 land on 18, I'm going to draw an 8 jump 9, shorter jump. I'm going to have these lined up, land on the 17. Then I might just step back and go, “Hmm. Like 17, that's almost where the 18 was.” Now if kids have noticed, if somebody used that first problem, then I'm going to say, “Well, tell us about that.” “Well, miss, we added 10 and that was 18, but now we're adding 1 less, so it's got to be 1 less.” And we go, “Well, is 17 one less than 18? Huh, sure enough.” Then I give the next set of problems. That might be 5 plus 10 and then 5 plus 9, and then I might do 7 plus 10. Maybe I'll do 9 next. 9 plus 10 and then 9 plus 9. Then I might end that string. The next problem, the last problem might be, “What is 7 plus 9?” Now notice I didn't give the helper. So in this case I might go, “Hey, I've kind of gave you plus 10. A lot of you use that to do plus 9. I gave you plus 10. Some of you use that to do plus 9, I gave you plus 10. Some of you used that plus 9. For this one, I'm not giving you a helper. I wonder if you could come up with your own helper.”  Now brilliantly, what we've done is say to students, “You've been using what I have up here, or not, but could you actually think, ‘What is the pattern that's happening?’ and create your own helper?” Now that's meta. Right? Now we're thinking about our thinking. I'm encouraging that pattern recognition in a different way. I'm asking kids, “What would you create?” We're going to share that helper. I'm not even having them solve the problem. They're just creating that helper and then we can move from there.  So that's an example of a young string that actually can grow up. So now I can be in a second grade class and I could ask a similar [question]: “Could you use something that's adding a bit too much to back up?” But I could do that with bigger numbers. So I could start with that 8 plus 10, 8 plus 9, but then the next pair might be 34 plus 10, 34 plus 9. But then the next pair might be 48 plus 20 and 48 plus 19. And the last problem of that string might be something like 26 plus 18. Mike: So in those cases, there's this mental scaffolding that you're creating. And I just want to mark this. I have a good friend who used to tell me that part of teaching mathematics is you can lead the horse to water, you can show them the water, they can look at it, but darn it, do not push their head in the water. And I think what he meant by that is “You can't force it,” right?  But you're not doing that with a string. You're creating a set of opportunities for kids to notice. You're doing all kinds of implicit things to make structure available for kids to attend to—and yet you're still allowing them the ability to use the strategies that they have. We might really want them to notice that, and that's beautiful about a string, but you're not forcing. And I think it's worth saying that because I could imagine that's a place where folks might have questions, like, “If the kids don't do the thing that I'm hoping that they would do, what should I do?” Pam: Yeah, that's a great question. Let me give you another example. And in that example I'll talk about that.  So especially as the kids get older, I'm going to use the same kind of relationship. It's maybe easier for people to hang on to if I stay with the same sort of relationship. So I might say, “Hey everybody. 7 times 8. That's a fact I'm noticing most of us just don't have [snaps] at our fingertips. Let's just work on that. What do you know?” I might get a couple of strategies for kids to think about 7 times 8. We all agree it's 56.  Then I might say, “What's 70 times 8?” And then let kids think about that. Now, this would be the first time I do that, but if we've dealt with scaling times 10 at all, if I have 10 times the number of whatever the things is, then often kids will say, “Well, I've got 10 times 7 is 70, so then 10 times 56 is 560.” And then the next problem might be, “I wonder if you could think about 69 times 8. If we've got 70 eights, can I use that to help me think about 69 eights?” And I'm saying that in a very specific way to help ping on prior knowledge. So then I might do something similar. Well, let's pick another often missed facts, I don't know, 6 times 9. And then we could share some strategies on how kids are thinking about that. We all agree it's 54. And then I might say, “Well, could you think about 6 times 90?” I'm going to talk about scaling up again. So that would be 540. Now I'm going really fast. But then I might say, “Could we use that to help us think about 6 times 89?” I don’t know if you noticed, but I sort of swapped. I'm not thinking about 90 sixes to 89 sixes. Now I'm thinking about 6 nineties to help me think about 6 eighty-nines. So that's a little bit of a—we have to decide how we're going to deal with that. I'll kind of mess around with that. And then I might have what we call that clunker problem at the end. “Notice that I've had a helper: 7 times 8, 70 times 8. A lot of you use that to help you think about 69 times 8. Then I had a helper: 6 times 9, 6 times 90. A lot of you use that to help you think about 6 times 89. What if I don't give you those helpers? What if I had something like”—now I'm making this up off the cuff here, like—“9 times 69. 9 times 69. Could you use relationships we just did?”  Now notice, Mike, I might've had kids solving all those problems using an algorithm. They might've been punching their calculator, but now I'm asking the question, “Could you come up with these helper problems?” Notice how I'm now inviting you into a different space. It's not about getting an answer. I'm inviting you into, “What are the patterns that we've been establishing here?” And so what would be those two problems that would be like the patterns we've just been using? That's almost like saying when you're out in the world and you hit a problem, could you say to yourself, “Hmm, I don't know that one, but what do I know? What do I know that could help me get there?” And that's math-ing. Mike: So, you could have had a kid say, “Well, I'm not sure about how—I don't know the answer to that, but I could do 9 times 60, right?” Or “I could do 10 times”—I'm thinking—“10 times 69.” Correct? Pam: Yes, yes. In fact, when I gave that clunker problem, 9 times 69, I said to myself, “Oh, I shouldn't have said 9 because now you could go either direction.” You could either “over” either way. To find 9 I can do 10, or to find 69 I can do 70. And then I thought, “Ah, we'll go with it because you can go either way.” So I might want to focus it, but I might not. And this is a moment where a novice could just throw it out there and then almost be surprised. “Whoa, they could go either direction.” And an expert could plan, and be like, “Is this the moment where I want lots of different ways to go? Or do I want to focus, narrow it a little bit more, be a little bit more explicit?” It's not that I'm telling kids, but I'm having an explicit goal. So I'm maybe narrowing the field a little bit. And maybe the problem could have been 7 times 69, then I wouldn't have gotten that other “over,” not the 10 to get 9. Does that make sense? Mike: It absolutely does. What you really have me thinking about is NCTM’s [National Council of Teachers of Mathematics’] definition of “fluency,” which is “accuracy, efficiency, and flexibility.” And the flexibility that I hear coming out of the kinds of things that kids might do with a string, it's exciting to imagine that that's one of the outcomes you could get from engaging with strings. Pam: Absolutely. Because if you're stuck teaching memorizing algorithms, there's no flexibility, like none, like zilch. But if you're doing strings like this, kids have a brilliant flexibility. And one of the conversations I'd want to have here, Mike, is if a kid came up with 10 times 69 to help with 9 times 69, and a different kid came up with 9 times 70 to help with 9 times 69, I would want to just have a brief conversation: “Which one of those do you like better, class, and why?” Not that one is better than the other, but just to have the comparison conversation. So the kids go, “Huh, I have access to both of those. Well, I wonder when I'm walking down the street, I have to answer that one: Which one do I want my brain to gravitate towards next time?” And that's mathematical behavior. That's mathematical disposition to do one of the strands of proficiency. We want that productive disposition where kids are thinking to themselves, “I own relationships. I just got to pick a good one here to—what's the best one I could find here?” And try that one, then try that one. “Ah, I'll go with this one today.” Mike: I love that.  As we were talking, I wanted to ask you about the design of the string, and you started to use some language like “helper problems” and “the clunker.” And I think that's really the nod to the kinds of features that you would want to design into a string. Could you talk about either a teacher who's designing their own string—what are some of the features?—or a teacher who's looking at a string that they might find in a book that you've written or that they might find in, say, the Bridges curriculum? What are some of the different problems along the way that really kind of inform the structure? Pam: So you might find it interesting that over time, we've identified that there's at least five major structures to strings, and the one that I just did with you is kind of the easiest one to facilitate. It's the easiest one to understand where it's going, and it's the helper-clunker structure. So the helper-clunker structure is all about, “I'm going to give you a helper problem that we expect all kids can kind of hang on.” They have some facility with, enough that everybody has access to. Then we give you a clunker that you could use that helper to inform how you could solve that clunker problem. In the first string I did with you, I did a helper, clunker, helper, clunker, helper, clunker, clunker. And the second one we did, I did helper, helper, clunker, helper, helper, clunker, clunker. So you can mix and match kind of helpers and clunkers in that, but there are other major structures of strings. If you're new to strings, I would dive in and do a lot of helper-clunker strings first. But I would also suggest—I didn't create my own strings for a long time. I did prewritten [ones by] Cathy Fosnot from the Netherlands, from the Freudenthal Institute. I was doing their strings to get a feel for the mathematical relationships for the structure of a string. I would watch videos of teachers doing it so I could get an idea of, “Oh, that move right there made all the difference. I see how you just invited kids in, not demand what they do.” The idea of when to have paper and pencil and when not, and just lots of different things can come up that if you're having to write the string as well, create the string, that could feel insurmountable.  So I would invite anybody out listening that's like, “Whoa, this seems kind of complicated,” feel free to facilitate someone else’s prewritten strings. Now I like mine. I think mine are pretty good. I think Bridges has some pretty good ones. [https://educate.mathlearningcenter.org/pdf/12698/28#page=28] But I think you'd really gain a lot from facilitating prewritten strings.  Can I make one quick differentiation that I'm running into more and more? So I have had some sharp people say to me, “Hey, sometimes you have extra problems in your string. Why do you have extra problems in your string?” And I'll say—well, at first I said, “What do you mean?” Because I didn't know what they were talking about. Are you telling me my string's bad? Why are you dogging my string? But what they meant was, they thought a string was the process a kid—or the steps, the relationships a kid used to solve the last problem. Does that make sense? Mike: It does. Pam: And they were like, “You did a lot of work to just get that one answer down there.” And I'm like, “No, no, no, no, no, no. A problem string or a number string, a string is an instructional routine. It is a lesson structure. It's a way of teaching. It's not a record of the relationships a kid used to solve a problem.” In fact, a teacher just asked—we run a challenge three times a year. It's free. I get on and just teach. One of the questions that was asked was, “How do we help our kids write their own strings?” And I was like, “Oh, no, kids don't write strings. Kids solve problems using relationships.” And so I think what the teachers were saying was, “Oh, I could use that relationship to help me get this one. Oh, and then I can use that to solve the problem.” As if, then, the lesson’s structure, the instructional routine of a string was then what we want kids to do is use what they know to logic their way through using mathematical relationships and connections to get answers and to solve problems. That record is not a string, that record is a record of their work. Does that make sense, how there's a little difference there? Mike: It totally does, but I think that's a good distinction. And frankly, that's a misunderstanding that I had when I first started working with strings as well. It took me a while to realize that the point of a string is to unveil a set of relationships and then allow kids to take them up and use them. And really it's about making these relationships or these problem solving strategies sticky, right? You want them to stick. We could go back to what you said. We're trying to high-dose a set of relationships that are going to help kids with strategies, not only in this particular string, but across the mathematical work they're doing in their school life. Pam: Yes, very well said. So for example, we did an addition “over” relationship in the addition string that I talked through, and then we did a multiplication “over” set of relationships and multiplication. We can do the same thing with subtraction. We could have a subtraction string where the helper problem is to subtract a bit too much. So something like 42 minus 20, and then the next problem could be 42 minus 19. And we're using that: I'm going to subtract a bit too much and then how do you adjust? And hoo, after you've been thinking about addition “over,” subtraction “over” is quite tricky. You're like, “Wait, why are we adding what we're subtracting?” And it's not about teaching kids a series of steps. It's really helping them reason. “Well, if I give you—if you owe me 19 bucks and I give you a $20 bill, what are we going to do?” “Oh, you’ve got to give me 1 back.” Now that's a little harder today because kids don't mess around with money. So we might have to do something that feels like they can—or help them feel money. That's my personal preference. Let's do it with money and help them feel money.  So one of the things I think is unique to my work is as I dove in and started facilitating other people's strings and really building my mathematical relationships and connections, I began to realize that many teachers I worked with, myself included, thought, “Whoa, there's just this uncountable, innumerable wide universe of all the relationships that are out there, and there's so many strategies, and anything goes, and they're all of equal value.” And I began to realize, “No, no, no, there's only a small set of major relationships that lead to a small set of major strategies.” And if we can get those down, kids can solve any problem that's reasonable to solve without a calculator, but in the process, building their brains to reason mathematically. And that's really our goal, is to build kids' brains to reason mathematically. And in the process we're getting answers. Answers aren't our goal. We'll get answers, sure. But our goal is to get them to build that small set of relationships because that small set of strategies now sets them free to logic their way through problems. And bam, we've got kids math-ing using the mental actions of math-ing. Mike: Absolutely. You made me think about the fact that there's a set of relationships that I can apply when I'm working with numbers Under 20. There's a set of relationships, that same set of relationships, I can apply and make use of when I'm working with multidigit numbers, when I'm working with decimals, when I'm working with fractions. It's really the relationships that we want to expose and then generalize and recognize this notion of going over or getting strategically to a friendly number and then going after that or getting to a friendly number and then going back from that. That's a really powerful strategy, regardless of whether you're talking about 8 and 3 or whether you're talking about adding unit fractions together. Strings allow us to help kids see how that idea translates across different types of numbers. Pam: And it's not trivial when you change a type of number or the number gets bigger. It's not trivial for kids to take this “over” strategy and to be thinking about something like 2,467 plus 1,995—and I know I just threw a bunch of numbers out, on purpose. It's not trivial for them to go, “What do I know about those numbers? Can I use some of these relationships I've been thinking about?” Well, 2,467, that's not really close to a friendly number. Well, 1,995 is. Bam. Let's just add 2,000. Oh, sweet. And then you just got to back up 5. It's not trivial for them to consider, “What do I know about these two numbers, and are they close to something that I could use?” That's the necessary work of building place value and magnitude and reasonableness. We've not known how to do that, so in some curriculum we create our whole extra unit that's all about place value reasonableness. Now we have kids that are learning to rote memorize, how to estimate by round. I mean there's all this crazy stuff that we add on when instead we could actually use strings to help kids build that stuff naturally kind of ingrained as we are learning something else.  