Ramsey Merritt, Improving Students' Turn & Talk Experience ROUNDING UP: SEASON 4 | EPISODE 5

Most educators know what a turn and talk is—but are your students excited to do them?

In this episode, we put turn and talks under a microscope. We'll talk with Ramsey Merritt from the Harvard Graduate School of Education about ways to revamp and better scaffold turn and talks to ensure your students are having productive mathematical discussions.

BIOGRAPHY

Ramsey Merritt is a lecturer in education at Brandeis University and the director of leadership development for Reading (MA) Public Schools. He has taught and coached at every level of the U.S. school system in both public and independent schools from New York to California. Ramsey also runs an instructional leadership consulting firm, Instructional Success Partners, LLC. Prior to his career in education, he worked in a variety of roles at the New York Times. He is currently completing his doctorate in education leadership at Harvard Graduate School of Education. Ramsey's book, Diving Deeper with Upper Elementary Math, will be released in spring 2026.

TRANSCRIPT

Mike Wallus: Welcome to the podcast, Ramsey. So great to have you on.

Ramsey Merritt: It is my pleasure. Thank you so much for having me.

Mike: So turn and talk's been around for a while now, and I guess I'd call it ubiquitous at this point. When I visit classrooms, I see turn and talks happen often with quite mixed results. And I wanted to start with this question: At the broadest level, what's the promise of a turn and talk? When strategically done well, what's it good for?

Ramsey: I think at the broadest level, we want students talking about their thinking and we also want them listening to other students' thinking and ideally being open to reflect, ask questions, and maybe even change their minds on their own thinking or add a new strategy to their thinking. That's at the broadest level.

I think if we were to zoom in a little bit, I think turn and talks are great for idea generation. When you are entering a new concept or a new lesson or a new unit, I think they're great for comparing strategies. They're obviously great for building listening skills with the caveat that you put structures in place for them, which I'm sure we'll talk about later. And building critical-thinking and questioning skills as well.

I think I've also seen turn and talks broadly categorized into engagement, and it's interesting when I read that because to me I think about engagement as the teacher's responsibility and what the teacher needs to do no matter what the pedagogical tool is. So no matter whether it's a turn and talk or something else, engagement is what the teacher needs to craft and create a moment. And I think a lot of what we'll probably talk about today is about crafting moments for the turn and talk. In other words, how to engage students in a turn and talk, but not that a turn and talk is automatically engagement.

Mike: I love that, and I think the language that you've used around crafting is really important. And it gets to the heart of what I was excited about in this conversation because a turn and talk is a tool, but there is an art and a craft to designing its implementation that really can make or break the tool itself.

Ramsey: Yeah. If we look back a little bit as to where turn and talk came from, I sort of tried to dig into the papers on this. And what I found was that it seems as if turn and talks may have been a sort of spinoff of the think-pair-share, which has been around a little bit longer. And what's interesting in looking into this is, I think that turn and talks were originally positioned as a sort of cousin of think-pair-share that can be more spontaneous and more in the moment. And I think what has happened is we've lost the "think" part. So we've run with it, and we've said, "This is great," but we forgot that students still need time to think before they turn and talk. And so what I see a lot is, it gets to be somewhat too spontaneous, and certain students are not prepared to just jump into conversations. And we have to take a step back and sort of think about that.

Mike: That really leads into my next question quite well because I have to confess that when I've attended presentations, there are points in time when I've been asked to turn and talk when I can tell you I had not a lot of interest nor a lot of clarity about what I should do. And then there were other points where I couldn't wait to start that conversation. And I think this is the craft and it's also the place where we should probably think about, "What are the pitfalls that can derail or have a turn and talk kind of lose the value that's possible?" How would you talk about that?

Ramsey: Yeah, it is funny that we as adults have that reaction when people say, "Turn and talk."

The three big ones that I see the most, and I should sort of say here, I've probably been in 75 to 100 buildings and triple or quadruple that for classrooms. So I've seen a lot of turn and talks, just like you said. And the three big ones for me, I'll start with the one that I see less frequently but still see it enough to cringe and want to tell you about it. And it's what I call the "stall" turn and talk. So it's where teachers will sometimes use it to buy themselves a little time. I have literally heard teachers say something along the lines of, "OK, turn and talk to your neighbor while I go grab something off the printer."

But the two biggest ones I think lead to turn and talk failure are a lack of specificity. And in that same vein too, what are you actually asking them to discuss? So there's a bit of vagueness in the prompting, so that's one of the big ones.

The other big one for me is, and it seems so simple, and I think most elementary teachers are very good at using an engaging voice. They've learned what tone does for students and what signals tone sends to them about, "Is now the time to engage? Should I be excited?" But I so often see the turn and talk launched unenthusiastically, and that leads to an engagement deficit. And that's what you're starting out with if you don't have a good launch: Students are already sort of against you because you haven't made them excited to talk.

Mike: I mean those things resonate. And I have to say there are some of them that I cringe because I've been guilty of doing, definitely the first thing when I've been unprepared. But I think these two that you just shared, they really go to this question of how intentionally I am thinking about building that sense of engagement and also digging into the features that make a turn and talk effective and engaging.

So let's talk about the features that make turn and talks effective and engaging for students. I've heard you talk about the importance of picking the right moment for a turn and talk. So what's that mean?

Ramsey: So for me, I break it down into three key elements. And one of them, as you say, is the timing. And this might actually be the most important element, and it goes back to the origin story, is: If you ask a question, and say you haven't planned a turn and talk, but you ask a question to a whole group and you see 12 hands shoot up, that is an ideal moment for a turn and talk. You automatically know that students are interested in this topic. So I think that's the sort of origin story, is: Instead of whipping around the room and asking all 12 students—because especially at the elementary level, if students don't get their chance to share, they are very disappointed. So I've also seen these moments drag out far too long. So it's kind of a good way to get everyone's voice heard. Maybe they're not saying it out to the whole group, but they get to have everyone's voice heard. And also you're buying into the engagement that's already there. So that would be the more spontaneous version, but you can plan in your lesson planning to time a turn and talk at a specific moment if you know your students well enough that you know can get them engaged in.

And so that leads to one of the other points is the launch itself. So then you're really thinking about, "OK, I think this could be an interesting moment for students. Let me think a little bit deeper about what the hook is." Almost every teacher knows what a hook is, but they typically think about the hook at the very top of their lesson. And they don't necessarily think about, "How do I hook students in to every part of my lesson?" And maybe it's not a full 1-minute launch, maybe it's not a full hook, but you've got to reengage students, especially now in this day and time, we're seeing students with increasingly smaller attention spans. So it's important to think about how you're launching every single piece of your lesson.

And then the third one, which goes against that origin story that I may or may not even be right about, but it goes against that sort of spontaneous nature of turn and talks, is: I think the best turn and talks are usually planned out in advance.

So for me it's planning, timing, and launching. Those are my elements to success when I'm coaching teachers on doing a turn and talk.

Mike: Another question that I wanted to unpack is: Talk about what. The turn and talk is a vehicle, but there's also content, right? So I'm wondering about that. And then I'm also wondering are there prompts or particular types of questions that educators can use that are more interesting and engaging, and they help draw students in and build that engagement experience you were talking about?

Ramsey: Yeah, and it's funny you say, "Talk about what" because that's actually feedback that I've given to teachers, when I say, "How did that go for you?" And they go, "Well, it went OK." And I say, "Well, what did you ask them to talk about? Talk about what is important to think about in that planning process." So I hate to throw something big out there, but I would actually argue that at this point, we have seen the turn and talk sort of devolve into something that is stigmatized that often is vague.

So what if instead of calling everything a turn and talk, you had specific types of turn and talks in your classroom. And these would take a little time to routinize; students would have to get used to them. But one idea I had is: What if you just called one "pick a side"? Pick a side, it tells the students right away what they need to do; it's extremely specific. So you're giving them one or two or—well not one, you're giving them two or three strategies, and you're telling them, "You have to pick one of these. And you're going to be explaining to your partner your rationale as to why you think that strategy works best or most efficiently." Or maybe it's an error analysis kind of thing. Maybe you plant one n as wrong, one n as right. And then you still ask them, "Pick a side here. Who do you agree with?" And then you also get a check for understanding because the students around the room who are picking the wrong one, you're picking up data on what they understand about the topic.

Another one you can do is, you could just call it "justify your thinking." Justify your thinking. So that just simply says to them, "I have to explain to the person next to me why I'm thinking the way that I'm thinking about this prompt or this problem."

So that could also be a "help their thinking." So maybe you put up someone's thinking on the board that is half baked, and now their job is to help that person. So that's a sort of deeper knowledge kind of thing too.

And then the last one is we can turn the "What do you notice? What do you wonder?" [activity] into a routine that is very similar to a turn and talk, where both people have an opportunity to share what they're wondering or what they're noticing.

But I think no matter what you call them, no matter how you routinize them, I think it's important to be more specific than "turn and talk."

Mike: You use the word routinized. It's making me think a lot about why we find routines to have value, right? Because once you teach a particular routine, kids know what it is to do said routine. They know what it is to show up when you're doing Which one doesn't belong? They know the role that they play. And I think part of what really jumps out is: If you had a series of more granular turn and talk experiences that you were trying to cultivate, kids actually have a sense of what it is to do a turn and talk if you are helping thinking, or if you are agreeing or disagreeing, or whatever the choice might be.

Ramsey: That's right. For me, everything, even when I'm working with middle and high school teachers, I say, "The more that you can put structures in place that remove those sort of barriers for thinking, the better off you're going to be." And so we could talk more too about how to differentiate and scaffold turn and talk. Sometimes that gets forgotten as well.

But I think the other piece I would love to point out here is around—you're right, turn and talk is so ubiquitous. And what that means, what I've seen in schools, if I've seen, I'll go into a school and I might watch four different teachers teach the same lesson and the turn and talk will look and feel differently in each room. So the other advantage to being more specific is that if a student—let's say they went to, because even in an elementary school you might go to a specialist, you might go to art class. And that teacher might use a turn and talk. And what happens is they sort of get this general idea around the turn and talk and then they come into your room with whatever the turn and talk was in the last class or however the teacher used it last year. So to me there's also a benefit in personalizing it to your room as well so that you can get rid of some of that stigma if it wasn't going well for the student before, especially if you then go in and scaffold it.

Mike: Let's talk a little bit about those scaffolds and maybe dig in a little bit deeper to some of the different kinds of routinized turn and talks. I'm wondering if you wanted to unpack anything in particular that you think would really be important for a teacher to think about as they're trying to take up the ideas that we've been discussing.

Ramsey: And one of the simplest ones to implement is the Partner A, Partner B routine. I think maybe many of your listeners will be like, "Yeah, I use that." But one of the pieces that's really important there is that you really hold students accountable to honoring Partner A's time. So when Partner A is speaking, Partner B needs to be trying to make—you know, not everybody can do the eye contact thing, but there are some things that you can recommend and suggest for them. Maybe they have something to take notes on. So this could be having whiteboards at your rug, it could be clipboards, it could be that they have a turn and talk thought-catcher notebook or folder.