Can I just say one other thing that we did in my new book? Developing Mathematical Reasoning: Avoiding the Trap of Algorithms. So I actually wrote it with my son, who is maybe the biggest impetus to me diving into the research and figuring out all of this math-ing and what it means. He said, as we were writing, he said, “I think we could make the point that algorithms don't help you learn a new algorithm.” If you learn the addition algorithm and you get good at it and you can do all the addition and columns and all the whatever, and then when you learn the subtraction algorithm, it's a whole new thing. All of a sudden it's a new world, and you're doing different—it looks the same at the beginning. You line those numbers still up and you're still working on that same first column, but boy, you're doing all sorts—now you're crossing stuff out. You're not just little ones, and what? Algorithms don't necessarily help you learn the next algorithm. It's a whole new experience. Strategies are synergistic. If you learn a strategy, that helps you learn the next set of relationships, which then refines to become a new strategy. I think that's really helpful to know, that we can—strategies build on each other. There's synergy involved. Algorithms, you got to learn a new one every time. Mike: And it turns out that memorizing the dictionary of mathematics is fairly challenging. Pam: Indeed [laughs], indeed. I tried hard to memorize that. Yeah. Mike: You said something to me when we were preparing for this podcast that I really have not been able to get out of my mind, and I'm going to try to approximate what you said. You said that during the string, as the teacher and the students are engaging with it, you want students' mental energy primarily to go into reasoning. And I wonder if you could just explicitly say, for you at least, what does that mean and what might that look like on a practical level? Pam: So I wonder if you're referring to when teachers will say, “Do we have students write? Do we not have them write?” And I will suggest: “It depends. It's not if they write; it's what they write that's important.”  What do I mean by that? What I mean is if we give kids paper and pencil, there is a chance that they're going to be like, “Oh, thou shalt get an answer. I'm going to write these down and mimic something that I learned last year.” And put their mental energy either into mimicking steps or writing stuff down. They might even try to copy what you've been representing strategies on the board. And their mental effort either goes into mimicking, or it might go into copying.  What I want to do is free students up [so] that their mental energy is, how are you reasoning? What relationships are you using? What's occurring to you? What's front and center and sort of occurring? Because we're high-dosing you with patterns, we're expecting those to start happening, and I'm going to be saying things, giving that helper problem. “Oh, that's occurring to you? It's almost like it's your idea—even though I just gave you the helper problem!” It's letting those ideas bubble up and percolate naturally and then we can use those to our advantage. So that's what I mean when [I say] I want mental energy into “Hmm, what do I know, and how can I use what I know to logic my way through this problem?” And that's math-ing. Those are the mental actions of mathematicians, and that's where I want kids' mental energy. Mike: So I want to pull this string a little bit further. Pun 100% intended there. Apologies to listeners.  What I find myself thinking about is there've got to be some do's and don'ts for how to facilitate a string that support the kind of reasoning and experience that you've been talking about. I wonder if you could talk about what you've learned about what you want to do as a facilitator when you're working with a string and maybe what you don't want to do. Pam: Yeah, absolutely. So a good thing to keep in mind is you want to keep a string snappy. You don't want a lot of dead space. You don't want to put—one of the things that we see novice, well, even sometimes not-novice, teachers do, that’s not very helpful, is they will put the same weight on all the problems.  So I'll just use the example 8 plus 10, 8 plus 9, they'll—well, let me do a higher one. 7 times 8, 70 times 8. They'll say, “OK, you guys, 7 times 8. Let's really work on that. That's super hard.” And kids are like, “It's 56.” Maybe they have to do a little bit of reasoning to get it, because it is an often missed fact, but I don't want to land on it, especially—what was the one we did before? 34 plus 10. I don't want to be like, “OK, guys, phew.” If the last problem on my string is 26 plus 18, I don't want to spend a ton of time. “All right, everybody really put all your mental energy in 36 plus 10” or whatever I said. Or, let's do the 7 times 8 one again. So, “OK, everybody, 7 times 8, how are you guys thinking about that?” Often we're missing it. I might put some time into sharing some strategies that kids use to come up with 7 times 8 because we know it's often missed. But then when I do 70 times 8, if I'm doing this string, kids should have some facility with times 10. I'm not going to be like, “OK. Alright, you guys, let's see what your strategies are. Right? Everybody ready? You better write something down on your paper. Take your time, tell your neighbor how….” Like, it's times 10. So you don't want to put the same weight—as in emphasis and time, wait time—either one on the problems that are kind of the gimmes, we're pretty sure everybody's got this one. Let's move on and apply it now in the next one. So there's one thing. Keep it snappy. If no one has a sense of what the patterns are, it's probably not the right problem string. Just bail on it, bail on it. You're like, “Let me rethink that. Let me kind of see what's going on.” If, on the other hand, everybody's just like, “Well, duh, it's this” and “duh, it's that,” then it's also probably not the right string. You probably want to up the ante somehow.  So one of the things that we did in our problem string books is we would give you a lesson and give you what we call the main string, and we would write up that and some sample dialogs and what the board could look like when you're done and lots of help. But then we would give you two echo strings. Here are two strings that get at the same relationships with about the same kind of numbers, but they're different and it will give you two extra experiences to kind of hang there if you're like, “Mm, I think my kids need some more with exactly this.” But we also then gave you two next-step strings that sort of up the ante. These are just little steps that are just a little bit more to crunch on before you go to the next lesson that's a bit of a step up, that's now going to help everybody increase. Maybe the numbers got a little bit harder. Maybe we're shifting strategy. Maybe we're going to use a different model. I might do the first set of strings on an area model if I'm doing multiplication. I might do the next set of strings in a ratio table. And I want kids to get used to both of those.  When we switch up from the 8 string to the next string, kind of think about only switching one thing. Don't up the numbers, change the model, and change the strategy at the same time. Keep two of those constant. Stay with the same model, maybe up the numbers, stay with the same strategy. Maybe if you're going to change strategies, you might back up the numbers a little bit, stick with the model for a minute before you switch the model before you go up the numbers. So those are three things to consider. Kind of—only change up one of them at a time or kids are going to be like, “Wait, what?” Kids will get higher dosed with the pattern you want them to see better if you only switch one thing at a time. Mike: Part of what you had me thinking was it's helpful, whether you're constructing your own string or whether you're looking at a string that's in a textbook or a set of materials, it's still helpful to think about, “What are the variables at play here?” I really appreciated the notion that they're not all created equal. There are times where you want to pause and linger a little bit that you don't need to spend that exact same amount of time on every clunker and every helper. There's a critical problem that you really want to invest some time in at one point in the string. And I appreciated the way you described, you're playing with the size of the number or the complexity of the number, the shift in the model, and then being able to look at those kinds of things and say, “What all is changing?” Because like you said, we're trying to kind of walk this line of creating a space of discovery where we haven't suddenly turned the volume up to 11 and made it really go from like, “Oh, we discovered this thing, now we're at full complexity,” and yet we don't want to have it turned down to, “It's not even discovery because it's so obvious that I knew it immediately. There's not really anything even to talk about.” Pam: Nice. Yeah, and I would say we want to be right on the edge of kids’ own proximal development, right on the edge. Right on the edge where they have to grapple with what's happening. And I love the word “grapple.” I've been in martial arts for quite a while, and grappling makes you stronger. I think sometimes people hear the word “struggle” and they're like, “Why would you ever want kids to struggle?” I don't know that I've met anybody that ever hears the word “grapple” as a negative thing. When you “grapple,” you get stronger. You learn. So I want kids right on that edge where they are grappling and succeeding. They're getting stronger. They're not just like, “Let me just have you guess what's in my head.” You're off in the field and, “Sure hope you figure out math, guys, today.” It's not that kind of discovery that people think it is. It really is: “Let me put you in a place where you can use what you know to notice maybe a new pattern and use it maybe in a new way. And poof! Now you own those relationships, and let's build on that.” And it continues to go from there.  When you just said—the equal weight thing, let me just, if I can—there's another, so I mentioned that there's at least five structures of problem strings. Let me just mention one other one that we like, to give you an example of how the weight could change in a string. So if I have an equivalent structure, an equivalent structure looks like: I give a problem, and an example of that might be 15 times 18. Now I'm not going to give a helper; I'm just going to give 15 times 18. If I'm going to do this string, we would have developed a few strategies before now. Kids would have some partial products going on. I would probably hope they would have an “over,” I would've done partial products over and probably, what I call “5 is half a 10.”  So for 15 times 18, they could use any one of those. They could break those up. They could think about twenty 15s to get rid of the extra two to have 18, 15. So in that case, I'm going to go find a partial product, an “over” and a “5 is half a 10,” and I'm going to model those. And I'm going to go, “Alright, everybody clear? Everybody clear on this answer?” Then the next problem I give—so notice that we just spent some time on that, unlike those helper clunker strings where the first problem was like a gimme, nobody needed to spend time on that. That was going to help us with the next one. In this case, this one's a bit of a clunker. We're starting with one that kids are having to dive in, chew on. Then I give the next problem: 30 times 9. So I had 15 times 18 now 30 times 9. Now kids get a chance to go, “Oh, that's not too bad. That's just 3 times 9 times 10. So that's 270. Wait, that was the answer to the first problem. That was probably just coincidence. Or was it?” And now especially if I have represented that 15 times 18, one of those strategies with an area model with an open array, now when I draw the 30 by 9, I will purposely say, “OK, we have the 15 by 18 up here. That's what that looked like. Mm, I'll just use that to kind of make sure the 30 by 9 looks like it should. How could I use the 15 by 18? Oh, I could double the 15? OK, well here's the 15. I'm going to double that. Alright, there's the 30. Well, how about the 9? Oh, I could half? You think I should half? OK. Well I guess half of 18. That's 9.”  So I've just helped them. I've brought out, because I'm inviting them to help me draw it on the board. They're thinking about, “Oh, I just half that side, double that side. Did we lose any area? Oh, maybe that's why the products are the same. The areas of those two rectangles are the same. Ha!” And then I give the next problem. Now I give another kind of clunker problem and then I give its equivalent. And again, we just sort of notice: “Did it happen again?” And then I might give another one and then I might end the string with something like 3.5 times—I'm thinking off the cuff here, 16. So 3.5 times 16. Kids might say, “Well, I could double 3.5 to get 7 and I could half the 16 to get 8, and now I'm landing on 7 times 8.” And that's another way to think about 3.5 times 16. Anyway, so, equivalent structure is also a brilliant structure that we use primarily when we're trying to teach kids what I call the most sophisticated of all of the strategies. So like in addition, give and take, I think, is the most sophisticated addition. In subtraction, constant difference. In multiplication, there's a few of them. There's doubling and having, I call it flexible factoring to develop those strategies. We often use the equivalent structure, like what's happening here? So there's just a little bit more about structure. Mike: There's a bit of a persona that I've noticed that you take on when you're facilitating a string. I'm wondering if you can talk about that or if you could maybe explain a little bit because I've heard it a couple different times, and it makes me want to lean in as a person who's listening to you. And I suspect that's part of its intent when it comes to facilitating a string. Can you talk about this? Pam: So I wonder if what you're referring to, sometimes people will say, “You're just pretending you don't know what we're talking about.” And I will say, “No, no, I'm actually intensely interested in what you're thinking. I know the answer, but I'm intensely interested in what you're thinking.” So I'm trying to say things like, “I wonder.” “I wonder if there's something up here you could use to help. I don't know. Maybe not. Mm. What kind of clunker could—or helper could you write for this clunker?”  So I don't know if that's what you're referring to, but I'm trying to exude curiosity and belief that what you are thinking about is worth hearing about. And I'm intensely interested in how you're thinking about the problem and there's something worth talking about here. Is that kind of what you're referring to? Mike: Absolutely.  OK. We're at the point in the podcast that always happens, which is: I would love to continue talking with you, and I suspect there are people who are listening who would love for us to keep talking. We’re at the end of our time. What resources would you recommend people think about if they really want to take a deeper dive into understanding strings, how they're constructed, what it looks like to facilitate them. Perhaps they're a coach and they're thinking about, “How might I apply this set of ideas to educators who are working with kindergartners and first graders, and yet I also coach teachers who are working in middle school and high school.” What kind of resources or guidance would you offer to folks? Pam: So the easiest way to dive in immediately would be my brand-new book from Corwin. It's called Developing Mathematical Reasoning: Avoiding the Trap of Algorithms. There's a section in there all about strings. We also do a walk-through where you get to feel a problem string in a K–2 class and a 3–5 [class]. And well, what we really did was counting strategies, additive reasoning, multiplicative reasoning, proportional reasoning, and functional reasoning. So there's a chapter in there where you go through a functional reasoning problem string. So you get to feel: What is it like to have a string with real kids? What's on the board? What are kids saying? And then we link to videos of those. So from the book, you can go and see those, live, with real kids, expert teachers, like facilitating good strings. If anybody's middle school, middle school coaches: I've got building powerful numeracy and lessons and activities for building powerful numeracy. Half of the books are all problem strings, so lots of good resources.  If you'd like to see them live, you could go to mathisfigureoutable.com/ps [http://mathisfigureoutable.com/ps], and we have videos there that you can watch of problem strings happening.  If I could mention just one more, when we did the K–12, Developing Mathematical Reasoning, Avoiding the Trap of Algorithms, that we will now have grade band companion books coming out in the fall of ’25. The K–2 book will come out in the spring of ’26. The [grades] 3–5 book will come out in the fall of ’26. The 6–8 book will come out and then six months after that, the 9–12 companion book will come out. And those are what to do to build reasoning, lots of problem strings and other tasks, rich tasks and other instructional routines to really dive in and help your students reason like math-y people reason because we are all math-y people. Mike: I think that's a great place to stop. Pam, thank you so much for joining us. It's been a pleasure talking with you. Pam: Mike, it was a pleasure to be on. Thanks so much. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center | www.mathlearningcenter.org

Kim Montague, I Have, You Need: The Utility Player of Instructional Routines ROUNDING UP: SEASON 4 | EPISODE 3 In sports, a utility player is someone who can play multiple positions competently, providing flexibility and adaptability. From my perspective, the routine I have, you need may just be the utility player of classroom routines. Today we're talking with Kim Montague about I have, you need and the ways it can be used to support everything from fact fluency to an understanding of algebraic properties.  BIOGRAPHY Kim Montague is a podcast cohost and content lead at Math is Figure-out-able™. She has also been a teacher for grades 3–5, an instructional coach, a workshop presenter, and a curriculum developer. Kim loves visiting classrooms and believes that when you know your content and know your kids, real learning occurs. RESOURCES Math is Figure-out-able!™ Podcast [https://podcast.mathisfigureoutable.com/] Math is FigureOutAble!™ Guide [https://www.mathisfigureoutable.com/youneed] (Download) Journey Coaching [https://www.mathisfigureoutable.com/journey-wait] TRANSCRIPT Mike Wallus: Welcome to the podcast, Kim. I am really excited to talk with you today.  So let me do a little bit of grounding. For listeners without prior knowledge, I'm wondering if you could briefly describe the I have, you need routine. How does it work, and how would you describe the roles that the teacher and the student play? Kim Montague: Thanks for having me, Mike. I'm excited to be here. I think it's an important routine.  So for those people who have never heard of I have, you need, it is a super simple routine that came from a desire that I had for students to become more fluent with partners of ten, hundred, thousand. And so it simply works as a call-and-response. Often I start with a context, and I might say, “Hey, we're going to pretend that we have 10 of something, and if I have 7 of them, how many would you need so that together we have those 10?” And so it's often prosed as a missing addend. With older students, obviously, I'm going to have some higher numbers, but it's very call-and-response. It's playful. It’s game-like. I'll lob out a question, wait for students to respond. I'm choosing the numbers, so it's a teacher-driven purposeful number sequence, and then students figure out the missing number. I often will introduce a private signal so that kids have enough wait time to think about their answer and then I'll signal everyone to give their response. Mike: OK, so there's a lot to unpack there. I cannot wait to do it.  One of the questions I've been asking folks about routines this season is just, at the broadest level, regardless of the numbers that the educator selects, how would you describe what you think I have, you need is good for? What's the routine good for? How can an educator think about its purpose or its value? You mentioned fluency. Maybe say a little bit more about that and if there's anything else that you think it's particularly good for. Kim: So I think one of the things that is really fantastic about I have, you need is that it's really simple. It's a simple-to-introduce, simple-to-facilitate routine, and it's great for so many different grade levels and so many different areas of content. And I think that's true for lots of routines. Teachers don't have time to reintroduce something brand new every single day. So when you find a routine that you can exchange pieces of content, that's really helpful. It's short, and it can be done anywhere. And like I said, it builds fluency, which is a hot topic and something that's important. So I can build fluency with partners of ten, partners of a hundred, partners of thousand, partners of one. I can build complementary numbers for angle measure and fractions. Lots of different areas depending on the grade that you're teaching and what you're trying to focus on. Mike: So one of the things that jumped out for me is the extent to which this can reveal structure. When we're talking about fluency, in some ways that's code for the idea that a lot of our combinations we're having kids think about—the structure of ten or a hundred or a thousand or, in the case of fractions, one whole and its equivalence. Does that make sense? Kim: Yeah, absolutely. So we have a really cool place value system. And I think that we give a lot of opportunities, maybe to place label, but we don't give a lot of opportunities to experience the structure of number. And so there are some very nice structures within partners of ten that then repeat themselves, in a way, within partners of a hundred and partners of a thousand and partners of one, like I mentioned. And if kids really deeply understand the way numbers form and the way they are fitting together, we can make use of those ideas and those experiences within other things like addition, subtraction. So this routine is not simply about, “Can you name a partner number?,” but it's laying foundation in a fun experience that kids then are gaining fluency that is going to be applied to other work that they're doing. Mike: I love that, and I think it's a great segue. My next question was going to be, “Could we talk a little bit about different sequences that you might use at different grade levels?” Kim: Sure. So younger students, especially in first grade, we're making a lot of use out of partners of 10 and working on owning those relationships. But then once students understand partners of 10, or when they're messing with partners of 10, the teacher can help make connections moving from partners of 10 to partners of 100 or partners of 20. So if you know that 9 plus 1 is 10, then there's some work to be done to help students understand that 9 tens and 1 ten makes 10 tens or 100. You can also use—capitalize on the idea of “9 and 1 makes 10” to understand that within 20, there are 2 tens. And so if you say “9” and I say “1,” and then you say “19,” and I say “1,” that work can help sharpen the idea that there's a ten within 20 and there's some tens within 30. So when we do partners of ten, it's a foundation, but we've got to be looking for opportunities to connect it to other relationships. I think that one of the things that's so great I have, you need is that we keep it game-like, but there's so many extensions, so many different directions that you can go, and we want teachers to purposefully record and draw out these relationships with their students. There's a bit to it where it's a call-and-response oral, but I think as we'll talk about further, there's a lot of nuance to number choice and there's a lot of nuance and when to record to help capitalize on those relationships. Mike: So I think the next best thing we could do is listen to a clip. I've got a clip of you working with a student, and I'm wondering if you could set the stage for what we're about to hear. Kim: Yeah, one of my very favorite things to do is to sit down with students and interview and kind of poke around in their head a little bit to find out where they currently are with the things that they're working on and where they can sharpen some content and where to take them next. So this is me sitting down with a student, Lanaya, who I didn't know very well, but I thought, let me start off by playing I have, you need with you, because that gives me a lot of insight into your number development. So this is me sitting down with her and saying, let's just play this game that I'd like to introduce to you. Kim (teacher): Oh, can I do one more thing with you? Can I play a game that I love?  Lanaya (student): Sure. Kim (teacher): OK, one more game. It's called I have, you need. And so it's a pretty simple game, actually. It just helps me think about or hear what kids are thinking. So it just is simply, if I say a number, you tell me how much more to get to 100. So if I have 50, you would say you need… Lanaya (student): 50. Kim (teacher): …so that together we would have 100. What if I said 92? Lanaya (student): 8. Kim (teacher): What if I said 75?  Lanaya (student): Um…25.  Kim (teacher): How do you know that one?  Lanaya (student): Because it's 30 to 70, so I just like minus 5 more. Kim (teacher): Oh, cool. What if I said 64?  Lanaya (student): Um…36. Kim (teacher): What if I said 27? Lanaya (student): Um…27…8—no, 72? No, 73. Kim (teacher): I don't remember what I said. [laughs] Did I say…? Lanaya (student): 27, I think. Kim (teacher): 27. So then you said 73, is that what you said? And you were about to say 80-something. Why were you going to say 80-something? Lanaya (student): Because 20 is like 80, like it’s the other half, but I just had to take away more. Kim (teacher): Perfect. I see. Three more. What if I said 32? Lanaya (student): Um…68.  Kim (teacher): What if I said 68?  Lanaya (student): 32.  Kim (teacher): [laughs] What if I said 79? Lanaya (student): Um…21. Kim (teacher): How do you know that one? Lanaya (student): Because…wait, wait, what was that one?  Kim (teacher): What if I said 79?  Lanaya (student): 79. Because 70 plus 30 is 100, but then I have to take away 9 more because the other half is 1, so yeah. Kim (teacher): Oh, you want to do it a little harder? Are you willing? Maybe I'll ask you that. Are you willing? Lanaya (student): Sure. Kim (teacher): OK. What if I said now our total is 1,000? What if I said 850? Lanaya (student): Um…250?  Kim (teacher): How do you know?  Lanaya (student): Or, actually, that'd be 150.  Kim (teacher): How do you know? Lanaya (student): Because, um…uh…800 plus 200 is 1,000. And so I would just have to take—what was the number again?  Kim (teacher): 850. Lanaya (student): I would have to add 50—er, have to minus 50 to that number. Kim (teacher): Um, 640. Lanaya (student): Uh, thir—360.  Kim (teacher): What about 545? Lanaya (student): 400…uh, you said 549? Kim (teacher): 545, I think is what I said. Lanaya (student): Um…that'd be 465. Kim (teacher): How do you know? Lanaya (student): Because the—I just took away the number of each one. So this is 5 to make 10, and then this is 6 to make 10, and then it's 5 again, I think, or no, it would be 465, right? Kim (teacher): 465. Lanaya (student): I don't… Kim (teacher): Not sure about that one. There's a lot of 5s in there. What if I give you another one? What if I said seven hundred and thirty…721? Lanaya (student): Uh, that'd be… Kim (teacher): If it helps to write it down, so you can see it, go ahead. Lanaya (student): 389, I think? Kim (teacher): Ah, OK. Because you wanna—you’re making a 10 in the… Lanaya (student): Yeah. Kim (teacher): …hundreds and a 10 in the middle and a 10 at the end.  Lanaya (student): Yeah.  Kim (teacher): Interesting. Mike: Wow. So there is a lot to unpack in that clip. Kim: There is, yeah. Mike: I want to ask you to pull the curtain back on this a little bit. Let's start with this question: As you were thinking about the sequence of numbers, what was going through your mind as the person who's facilitating? Kim: Yeah, so as I said, I don't really know Lanaya much at this point, so I'm kind of guessing in the beginning, and I just want her comfortable with the routine, and I'm going to give her maybe what I think might be a simple entry. So I asked [her about] 50 and then I asked [about] 92. Just gives a chance to see kind of where she is. Is she comfortable with those size of numbers? You'll notice that I did 50 and 92 and then I did 75. 75, often, if—I might hear a student talk about quarters with 75, and she didn't, but I did ask her her strategy, and throughout she uses the same strategy, which is interesting.  But I changed the number choices up and you'll see—if you were to write down the numbers that I did— [I] kind of backed away from the higher numbers. I went to 64 and then 27 and then 32. So getting further and further away from the target number. If I have students who are counting a lot, then it becomes cumbersome for them to count and they might be nudged away from accounting strategy into something a little bit more sophisticated. At one point I asked her [about] 32, and then I asked her [about] the turnaround of that, 68. Just checking to see what she knows about the commutative property.  Eventually I moved into 1,000. And I mentioned earlier that [with] young students, you start with 10 and maybe combinations of 100, multiples of 10. But I didn't mention that with older grades, we might do hundreds by 1 or thousands by multiples of 100 and then by 5s. So I did that with Lanaya. She seemed to feel very comfortable with the two-digit numbers, and I thought, “Well, let's take it to the thousands.” But if you notice, I did 850, 640, some multiples of 10 still. She seemed comfortable with those, but [she] is still using the strategy of, “Let me go a little bit over. Let me add all the hundreds I need and then make adjustments.”  Mike: Mm-hmm. Kim: And so then I decided to do 545 and see what happened in that moment because at that point she's having to readjust more than one digit. Mike: Yep. Kim: And when I said the number 545, I thought, “Oh man, this is a poor choice because there's a lot of 5s and 4s.” And so when she kind of maybe fumbled a little bit, I thought, “Is this because I did a poor number choice and there are lots of 4s and 5s, or is it because she's using a particular strategy that is a little more cumbersome?” So I gave her a final problem of 721, and again, that was a little bit more to adjust. So in that moment, I thought, “OK, I know where we need to work. And I need to work with her on some different strategies that aren't always about making tens.” Because as she gets larger numbers or she's getting numbers that are by 1s, that becomes less sophisticated. It becomes more cumbersome. It becomes more adjustment than you maybe are even able to hold.  It's not about holding it in your head. We could have been writing some things down and we did towards the end. But it's just a lot of adjustment to make, and the strategies that she's using really aren't going to be ones that help later in addition or in subtraction. So it's just kind of playing with number, and she's pretty strong with what she's working on, but there is some work to do there that I would want to do with her. Mike: It was fascinating because as I was attending to the choices you were making and what she was doing and the back and forth, I found myself thinking a bit about this notion of fluency, that part of it is the ability to be efficient, but also to be flexible at the same time. And I really connect that with what you said because she had a strategy that was working for her, but you also made a move to kind of say, “Let's see what happens if we give a set of numbers where that becomes more cumbersome.” And it kind of exposed— there's this space where, again, as you said, “Now I know where we need to work.” So it's a bit like a formative assessment too. Kim: Yeah, yeah. Interviewing students, like I said, is my very favorite thing to do. And it's tough because we want kids to be successful, which is a great goal, but I think it's often unfortunate that we leave students with a strategy that we think, “Oh, that's great. They have a strategy and it works for them,” but we aren't really thinking about the long game. We're not thinking about, “Will this thing that they're doing support their needs as the size of the numbers increase, as the type of the numbers change?” And we want them to have choice. And again, I have, you need is fantastic because within this game, this simple routine, you can share strategies. There's a handful of strategies that kids generally use, and in the routine in the game, we get to talk about those strategies. So we have a student who's using the kind of same strategy over and over and it stops working because it's less sophisticated, it's less efficient, it's more cumbersome. Then in the routine, we get to expose other strategies that they can try on and see what works for them based on the numbers that they're being given. Mike: You made me think about something that, I'm not sure how you could even put my finger on why, but sometimes people are wonky about this notion that students should have a choice of their strategies. In some ways, it makes me think that what you're really suggesting is part of this work around flexibility is building options, right? You're not trapped in a strategy if suddenly the numbers don't make it something that's efficient. You have options, and I think that really jumps out when you think about what happened with Lanaya, but just generally what you're trying to build when you're using this routine. Kim: Yeah, I mean we are big fans of building relationships, so that strategies are natural outcomes. And I think if you are new to numeracy or you didn't grow up playing with number, it can feel like, “I'm just going to offer multiple and kids have to own them all, and now there's too many things and they don't know how to pick.” But when we really focus on relationship in number, then we strengthen those relationships like in a routine with I have you, need. I grew up messing with number, and the strategies don't feel like a bunch of new things I have to memorize. I've strengthened partners of ten and hundred and thousand, and I understand doubles, and I understand the fact that you can add a little too much and back up. And so those relationships just get used in the way that I solve problems, and that's what we want for kids. Mike: I love that.  