And it doesn't matter what it is, but not everyone has the same processing skills. So we think about turn and talk sometimes as spontaneous, but we're forgetting that 12 students raised their hand and they were eager. What about the other 12 or 15? If they didn't raise their hand, it could be that they're shy but they have something on their mind. But it also could be that you just threw out a prompt and they haven't fully processed it yet. We know kids process things at different times and at different speeds. So incorporating in that—maybe it's even a minute up top. Everybody's taking their silent and solo minute to think about this prompt. Then Partner A is going to go. It's about equity and voice across the room. It's about encouraging listening, it's about giving think time.

Mike: Well, I want to stop and mark a couple things.

What occurs to me is that in some ways a podcast interview like this is one long turn and talk in the sense that you and I are both listening and talking with one another. And as you were talking, one of the things I realized is I didn't have a piece of paper with me. And what you were saying really connects deeply because even if it's just jotting down a word or two to help me remember that was a salient point or this is something that I want to follow up on, that's really critical. Otherwise, it really can feel like it can evaporate and then you're left not being able to explore something that might've been really important.

I think the other thing that jumps out is the way that this notion of having a notepad or something to jot is actually a way to not necessarily just privilege spoken communication. That if I'm going to process or if I'm going to try to participate, having something like that might actually open up space for a kid whose favorite thing to do isn't to talk and process as they're talking. Does that make sense?

Ramsey: Totally. I had a student in a program I was working with this summer who was 13 years old but was selectively mute. And the student teachers who were working in this room wanted to still be able to do a turn and talk. And they had her still partner with people, but she wrote down sentences and she literally held up her whiteboard and then the other student responded to the sentence that she wrote down on her whiteboard. So that's real.

And to your other point about being able to jot down so you can remember—yeah, we have to remember we're talking about six-, seven-, eight-, nine-year olds. We're fully functioning adults and we still need to jot things down. So imagine when your brain is not even fully developed. We can't expect them to remember something from when they haven't been allowed to interrupt the other. And so I think going on now what you're saying is, that then makes me think about the Partner A, Partner B thing could also sort of tamper down the excitement a little bit if you make another student wait. So you also have to think about maybe that time in between, you might need to reengage. That's my own thinking right now, evolving as we're talking.

Mike: So in some ways this is a nice segue to something else that you really made me think about. When we were preparing for this interview, much of what I was thinking about is the role of the teacher in finding the moment, as you said, where you can build excitement and build engagement, or thinking about the kind of prompts that have a specificity and how that could impact the substance of what kids are talking about. But what really jumped out from our conversation is that there's also a receptive side of turn and talk, meaning that there are people who are talking, but we also don't want the other person to just be passive. What does it look like to support the listening side of turn and talk? And I would love it if you would talk about the kinds of things you think it's important for educators to think about when they're thinking about that side of turn and talk.

Ramsey: I would say don't forget about sentence starters that have to do with listening. So often when we're scaffolding, we're thinking about, "How do I get them to share out? How do I get them to be able to address this prompt?"

But one of the easiest scaffolds that I've heard for listening—and it works very, very well—is, "What I heard you say is, blank." And so then the receptive student knows that a—tells them they have to be listening pretty carefully because they're about to be asked to repeat what the other person said. And this is an age-old elementary school sort of piece of pedagogy, is a call and response situation. But then we want to give them a stem that allows them maybe to ask a question. So it's, "What I heard you say was, blank. What I'm wondering is, blank." So that takes it to the next thinking level. But again, it's about being really specific and very intentional with your students and saying, "When it's Partner B's turn, you must lead with, 'What I heard you say is,' and only then can you get to your thinking or asking questions."

Mike: That's huge. I think particularly when you think about the fact that there may be status issues between Partner A and Partner B. If Partner A is seen as or sees themselves as someone who's good at math and that's less true for Partner B, the likelihood of actually listening in a productive way seems like it's in danger at the very least. So I see these as tools that really do, one, build a level of accountability responsibility, but also level the playing field when it comes to things like status between two students.

Ramsey: I would agree with that, yeah.

I think, too, we always want to be mixing our groups. I think sometimes you get, when I think about those sort of people or those students who—you can walk into any classroom and you right away can look around the room, if you've seen enough math teaching, you can see the students who have the most confidence in math.

So another piece to sort of leveling that field is making sure that your turn and talks are not always built on skill or high-level conceptual understanding. So that's where it might be helpful to have a more low-floor task, like a What do you notice? What do you wonder? But using the turn and talk routine of that. So it gives people more of a chance to get involved even when they don't have the highest level. It's kind of like the same idea with a Which one doesn't belong? [task] or a typical number talk. But, so you as the teachers have to be thinking about, "OK, yesterday we did one that was comparing two people's strategies, and I know that some of my students didn't quite understand either one of them. So today, in order to rebuild some of that confidence, I might do a version of a turn and talk that is much more open to different kinds of thinking."

Mike: You started to go there in this last conversation we had about supporting the receptive side of turn and talk. I did want to ask if we can go a little bit deeper and think about tools like anchor charts. And you already mentioned sentence prompts, but sentence frames. To what extent do you feel like those can be helpful in building the kinds of habits we're talking about, and do you have any thoughts about those or any other resources that you think are important scaffolds?

Ramsey: Yeah. I have seen some really, really wonderful teachers bring in such a simple way of activating an anchor chart and that is especially—it's easier to do an inquiry-based learning, but I think you can do it in any kind of classroom—is, when a student presents their thinking early on in a unit, and let's say we're talking about comparing fractions. And they say, "This is how I compared fractions," and you're annotating and you're charting it up for them as the teacher, you can call that strategy, "Maya's strategy." And so now it has a little bit more stickiness for both the students and for you. Now you know that there's a specific mathematical name for that strategy, but the students don't necessarily need to know that. You could put it in parentheses if you want. But I have seen that be really effective, and I've actually heard other students go, "I'm going to use Maya's strategy for this one," and able to then look and reference it.

I think what happens sometimes with the anchor charts is, we still live in a sort of Pinterest world, and some people want those anchor charts to be beautiful, but they're not actually useful because it was drawn up perfectly and it's lovely and it's pretty, but the students don't have a real connection to it. So the other piece to that is the cocreation of the anchor chart. So it's not just naming the student; it's also going through it step by step. Maybe they're leading through it, maybe you're guiding it. Maybe you're asking probing questions. Maybe you throw in a turn and talk in the middle of that sort of exploration. And then students have a connection to that piece of paper. Anchor charts that have been created during your prep period, I guarantee you will have very little effect. So that's how I feel about those.

I also love, I call them like mini anchor charts, but they sit on tables. In recent years I've seen more and more, especially in elementary classrooms—and I've encouraged them at the middle school and the high school level—of putting in a little, I don't really know the best way to describe it for listeners, but it sits on the tabletop, and it's almost like a placard holder. And inside of that you put a mini version of an anchor chart that sits at the students' tables. So if you're doing turn and talks at their desks, and they're sitting in desks of four, and that's right there in front of them with some sentence starters or maybe your very specific routines—pick a side!—and then you have the three steps to picking aside underneath. If that's sitting on the table right in front of them, they are much more likely to reference it than if it's on the wall across the room. That gets a little trickier if you're down at the rug if you're doing turn and talks down at the rug, but hey, you can get a slightly bigger one and stick a few down on the rug around them too if you really need to.

Mike: I love that. That seems powerful and yet imminently practical.

Ramsey: I've seen it work.

Mike: Well, this happens to me every time I do a podcast. I have a lovely conversation, and we get close to the end of it, and I find myself asking: For listeners, what recommendations do you have for people who either want to learn more or would like to get started implementing some of the ideas we discussed today?

Ramsey: Sure. I mean the biggest one that I tell both new teachers and veterans when you're looking to sort of improve on your practice is to go watch someone else teach. So it's as simple as asking a colleague, "Hey, do you know anybody who does this really well?" In fact, I've led some [professional development trainings] at schools where I've said, "Who in the room is great at this?" And a few people will throw their hands up, and I go, "Great. Instead of me explaining it, I'm going to have you tell us why you're so successful at that." So the easiest one is to go watch someone who has this down.

But for some of the things that I've mentioned, I would think about not biting off too much. So if you are someone who your turn and talks, you readily admit that they're not specific, they're fully routinized, and they don't go well for you, I would not recommend putting in four new routines tomorrow, the A/B partner thing, and making the anchor charts for the tables all at once.

What I always say is try one thing and also be transparent with the students. It goes a really long way, even with seven-year-olds, when you say, "Alright guys, we're going to do a new version of the turn and talk today because I've noticed that some of you have not been able to share as much as I would like you to. So we're going to try this, which is for me, I hope it allows both people to share and afterwards you can let me know how that felt." Students really appreciate that gesture, and I think that's really important if you are going to try something new to sort of be transparent about it. Oftentimes when teachers implement something new, it can feel like, not a punishment, but it's almost like a, "Ooh, why is she changing this up on us?" So letting them know also creates a warmer space too, and it shows them that you're learning, you're growing.

Mike: I love that, and I think that's a great place to stop.

Ramsey, thank you so much. It has really been a pleasure talking with you.

Ramsey: Thank you. Like you said, I could do it all day, so I really appreciate it. I wish everyone out there well, and thanks again.

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

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 by Catherine Fosnot and Maarten Dolk

Procedural fluency in mathematics: Reasoning and decision-making, not rote application of procedures position by the National Council of Teachers of Mathematics

Bridges number string example from Grade 5, Unit 3, Module 1, Session 1 (BES login required)

Developing Mathematical Reasoning: Avoiding the Trap of Algorithms by Pamela Weber Harris and Cameron Harris

Math is Figure-out-able!™ Problem Strings

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. 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, 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

Math is FigureOutAble!™ Guide (Download)

Journey Coaching

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, 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, so you can see something that would be helpful. And we have, at Math is Figure-Out-Able, an online coaching support called 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

Looney Math

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, 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. 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.

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?"

Talking Math With Your Kids by Christopher Danielson

Math On-A-Stick

5 Practices for Orchestrating Productive Mathematics Discussion by Margaret (Peg) Smith & Mary Kay Stein

How Many?: A Counting Book by Christopher Danielson

How Did You Count? A Picture Book 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 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

National Council of Teachers of Mathematics

Mathematics Teacher: Learning and Teaching PK– 12

English Learners Success Forum

SDSU-ELSF Video Cases for Professional Development

The Math Learning Center materials

Bridges in Mathematics curriculum

Bridges in Mathematics Teachers Guides [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] 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] provides a number of really useful resources. And there are articles, for example, in the [NCTM journal] Mathematics Teacher: Learning and Teaching PK– 12 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 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], 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, the Bridges in 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 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

Tisha Jones, Assessment as a Shared Journey: Cultivating Partnerships with Families & Caregivers ROUNDING UP: SEASON 3 | EPISODE 16

Families and caregivers play an essential role in students' success in school and in shaping their identities as learners. Therefore, establishing strong partnerships with families and caregivers is crucial for equitable teaching and learning. This episode is designed to help educators explore the importance of collaborating with families and caregivers and learn strategies for shifting to asset-based communication.

BIOGRAPHY

Tisha Jones is the senior manager of assessment at The Math Learning Center. Previously, Tisha taught math to elementary and middle school students as well as undergraduate and graduate math methods courses at Georgia State University.