We've spent a fair amount of time talking about this connection between building fluency and helping kids see and make use of structure. I'm also really taken by some of the properties that jump out of this routine. They're not formal, meaning they come up organically, and I found myself thinking a lot about algebraic reasoning or setting kids up for algebra. Could you just talk a little bit about some of that part of the work? Kim: I think that when we want kids to own and use properties, one way to go about it is to say, “Today we're going to talk about the commutative property.” And you define it and you verbalize it and you write it down. You might make a poster. But more organically is the opportunity to use it and then name it as it's occurring. So in the routine, if I say “68” and she says “32” and then I say “32” and she says “68,” then we are absolutely using the idea of “68 plus something is 100” and then “32 plus something is 100.” There is something natural about you just [knowing] it's the other addend. In some of the other strategies that we develop through I have, you need, it's about breaking apart numbers in such a way that they are reassociating. And so when that happens for students, then we can name it afterward and say, “Oh, that's just this thing.” And whether we name the property to students or not, it's more important that they're using them. And so we put it in a game, we put it in a form that we just say, “Oh, that's just where you're breaking apart numbers and finding friendly addends to go together.” And I think it's really more important that teachers really understand the strategies that work so that they invite students to participate in experiences where they're using them. Mike: Yeah, I mean, what hits me about that is there's something about making use of a relationship, fleshing it out through this process of I have, you need, and then at the end coming back and saying, “Oh, we have a formal name for that.” That's different than saying, “Here's the thing, here's the definition. Remember the definition, remember the name.” It just works so much more smoothly and sensibly because I've been able to apply that relationship and it feels like it's inside of me now. I have an understanding and now I've just attached a name to that thing. That just feels really, really different. Kim: Yeah, I mean, if we give students the right experiences, then they have those experiences to draw on. And I'm a big fan of saying that some kids just have more experiences than others. And all kids can, but it's our job to provide the right experiences for students that they can use and that they can think back on and that they can connect to other experiences that they have. Using the relationships of number is so powerful, and I think we just need to do more and more so that kids are just stronger in the properties and stronger in connections and relationships so that then when they go solve problems, they're using what they know. Mike: Nice. So something that I want to call out for listeners who, again, this might be new for them, is there's really two parts to this routine. There's the call-and-response, whether it's with an individual student or whether it's with a whole class of students. And then there's what happens after that call-and-response. So how do you think about the choices a teacher has after they've called a number and kids have responded? What are some of the choices available to a teacher in that moment? Kim: Well, I think if you're playing, then you are kind of on a mission to learn more about students. For me, I'm always trying to figure out where students are and what they know and what they're tinkering with right now so that then I can make informed choices about what to do next. So I might make choices that are about my entire class. I might make choices based on, I'm watching particular students as we play to see where are they kind of dropping off. Where—you know, if I'm watching a video of myself playing this routine with a class, I'm scanning to, say, those students wait a little bit longer and I want to strengthen some work when we do multiples of 5 because they're chiming in just a little bit late. So I'm looking for who's fluent, who's not, who's counting on by 1s, who needs another nudge. I'm ready to bump them a little bit further along. It's not about speed. This isn't a speed routine. I absolutely think we give kids some time to wait, but just enough. So like I said, we introduce a private signal, then they let me know when the majority of class is ready. Then I call for everyone to reply. But there is some bit of this where if you're counting by 1s to get up from 68 to 100, then there's some intervention [needed]. There's some work that we can do to strengthen you.  So it's important to give some think time, it's important to use the private signal, and it's about the teacher being responsive to what they notice. “Am I pulling a small group to give some students more experience, making connections?” “Am I moving some students to another set of numbers?” “Am I purposefully pairing students to give them what they need while I'm working with somebody else?” So it's an information-finding routine if I'm noticing and I'm aware of what's going on. Mike: I noticed with Lanaya, there were points where you called, she responded, and you went right in and you called after and she responded—and there were other points where you decided to say something equivalent to, “Tell me how you know.” How do you think about the points where you just keep on rolling or you pause and you ask that probing question? Kim: That's a great question. So when I make a shift is often a time that I will ask, “How do you know?” First of all, it's super important to ask, “How do you know?” when students have both right and wrong answers. We have a lot of kids who are only asked, “How do you know?” when it's wrong. And then they backpedal, right? And then they just pick a new answer. And I think giving kids confidence to commit to their answer and say, “Yeah, I know it's that, and here's how I know.” We continue to build that in students, that we are not the ones who hold all the answers when we question. And so, in a shift is often when I think about making a change. So if I'm asking about combinations of 10 and then I shift to a 5, multiples of 5, maybe the first or second time I ask them how they know. I think about, “Have kids had a chance to verbalize their thinking?” There are moments where you completely understand what Lanaya is saying. And then there's a few where maybe if you're not a careful listener of students, you might think, “I'm not sure she knows what she's saying.” But over time, when you're a practiced listener of students, even though their words may not be fantastic, they're kind of sharing their thinking. And so it will bog it down to ask, “How do you know?” every single time. But in those shifts where I want to know, “Are you changing your strategy up?,” “Are you continuing to do the same thing every time?,” I think it's important to ask. Mike: So I have one last practitioner question before we move on from this. I'm wondering about annotation and the extent to which it's important and whether there are different points in time where it is, where it's not. How do you think about that? Kim: Yeah, I think that's a really important question. You can very easily hear something like this interview with Lanaya and think, “Oh, I'm just [doing] call-and-response.” Which—there can be moments of that, but an important piece is annotation to draw out strategies that kids are using. So I might introduce this routine to a class and I might [do] call-and-response a day or two or a couple of times, depending on how many times that week or how often we get to play.  But at some moment there's a chance to say, “Hang on a second. How did you think about that?” If I say “65” and some kids call it back, I'll say, “How did you come up with that?” And then I ask students to share their strategies, and this is the sharing part. This is the part where students get to learn from each other. And so a kid might say, “I added 5 to get to 70 and then I added 30 more to get to 100.” And some kid will listen and I'm going to record that on a number line, making the jumps that they say out loud. And another student might say, “Wait a second, that's not what I did.” And so there's this opportunity to share strategy, and then we can say, “Well, try that on.” But if I'm not representing what students are saying on a number line, it could be really hard for others to hold onto it. It's not about [holding] everything in your head. So I often record on a number line as we're starting to share strategies or if I want to uncover a mistake that somebody makes, or if I see the kids all using one strategy, I want to draw attention back. Another really important thing is that I might want to lighten the mental load by recording the number that I said. If I'm saying, “721” and I'm not writing anything down, you might be trying to hold “7-2-1” or “720 and 1” at the same time that you're trying to do some figuring, and it's not about who can hold more. So depending on the age, the size of the numbers, I might just [quickly] sketch the number that I said because they can stare at the number while they're also doing some figuring. Or they might write the number down on their notebooks so that they can do some figuring. Mike: One of the things that jumped out is the fact that you talked about when you stop to annotate, one of the ways that you do it is to annotate on a number line as opposed to—I think what I had in my mind initially is a set of equations. Which is not to say that you couldn't do that, but I thought it was interesting that you said, “Actually, I will go to a number line for my annotations.” Kim: So I think making thinking visible is hugely helpful. And if a student says—let's say I give the number 89. If somebody says, “Well, I thought about adding 1 to get to 90 and then I added 10 more to get to 100,” then their strategy of adding 1 more to get to that next friendly number is one of the major strategies that we would want to develop in students when they're adding. But another student might say, “Oh, that's interesting. I started at 89 and I added 10 first to get to 99, and then I added the 1.” And that's a different major strategy that we want to develop. And when you put them both up on a number line, you can see that that missing addend, that missing part is 11, but they're handling it in two different ways. And so it's a beautiful representation of thinking of things in different ways, but that they're equivalent and that you can talk about it when you see it on the board. Equations are fantastic ways to represent, but I have an affinity for number lines to represent student thinking. Mike: Love it.  As a fellow podcaster, you know that the challenge of hosting one of these is we have a short amount of time to talk about something that I suspect we could talk about for hours. Talk to folks who want to keep learning about I have, you need and any other resources you would recommend for people thinking about their practice. Where could someone go if they wanted to continue this journey? Kim: They could listen to the Math is Figure-Out-Able podcast [https://podcast.mathisfigureoutable.com/], first of all. We have had several episodes where we talk about this routine and revisit it over and over again because it's super powerful. We also have a free download that I think you're going to share. It's mathisfigureoutable.com/youneed [http://mathisfigureoutable.com/youneed], so you can see something that would be helpful. And we have, at Math is Figure-Out-Able, an online coaching support called Journey [https://www.mathisfigureoutable.com/journey], where we just get to work with teachers on a regular basis to unpack the practices and the routines that you're using and spend a lot of time working with teachers and students in the classroom to develop these kinds of things that are more bang for your buck, to make the most that you can in the time that you have with your students. Mike: That's awesome. And yes, for listeners, we will include links to everything that Kim just mentioned.  I wish that we could keep going. I think this is probably a good place to stop, Kim. Thank you so much for joining us. It's been a pleasure. Kim: Oh, Mike, thank you. Appreciate you having me. Mike: Absolutely.  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. © 2025 The Math Learning Center | www.mathlearningcenter.org

Sue Looney, Same but Different: Encouraging Students to Think Flexibly ROUNDING UP: SEASON 4 | EPISODE 2 Sometimes students struggle in math because they fail to make connections. For too many students, every concept feels like its own entity without any connection to the larger network of mathematical ideas.  On the podcast today, we’re talking with Dr. Sue Looney about the powerful same and different routine. We explore the ways that teachers can use this routine to help students identify connections and foster flexible reasoning. BIOGRAPHY Sue Looney holds a doctorate in curriculum and instruction with a specialty in mathematics from Boston University. Sue is particularly interested in our most vulnerable and underrepresented populations and supporting the teachers that, day in and day out, serve these students with compassion, enthusiasm, and kindness. RESOURCES Same but Different Math [http://samebutdifferentmath.com] Looney Math [http://looneymath.com] TRANSCRIPT Mike Wallus: Students sometimes struggle in math because they fail to make connections. For too many students, every concept feels like its own entity without any connection to the larger network of mathematical ideas.  Today we're talking with Sue Looney about a powerful routine called same but different and the ways teachers can use it to help students identify connections and foster flexible reasoning.  Well, hi, Sue. Welcome to the podcast. I'm so excited to be talking with you today. Sue Looney: Hi Mike. Thank you so much. I am thrilled too. I've been really looking forward to this. Mike: Well, for listeners who don't have prior knowledge, I'm wondering if we could start by having you offer a description of the same but different routine. Sue: Absolutely. So the same but different routine is a classroom routine that takes two images or numbers or words and puts them next to each other and asks students to describe how they are the same but different. It's based in a language learning routine but applied to the math classroom. Mike: I think that's a great segue because what I wanted to ask is: At the broadest level—regardless of the numbers or the content or the image or images that educators select—how would you explain what [the] same but different [routine] is good for? Maybe put another way: How should a teacher think about its purpose or its value? Sue: Great question. I think a good analogy is to imagine you're in your ELA— your English language arts—classroom and you were asked to compare and contrast two characters in a novel. So the foundations of the routine really sit there. And what it's good for is to help our brains think categorically and relationally. So, in mathematics in particular, there's a lot of overlap between concepts and we're trying to develop this relational understanding of concepts so that they sort of build and grow on one another. And when we ask ourselves that question—“How are these two things the same but different?”—it helps us put things into categories and understand that sometimes there's overlap, so there's gray space. So it helps us move from black and white thinking into this understanding of grayscale thinking.  And if I just zoom out a little bit, if I could, Mike—when we zoom out into that grayscale area, we're developing flexibility of thought, which is so important in all aspects of our lives. We need to be nimble on our feet, we need to be ready for what's coming. And it might not be black or white, it might actually be a little bit of both.  So that's the power of the routine and its roots come in exploring executive functioning and language acquisition. And so we just layer that on top of mathematics and it's pure gold. Mike: When we were preparing for this podcast, you shared several really lovely examples of how an educator might use same but different to draw out ideas that involve things like place value, geometry, equivalent fractions, and that's just a few. So I'm wondering if you might share a few examples from different grade levels with our listeners, perhaps at some different grade levels. Sue: Sure. So starting out, we can start with place value. It really sort of pops when we look in that topic area. So when we think about place value, we have a base ten number system, and our numbers are based on this idea that 10 of one makes one group of the next. And so, using same but different, we can help young learners make sense of that system.  