TRANSCRIPT

Mike Wallus: As educators, we know that families and caregivers play an essential role in our students' success at school. With that in mind, what are some of the ways we can establish strong partnerships with caregivers and communicate about students' progress in asset-based ways? We'll explore these questions with MLC's [senior] assessment manager, Tisha Jones, on this episode of Rounding Up.

Welcome back to the podcast, Tisha. I think you are our first guest to appear three times. We're really excited to talk to you about assessment and families and caregivers.

Tisha Jones: I am always happy to talk to you, Mike, and I really love getting to share new ideas with people on your podcast.

Mike: So, we've titled this episode "Assessment as a Shared Journey with Families & Caregivers," and I feel like that title—especially the words "shared journey"—say a lot about how you hope educators approach this part of their practice.

Tisha: Absolutely.

Mike: So, I want to start by being explicit about how we at The Math Learning Center think about the purpose of assessment because I think a lot of the ideas and the practices and the suggestions that you're about to offer flow out of that way that we think about the purpose.

Tisha: When we think about the purpose of assessment at The Math Learning Center, what sums it up best to me is that all assessment is formative, even if it's summative, which is a belief that you'll find in our Assessment Guide. And what that means is that assessment really is to drive learning. It's for the purpose of learning. So, it's not just to capture, "What did they learn?," but it's, "What do they need?," "How can we support kids?," "How can we build on what they're learning?" over and over and over again. And so, there's no point where we're like, "OK, we've assessed it and now the learning of that is in the past." We're always trying to build on what they're doing, what they've learned so far.

Mike: You know, I've also heard you talk about the importance of an asset-focused approach to assessment. So, for folks who haven't heard us talk about this in the past, what does that mean, Tisha?

Tisha: So that means starting with finding the things that the kids know how to do and what they understand instead of the alternative, which is looking for what they don't know, looking for the deficits in their thinking. We're looking at, "OK, here's the evidence for all the things that they can do," and then we're looking to think about, "OK, what are their opportunities for growth?"

Mike: That sounds subtle, but it is so profound a shift in thinking about what is happening when we're assessing and what we're seeing from students. How do you think that change in perspective shifts the work of assessing, but also the work of teaching?

Tisha: When I think about approaching assessment from an asset-based perspective—finding the things that kids know how to do, the things that kids understand—one, I am now on a mission to find their brilliance. I am just this brilliance detective. I'm always looking for, "What is that thing that this kid can shine at?" That's one, and a different way of thinking about it just to start with.

And then I think the other thing, too, is, I feel like when you find the things that they're doing, I can think about, "OK, what do I need to know? What can I do for them next to support them in that next step of growth?"

Mike: I think that sounds fairly simple, but there's something very different about thinking about building from something versus, say, looking for what's broken.

Tisha: For sure. And it also helps build relationships, right? If you approach any relationship from a deficit perspective, you're always focusing on the things that are wrong. And so, if we're talking about building stronger relationships with kids, coming from an asset-based perspective helps in that area too.

Mike: That's a great pivot point because if we take this notion that the purpose of assessment is to inform the ways that we support student learning, it really seems like that has a major set of implications for how and what and even why we would communicate with families and caregivers.

So, while I suspect there isn't a script for the type of communication, are there some essential components that you'd want to see in an asset-focused assessment conversation that an educator would have with a family or with their child's caregivers?

Tisha: Well, before thinking about a singular conversation, I want to back it up and think about—over the course of the school year. And I think that when we start the communication, it has to start before that first assessment. It has to start before we've seen a piece of kids' work. We have to start building those relationships with families and caregivers. We need to invite them into this process. We need to give them an opportunity to understand what we think about assessment. How are we approaching it? When we send things home, and they haven't heard of things like "proficiency" or "meeting current expectations"—those are common words that you'll see throughout the Bridges assessment materials—if parents haven't seen that, if families and caregivers haven't heard from you on what that means for you in your classroom at your school, then they have questions. It feels unfamiliar. It feels like, "Wait, what does this mean about how my child is doing in your class?"

And so, we want to start this conversation from the very beginning of the school year and continue it on continuously. And it should be this open invitation for them to participate in this process too, for them to share what they're seeing about their student at home, when they're talking about math or they're hearing how their student is talking about math. We want to know those things because that informs how we approach the instruction in class.

Mike: Let's talk about that because it really strikes me that what you're describing in terms of the meaning of proficiency or the meaning of meeting expectations—that language is likely fairly new to families and caregivers.

And I think the other thing that strikes me is, families and caregivers have their own lived experience with assessment from when they were children, perhaps with other children. And that's generally a mixed bag at best. Folks have this set of ideas about what it means when the teacher contacts them and what assessment means. So, I really hear what you're saying when you're talking about, there's work that educators need to do at the start of the year to set the stage for these conversations.

Let's try to get a little bit specific, though. What are some of the practices that you'd want teachers to consider when they're thinking about their communication?

Tisha: So, I think that starting at the very beginning of the year, most schools do some sort of a curriculum night. I would start by making sure that assessment is a part of that conversation and making sure that you're explaining what assessment means to you. Why are you assessing? What are the different ways that you're assessing? What are some things that [families and caregivers] might see coming home? Are they going to see feedback? Are they going to see scores from assessments? But how were you communicating progress? How do they know how their student is doing? And then also that invitation, right then and there, to be a part of this process, to hear from them, to hear their concerns or their ideas around feedback or the things that they've got questions about.

I would also suggest … really working hard to have that asset-based lens apply to parents and families and caregivers. I know that I have been that parent that was the last one to sign up for the parent teacher conferences, and I'm sending the apologetic email, and I'm begging for a special time slot. So, it didn't mean that I didn't care about my kids. It didn't mean that I didn't care about what they were doing. I was swamped. And so, I think we want to keep finding that asset-based lens for parents and caregivers in the same way that we do for the students.

And then making sure that you're giving them good news, not just bad news. And then making sure when you're sending any communication about how a student is doing, try to be concrete about what you're seeing, right? So, trying to say, "These are the things where I see your child's strengths. These are the strengths that I'm seeing from your student. And these are the areas where we're working on to grow. And this is what we're doing here at school, and this is what you can do to support them at home."

Mike: I was really struck by a piece of what you said, Tisha, when you really made the case for not assuming that the picture that you have in your mind as an educator is clear for families when it comes to assessment. So, really being transparent about how you think about assessment, why you're assessing, and the cadence of when parents or families or caregivers could expect to hear from you and what they could expect as well.

I know for a fact that if my teacher called my family when I was a kid, generally there was a look that came across their face when they answered the phone. And even if it was good news, they didn't think it was good news at the front end of that conversation.

Tisha: I've been there. I had my son's fifth grade teacher call me last year, and I was like, "Oh, what is this?" [laughs]

Mike: One of the things that I want to talk about before we finish this conversation is homework. I want to talk a little bit about the purpose of homework. We're having this conversation in the context of Bridges in Mathematics, which is the curriculum that The Math Learning Center publishes. So, while we can't talk about how all folks think about homework, we can talk about the stance that we take when it comes to homework: what its purpose is, how we imagine families and caregivers can engage with their students around it.

Can you talk a little bit about our perspective on homework? How we think about its value, how we think about its purpose? And then we can dig a little bit into what it might look like at home, but let's start with purpose and intent.

Tisha: So, we definitely recognize that there are lots of different ideas about homework, and I think that shows in how we've structured homework through our Bridges units. Most of the time, it's set up so that there's a homework [assignment] that goes with every other session, but it's still optional. So, there's no formal expectation in our curriculum that homework is given on a nightly basis or even on an every-other-night basis. We really have left that up to the schools to determine what is best practice for their population. And I think that is actually what's really the most important thing is, understanding the families and caregivers and the situations that are in your building, and making determinations about homework that makes sense for the students that you're serving. And so, I think we've set homework up in a way that makes it so that it's easy for schools to make those decisions.

Mike: One of the things that I'm thinking about is that—again, I'm going to be autobiographical—when I was a kid, homework went back, it was graded, and it actually counted toward my grade at the end of the semester or the quarter or what have you. And I guess I wonder if a school or a district chose to not go about that, to not have homework necessarily be graded, I wonder if some families and caregivers might wonder, "What's the purpose?" I think we know that there can be a productive and important purpose—even if educators aren't grading homework and adding it to a percentage that is somehow determining students' grades, that it can actually still have purpose. How do you think about the purpose of homework, regardless of whether it's graded or not?

Tisha: So first off, I would just like to advocate not grading homework if I can.

Mike: You certainly can, yeah.

Tisha: [laughs]

Mike: Let's talk about that.

Tisha: I think that, one, if we're talking about this idea of putting this score into an average grade or this percentage grade, I think that this is something that has so many different circumstances for kids at home. You have some students who get lots and lots of help. You get some students who do not have help available to them.

Another experience that has been very common when I was teaching was that I would get messages where it was like, "We were doing homework. The kid was in tears, I was in tears. This was just really hard." And that's just not—I don't ever want that scenario for any student, for any family, for any caregiver, for anybody trying to support a child at home. I used to tell them, "If you are getting to the point where it's that level of frustration, please just stop and send me a message, write it on the homework. Just communicate something that [says,] 'This was too hard' because that's information now that I can use."

And so, for me, I think about [how] homework can be an opportunity for students to practice some skills and concepts and things that they've learned at home. It's an opportunity for parents, families, caregivers to see some of the things that the kids are working on at school.

Mike: What do you think is meaningful for homework? And I have kind of two bits to that. What do you think is meaningful for the child? And then, what do you think might be meaningful for the interaction between the child and their family or caregiver? What's the best case for homework? When you imagine a successful or a productive or a meaningful experience with homework at home between child and family and caregiver, what's that look like?

Tisha: Well, one of the things that I've heard families say is, "I don't know how to help my child with blank." So, then I think it is, "Well, how do we support families and caregivers in knowing what [to] do with homework when we don't know how to tell them what to do?"

So, to me, it's about, how can we restructure the homework experience so that it's not this, "I have to tell you how to do it so you can get the right answer so you can get the grade." But it's like, "How can I get at more of your thinking? How can I understand then what is happening or what you do know?" So, "We can't get to the answer. OK. So tell me about what you do know, and how can we build from there? How can we build understanding?" And that way it maybe will take some of the pressure off of families and caregivers to help their child get to the right answer.

Mike: What hits me is we've really come full circle with that last statement you made because you could conceivably have a student who really clearly understands a particular problem that might be a piece of homework, [who] might have some ideas that are on the right track, but ultimately perhaps doesn't get to a fully clear answer that is perfect. And you might have a student who at a certain point in time, maybe [for them] the context or the problem itself is profoundly challenging.

And in all of those cases, the question, "Tell me what you do know" or "Tell me what you're thinking" is still an opportunity to draw out the students' ideas and to focus on the assets. Even if the work as you described it is to get them to think about, "What are the questions that are really causing me to feel stuck?" That is a productive move for a family and a caregiver and a student to engage in, to kind of wonder about, "What's going on here that's making me feel stuck?" Because then, as you said, all assessment is formative.

Tisha: Mm-hmm.

Mike: That homework that comes back is functioning as a formative assessment, and it allows you to think about your next moves, how you build on what the student knows, or even how you build on the questions that the student is bringing to you.