So, for example, we could look at an image that shows a 10-stick. So maybe that's made out of Unifix cubes. There's one 10-stick a—stick of 10—with three extras next to it and next to that are 13 separate cubes. And then we ask, “How are they the same but different?” And so helping children develop that idea that while I have 1 ten in that collection, I also have 10 ones. Mike: That is so amazing because I will say as a former kindergarten and first grade teacher, that notion of something being a unit of 1 composed of smaller units is such a big deal. And we can talk about that so much, but the way that I can visualize this in my mind with the stick of 10 and the 3, and then the 13 individuals—what jumps out is that it invites the students to notice that as opposed to me as the teacher feeling like I need to offer some kind of perfect description that suddenly the light bulb goes off for kids. Does that make sense? Sue: It does. And I love that description of it. So what we do is we invite the students to add their own understanding and their own language around a pretty complex idea. And they're invited in because it seems so simple: “How are these the same but different?” “What do you notice?” And so it's a pretty complex idea, and we gloss over it. Sometimes we think our students understand that and they really don't. Mike: Is there another example that you want to share? Sue: Yeah, I love the fraction example. So equivalence—when I learned about this routine, the first thing that came to mind for me when I layered it from thinking about language into mathematics was, “Oh my gosh, it's equivalent fractions.”  So if I were to ask listeners to think about—put a picture in your head of one-half, and imagine in your mind's eye what that looks like. And then if I said to you, “OK, well now I want you to imagine two-fourths. What does that look like?” And chances are those pictures are not the same.  Mike, when you imagine, did you picture the same thing or did you picture different things? Mike: They were actually fairly different. Sue: Yeah. So when we think about one-half as two fourths, and we tell kids those are the same—yes and no, right? They have the same value that, if we were looking at a collection of M&M’S or Skittles or something, maybe half of them are green, and if we make four groups, [then] two-fourths are green. But contextually it could really vary. And so helping children make sense of equivalence is a perfect example of how we can ask the question, same but different. So we just show two pictures. One picture is one-half and one picture is two-fourths, and we use the same colors, the same shapes, sort of the same topic, but we group them a little differently and we have that conversation with kids to help make sense of equivalence. Mike: So I want to shift because we've spent a fair amount of time right now describing two instances where you could take a concept like equivalent fractions or place value and you could design a set of images within the same but different routine and do some work around that.  But you also talked with me, as we were preparing, about different scenarios where same but different could be a helpful tool. So what I remember is you mentioned three discrete instances: this notion of concepts that connect; things learned in pairs; and common misconceptions—or, as I've heard you describe them, naive conceptions. Can you talk about each of those briefly? Sue: Sure. As I talk about this routine to people, I really want educators to be able to find the opportunities—on their own, authentically—as opportunities arise. So we should think about each of these as an opportunity.  So I'll start with concepts that connect. When you're teaching something new, it's good practice to connect it to, “What do I already know?” So maybe I'm in a third grade classroom, and I want to start thinking about multiplication. And so I might want to connect repeated addition to multiplication. So we could look at 2 plus 2 plus 2 next to 2 times 3. And it can be an expression, these don't always have to be images. And a fun thing to look at might be to find out, “Where do I see 3 and 2 plus 2 plus 2?” So what's happening here with factors? What is happening with the operations? And then of course they both yield the same answer of 6. So concepts that connect are particularly powerful for helping children build from where they know, which is the most powerful place for us to be. Mike: Love that. Sue: Great. The next one is things that are learned in pairs. So there's all sorts of things that come in pairs and can be confusing. And we teach kids all sorts of weird tricks and poems to tell themselves and whatever to keep stuff straight. And another approach could be to—let's get right in there, to where it's confusing.  So for example, if we think about area and perimeter, those are two ideas that are frequently confusing for children. And we often focus on, “Well, this is how they're different.” But what if we put up an image, let's say it's a rectangle, but [it] wouldn't have to be. And we've got some dimensions on there. We're going to think about the area of one and then the perimeter on the other. What is the same though, right? Because where the confusion is happening. So just telling students, “Well, perimeter’s around the outside, so think of ‘P’ for ‘pen’ or something like that, and area’s on the inside.” What if we looked at, “Well, what's the same about these two things?” We're using those same dimensions, we've got the same shape, we're measuring in both of those. And let students tell you what is the same and then focus on that critical thing that's different, which ultimately leads to understanding formula for finding both of those things. But we've got to start at that concept level and link it to scenarios that make sense for kids. Mike: Before we move on to talking about misconceptions, or naive conceptions, I want to mark that point: this idea that confusion for children might actually arise from the fact that there are some things that are the same as opposed to a misunderstanding of what's different.  I really think that's an important question that an educator could consider when they're thinking about making this bridging step between one concept or another or the fact that kids have learned how whole numbers behave and also how fractions might behave. That there actually might be some things that are similar about that that caused the confusion, particularly on the front end of exploration, as opposed to, “They just don't understand the difference.” Sue: And what happens there is then we aid in memory because we've developed these aha moments and painted a more detailed picture of our understanding in our mind's eye. And so it's going to really help children to remember those things as opposed to these mnemonic tricks that we give kids that may work, but it doesn't mean they understand it. Mike: Absolutely. Well, let's talk about naive conceptions and the ways that same and [different] can work with those. Sue: So, I want to kick it up to maybe middle school, and I was thinking about what example might be good here, and I want to talk about exponents. So if we have 2 raised to the third power, the most common naive conception would be, like, “Oh, I just multiply that. It's just 2 times 3.”  So let's talk about that. So if I am working on exponents, I notice a lot of my students are doing that, let's put it right up on the board: “Two rays to the third power [and] 2 times 3. How are these the same but different?” And the conversation’s a bit like that last example, “Well, let's pay attention to what's the same here.” But noticing something that a lot of children have not quite developed clearly and then putting it up there against where we want them to go and then helping them—I like that you use the word “bridge”—helping them bridge their way over there through this conversation is especially powerful. Mike: I think the other thing that jumps out for me as you were describing that example with exponents is that, in some ways, what's happening there when you have an example like “2 times 3” next to “2 to the third power” is you're actually inviting kids to tell you, “This is what I know about multiplication.” So you're not just disregarding it or saying, “We're through with that.” It's in the exploration that those ideas come out, and you can say to kids, “You are right. That is how multiplication functions. And I can see why that would lead you to think this way.” And it's a flow that's different. It doesn't disregard kids' thinking. It actually acknowledges it. And that feels subtle, but really important. Sue: I really love shining a light on that. So it allows us to operate from a strength perspective. So here's what I know, and let's build from there. So it absolutely draws out in the discussion what it is that children know about the concepts that we put in front of them. Mike: So I want to shift now and talk about enacting same but different. I know that you've developed a protocol for facilitating the same but different routine, and I'm wondering if you could talk us through the protocol, Sue. How should a teacher think about their role during same but different? And are there particular teacher moves that you think are particularly important? Sue: Sure. So the protocol I've worked out goes through five steps, and it's really nice to just kind of think about them succinctly. And all of them have embedded within them particular teacher moves. They are all based on research of how children learn mathematics and engage in meaningful conversation with one another.  So step 1 is to look. So if I'm using this routine with 3- and 4-year-olds, and I'm putting a picture in front of them, learning that to be a good observer, we've got to have eyes on what it is we're looking at. So I have examples of counting, asking a 4-year-old, “How many things do I have in front of me?” And they're counting away without even looking at the stuff. So teaching the skill of observation. Step 1 is look. Step 2 is silent think time. And this is so critically important. So giving everybody the time to get their thoughts together. If we allow hands to go in the air right away, it makes others that haven't had that processing time to figure it out shut down quite often. And we all think at different speeds with different tasks all the time, all day long. So, we just honor that everyone's going to have generally about 60 seconds in which to silently think, and we give students a sentence frame at that time to help them. Because, again, this is a language-based learning routine. So we would maybe put on the board or practice saying out loud, “I'd like you to think about: ‘They are the same because blank; they are different because blank.’” And that silent think time is just so important for allowing access and equitable opportunities in the classrooms. Mike: The way that you described the importance of giving kids that space, it seems like it's a little bit of a two-for-one because we're also kind of pushing back on this notion that to be good at math, you have to have your hand in the air first, and if you don't have your hand in the air first or close to first, your idea may be less valuable. So I just wanted to shine a light on the different ways that that seems important for children, both in the task that they're engaging with and also in the culture that you're trying to build around mathematics. Sue: I think it's really important. And if we even zoom out further just in life, we should think before we speak. We should take a moment. We should get our thoughts together. We should formulate what it is that we want to say. And learning how to be thoughtful and giving the luxury of what we're just going to all think for 60 seconds. And guess what? If you had an idea quickly, maybe you have another one. How else are they the same but different? So we just keep that culture that we're fostering, like you mentioned, we just sort of grow that within this routine. Mike: I think it's very safe to say that the world might be a better place if we all took 60 seconds to think about [laughs] what we wanted to say sometimes. Sue: Yes, yes. So as teachers, we can start teaching that and we can teach kids to advocate for that. “I just need a moment to get my thoughts together.”  All right, so the third step is the turn and talk. And it's so important and it's such an easy move. It might be my favorite part. So during that time, we get to have both an experience with expressive language and receptive language—every single person. So as opposed to hands in the air and I'm playing ball with you, Mike, and you raise your hand and you get to speak and we're having a good time. When I do a turn and talk, everybody has an opportunity to speak. And so taking the thoughts that are in their head and expressing them is a big deal. And if we think about our multilingual learners, our young learners, even our older learners, and it's just a brand new concept that I've never talked about before. And then on the other side, the receptive learning. So you are hearing from someone else and you're getting that opportunity of perspective taking. Maybe they notice something you hadn't noticed, which is likely to happen to somebody within that discussion. “Wow, I never thought about it that way.” So the turn and talk is really critical. And the teacher's role during this is so much fun because we are walking around and we're listening. And I started walking around with a notebook. So I tell students, “While you are talking, I'm going to collect your thinking.” And so I'm already imagining where this is going next. And so I'm on the ground if we're sitting on the rug, I'm leaning over, I'm collecting thoughts, I'm noticing patterns, I'm noticing where I want to go next as the facilitator of the conversation that's going to happen whole group. So that's the third component, turn and talk. The fourth component is the share. So if I've walked around and gathered student thinking, I could say, “Who would like to share their thinking?” and just throw it out there. But I could instead say—let's say we're doing the same but different with squares and rectangles. And I could say, “Hmm, I noticed a lot of you talking about the length of the sides. Is there anyone that was talking about the lengths of the sides that would like to share what either you or your partner said?” So I know that I want to steer it in that direction. I know a lot of people talked about that, so let's get that in the air. But the share is really important because these little conversations have been happening. Now we want to make it public for everybody, and we're calling on maybe three or four students. We're not trying to get around to everybody. We're probably hopefully not going to [be] drawing Popsicle sticks and going random. At this point, students have had the opportunity to talk, to listen, to prepare. They've had a sentence stem. So let's see who would like to share and get those important ideas out. Mike: I think what strikes me is there's some subtlety to what's happening there because you are naming some themes that you heard. And as you do that, and you name that, kids can say, “That's me,” or, “I thought about that,” or, “My partner thought about that. You're also clearly acting with intention. As an educator, there are probably some ideas that you either heard that you want to amplify or that you want kids to attend to, and yet you're not doing it in a way that takes away from the conversations that they had. You're still connecting to what they said along the way. And you're not suddenly saying, “Great, you had your turn and talk, but now let's listen to David over here because we want to hear what he has to share.” Sue: Yes. And I don't have to be afraid of calling out a naive conception. Maybe a lot of people were saying, “Well, I think the rectangles have two long [sides and] two short.” And they're not seeing that the square is also a rectangle. And so maybe I'm going to use that language in the conversation too, so that yeah, the intentionality is exactly it. Building off of that turn and talk to the share. The last step is the summary. So after we've shared, we have to put a bow on that, right? So we've had this experience. They generally are under 15 minutes, could be 5 minutes, could be 10 minutes. But we've done something important all together. And so the teacher's role here is to summarize, to bring that all together and to sort of say, “OK, so we looked at this picture here, and we noticed”—I'll stick with the square/rectangle example—“that both shapes have four sides and four square corners. They're both rectangles, but this one over here is a special one. It's a square and all four sides are equal and that's what makes it special.” Or something like that. But we want to succinctly nail that down in a summary.  