Tisha: And that's such a great point, too, is there's really more value in them coming back with an incomplete assignment or there's, I don't know, maybe "more value" is not the right way to say it. But there is value in kids coming back with an incomplete assignment or an attempted assignment, but they weren't sure how to get through all the problems—as opposed to a parent who has told their student what to do to get to all of the right answers. And so, now they have all these right answers, but it doesn't really give you a clear picture of what that student actually does understand.

So, I'd much rather have a student attempt the homework and stop because they got too stuck, because now I know that, than having a family [member] or a caregiver—somebody working with that student—feel like if they don't have all of the right answers, then it's a problem.

Mike: I think that's really great guidance, both for teachers as they're trying to set expectations and be transparent with families. But also I think it takes that pressure off of families or caregivers who feel like their work when homework shows up, is to get to a right answer. It just feels like a much more healthy relationship with homework and a much more healthy way to think about the value that it has.

Tisha: Well, in truth, it's a healthier relationship with math overall, right? That math is a process. It's not just—the value is not in just this one right answer or this paper of right answers, but it's really in, "How do we deepen our understanding?," "How do we help students deepen their understanding and have this more positive relationship with math?" And I think that creating these homework struggles between families and caregivers and the children does not support that end goal of having a more positive relationship with math overall.

Mike: Which is a really important part of what we're looking for in a child's elementary experience.

Tisha: Absolutely.

Mike: I think that's a great place to stop. Tisha Jones, thank you so much for joining us. We would love to have you back at some time. It has been a pleasure talking with you.

Tisha: It's been great talking to you, too, Mike.

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

Ryan Flessner, What If I Don't Understand Their Thinking? ROUNDING UP: SEASON 3 | EPISODE 15

"What do I do if I don't understand my student's strategy?" This is a question teachers grapple with constantly, particularly when conferring with students during class. How educators respond in moments like these can have a profound impact on students' learning and their mathematical identities.

In this episode, we talk with Ryan Flessner from Butler University about what educators can say or do when faced with this situation.

BIOGRAPHY

Ryan Flessner is a professor of teacher education in the College of Education at Butler University in Indianapolis, Indiana. He holds a PhD in curriculum and instruction with an emphasis in teacher education from the University of Wisconsin–Madison; a master of arts in curriculum and teaching from Teachers College, Columbia University; and a bachelor of science in elementary education from Butler University. Prior to his time at the university level, he taught grades 3–7 in Indianapolis; New York City; and Madison, Wisconsin.

RESOURCES

Nearpod

Pear Deck

GeoGebra

Magma Math

TRANSCRIPT

Mike Wallus: "What do I do if I don't understand my student's strategy?" This is a question teachers grapple with constantly, particularly when conferring with students during class. How we respond in moments like these can have a profound impact on our students' learning and their mathematical identities. Today we'll talk with Ryan Flessner from Butler University about what educators can say or do when faced with this very common situation.

Welcome to the podcast, Ryan. Really excited to talk to you today.

Ryan Flessner: Thanks, Mike. I'm flattered to be here. Thank you so much for the invitation.

Mike: So, this experience of working with a student and not being able to make sense of their solution feels like something that almost every teacher has had. And I'll speak for myself and say that when it happens to me, I feel a lot of anxiety. And I just want to start by asking, what would you say to educators who are feeling apprehensive or unsure about what to do when they encounter a situation like this?

Ryan: Yeah, so I think that everybody has that experience. I think the problem that we have is that teachers often feel the need to have all of the answers and to know everything and to be the expert in the room. But as an educator, I learned really quickly that I didn't have all the answers. And to pretend like I did put a lot of pressure on me and made me feel a lot of stress and would leave me answering children by saying, "Let me get back to you on that." And then I would scurry and try and find all the answers so I could come back with a knowledgeable idea. And it was just so much more work than to just simply say, "I don't know. Let's investigate that together." Or to ask kids, "That's something interesting that I'm seeing you do. I've never seen a student do that before. Can you talk to me a little bit about that?"

And just having that ability to free myself from having to have all the answers and using that Reggio-inspired practice—for those who know early childhood education—to follow the child, to listen to what he or she or they say to us and try to see. I can usually keep up with a 7- or an 8-year-old as they're explaining math to me. I just may never have seen them notate something the way they did. So, trying to ask that question about, "Show me what you know. Teach me something new." The idea that a teacher could be a learner at the same time I think is novel to kids, and I think they respond really well to that idea.

Mike: So, before we dig in a little bit more deeply about how teachers respond to student strategies if they don't understand, I just want to linger and think about the assumptions that many educators, myself included, might bring to this situation. Assumptions about their role, assumptions about what it would mean for a student if they don't know the answer right away. How do you think about some of the assumptions that are causing some of that anxiety for us?

Ryan: Yeah. When the new generation of standards came out, especially in the field of math, teachers were all of a sudden asked to teach in a way that they themselves didn't learn. And so, if you have that idea that you have to have all the answers and you have to know everything, that puts you in a really vulnerable spot because how are we supposed to just magically teach things we've never learned ourselves? And so, trying to figure out ways that we can back up and try and make sense of the work that we're doing with kids, for me that was really helpful in understanding what I wanted from my students. I wanted them to make sense of the learning. So, if I hadn't made sense of it yet, how in the world could I teach them to make sense of it? And so we have to have that humility to say, "I don't know how to do this. I need to continue my learning trajectory and to keep going and trying to do a little bit better than the day that I did before."

I think that teachers are uniquely self-critical and they're always trying to do better, but I don't know if we necessarily are taught how to learn once we become teachers. Like, "We've already learned everything we have to do. Now we just have to learn how to teach it to other people." But I don't think we have learned everything that we have to learn. There's a lot of stuff in the math world that I don't think we actually learned. We just memorized steps and kind of regurgitated them to get our A+ on a test or whatever we did.

So, I think having the ability to stop and say, "I don't know how to do this, and so I'm going to keep working at it, and when I start to learn it, I'm going to be able to ask myself questions that I should be asking my students." And just being really thoughtful about, "Why is the child saying the thing that she is?," "Why is she doing it the way that she's doing it?," "Why is she writing it the way that she's writing it?" And if I can't figure it out, the expert on that piece of paper is the child [herself], so why wouldn't I go and say, "Talk to me about this."? I don't have to have all the answers right off the cuff.

Mike: In some ways, what you were describing just there is a real nice segue because I've heard you say that our minds and our students' minds often work faster than we can write, or even in some cases faster than we can speak. I'm wondering if you can unpack that. Why do you think this matters, particularly in the situation that we're talking about?

Ryan: Yeah, I think a lot of us, especially in math, have been conditioned to get an answer. And nobody's really asked us "Why?" in the past. And so, we've done all of the thinking, we give the answer, and then we think the job is done. But with a lot of the new standards, we have to explain why we think that way. And so, all those ideas that just flurried through our head, we have to now articulate those either in writing on paper or in speech, trying to figure out how we can communicate the mathematics behind the answer.

And so, a lot of times I'll be in a classroom, and I'll ask a student for an answer, and I'll say, "How'd you get that?" And the first inclination that a lot of kids have is, "Oh, I must be wrong if a teacher is asking me why." So, they think they're wrong. And so I say, "No, no, no. It's not that you're wrong. I'm just curious. You came to that answer, you stopped and you looked up at the ceiling for a while and then you came to me and you said the answer is 68. How did you do that?" A child will say something like, "Well, I just thought about it in my head." And I say, "Well, what did you think about in your head?" "Well, my brain just told me the answer was 68."

And we have to actually talk to kids. And we have to teach them how to talk to us—that we're not quizzing them or saying that they're wrong or they didn't do something well enough—that we just want them to communicate with us how they're going about finding these things, what the strategies are. Because if they can communicate with us in writing, if they can communicate on paper, if they can use gestures to explain what they're thinking about, all of those tell us strengths that they bring to the table. And if I can figure out the strengths that you have, then I can leverage those strengths as I address needs that arise in my classroom. And so, I really want to create this bank of information about individual students that will help me be the best teacher that I can be for them. And if I can't ask those questions and they can't answer those questions for me, how am I going to individualize my instruction in meaningful ways for kids?

Mike: We've been talking a little bit about the teacher experience in this moment, and we've been talking about some of the things that a person might say.

One of the things that I'm thinking about before we dig in a little bit deeper is, just, what is my role? How do you think about the role of a teacher in the moment when they encounter thinking from a student that they don't quite understand […] yet? Part of what I'm after is, how can a teacher think about what they're trying to accomplish in that moment for themselves as a learner and also for the learner in front of them? How would you answer that question?

Ryan: When I think about an interaction with a kid in a moment like that, I try to figure out, as the teacher, my goal is to try and figure out what this child knows so that I can continue their journey in a forward trajectory. Instead of thinking about, "They need to go to page 34 because we're on page 33," just thinking about, "What does this kid need next from me as the teacher?"

What I want them to get out of the situation is I want them to understand that they are powerful individuals, that they have something to offer the conversation and not just to prove it to the adult in the room. But if I can hear them talk about these ideas, sometimes the kids in the classroom can answer each other's questions. And so, if I can ask these things aloud and other kids are listening in, maybe because we're in close proximity or because we're in a small-group setting, if I can get the kids to verbalize those ideas sometimes one kid talking strikes an idea in another kid. Or another kid will say, "I didn't know how to answer Ryan when he asked me that question before, but now that I hear what it sounds like to answer that type of a question, now I get it, and I know how I would say it if it were my turn."

So, we have to actually offer kids the opportunity to learn how to engage in those moments and how to share their expertise so others can benefit from their expertise and use that in a way that's helpful in the mathematical process.

Mike: One of the most practical—and, I have to say, freeing—things that I've heard you recommend when a teacher encounters student work and they're still trying to make sense of it, is to just go ahead and name it. What are some of the things you imagine that a teacher might say that just straight out name the fact that they're still trying to understand a student's thinking? Tell me a little bit about that.

Ryan: Well, I think the first thing is that we just have to normalize the question "Why?" or "Tell me how you know that." If we normalize those things—a lot of times kids get asked that question when they're wrong, and so it's an [immediate] tip of the hat that "You're wrong, now go back and fix it. There's something wrong with you. You haven't tried hard enough." Kids get these messages even if we don't intend for them to get them. So, if we can normalize the question "Tell me why you think that" or "Explain that to me"—if we can just get them to see that every time you give me an answer whether it's right or wrong, I'm just going to ask you to talk to me about it, that takes care of half of the problem.

But I think sometimes teachers get stuck because—and myself being one of them—we get stuck because we'll look at what a student is doing and they do something that we don't anticipate. Or we say, "I've shown you three different ways to get at this problem, different strategies you can use, and you're not using any of them." And so, instead of getting frustrated that they're not listening to us, how do we use that moment to inquire into the things that we said obviously aren't useful, so what is useful to this kid? How is he attacking this on his paper?

So, I often like to say to a kid, "Huh, I noticed that you're doing something that isn't up on our anchor chart. Tell me about this. I haven't seen this before. How can you help me understand what you're doing?" And sometimes it's the exact same thinking as other strategies that kids are using. So, I can pair kids together and say, "Huh, you're both talking about it in the same way, but you're writing it differently on paper." And so, I think about how I can get kids just to talk to me and tell me what's happening so that I can help give them a notation that might be more acceptable to other mathematicians or to just honor the fact that they have something novel and interesting to share with other kids.

Other questions I talk about are, I will say, "I don't understand what's happening here, and that's not your fault, that's my fault. I just need you to keep explaining it to me until you say something that strikes a chord." Or sometimes I'll bring another kid in, and I'll have the kids listen together, and I'll say, "I think this is interesting, but I don't understand what's going on. Can you say it to her? And then maybe she'll say it in a way that will make more sense to me." Or I'll say, "Can you show me on your paper—you just said that—can you show me on your paper where that idea is?" Because a lot of times kids will think things in their head, but they don't translate it all onto the paper. And so, on the paper, it's missing a step that isn't obvious to the viewer of the paper. And so, we'll say, "Oh, I see how you do that. Maybe you could label your table so that we know exactly what you're talking about when you do this. Or maybe you could show us how you got to 56 by writing 8 times 7 in the margin or something."

Just getting them to clarify and try to help us understand all of the amazing things that are in their head. I will often tell them too, "I love what you're saying. I don't see it on your paper, so I just want you to say it again. And I'm going to write it down on a piece of paper that makes sense to me so that I don't forget all of the cool things that you said." And I'll just write it using more of a standard notation, whether that's a ratio table or a standard US algorithm or something. I'll write it to show the kid that thing that you're doing, there's a way that people write that down. And so, then we can compare our notations and try and figure out "What's the thing that you did?," "How does that compare to the thing that I did?," "Do I understand you clearly now?" to make sure that the kid has the right to say the thing she wants to say in the way that she wants to say it, and then I can still make sense of it in my own way. It's not a problem for me to write it differently as long as we're speaking the same language.

Mike: I want to mark something really important, and I don't want it to get lost for folks. One of the things that jumped out is the moves that you were describing. You could potentially take up those moves if you really were unsure of how a student were thinking, if you had a general notion but you had some questions, or if you totally already understood what the student was doing. Those are questions that aren't just reserved for the point in time when you don't understand—they're actually good questions regardless of whether you fully understand it or don't understand it at all. Did I get that right?

Ryan: Yes. I think that's exactly the point. One thing that I am careful of is, sometimes kids will ask me a question that I know the answer to, and there's this thing that we do as teachers where we're like, "I'm not sure. Why don't you help me figure that out?"—when the kid knows full well that you know the answer.

And so, trying not to patronize kids with those questions, but to really show that I'm asking you these questions, not because I'm patronizing you. I'm asking these questions because I am truly curious about what you're thinking inside and all of the ideas that surround the things that you've written on your paper, or the things that you've said to your partner, to truly honor that the more I know about you, the better teacher I can be for you.

Mike: So, in addition to naming the situation, one of the things that jumped out for me—particularly as you were talking about the students—is, what do you think the impact is on a student's thinking? But also their mathematical identity, or even the set of classroom norms, when they experience this type of questioning or these [types] of questions?

Ryan: So, I think I talked a little bit about normalizing the [questions] "Why?" or "How do you know that?" And so, just letting that become a classroom norm I think is a sea-changing moment for a lot of classrooms—that the conversation is just different if the kids know they have to justify their thinking whether they're right or wrong. Half the time, if they are incorrect, they'll be able to correct themselves as they're talking it through with you. So, kids can be freed up when they're allowed to use their expertise in ways that allow them to understand that the point of math is to truly make sense of it so that when you go out into the world, you understand the situation, and you have different tools to attack it.

So, what's the way that we can create an environment that allows them to truly see themselves as mathematical thinkers? And to let them know that "Your grades in other classes don't tell me much about you as a mathematician. I want to learn what really works for you, and I want to try and figure out where you struggle. And both of those things are important to me because we can use them in concert with each other. So, if I know the things you do well, I can use those to help me build a plan of instruction that will take you further in your understandings."

I think that one of the things that is really important is for kids to understand that we don't do math because we want a good grade. I think a lot of people think that the point of math is to get a good grade or to pass a test or to get into the college that you want to get into, or because sixth grade teachers want you to know this. I really want kids to understand that math is a fantastic language to use out in the world, and there are ways that we can interpret things around us if we understand some pretty basic math. And so how do we get them to stop thinking that math is about right answers and next year and to get the job I want? Well, those things may be true, but that's not the real meaning of math. Math is a way that we can live life. And so, if we don't help them understand the connections between the things that they're doing on a worksheet or in a workbook page, if we don't connect those things to the real world, what's the meaning? What's the point for them? And how do we keep them engaged in wanting to know more mathematics?

So, really getting kids to think about who they are as people and how math can help them live the life that they want to live. Creating classroom environments that have routines in place that support kids in thinking in ways that will move them forward in their mathematical understanding. Trying to help them see that there's no such thing as "a math person" or "not a math person." That everybody has to do math. You do math all the time. You just might not even know that you're doing math. So, I think all of those ideas are really important. And the more curious I can be about students, maybe the more curious they'll be about the math.

Mike: You're making me think that this experience of making sense of someone else's reasoning has a lot of value for students. And I'm wondering how you've seen educators have students engage and make sense of their peer strategies.

Ryan: Yeah. One of the things that I love to see teachers doing is using students' work as the conversation starter. I often, in my classroom, when I started doing this work, I would bring children up to the overhead projector or the document camera. And they would kind of do a show and tell and just say, "I did this and then I did this, and then I did this thing next." And I would say, "That's really great, thank you." And I'd bring up the next student. And it kind of became a show-and-tell-type situation. And I would look at the faces of the other kids in the room, and they would kind of just either be completely checked out or sitting there like raising their hand excitedly—"I want to share mine, I want to share mine." And what I realized was, that there was really only one person who was engaged in that show-and-tell manner, and that was the person who was sharing their work.

And so, I thought, "How can I change that?" So, I saw a lot of really amazing teachers across my career. And the thing that I saw that I appreciated the most is that when a piece of student work is shared, the person who really shouldn't talk is the person who created the work because they already know the work. What we need to do as a group is we need to investigate, "What happened here on this paper?" "Why do you think they made the moves that they made? And how could that help us understand math, our own math, in a different way?" And so, getting kids to look in at other kids' work, and not just saying, "Oh, Mike, how do you understand Ryan's work?" It's "Mike, can you get us started?" And then you say the first thing, and then I say, "OK, let's stop. Let's make sure that we've got this right." And then we go to the kid whose work it is and say, "Are we on the right track? Are we understanding what you're …?" So, we're always checking with that expert. We're making sure they have the last word, because It's not my strategy. I didn't create it. Just because I'm the teacher doesn't mean you should come and ask me about this because this is Mike's strategy. So go and ask the person who created that.

So, trying to get them to understand that we all need to engage in each other's work. We all need to see the connections. We can learn from each other. And there's an expectation that everyone shares, right? So, it's not just the first kid who raises his hand. It's "All of you are going to get a chance to share." And I think the really powerful thing is I've done this work even with in-service teachers. And so, when we look at samples of student work, what's fascinating is it just happens naturally because the kid's not in the room. We can't have that kid do a show and tell. We have to interpret their work. And so, trying to look at the kid's work and imagine, "What are the types of things we think this child is doing?," "What do we think the strengths are on this paper?," "What questions would you ask?," "What would you do next?," is such an interesting thing to do when the child isn't in the room. But when I'm with students, it's just fascinating to watch the kid whose work is on display just shine, even though they're not saying a word, because they just say, "Huh." They get it. They understand what I did and why I did it.

I think that it's really important for us not just to have kids walk up to the board and do board work and just solve a problem using the steps that they've memorized or just go up and do a show and tell, [but] to really engage everyone in that process so that we're all learning. We're not just kind of checking out or waiting for our turn to talk.

Mike: OK, you were talking about the ways that an educator can see how a student was thinking or the ways that an educator could place student work in front of other students and have them try to make sense of it. I wonder if there are any educational technology tools that you've seen that might help an educator who's trying to either understand their students' thinking or put it out for their students to understand one another's thinking.

Ryan: Yeah, there's so many different pieces of technology and things out there. It's kind of overwhelming to try and figure out which one is which. So, I mean, I've seen people use things like Nearpod or Pear Deck—some of those kind of common technologies that you'll see when people do an educational technology class or a workshop at a conference or something. I've seen a lot of people lately using GeoGebra to create applets that they can use with their kids. One that I've started using a lot recently is Magma Math. Magma Math is great. I've used this with teachers and professional development situations to look at samples of student work because the thing that Magma has that I haven't seen in a lot of other technologies is there's a playback function. So, I can look at a static piece of finished work, but I can also rewind, and as the child works in this program, it records it. So, I can watch in real time what the child does. And so, if I can't understand the work because things are kind of sporadically all over the page, I can just rewatch the order that the child put something onto the page. And I think that's a really great feature.

There's just all these technologies that offer us opportunities to do things that I couldn't do at the beginning of my career or I didn't know how to do. And the technology facilitates that. And it's not just putting kids on an iPad so they can shoot lasers at the alien that's invading by saying, "8 times 5 is 40," and the alien magically blows up. How does that teach us anything? But some of these technologies really allow us to dig deeply into a sample of work that students have finished or inquire into, "How did that happen and why did that happen?" And the technologies are just getting smarter and smarter, and they're listening to teachers saying, "It would be really helpful if we could do this or if we could do that." And so, I think there are a lot of resources out there—sometimes too many, almost an embarrassment of riches. So, trying to figure out which ones are the ones that are actually worth our time, and how do we fund that in a school district or in a school so that teachers aren't paying for these pieces out of their pocket.

Mike: You know what? I think that's a great place to stop. Ryan, thank you so much for joining us. It has been an absolute pleasure talking with you.

Ryan: It's always great to talk to you, Mike. Thanks for all you do.

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

Dr. Cathery Yeh, Supporting Neurodiverse Students in Elementary Mathematics Classrooms ROUNDING UP: SEASON 3 | EPISODE 14

What meaning does the term neurodiverse convey and how might it impact a student's learning experience?

And how can educators think about the work of designing environments and experiences that support neurodiverse students learning mathematics?

In this episode, we discuss these questions with Dr. Cathery Yeh, a professor in STEM education from the University of Texas at Austin.

BIOGRAPHY

Dr. Cathery Yeh is an assistant professor in STEM education and a core faculty member in the Center for Asian American Studies from the University of Texas at Austin. Her research examines the intersections of race, language, and disability to provide a nuanced analysis of the constructions of ability in mathematics classrooms and education systems.

TRANSCRIPT

Mike Wallus: What meaning does the term neurodiverse convey and how might that language impact a student's learning experience? In this episode, we'll explore those questions. And we'll think about ways that educators can design learning environments that support all of their students. Joining us for this conversation is Dr. Cathery Yeh, a professor in STEM education from the University of Texas at Austin.

Welcome to the podcast, Cathery. It's really exciting to have you with us today.

Cathery Yeh: Thank you, Mike. Honored to be invited.

Mike: So, I wonder if we can start by offering listeners a common understanding of language that we'll use from time to time throughout the episode. How do you think about the meaning of neurodiversity?