If you do a same but different and nobody gets there, and so you chose this with intention, you said, “This is what we need to talk about today,” and all of a sudden you're like, “Oh, boy,” then your summary might not sound like that. It might sound like, “Some of you noticed this and some of you noticed that, and we're going to come back to this after we do an activity where we're going to be sorting some shapes.” So it's an opportunity for formative assessment. So summary isn't, “Say what I really wanted to say all along,” even though I do have something I want to say; it's a connection to what happened in that conversation. And so almost always it comes around to that. But there are those instances where you learn that we need to do some more work here before I can just nicely put that bow on it. Mike: You're making me think about what one of my longtime mentors used to say, and the analogy he would use is, “You can definitely lead the horse to water, but it is not your job to shove the horse's face in the water.” And I think what you're really getting at is, I can have a set of mathematical goals that I'm thinking about as I'm going into a same and different. I can act with intention, but there is still kind of this element of, “I don't quite know what's going to emerge.” And if that happens, don't shove the metaphorical horse's head in the water, meaning don't force that there. If the kids haven't made the connection yet or they haven't explored the gray space that's important. Acknowledge that that's still in process. Sue: Exactly. There is one last optional step which relates to summary. So if you have time and you're up for an exploration, you can now ask your students to make one of their own. And that's a whole other level of sophistication of thought for students to recognize, “Oh, this is how those two were same but different. I'm going to make another set that are the same but different in the same way.” It's actually a very complex task. We could scaffold it by giving students, “If this was my first image, what would the other one be?” That would be like what we just did. Very worthwhile. Obviously now we're not within the 10-minute timeframe. It's a lot bigger. Mike: What I found myself thinking about, the more that we talk through intent, purpose, examples, the protocol steps, is the importance of language. And it seemed like part of what's happening is that the descriptive language that's accessed over the course of the routine that comes from students, it really paves the way for deeper conceptual understanding. Is that an accurate understanding of the way that same and different can function? Sue: A hundred percent. So it's really the way that we think as we're looking at something. We might be thinking in mental pictures of things, but we might also be thinking in the words. And if we're going to function in a classroom and in society, we have to have the language for what it is that we're doing. And so yes, we're playing in that space of language acquisition, expressive language, receptive language, all of it, to help us develop this map of what that is really deeply all about so that when I see that concept in another context, I have this rich database in my head that involves language that I can draw on to now do the next thing with it. Mike: That's really powerful. Listeners have heard me say this before, but we've just had a really insightful conversation about the structure, the design, the implementation, and the impact of same and different. And yet we're coming to the end of the podcast. So I want to offer an opportunity for you to share any resources, any websites, any tools that you think a listener who wanted to continue learning about same but different, where might they go? What might you recommend, Sue? Sue: Sure. So there's two main places to find things, and they actually do exist in both. But the easiest way to think about this, there is the website, which is samebutdifferentmath.com [http://samebutdifferentmath.com], and it's important to get the word “math” in there. And that is full of images from early learning, really even up through high school. So that's the first place, and they are there with a creative common licensing.  And then you mentioned tools. So there are some tools, and if we wanted to do deeper learning, and I think the easiest way to access those is my other website, which is just looneymath.com [http://looneymath.com]. And if you go up at the top under Books, there's a children's book that you can have kids reading and enjoying it with a friend. There's a teacher book that talks about in more detail some of the things we talked about today. And then there are some cards where students can sit in a learning center and turn over a card that presents them with an opportunity to sit shoulder to shoulder. And so those are all easily accessed really on either one of those websites, but probably easiest to find under the looneymath.com [http://looneymath.com]. Mike: Well, for listeners, we'll put a link to those resources in the show notes to this episode.  Sue, I think this is probably a good place to stop, but I just want to say thank you again. It really has been a pleasure talking with you today. Sue: You're welcome, Mike. It's one of my favorite things to talk about, so I really appreciate the opportunity. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center | www.mathlearningcenter.org

Christopher Danielson, Which One Doesn’t Belong? Routine: Fostering Flexible Reasoning ROUNDING UP: SEASON 4 | EPISODE 1 The idea of comparing items and looking for similarities and differences has been explored by many math educators. Christopher Danielson has taken this idea to new heights. Inspired by the Sesame Street song “One of These Things (Is Not Like the Others),” Christopher wrote the book Which One Doesn't Belong? In this episode, we'll ask Christopher about the routine of the same name and the features that make it such a powerful learning experience for students.  BIOGRAPHY Christopher Danielson started teaching in 1994 in the Saint Paul (MN) Public Schools. He earned his PhD in mathematics education from Michigan State University in 2005 and taught at the college level for 10 years after that. Christopher is the author of Which One Doesn’t Belong?, How Many?, and How Did You Count? Christopher also founded Math On-A-Stick, a large-scale family math playspace at the Minnesota State Fair. RESOURCES What Is “Which One Doesn’t Belong?” [https://talkingmathwithkids.com/wodb-about/] Talking Math With Your Kids [https://talkingmathwithkids.com/] by Christopher Danielson Math On-A-Stick [https://www.mnstatefair.org/location/math-on-a-stick/] 5 Practices for Orchestrating Productive Mathematics Discussion [http://afmboordcoldfmd9fo67spqqukne4vxevi8hrxtyfk23p6bfyefanwru] by Margaret (Peg) Smith & Mary Kay Stein How Many?: A Counting Book [https://www.routledge.com/How-Many-A-Counting-Book/Danielson/p/book/9781625311825?srsltid=AfmBOorIFjAgrjwlQe3nrOiyU5hFKbatwWeYQfXRXn6KBT3xWB1J6L-I] by Christopher Danielson How Did You Count? A Picture Book [https://www.routledge.com/How-Did-You-Count-Picture-Book/Danielson/p/book/9781032898353] by Christopher Danielson TRANSCRIPT Mike Wallus: The idea of comparing items and looking for similarities and differences has been explored by many math educators. That said, Christopher Danielson has taken this idea to new heights. Inspired by Sesame Street’s [song] “One of These Things (Is Not Like the Others),” Christopher wrote the book Which One Doesn't Belong? In this episode, we'll ask Christopher about the Which one doesn't belong? routine and the features that make it such a powerful learning experience for students.  Well, welcome to the podcast, Christopher. I'm excited to be talking with you today.  Christopher Danielson: Thank you for the invitation. Delightful to be invited.  Mike: I would love to chat a little bit about the routine Which one doesn't belong? So, I'll ask a question that I often will ask folks, which is: If I'm a listener, and I don't have prior knowledge of that routine, how would you describe it for someone? Christopher: Yeah. Sesame Street, back in the day, had a routine called Which one doesn't belong? There was a little song that went along with it. And for me, the iconic Sesame Street image is [this:] Grover is on the stairs up to the brownstone on the Sesame Street set, and there are four circles drawn in a 2-by-2 grid in chalk on the wall. And there are a few of the adults and a couple of the puppets sitting around, and they're asking Grover and singing the song, “Which One of Them Doesn't Belong?” There are four circles. Three of them are large and one is small—or maybe it's the other way around, I don't remember. So, there's one right answer, and Grover is thinking really hard—"think real hard” is part of the song. They're singing to him. He's under kind of a lot of pressure to come up with which one doesn't belong and fortunately, Grover succeeds. Grover's a hero.  But what we're wanting kids to attend to there is size. There are three things that are the same size. All of them are the same shape, three that are the same size, one that has a different size. They're wanting to attend to size. Lovely. This one doesn't belong because it is a different size, just like my underwear doesn't belong in my socks drawer because it has a different function. I mean, it's not—for me there is, we could talk a little bit about this in a moment. The belonging is in that mathematical and everyday sense of objects and whether they belong.  So, that's the Sesame Street version. Through a long chain of math educators, I came across a sort of tradition that had been flying along under the radar of rethinking that, with the idea being that instead of there being one property to attend to, we're going to have a rich set of shapes that have rich and interesting relationships with each other. And so Which one doesn't belong? depends on which property you're attending to.  So, the first page of the book that I published, called Which One Doesn't Belong?, has four shapes on it. One is an equilateral triangle standing on a vertex. One is a square standing on a vertex. One is a rhombus, a nonsquare rhombus standing on its vertex, and it's not colored in. All the other shapes are colored in. And then there is the same nonsquare thrombus colored in, resting on a side. So, all sort of simple shapes that offer simple introductory properties, but different people are going to notice different things. Some kids will hone in on that. The one in the lower left doesn't belong because it's not colored in. Other kids will say, “Well, I'm counting the number of sides or the number of corners. And so, the triangle doesn't belong because all the others have four and it has three.” Others will think about angle measure, they'll choose a square. Others will think about orientation. I've been taken to task by a couple of people about this. Kindergartners are still thinking about orientation as one of the properties. So, the shape that is in the lower right on that first page is a rhombus resting on a side instead of on a vertex. And kids will describe it as “the one that feels like it's leaning over” or that “has a flat bottom” or “it's pointing up and to the right” and all the others are pointing straight up and down. So that's the routine. And then things, as with “How Did You Count?” as with “How Many?” As you page your way through the book, things get more sophisticated. And for me, the entry was a geometry book because when my kids were small, we had sort of these simplistic shapes books, but really rich narrative stories in picture books that we could read. And it was always a bummer to me that we'd read these rich stories about characters interacting. We'd see how their interactions, their conflicts relate to our own lives, and then we'd get to the math books, and it would be like, “triangle: always equilateral, always on a side.” “Square: never a square on the rectangle page.” Rectangle gets a different page from square. And so, we understand culturally that children can deal with and are interested in and find fascinating and imaginative rich narratives, but we don't understand as a culture that children also have rich math minds.  So, for a long time I wanted there to be a better shapes book, and there are some better shapes books. They're not all like that, but they're almost all like that. And so, I had this idea after watching one of my colleagues here in Minnesota, Terry Wyberg. This routine, he was doing it with fractions, but about a week later I thought to myself, “Hey, wait a minute, what if I took Terry's idea about there not being one right answer, but any of the four could be, and combine that with my wish for a better shapes book?” And along came Which One Doesn't Belong? as a shapes book. So, there's a square and a rectangle on the same page. There are shapes with curvy sides and shapes with straight sides on the same page, and kids have to wrestle with or often do wrestle with: What does it mean to be a vertex or a corner? A lot of really rich ideas can come out of some well-chosen, simple examples. I chose to do it in the field of geometry, but there are lots of other mathematical objects as well as nonmathematical objects you could apply the same mathematical thinking to.  Mike: So, I think you have implicitly answered the question that I'm going to ask. If you were to say at the broadest level, regardless of whether you're using shapes, numbers, images—whatever the content is that an educator selects to put into the 2-by-2, that is structurally the way that Which one doesn't belong? is set up—what's it good for? What should a teacher think about in terms of “This will help me or will help my students…,” fill in the blank. How do you think about the value that comes out of this Which one doesn't belong? structure and experience?  Christopher: Multidimensional for me. I don't know if I'll remember to say all of the dimensions, so I'll just try to mention a couple that I think are important.  One is that I'm going to make you a promise that whatever mathematical ideas you bring to this classroom during this routine are going to be valued. The measure of what's right, what counts as a right answer here, is going to be what's true—not what I thought of when I was setting up this set. I think there is a lot of power in making that promise and then in holding that promise. It is really, really easy—all of us have been there as teachers—[to] make an instructional promise to kids, [but] then there comes a time where it either inadvertently or we make a decision to break that promise. I think there's a lot of costs to that. I know from my own experience as a learner, from my own experiences as a teacher, that there can be a high cost to that. So valuing ideas, I think this is a space. I love having Which one doesn't belong? as a time that we can set aside for the measure of “what's right is what's true.” So, when children are making claims about this one in the upper right doesn't belong, I want you to for a moment try to think like that person, even if you disagree that that's important. And so, teachers have to play that role also.  Where that comes up a lot is in, especially when I'm talking with adults, if I'm talking to parents about Which one doesn't belong?, often parents who don't identify as math people or who explicitly identify as nonmath people, will say, “That one in the lower left, it's not colored in. But I don't think that really counts.” In that moment, kids are less likely to make that apology, but adults will make that apology all the time. And in that moment, I have to both bring the adult in as a mathematical thinker but also model for them: What does it look like when their kid chooses something that the parent doesn't think counts? So, for me, the real thing that Which one doesn't belong? is doing is teaching children, giving children practice and expertise—therefore learning—about a particular mathematical practice, which is abstraction. That when we look at these sets of shapes, there are lots of properties. And so, we have to for a moment, just think about number of sides. And if we do that, then the triangle doesn't belong because of the other four. But as soon as we shift the property and say, “Well, let's think about angle measures,” then the ways that we're going to sort those shapes, the relationships that they have with each other, changes. And that's true with all mathematical objects.  And you can do that kind of mathematical thinking with non-mathematical objects. One of my favorite Which one doesn't belong? sets is: There's a doughnut, a chocolate doughnut; there's a coffee cup, one of those speckled blue camping metal coffee cups; there's half a hamburger bun with a bunch of seeds on top; and then there is a square everything bagel. And so, as kids start thinking about that, they're like, “Well, if we're thinking about holes, the hamburger bun doesn't have a hole. If we're thinking about speckling, the chocolate doughnut isn't speckled. If we're thinking about whether it's an edible substance, the coffee cup is not edible.” And so that's that same abstraction. If we pay attention to just this one property, that forces a sort. If we pay attention to a different property, we're going to get a different sort. And that's one of the practices of mathematicians on a regular basis. So regular that often when we're doing mathematics, we don't even notice that we're doing it. We don't notice that we're asking kids to ignore all the other properties of the number 2 except for its evenness right now. If you do that, then 2 and 4 are like each other. But if we're supposed to be paying attention to primality as to a prime number, then 2 and 4 are not like each other. All mathematical objects, all mathematicians have to do that kind of sort on the objects that they're working with.  I had a college algebra class at the community college while I was working on Which One Doesn't Belong?, and so, I was test-driving this with graphs and my students. I can still see Rosalie in the middle of the room—a room full of 45 adults ranging from 17 to 52, and I'm this 45-year-old college instructor—and we have three parabolas and one absolute value function. So, a parabola is “y equals x squared.” It's that nice curving swooping thing that goes up at one end down to a nice bowl and then up again. There was one that's upside down. I think there was one pointing sideways. And then an absolute value function is the same idea, except it's two lines coming together to make a bowl, sort of a very sharp bowl, instead of being curved. And we got this lovely Which one doesn't belong?, right? So, we've got this lovely collection of them. And Rosalie, her eyebrows are getting more and more knitted as this conversation goes on. So finally, she raises her hand. I call on her, and she says, “Mr. Danielson, I get that all of these things are true about these, but which ones matter?” Which is a fabulous question that within itself holds a lot of tensions that Rosalie is used to being in math class and being told what things she's supposed to pay attention to.  And so, in some ways it's sort of disturbing to have me up there, and I get that, up there in front of the classroom valuing all these different ways of viewing these graphs because she's like, “Which one is going to matter when you ask me this question about something on an exam? Which ones matter?”  But truly, the only intellectually honest answer to her question is, “Well, it depends. Are we paying attention to direction of concavity? Then the one that's pointing sideways doesn't count.” Any one of these is, it depends on whether you're studying algebra or whether you're studying geometry or topology. And I did give her, I think—I hope—what was a satisfying answer after giving her the true but not very satisfying answer of “It depends,” which is something like, “Well, in the work we're about to do with absolute value functions, the direction that they open up and how steeply they open up are going to be the things that we're really attending to, and we're not going to be attending as much to how they are or are not like parabolas. But seeing how they have some properties in common with these parabolas is probably going to be really useful for us.  Mike: That actually makes me think of, one, a statement of what I think is really powerful about this. And then, two, a pair of questions that I think are related.  It really struck me—Rosalie's question—how different the experience of engaging with a Which one doesn't belong? is from what people have traditionally considered math tasks where there is in fact an answer, right? There's something that the teacher's like, “Yep, that's the thing.” Even if it's perhaps obscured by the task at first, ultimately, oftentimes there is a thing and a Which one doesn't belong? is a very, very different type of experience. So that really does lead me to two questions. One is: What is important to think about when you're facilitating a Which one doesn't belong? experience? And then, maybe even the better question to start with is: What's important to think about when you're planning for that experience?  Christopher: Facilitating is going to be about making a promise to kids. That measure of “what's right is what's true.” I'm interested in the various ways that you're thinking and doing all the kind of work that we discussed but now in this context of geometry, or in my case in the college algebra classroom, in the context of algebraic representations.  Planning. I have been so deeply influenced by the work of Peg Smith [https://www.corwin.com/books/5-practices-262956?srsltid=AfmBOorDCOLDFMd9fo67spQqUkNE4VxEVI8hRXtyfk23p6bFYeFANwrU] and her colleagues and the five practices for facilitating mathematical conversations. And in particular, I think in planning for these conversations, planning a set—when I'm deciding what shapes are going to go in the set, or how I'm going to arrange the eggs in the egg carton, or how many half avocados am I going to put on the cutting board—I'm anticipating one of those practices: What is it that kids are likely to do with this? And if I can't anticipate anything interesting that they're going to do with it, then either my imagination isn't good enough, and I better go try it out with kids or my imagination is absolutely good enough and it's just kind of a junky thing that's not going to take me anywhere, and I should abandon it. So over time, I've gotten so much better at that anticipating work because I have learned, I've become much more expert at what kids are likely to see. But I also always get surprised. In a sufficiently large group of kids, somebody will notice something or have some way of articulating differences among the shapes, even these simple shapes on the first page, that I haven't encountered before. And I get to file that away again for next time. That's learning that gets fed back into the machine, both for the next time I'm going to work with a group of kids, but also for the next time I'm sitting down to design an experience.  Mike: You have me thinking about something else, which is what closure might look like in an experience like this. Because I'm struck by the fact that there might be some really intentional choices of the items in the Which one doesn't belong? So, the four items that end up being there, [they] may be designed to drive a conversation around a set of properties or a set of relationships—and yet at the same time be open enough to allow lots of kids to be right in the things that they're noticing.  And so, if I've got a Which one doesn't belong? that kind of is intended to draw out some ideas or have kids notice some of those ideas and articulate them, what does closure look like? Because I could imagine you don't know what you're going to get necessarily from kids when you put a Which one doesn't belong? in front of them. So, how do you think about different ways that a routine or experience like this might close for a teacher and for students?  Christopher: Yeah, I think one of the best roles that a teacher can play at the end of a Which one doesn't belong? conversation is going back and summarizing the various properties that kids attended to. Because as they're being presented and maybe annotated, we're noticing them sort of one by one. And we might not have a moment to set them aside. It might take a minute for a kid to draw out their ideas about the orientation of this shape. And it might take a little bit and some clarification with another kid about how they were counting sides. They might not have great words for “sides” or “corners,” and [instead they use] gestures, and we're all trying to figure things out. And so, by the time we figured that out, we've forgotten about the orientation answer that we had before.  So I think a really powerful move, one of many that are in teachers' toolkits, is to come back and say, “All right, so we looked at these four shapes, and what we noticed is that if you're paying attention to how this thing is sitting on the page, to its orientation, which direction it's pointing, then this one didn't belong, and Susie gave us that answer. And then another thing you might pay attention to, another property could be the number of sides. If you're paying attention to the number of sides the triangle doesn't belong, and we got that one from Brent, right?” And so run through some of the various properties.  Also, noticing along the way that there were two reasons to pick the triangle as the one that doesn't belong. It might be the sides, and it might be, you might have some other reason for picking it that isn't the number of sides. For kindergartners, the number of corners, or vertices, and the number of sides are not yet obviously the same as each other. So, for a lot of kindergartners that feels like two answers rather than one. Older audiences are more likely to know that that's going to be the same.  So yeah, I think that being able to come back and state succinctly after we've had this conversation—valuing each of the contributions that came along, but also being able to compare them, maybe we're writing them down as part of our annotation. There might be other ways that we do that. But I think summarizing so that we can look at this set of ideas that's been brought out altogether, I think is a really powerful way.  One other quick thing about designing, which is—I hear this a lot from teachers, they're saying, “OK, so we're studying quadrilaterals. So, I made a Which one doesn't belong? with four quadrilaterals. And nobody noticed that they were all quadrilaterals.” To which I say, “They didn't notice because you didn't contrast that property.” So, if there's a property you want to bring out, you better make sure, I think, that you have three things that have it and one that doesn't. Or vice versa—three that don't, and one that does—because then that's a thing for kids to notice. They're not going to notice what they all have in common because that's not the task we're asking them. So, if you want to make one about quadrilaterals, throw a pentagon in there.  Mike: Love it.  So, the question that I typically will ask any guest before the close of the interview is, what are some resources that educators might grab onto, be they yours or other work in the field that you think is really powerful, that supports the kind of work that we've been talking about? What would you offer to someone who's interested in continuing to learn and maybe to try this out?  Christopher: So, we've referred to number talks. “Dot talks” and “number talks,” those are both phrases that can be googled. There are three books, Which One Doesn't Belong?, How Many?, How Did You Count?—all published by Stenhouse, all available as a hardcover book, hardcover student book, or home picture book.  Mike: So, for listeners, just so you know, we're going to add links to the resources that Christopher referred to in all of our show notes for folks’ convenience.  Christopher, I think this is probably a good place to stop. Thank you so much for joining us. It's absolutely been a pleasure chatting with you.  Christopher: Yeah, thank you for the invitation, for your thoughtful prep work, and support of both the small and the larger projects along the way. I appreciate that. I appreciate all of you at Bridges and The Math Learning Center. You do fabulous work.  Mike: This concludes part one of our discussion with Christopher Danielson. Christopher is going to join us again later this season, where we'll have a conversation about the nature of counting and how an expanded definition of counting might help support students later in their mathematical journey. I hope that you'll join us for this conversation.    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. © 2025 The Math Learning Center | www.mathlearningcenter.org

William Zahner, Understanding the Role of Language in Math Classrooms ROUNDING UP: SEASON 3 | EPISODE 17 How can educators understand the relationship between language and the mathematical concepts and skills students engage with in their classrooms? And how might educators think about the mathematical demands and the language demands of tasks when planning their instruction?  In this episode, we discuss these questions with Bill Zahner, director of the Center for Research in Mathematics and Science Education at San Diego State University. BIOGRAPHY Bill Zahner is a professor in the mathematics department at San Diego State University and the director of the Center for Research in Mathematics and Science Education. Zahner's research is focused on improving mathematics learning for all students, especially multilingual students who are classified as English Learners and students from historically marginalized communities that are underrepresented in STEM fields. RESOURCES Teaching Math to Multilingual Learners, Grades K–8 by Kathryn B. Chval, Erin Smith, Lina Trigos-Carrillo, and Rachel J. Pinnow [https://www.corwin.com/books/multilingual-success-in-math-272643?srsltid=AfmBOor4yTyy-IbOxAK2bzJBcINQZ3WdOn5V1lRvDMnV3bbcWuHnbKZz] National Council of Teachers of Mathematics [https://www.nctm.org/] Mathematics Teacher: Learning and Teaching PK– 12 [https://pubs.nctm.org/view/journals/mtlt/mtlt-overview.xml] English Learners Success Forum [https://www.elsuccessforum.org/] SDSU-ELSF Video Cases for Professional Development [https://elsf.sdsu.edu/] The Math Learning Center materials [https://store.mathlearningcenter.org/s/] Bridges in Mathematics curriculum [https://www.mathlearningcenter.org/curriculum/bridges-mathematics] Bridges in Mathematics Teachers Guides [https://teach.mathlearningcenter.org/curriculum/brk] [BES login required] TRANSCRIPT Mike Wallus: How can educators understand the way that language interacts with the mathematical concepts and skills their students are learning? And how can educators focus on the mathematics of a task without losing sight of its language demands as their planning for instruction? We'll examine these topics with our guest, Bill Zahner, director of the Center for Research in Mathematics and Science Education at San Diego State University.  Welcome to the podcast, Bill. Thank you for joining us today. Bill Zahner: Oh, thanks. I'm glad to be here. Mike: So, I'd like to start by asking you to address a few ideas that often surface in conversations around multilingual learners and mathematics. The first is the notion that math is universal, and it's detached from language. What, if anything, is wrong with this idea and what impact might an idea like that have on the ways that we try to support multilingual learners? Bill: Yeah, thanks for that. That's a great question because I think we have a common-sense and strongly held idea that math is math no matter where you are and who you are. And of course, the example that's always given is something like 2 plus 2 equals 4, no matter who you are or where you are. And that is true, I guess [in] the sense that 2 plus 2 is 4, unless you're in base 3 or something. But that is not necessarily what mathematics in its fullness is. And when we think about what mathematics broadly is, mathematics is a way of thinking and a way of reasoning and a way of using various tools to make sense of the world or to engage with those tools [in] their own right. And oftentimes, that is deeply embedded with language.  Probably the most straightforward example is anytime I ask someone to justify or explain what they're thinking in mathematics. I'm immediately bringing in language into that case. And we all know the old funny examples where a kid is asked to show their thinking and they draw a diagram of themselves with a thought bubble on a math problem. And that's a really good case where I think a teacher can say, “OK, clearly that was not what I had in mind when I said, ‘Show your thinking.’”  And instead, the demand or the request was for a student to show their reasoning or their thought process, typically in words or in a combination of words and pictures and equations. And so, there's where I see this idea that math is detached from language is something of a myth; that there's actually a lot of [language in] mathematics. And the interesting part of mathematics is often deeply entwined with language. So, that's my first response and thought about that.  And if you look at our Common Core State Standards for Mathematics, especially those standards for mathematical practice, you see all sorts of connections to communication and to language interspersed throughout those standards. So, “create viable arguments,” that's a language practice. And even “attend to precision,” which most of us tend to think of as, “round appropriately.” But when you actually read the standard itself, it's really about mathematical communication and definitions and using those definitions with precision. So again, that's an example, bringing it right back into the school mathematics domain where language and mathematics are somewhat inseparable from my perspective here. Mike: That's really helpful. So, the second idea that I often hear is, “The best way to support multilingual learners is by focusing on facts or procedures,” and that language comes later, for lack of a better way of saying it. And it seems like this is connected to that first notion, but I wanted to ask the question again: What, if anything, is wrong with this idea that a focus on facts or procedures with language coming after the fact? What impact do you suspect that that would have on the way that we support multilingual learners? Bill: So, that's a great question, too, because there's a grain of truth, right? Both of these questions have simultaneously a grain of truth and simultaneously a fundamental problem in them. So, the grain of truth—and an experience that I've heard from many folks who learned mathematics in a second language—was that they felt more competent in mathematics than they did in say, a literature class, where the only activity was engaging with texts or engaging with words because there was a connection to the numbers and to symbols that were familiar. So, on one level, I think that this idea of focusing on facts or procedures comes out of this observation that sometimes an emergent multilingual student feels most comfortable in that context, in that setting.  But then the second part of the answer goes back to this first idea that really what we're trying to teach students in school mathematics now is not simply, or only, how to apply procedures to really big numbers or to know your times tables fast. I think we have a much more ambitious goal when it comes to teaching and learning mathematics. That includes explaining, justifying, modeling, using mathematics to analyze the world and so on. And so, those practices are deeply tied with language and deeply tied with using communication. And so, if we want to develop those, well, the best way to do that is to develop them, to think about, “What are the scaffolds? What are the supports that we need to integrate into our lessons or into our designs to make that possible?”  