Cathery: Thank you for this thoughtful question. Language matters a lot. For me, neurodiversity refers to the natural variation in our human brains and our neurocognition, challenging this idea that there's a normal brain. I always think of… In Texas, we just had a snow day two days ago. And I think of, just as, there's no two snowflakes that are the same, there's no two brains that are exactly the same, too. I also think of its meaning from a personal perspective. I am not a special educator. I was a bilingual teacher and taught in inclusive settings. And my first exposure to the meaning of neurodiversity came from my own child, who—she openly blogs about it—as a Chinese-American girl, it was actually really hard for her to be diagnosed. Asian Americans, 1 out of 10 are diagnosed—that's the lowest of any ethnic racial group. And I'll often think about when… She's proud of her disabled identity. It is who she is. But what she noticed that when she tells people about her disabled identity, what do you think is the first thing people say when she says, "I'm neurodivergent. I have ADHD. I have autism." What do you think folks usually say to her? The most common response?

Mike: I'm going to guess that they express some level of surprise, and it might be associated with her ethnic background or racial identity.

Cathery: She doesn't get that as much. The first thing people say is, they apologize to her. They say, "I'm sorry."

Mike: Wow.

Cathery: And that happens quite a lot. And I say that because–and then I connected back to the term neurodiversity—because I think it's important to know its origins. It came about by Judy Singer. She's a sociologist. And about 30 years ago, she coined the term neurodiversity as an opposition to the medical model of understanding people and human difference as deficits. And her understanding is that difference is beautiful. All of us think and learn and process differently, and that's part of human diversity. So that original definition of neurodiversity was tied to the autism rights movement. But now, when we think about the term, it's expanded to include folks with ADHD, dyslexia, dyscalculia, mental health, conditions like depression, anxiety, and other neuro minorities like Tourette syndrome, and even memory loss. I wanted to name out all these things because sometimes we're looking for a really clean definition, and definitions are messy. There's a personal one. There's a societal one of how we position neurodiversity as something that's deficit, that needs to be fixed. But it's part of who one is. But it's also socially constructed. Because how do you decide when a difference becomes a difference that counts where you qualify as being neurodiverse, right? So, I think there's a lot to consider around that.

Mike: You know, the answer that you shared is really a good segue because the question I was going to ask you involves something that I suspect you hear quite often is people asking you, "What are the best ways that I can support my neurodiverse students?" And it occurs to me that part of the challenge of that question is it assumes that there's this narrow range of things that you do for this narrow range of students who are different. The way that you just talked about the meaning of neurodiversity probably means that you have a different kind of answer to that question when people ask it.

Cathery: I do get this question quite a lot. People email it to me, or they'll ask me. That's usually the first thing people ask. I think my response kind of matches my pink hair question. When they ask me the question, I often ask a question back. And I go, "How would you best educate Chinese children in math?" And they're like, "Why would you ask that?" The underlining assumption is that all Chinese children are the same, and they learn the same ways, they have the same needs, and also that their needs are different than the research-based equity math practices we know and have done 50–60 years of research that we've highlighted our effective teaching practices for all children. We've been part of NCTM for 20 years. We know that tasks that promote reasoning and problem solving have been effectively shown to be good for all. Using a connecting math representation—across math representations in a lesson—is good for all. Multimodal math discourse, not just verbal, written, but embodied in part who we are and, in building on student thinking, and all those things we know. And those are often the recommendations we should ask. But I think an important question is how often are our questions connecting to that instead? How often are we seeing that we assume that certain students cannot engage in these practices? And I think that's something we should prioritize more. I'm not saying that there are not specific struggles or difficulties that the neurodiversity umbrella includes, which includes ADHD, dyslexia, autism, bipolar disorder, on and on, so many things. I'm not saying that they don't experience difficulties in our school environment, but it's also understanding that if you know one neurodiverse student—you know me or my child—you only know one. That's all you know. And by assuming we're all the same, it ignores the other social identities and lived experiences that students have that impact their learning.

So, I'm going to ask you a question.

Mike: Fire away.

Cathery: OK. What comes to your mind when you hear the term "neurodiverse student"? What does that student look like, sound like, appear like to you?

Mike: I think that's a really great question. There's a version of me not long ago that would have thought of that student as someone who's been categorized as special education, receiving special education services, perhaps a student that has ADHD. I might've used language like "students who have sensory needs or processing." And I think as I hear myself say some of those things that I would've previously said, what jumps out is two things: One is I'm painting with a really broad brush as opposed to looking at the individual student and the things that they need. And two is the extent to which painting with a broad brush or trying to find a bucket of strategies that's for a particular group of students, that that really limits my thinking around what they can do or all the brilliance that they may have inside them.

Cathery: Thank you for sharing that because that's a reflection I often do. I think about when I learned about my child, I learned about myself. How I automatically went to a deficit lens of like, "Oh, no, how are we going to function in the world? How's she going to function in the world?" But I also do this prompt quite a lot with teachers and others, and I ask them to draw it. When you draw someone, what do you see? And I'll be honest, kind of like drawing a scientist, we often draw Albert Einstein. When I ask folks to draw what a neurodiverse student looks like, they're predominantly white boys, to be honest with you. And I want to name that out. It's because students of color, especially black, brown, native students—they're disproportionately over- and under-identified as disabled in our schooling. Like we think about this idea that when most of us associate autism or ADHD mainly as part of the neurodiversity branch and as entirely within as white boys, which often happens with many of the teachers that I talk to and parents. We see them as needing services, but in contrast, when we think about, particularly our students of color and our boys—these young men—there's often a contrast of criminalization in being deprived of services for them. And this is not even what I'm saying. It's been 50 years of documented research from the Department of Ed from annual civil rights that repeatedly shows for 50 years now extreme disproportionality for disabled black and Latinx boys, in particular from suspension, expulsion, and in-school arrests. I think one of the most surprising statistics for me that I had learned recently was African-American youth are five times more likely to be misdiagnosed with conduct disorder before receiving the proper diagnosis of autism spectrum disorder. And I appreciate going back to that term of neurodiversity because I think it's really important for us to realize that neurodiversity is an asset-based perspective that makes us shift from looking at it as the student that needs to be fixed, that neurodiversity is the norm, but for us to look at the environment.

And I really believe that we cannot have conversations about disability without fully having conversations about race, language, and the need to question what needs to be fixed, particularly not just our teaching, but our assessment practices. For example, we talk about neurodiversities around what we consider normal or abnormal, which is based on how we make expectations around what society thinks. One of the things that showed up in our own household—when we think about neurodiversity or assessments for autism—is this idea of maintaining eye contact. That's one of the widely considered autistic traits. In the Chinese and in the Asian household, and also in African communities, making eye contact to an adult or somebody with authority? It is considered rude. But we consider that as one of the characteristics when we engage in diagnostic tools. This is where I think there needs to be more deep reflection around how one is diagnosed, how a conversation of disability is not separate from our understanding of students and their language practices, their cultural practices. What do we consider normative? Because normative is highly situated in culture and context.

Mike: I would love to stay on this theme because one of the things that stands out in that last portion of our conversation was this notion that rather than thinking about, "We need to change the child." Part of what we really want to think about is, "What is the work that we might do to change the learning environment?" And I wonder if you could talk a bit about how educators go about that and what, maybe, some of the tools could be in their toolbox if they were trying to think in that way.

Cathery: I love that question of, "What can we as teachers do? What's some actionable things?" I really appreciate Universal Design for Learning framework, particularly their revised updated version, or 3.0 version, that just came out, I think it was June or July of this year. Let me give you a little bit of background about universal design. And I'm sure you probably already know. I've been reading a lot around its origins. It came about [in the] 1980s, we know from cast.org. But I want to go further back, and it really builds from universal design and the work of architecture. So universal design was coined by a disabled architect. His name was Ronald Mace. And as I was reading his words, it really helped me better understand what UDL is. We know that UDL— Universal Design for Learning and universal design—is about access. Everybody should have access to curriculum. And that sounds great, but I've also seen classrooms where access to curriculum meant doing a different worksheet while everybody else is engaging in small group, whole group problem-based learning.

Access might mean your desk is in the front of the room where you're self-isolated—where you're really close to the front of the board so you can see it really well—but you can't talk to your peers. Or that access might mean you're in a whole different classroom, doing the same set of worksheets or problems, but you're not with your grade-level peers.

And when Ronald Mace talks about access, he explained that access in architecture had already been a focus in the late 1900s, around 1998, I think. But he said that universal design is really about the longing. And I think that really shifted the framing. And his argument was that we need to design a place, an environment where folks across a range of bodies and minds feel a sense of belonging there. That we don't need to adapt—the space was already designed for you. And that has been such a transformative perspective: That it shouldn't be going a different route or doing something different, because by doing that, you don't feel like you belong. But if the space is one where you can take part equally and access across the ways you may engage, then you feel a sense of belonging.

Mike: The piece of what you said that I'm really contemplating right now is this notion of belonging. What occurs to me is that approaching design principles for a learning environment or a learning experience with belonging in mind is a really profound shift. Like asking the question, "What would it mean to feel a sense of belonging in this classroom or during this activity that's happening?" That really changes the kinds of things that an educator might consider going through a planning process. I'm wondering if you think you might be able to share an example or two of how you've seen educators apply universal design principles in their classrooms in ways that remove barriers in the environment and support students' mathematical learning.

Cathery: Oh gosh, I feel so blessed. I spend… Tomorrow I'm going to be at a school site all day doing this. UDL is about being responsive to our students and knowing that the best teaching requires us to listen deeply to who they are, honor their mathematical brilliance, and their agency. It's about honoring who they are. I think where UDL ups it to another level, is it asks us to consider who makes the decision. If we are making all the decisions of what is best for that student, that's not fully aligned with UDL. The heart of UDL, it's around multiple ways for me to engage, to represent and express, and then students are given choice. So, one of the things that's an important part of UDL is honoring students' agency, so we do something called "access needs." At the start of a lesson, we might go, "What do you need to be able to fully participate in math today?" And kids from kindergarten to high school or even my college students will just write out what they need. And usually, it's pretty stereotypical: "I want to talk to someone when I'm learning." "I would like to see it and not just hear it." And then you continually go back and you ask, "What are your access needs? What do you need to fully participate?"

So students are reflecting on their own what they need to be fully present and what they believe is helpful to create a successful learning environment. So that's a very strong UDL principle—that instead of us coming up with a set of norms for our students, we co-develop that. But we're co-developing it based on students reflecting on their experience in their environment. In kindergarten, we have children draw pictures. As they get older, they can draw, they can write. But it's this idea that it's an ongoing process for me to name out what I need to be fully present. And oftentimes, they're going to say things that are pretty critical. It's almost always critical, to be honest with you, but that's a… I would say that's a core component of UDL. We're allowing students to reflect on what they need so they can name it for themselves, and then we can then design that space together. And along the way, we have kids that name, "You know what? I need the manipulatives to be closer." That would not come about at the start of me asking about access needs. But if we did a lesson, and it was not close by, they'll tell me. So it's really around designing an environment where they can fully participate and be their full selves and feel a sense of belonging. So, that's one example.

Another one that we've been doing is teachers and kids who have traditionally not participated the most in our classrooms or have even engaged in pullout intervention. And we'll have them walk around school, telling us about their day. "Will you walk me through your day and tell me how you feel in each of these spaces, and what are your experiences like?" And again, we're allowing the students to name out what they need. And then they're naming out… Oftentimes, with the students that we're at, where I'm working in mostly multilingual spaces, they'll say, "Oh, I love this teacher because she allows us to speak in Spanish in the room. It's OK." So that's going back to ideas of action, expression, engagement, where students are allowed a trans language. That's one of the language principles.

But we're allowing students and providing spaces and really paying close attention to: "How do we decide how to maximize participation for our students with these set of UDL guidelines? How we are able to listen and make certain decisions on how we can strengthen their participation, their sense of belonging in our classrooms."

Mike: I think what's lovely about both of those examples—asking them to write or draw what they need or the description of, "Let's walk through the day. Let's walk through the different spaces that you learn in or the humans that you learn with"—is one, it really is listening to them and trying to make meaning of that and using that as your starting point. I think the other piece is that it makes me think that it's something that happens over time. It might shift, you might gain more clarity around the things that students need or they might gain more clarity around the things that they need over time. And those might shift a little bit, or it might come into greater focus. Like, "I thought I needed this" or "I think I needed this, but what I really meant was this." There's this opportunity for kids to refine their needs and for educators to think about that in the designs that they create.

Cathery: I really appreciate you naming that because it's all of that. It's an ongoing process where we're building a relationship with our students for us to co-design what effective teaching looks like—that it's not a one size fits all. It's disrupting this idea that what works for one works for all. It's around supporting our students to name out what they need. Now, I'm almost 50. I struggle to name out what I need sometimes, so it's not going to happen in, like, one time. It's an ongoing process. And what we need is linked to context, so it has to be ongoing. But there's also in the moments as well. And it's the heart of good teaching in math, when you allow students to solve problems in the ways that make sense to them, that's UDL by design. That's honoring the ideas of multiplicity in action, expression. When you might give a context-based problem and you take the numbers away and you give a set of number choices that students get to choose from. That is also this idea of UDL because there's multiple ways for them to engage. So there are also little things that we do that… note how they're just effective teaching. But we're honoring this idea that children should have agency. All children can engage in doing mathematics. And part of learning mathematics is also supporting our students to see the brilliance in themselves and to leverage that in their own teaching and learning.

Mike: Yeah. Something else that really occurred to me as we've been talking is the difference between the way we've been talking about centering students' needs and asking them to help us understand them and the process that that kind of kicks off. I think what strikes me is that it's actually opening up the possibilities of what might happen or the ways that a student could be successful as opposed to this notion that "You're neurodiverse, you fit in this bucket. There's a set of strategies that I'm going to do just for you," and those strategies might actually limit or constrict the options you have. For example, in terms of mathematics, what I remember happening very often when I was teaching is, I would create an open space for students to think about ways that they could solve problems. And at the time, often what would happen is kids who were characterized as neurodiverse wouldn't get access to those same strategies. It would be kind of the idea that "This is the way we should show them how to do it." It just strikes me how different that experience is. I suspect that that was done with the best of intentions, but I think the impact unfortunately probably really didn't match the intent.

Cathery: I love how you're being honest. I did the same thing when I was teaching, too, because we were often instructed to engage in whole-group instruction and probably do a small-group pullout. That was how I was taught. And when the same kids are repeatedly pulled out because we're saying that they're not able to engage in the instruction. I think that part of UDL is UDL is a process, realizing that if students are not engaging fully in the ways that we had hoped, instead of trying to fix the child, we look at the environment and think about what changes we need to make in tier one. So whole-group instruction, whole-group participation first to see how we can maximize their participation. And it's not one strategy, because it depends; it really depends. I think of, for example, with a group of teachers in California and Texas now, we've been looking at how we can track participation in whole-group settings. And we look at them across social demographics, and then we started to notice that when we promote multimodal whole-group participation, like kids have access to manipulatives even during whole-group share out. Or they have visuals that they can point to, their participation and who gets to participate drastically increase.

So there's many ways in which, by nature, we engage in some narrow practices because, too, oftentimes whole group discussion is almost completely verbal and, at times, written, and usually the teacher's writing. So it's going back to the idea of, "Can we look at what we want our students to do at that moment? So starting on the math concept and practices, but then looking at our students and when they're not participating fully, it's not them. What are the UDL principles and things that I know and strategies that I have with my colleagues that I can make some small shifts?"

Mike: You know, one of the things that I enjoy most about the podcast is that we really can take a deep dive into some big ideas, and the limitation is we have 20 minutes to perhaps a half hour. And I suspect there are a lot of people who are trying to make meaning of what we're talking about and thinking about, "How might I follow up? How might I take action on some of the ideas?" So I want to turn just for a little while to resources, and I'm wondering if there are resources that you would suggest for a listener who wants to continue learning about universal design in a mathematics classroom?

Cathery: Oh, my goodness, that's such a hard question because there's so many. Some good ones overall: I would definitely encourage folks to dive into the UDL guidelines—the 3.0 updates. They're amazing. They're so joyful and transformative that they even have, one of the principles is centering joy in play, and for us to imagine that, right?

Mike: Yes!

Cathery: What does that mean to do that in a math classroom? We can name out 50 different ways. So how often do we get to see that? So, I would highly encourage folks to download that, engage in deep discussion because it was a 2.2 version for, I think, quite a few years. I would also lean into a resource that I'm glad to email later on so it's more easily accessible. I talked about access needs, this idea of asking students, asking community members, asking folks to give this opportunity to name out what they need. It's written by a colleague, Dr. Daniel Reinholz and Dr. Samantha Ridgway. It's a lovely reading, and it focuses specifically in STEM but I think it's a great place to read. I would say that Dr. Rachel Lambert's new book on UDL math is an excellent read. It's a great joyful read to think about. I'm going to give one shout out to the book called the Year of the Tiger: An Activist's Life. It's by Alice Wong. I encourage that because how often do we put the word activism next to disability? And Alice Wong is one of the most amazing humans in the world, and it's a graphic novel. So it's just joyful. It's words with poetry and graphic novel mixed together to see the life of what it means to be a disabled activist and how activism and disability goes hand in hand. Because when you are disabled and multi-marginalized, you are often advocating for yourself and others. It's amazing. So I'll stop there. There's endless amounts.

Mike: So for listeners, we'll link the resources that Cathery was talking about in our show notes. I could keep going, but I think this is probably a great place to stop. I want to thank you so much for joining us. It's really been a pleasure talking with you.

Cathery: Thank you. Thank you.

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

Assessment in the Early Years Guest: Shelly Scheafer

ROUNDING UP: SEASON 3 | EPISODE 13

Mike: How is the work of assessing young children different from assessing students in upper elementary grades or in grades six through 12? And what actions can we take to ensure we understand our youngest learners' thinking? Today we're talking with Shelly Schafer, senior manager of Content Development with the math Learning Center about the ways educators can understand and advance the mathematical thinking of our youngest learners. Welcome to the podcast, Shelly. Thank you so much for joining us today.

Shelly: Thank you, Mike, for having me.

Mike: So I'd like to start with this question. What makes the work of assessing younger children, particularly students in grades K through two, different from assessing students in upper elementary grades or even beyond?

Shelly: Wow, there's a lot to that question, Mike. I think there's some obvious things. So effective assessment of our youngest learners is different because obviously our pre-K, our first, even our second grade students are developmentally different from fourth and fifth graders. So when we think about assessing these early primary students, we need to use appropriate assessment methods that match their stage of development. For example, when we think of typical paper pencil assessments and how we often ask students to show their thinking with pictures, numbers and words. Our youngest learners, our just starting to connect symbolic representations to mathematical ideas, let alone put letters together to make words. So we need to take into consideration that primary students are in the early stages of development with their language, their reading, and their writing skills. And this makes it challenging for them to fully articulate, write, sketch, any of their mathematical thinking. So we often find that with young children, interviews can be really helpful, but even then there's some drawbacks. Some children find it challenging to show in the moment what they know. Others just aren't fully engaged or interested because you've called them over from something that they're busy doing, or maybe they're not yet comfortable with the setting or even the person doing the interview. So when we work within children, we need to recognize all of these little peculiarities that come with working with that age. We also need to understand that their mathematical development is fluid, it's continually evolving, and this is why they often or some may respond differently to the same prompter question, especially if the setting of the context has changed. We may find that a kindergarten student who counts to nine on Monday may count to 69 or even a hundred later in the week, depending on what's going on in their mind at the time. So this means that assessment with young children needs to be frequent, formative, and ongoing. So we're not necessarily waiting for the end of the unit to see, aha, did they get this? What do we do? We're looking at their work all of the time. And fortunately, some of the best assessments on young children are the observations in their natural setting, like times when maybe they're playing a math game or working with a center activity or even during just your classroom routines. And it's these authentic situations that we can look at as assessments help us capture a more accurate picture of their abilities because we not only get to hear what they say or see what they write on paper, we get to watch them in action. We get to see what they do when they're engaged in small group activities or playing games with friends.

Mike: So I want to go back to something you said and the way that you said it. You were talking about watching or noticing what students can do and you really emphasize the words. Do talk a little bit about what you were trying to convey with that Shelly

Shelly: Young children are doers when they work on a math pass, they show their thinking and their actions with finger formations and objects. And we can see if a student has one-to-one correspondence when they're counting, if they group their objects, how they line 'em up, do they tag them? Do they move them as they count them? They may not always have the verbal skills to articulate their thinking, but we can also attend to things like head nodding, finger counting, and even how they cluster or match objects. I'm going to give you an example. Let's say that I'm watching some early first graders and they're solving the expression six plus seven, and the first student picks up a number rack, and if you're not familiar with a number rack, it's a tool with two rows of beads. And on the first row there are five red beads and five white beads. And on the second row there's five red beads and five white beads. And the students solving six plus seven begins by pushing over five red beads in one push and then one more bead on the top row. And then they do the same thing for the seven. They push over five red beads and two white beads, and they haven't said a word to me, I'm just watching their actions and I'm already able to tell, hmm, that student could subitize a group of five because I saw 'em push over all five beads in one push. And that they know that six is composed of five in one and seven is composed of five and two, and they haven't said a word. I'm just watching what they're doing. And then I might watch the student and I see 'em pause, nothing's being said, but I start to notice this slight little head nodding. And then they say 13, and they give me the answer and they're really pleased. I didn't get a lot of language from them, but boy did I get a lot from watching how they solve that problem. And I want to contrast that observation with a student who might be solving the same expression, six plus seven, and they might go, Hmm, six. And then they start popping up one finger at a time while counting 7, 8, 9, 10, 11, 12, 13. And when they get seven fingers held up, they say 13. Again, they've approached that problem quite differently, but again, I get that information that they understood the equation they were able to count on starting with six, and they kept track of their count with their fingers and they knew to stop when seven fingers were raised. And I might even have a different student that might start talking to me and they say, well, six plus six is 12, and seven is one more than six. So the answer is 13. And if this were being done on a paper pencil as an assessment item or they were answering on some kind of a device, all I would know about my students is that they were able to get the correct answer. I wouldn't really know a lot about how they got the answer, what skills do they have? What was their thinking? And there's not a lot that I can work with to plan my instruction. Does that kind of make sense?