And so, that might be the takeaway there, is that if you simply look at mathematics as calculations, then this could be true. But I think our vision of mathematics is much broader than that, and that's where I see this potential. Mike: That's really clarifying. I think the way that you unpack that is if you view mathematics as simply a set of procedures or calculations, maybe? But I would agree with you. What we want for students is actually so much more than that.  One of the things that I heard you say when we were preparing for this interview is that at the elementary level, learning mathematics is a deeply social endeavor. Tell us a little bit about what you mean by that, Bill. Bill: Sure. So, mathematics itself, maybe as a premise, is a social activity. It's created by humans as a way of engaging with the world and a way of reasoning. So, the learning of mathematics is also social in the sense that we're giving students an introduction to this way of engaging in the world. Using numbers and quantities and shapes in order to make sense of our environment.  And when I think about learning mathematics, I think that we are not simply downloading knowledge and sticking it into our heads. And in the modern day where artificial intelligence and computers can do almost every calculation that we can imagine—although your AI may do it incorrectly, just as a fair warning [laughs]—but in the modern day, the actual answer is not what we're so focused on. It's actually the process and the reasoning and the modeling and justification of those choices. And so, when I think about learning mathematics as learning to use these language tools, learning to use these ways of communication, how do we learn to communicate? We learn to communicate by engaging with other people, by engaging with the ideas and the minds and the feelings and so on of the folks around us, whether it's the teacher and the student, the student and the student, the whole class and the teacher. That's where I really see the power. And most of us who have learned, I think can attest to the fact that even when we're engaging with a text, really fundamentally we're engaging with something that was created by somebody else. So, fundamentally, even when you're sitting by yourself doing a math word problem or doing calculations, someone has given that to you and you think that that's important enough to do, right?  So, from that stance, I see all of teaching and learning mathematics is social. And maybe one of our goals in mathematics classrooms, beyond memorizing the times tables, is learning to communicate with other people, learning to be participants in this activity with other folks. Mike: One of the things that strikes me about what you were saying, Bill, is there's this kind of virtuous cycle, right? That by engaging with language and having the social aspect of it, you're actually also deepening the opportunity for students to make sense of the math. You're building the scaffolds that help kids communicate their ideas as opposed to removing or stripping out the language. That's the context in some ways that helps them filter and make sense. You could either be in a vicious cycle, which comes from removing the language, or a virtuous cycle. And it seems a little counterintuitive because I think people perceive language as the thing that is holding kids back as opposed to the thing that might actually help them move forward and make sense. Bill: Yeah. And actually that's one of the really interesting pieces that we've looked at in my research and the broader research is this question of, “What makes mathematics linguistically complex?” is a complicated question. And so sometimes we think of things like looking at the word count as a way to say, “If there are fewer words, it's less complex, and if there are more words, it's more complex.” But that's not totally true. And similarly, “If there's no context, it's easier or more accessible, and if there is a context, then it's less accessible.”  And I don't see these as binary choices. I see these as happening on a somewhat complicated terrain where we want to think about, “How do these words or these contexts add to student understanding or potentially impede [it]?” And that's where I think this social aspect of learning mathematics—as you described, it could be a virtuous cycle so that we can use language in order to engage in the process of learning language. Or, the vicious cycle is, you withhold all language and then get frustrated when students can't apply their mathematics. That’s maybe the most stereotypical answer: “My kids can do this, but as soon as they get a word problem, they can't do it.” And it's like, “Well, did you give them opportunities to learn how to do this? [laughs] Or is this the first time?” Because that would explain a lot. Mike: Well, it's an interesting question, too, because I think what sits behind that in some ways is the idea that you're kind of going to reach a point, or students might reach a point, where they're “ready” for word problems.  Bill: Right. Mike: And I think what we're really saying is it's actually through engaging with word problems that you build your proficiency, your skillset that actually allows you to become a stronger mathematician. Bill: Mm-hmm. Right. Exactly. And it's a daily practice, right? It's not something that you just hold off to the end of the unit, and then you have the word problems, but it's part of the process of learning. And thinking about how you integrate and support that. That's the key question that I really wrestle with. Not trivial, but I think that's the key and the most important part of this. Mike: Well, I think that's actually a really good segue because I wanted to shift and talk about some of the concrete or productive ways that educators can support multilingual learners. And in preparing for this conversation, one of the things that I've heard you stress is this notion of a consistent context. So, can you just talk a little bit more about what you mean by that and how educators can use that when they're looking at their lessons or when they're writing lessons or looking at the curriculum that they're using? Bill: Absolutely. So, in our past work, we engaged in some cycles of design research with teachers looking at their mathematics curriculum and opportunities to engage multilingual learners in communication and reasoning in the classroom. And one of the surprising things that we found—just by looking at a couple of standard textbooks—was a surprising number of contexts were introduced that are all related to the same concept. So, the concept would be something like rate of change or ratio, and then the contexts, there would be a half dozen of them in the same section of the book. Now, this was, I should say, at a secondary level, so not quite where most of the Bridges work is happening. But I think it's an interesting lesson for us that we took away from this. Actually, at the elementary level, Kathryn Chval has made the same observation.  What we realized was that contexts are not good or bad by themselves. In fact, they can be highly supportive of student reasoning or they can get in the way. And it's how they are used and introduced. And so, the other way we thought about this was: When you introduce a context, you want to make sure that that context is one that you give sufficient time for the students to understand and to engage with; that is relatable, that everyone has access to it; not something that's just completely unrelated to students' experiences. And then you can really leverage that relatable, understandable context for multiple problems and iterations and opportunities to go deeper and deeper.  To give a concrete example of that, when we were looking at this ratio and rate of change, we went all the way back to one of the fundamental contexts that's been studied for a long time, which is motion and speed and distance and time. And that seemed like a really important topic because we know that that starts all the way back in elementary school and continues through college-level physics and beyond. So, it was a rich context. It was also something that was accessible in the sense that we could do things like act out story problems or reenact a race that's described in a story problem. And so, the students themselves had access to the context in a deep way.  And then, last, that context was one that we could come back to again and again, so we could do variations [of] that context on that story. And I think there's lots of examples of materials out there that start off with a core context and build it out. I’m thinking of some of the Bridges materials, even on the counting and the multiplication. I think there's stories of the insects and their legs and wings and counting and multiplying. And that's a really nice example of—it's accessible, you can go find insects almost anywhere you are. Kids like it. [Laughs] They enjoy thinking about insects and other icky, creepy-crawly things. And then you can take that and run with it in lots of different ways, right? Counting, multiplication, division ratio, and so on. Mike: This last bit of our conversation has me thinking about what it might look like to plan a lesson for a class or a group of multilingual learners. And I know that it's important that I think about mathematical demands as well as the language demands of a given task. Can you unpack why it's important to set math and language development learning goals for a task, or a set of tasks, and what are the opportunities that come along with that, if I'm thinking about both of those things during my planning? Bill: Yeah, that's a great question. And I want to mark the shift, right? We've gone from thinking about the demands to thinking about the goals, and where we're going to go next.  And so, when I think about integrating mathematical goals—mathematical learning goals and language learning goals—I often go back to these ideas that we call the practices, or these standards that are about how you engage in mathematics. And then I think about linking those back to the content itself. And so, there's kind of a two-piece element to that. And so, when we're setting our goals and lesson planning, at least here in the great state of California, sometimes we'll have these templates that have, “What standard are you addressing?,” [Laughs] “What language standard are you addressing?,” “What ELD standard are you addressing?,” “What SEL standard are you addressing?” And I've seen sometimes teachers approach that as a checkbox, right? Tick, tick, tick, tick, tick. But I see that as a missed opportunity—if you just look at this like you're plugging things in—because as we started with talking about how learning mathematics is deeply social and integrated with language, that we can integrate the mathematical goals and the language goals in a lesson. And I think really good materials should be suggesting that to the teacher. You shouldn't be doing this yourself every day from scratch. But I think really high-quality materials will say, “Here's the mathematical goal, and here's an associated language goal,” whether it's productive or receptive functions of language. “And here's how the language goal connects the mathematical goal.”  Now, just to get really concrete, if we're talking about an example of reasoning with ratios—so I was going back to that—then it might be generalized, the relationship between distance and time. And that the ratio of distance and time gives you this quantity called speed, and that different combinations of distance and time can lead to the same speed. And so, explain and justify and show using words, pictures, diagrams. So, that would be a language goal, but it's also very much a mathematical goal.  And I guess I see the mathematical content, the practices, and the language really braided together in these goals. And that I think is the ideal, and at least from our work, has been most powerful and productive for students. Mike: This is off script, but I'm going to ask it, and you can pass if you want to.  Bill: Mm-hmm. Mike: I wonder if you could just share a little bit about what the impact of those [kinds] of practices that you described [have been]—have you seen what that impact looks like? Either for an educator who has made the step and is doing that integration or for students who are in a classroom where an educator is purposely thinking about that level of integration? Bill: Yeah, I can talk a little bit about that. In our research, we have tried to measure the effects of some of these efforts. It is a difficult thing to measure because it's not just a simple true-false test question type of thing that you can give a multiple-choice test for.  But one of the ways that we've looked for the impact [of] these types of intentional designs is by looking at patterns of student participation in classroom discussions and seeing who is accessing the floor of the discussion and how. And then looking at other results, like giving an assessment, but deeper than looking at the outcome, the binary correct versus incorrect. Also looking at the quality of the explanation that's provided. So, how [do] you justify an answer? Does the student provide a deeper or a more mathematically complete explanation?  That is an area where I think more investigation is needed, and it's also very hard to vary systematically. So, from a research perspective—you may not want to put this into the final version [laughs]—but from a research perspective, it's very hard to fix and isolate these things because they are integrated. Mike: Yeah. Yeah. Bill: Because language and mathematics are so deeply integrated that trying to fix everything and do this—“What caused this water to taste like water? Was it the hydrogen or the oxygen?”—well, [laughs] you can't really pull those apart, right? The water molecule is hydrogen and oxygen together. Mike: I think that's a lovely analogy for what we were talking about with mathematical goals and language goals. That, I think, is really a helpful way to think about the extent to which they're intertwined with one another. Bill: Yeah, I need to give full credit to Vygotsky, I think, who said that. Mike: You’re— Bill: Something. Might be Vygotsky. I'll need to check my notes. Mike: I think you're in good company if you're quoting Vygotsky.  Before we close, I'd love to just ask you a bit about resources. I say this often on the podcast. We have 20 to 25 minutes to dig deeply into an idea, and I know people who are listening often think about, “Where do I go from here?” Are there any particular resources that you would suggest for someone who wanted to continue learning about what it is to support multilingual learners in a math classroom? Bill: Sure. Happy to share that.  So, I think on the individual and collective level—so, say, a group of teachers—there's a beautiful book by Kathryn Chval and her colleagues [Teaching Math to Multilingual Learners, Grades K–8 [https://www.corwin.com/books/multilingual-success-in-math-272643?srsltid=AfmBOor4yTyy-IbOxAK2bzJBcINQZ3WdOn5V1lRvDMnV3bbcWuHnbKZz]] about supporting multilingual learners and mathematics. And I really see that as a valuable resource. I've used that in reading groups with teachers and used that in book studies, and it's been very productive and powerful for us. Beyond that, of course, I think the NCTM [National Council of Teachers of Mathematics [https://www.nctm.org/]] provides a number of really useful resources. And there are articles, for example, in the [NCTM journal] Mathematics Teacher: Learning and Teaching PK– 12 [https://pubs.nctm.org/view/journals/mtlt/mtlt-overview.xml] that could make for a really wonderful study or opportunity to engage more deeply.  And then I would say on a broader perspective, I've worked with organizations like the English Learners Success Forum [https://www.elsuccessforum.org/] and others. We've done some case studies and little classroom studies that are accessible on my website [SDSU-ELSF Video Cases for Professional Development [https://elsf.sdsu.edu/]], so you can go to that. But there's also from that organization some really valuable insights, if you're looking at adopting new materials or evaluating things, that gives you a principled set of guidelines to follow. And I think that's really helpful for educators because we don't have to do this all on our own. This is not a “reinvent the wheel at every single site” kind of situation. And so, I always encourage people to look for those resources.  And of course, I will say that the MLC materials, [https://store.mathlearningcenter.org/s/] the Bridges in Mathematics [https://www.mathlearningcenter.org/curriculum/bridges-mathematics] [curriculum], I think have been really beautifully designed with a lot of these principles right behind them. So, for example, if you look through the Teachers Guides [https://teach.mathlearningcenter.org/curriculum/brk] on the Bridges in Mathematics [BES login required], those integrated math and language and practice goals are a part of the design. Mike: Well, I think that's a great place to stop. Thank you so much for joining us, Bill. This has been insightful, and it's really been a pleasure talking with you. Bill: Oh, well, thank you. I appreciate it. Mike: And that's a wrap for Season 3 of Rounding Up. I want to thank all of our guests and the MLC staff who make these podcasts possible, as well as all of our listeners for tuning in. Have a great summer, and we'll be back in September for Season 4.  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. © 2025 The Math Learning Center | www.mathlearningcenter.org