Mike: Absolutely. I think the way that you described this, attending to behaviors, to gestures, to the way that kids are interacting with manipulatives, the self-talk that's happening, it makes a ton of sense. And I think for me, when I think back to my own practice, I wish I could wind the clock back because I think I was attending a lot to what kids were saying and sometimes their written communication. And there was a lot that I could have also taken in if I was attending to those things in a little bit more depth. It also strikes me that this might feel a bit overwhelming for an educator. How could an educator know what they're looking for?

Shelly: I do think it can feel overwhelming at first, but as teachers begin to make informal observations, really listening and watching students actions as part of just their daily practice, something that they're doing on a normal basis, they start to develop these kind of intuitive understandings of how children learn, what to expect them to do, what they might say next if they see a certain action. And after several years, let's say, of teaching kindergarten, if you've been a kindergarten teacher for four or 5, 6, 20 plus years, you start to notice these patterns of behavior, things that five and six year olds seem to say and think and do on a fairly consistent basis. And that kind of helps you know what you're looking at. And fortunately, we have several researchers that have been, let's say, kid watching for 40 years, and they have identified stages through which most children pass as they develop their counting skills or maybe strategies for solving addition and subtraction problems. And these stages are laid out as progressions of thinking or actions that students exhibit as they develop understanding over periods of time. And listeners might know these as learning progressions or learning trajectories. And these are ways to convey an idea of concept in little bits of understanding. So when I was sharing the thinking and actions of three students solving six plus seven listeners familiar with cognitively guided instruction, CGI, they might've recognized the sequence of strategies that children go through when they're solving addition and subtraction problems. So in my first student, they didn't say anything but gave me an answer was using direct modeling. We saw them push over five and one beads for six and then five and two beads for seven, and then kind of pause at their model. And I could tell with their head nodding that they were counting quietly in their head counting all the beads to get the answer. And that's one of those first stages that we see and recognize with direct modeling. And that gives me information on what I might do with a student coming next time. I might work on the second strategy that I conveyed with my second student where they were able to count on, they started with that six, and then they counted seven more using their fingers to keep track of their count and got the answer. And then that third kind of level in that progression as we're moving of understanding was shown with my third student when they were able to use a derived fact strategy. The student said, oh, well I know that six plus six is 12. I knew my double fact, and then I use that relationship of knowing that seven is one more than six. And so that's kind of how we move kids through. And so when I'm watching them, I can kind of pinpoint where they are and where they might go next, and I can also think about what I might do. And so it's this knowledge of development and progressions and how children, number concepts that can help teachers recognize the skills as they emerge, as they begin to see them with their students, and they can use those to guide their instruction for that student or look at the class overall and plan their instruction or think about more open-ended kinds of questions that they can ask that recognize these different levels that students are working with.

Mike: As a K one teacher, I remember that I spent a lot of my time tracking students with things like checklists. So I note if students had or didn't have a skill, and as I hear you talk, that feels fairly oversimplified when we think about this idea of developmental progressions. How do you suggest that teachers approach capturing evidence of student learning, Shelly?

Shelly: Well, we have to really think about assessment and children's learning is something that is ongoing and involving, and if we do, it becomes part of what we can do every day. We can look for opportunities to observe students' skills in authentic settings. Maybe it's something that we're having them write down on their whiteboard, or maybe it's something where they're showing the answer with finger formations or we're giving a thumbs up or a thumbs down to check in on their understanding. We might not be checking on every student, but we're capturing a few and we can take note because we're doing this on a daily basis of who we want to check in with, what do we want to see? We can also do a little more formal planning when we draw from what we're going to do already in our lesson. Let's say for example that our lesson today includes a.talk or a number talk, something that we're going to write down. We're going to record student thinking. And during the lesson, the teacher's going to be busy facilitating the discussion, recording the student's thinking and making all of those notes. But if we write the child's name, honor their thinking and give it that caption on that public record at the end of the lesson, we can capture a picture, just use an iPad quickly, take a picture of that student's thinking, and then we can record that where we're keeping track of our students. So we have, okay, another moment in time. And it's this collection of evidence that we keep growing. We can also by capturing these public records note, whose voice and thinking we're elevating in the classroom. So it gives us how are they thinking and who are we listening to and making sure that we're spreading that out. And hearing everyone, I think like you mentioned checklists that you use.

Mike: I did,

Shelly: Yeah. And even checklists can play a role in observation and assessments when they have a focus and a way to capture students' thinking. One of the things we did in third edition is we designed additional tools for gathering and recording information during workplaces. That's a routine where students are playing games and or engaged with partners doing some sort of a math activity. And we designed these based on what we might see students do at these different games and activities. And we didn't necessarily think about this is something you're going to do with every student or even in one day because these are spanned out over a period of four to six weeks where that they can go to these games and we might see the students go to these activities multiple times. And so let's say that kindergarten students are playing something like the game beat you to 10 where they're spinning a spinner, they're counting cubes, and they're trying to race their partner to collect 10 cubes. And with an activity like that, I might want to focus on students who I still want to see, do they have one-to-one correspondence? Are they developing cardinality? Are they able to count out a set? And those might be kinds of skills that you might've had typically on a checklist, right, Mike for kindergarten. But I could use this activity to gather that note and make any comment. So just for those kids I'm looking at, or maybe first graders are playing a game like sort the sum where they're drawing two different dominoes and they're supposed to find how many they have in all. So with a game like that, I might focus on what are their strategies? Are they counting all the dots? Are they counting on from one set of the dots on one side and then counting on the other? Are they starting with the greater number or the most dots? Are they starting with the one always on the left? Or I might even see they might instantly recognize some of those. So I might know the skills that I want to look for with those games and be making notes, which kind of feels checklist, but I can target that time to do it on students. I want that information by thinking ahead of time, what can I get by watching observing these students at these games? I mean, as you know, young children love it. Older children love it when the teacher goes over and wants to watch them play or even better wants to engage in the gameplay with them. But I can still use that as an assessment.

Mike: That's helpful, Shelly, for a couple of reasons. One of the things that you said was really powerful is thinking about not just the assessment tools that might be within your curriculum, but looking at the task itself that you're going to have students engage with, be it a game or a project or some kind of activity and really thinking, what can I get from this as a person who's trying to make sense of students' thinking? And I think my checklist suddenly feels really different when I've got a clear vision of what can I get from this task or this game that students are playing and looking for evidence of that versus feeling like I was pulling kids over one-on-one, which I think I would still do because there's some depth that I might want to capture, but it changes the way that I think about what I might do and also what I might get out of a task that really resonates for me. The other thing that you made me think about is the extent to which I remember thinking is I need to make sure if a student has got it or not. Got it. What you're making me think can really come out of this experience of observing students when they're working on a task or with a partner is that I can gather more evidence about the application of that idea. I can see the extent to which students are doing something like counting on in the context of a game or a task, and that adds to the evidence that I gather in a one-on-one interview with them. But it gives me a chance to see, is this way of thinking something that students are applying in different contexts or did it just happen at that one particular moment in time when I was with 'em? So that really helps me think about how those two different ways of assessing students be it one-on-one or observing them and seeing what's happening, support one another.

Shelly: And I think you also made me think it really hit on this idea that students, like I said, their learning is evolving over time and it might change with the context so that they show us that they know something in one context with these numbers or this scenario, but they don't necessarily always see that it applies across the board. They don't make generalizations. That's something that we really have to work with students to develop. And they're also young children. Think about how quickly a three-year-old and a four-year-old change the same five to six, six to seven, I mean, they're evolving all the time. And so we want to get this information for them on a regular basis. A unit of instruction may be a month or more long, and a lot can happen in that time. So we want to make sure that we continue to check in with them and help them to develop if needed or that we advance them, we nudge them along, we challenge them with maybe a question, will that apply to every number? So a student discovers when we add one to every number, it's like saying the next number, so six and one more, seven and eight and one more is nine. And you can challenge them. Ooh, does that always work? What if the number was 22? What if it was 132? Would it always work? So when you're checking in with kids, you have those opportunities to keep them thinking, to help them grow.

Mike: I want to pick up on something that we haven't necessarily said aloud, but I'd like to explore it. Looking at young students' work from an asset-based perspective, particularly with younger students, I've had points in time where there felt like so much that I needed to teach them, and sometimes I felt myself focusing on what they couldn't do. Looking back, I wish I had thought about my work as noticing the assets, the strategies, the ways of thinking that they were accumulating. Are there practices you think support an asset-based approach to assessment with young learners?

Shelly: I think probably the biggest thing we can do is broaden our thinking about assessment. The National Council of Teachers of Mathematics wrote in catalyzing change in early childhood and elementary mathematics that the primary purpose of assessment is to gather evidence of children's thinking, understanding and reasoning to inform both instructional decisions and student in teaching learning. If we consider assessments and observations as tools to inform our instruction, we need to pay attention to the details of the child's thinking. And when we're paying attention to the details, what the child is bringing to the table, what they can do, that's where our focus goes. So the question becomes, what is the student understanding? What assets do they bring to the task? It's no longer can they do it or can they not do it? And when we know, when we focusing on just what that student can do and we have some understanding of understanding learning progressions, how students learn, then we can place what they're doing kind of on that trajectory in that progression, and that becomes knowledge. And with that knowledge then we can help students move along the progression to more developed understanding. For example, again, if I go back to my six plus seven and we notice that a student is direct modeling, they're counting out each of the sets and counting all, we can start to nudge them toward counting on. We might cover, they're using that number rack. We might cover the first row and say, Ooh, you just really showed me a good physical representation of six plus seven, and I noticed that you were counting the beats to see how many were there. I'm wondering if I cover this first row, how many beats am I covering? Six. I wonder, could you start your counting at six? We can work with what they know, and I can do that because I've focused on where they are in that progression and where that development is going. And I have a goal of where I want students to go to further their thinking, not that being any one place is right or wrong, or yes, they can do it. No, they can't. It's my understanding of what assets they bring that I can build on. Is that kind of what you were after?

Mike: It is. And I think you also addressed something that again has gone unsaid, but I think you unpacked it there, which is assessment is really designed to inform my instruction. And I think the example you offered is a really lovely one where we have a student who's direct modeling and they're making sense of number in a certain way, and their strategy reflects that. And that helps us think about the kinds of nudges we can offer that might shift that thinking or press them to make sense of numbers in a different way. That really the assessment is it is a moment in time, but it also informs the way that you think about what you're going to do next to keep nudging that student's thinking.

Shelly: Exactly. And we have to know that if we have 20 students, they all have 20 little plans, that they're on 20 little pathways of their learning. And so we need to think about everybody. So we're going to ask questions that help 'em do 'em, and we're going to honor their thinking. So again, I'm going to go back to doing that.talk with those students. And so I'm honoring all these different ways that students are finding the total number of dots, and then I'm asking them to look for what's the same within their thinking so that other students also can serve to nudge kids, have them let them try and explore a different idea or, Ooh, can we try that Mike's way and see if we can do that? Or what do you notice about how Mike solve the problem and how Shelly solve the problem? Where is their thinking the same? Where is it different? And so we're honoring everybody's place of where they're at, but they're still learning from each other.

Mike: You have made multiple mentions to this idea of progressions or trajectories, and I'm wondering, what are some of the resources that helped you build an understanding of children's developmental progression? Shelly?

Shelly: Honestly, I can say that I learned a lot from the students I taught in my classroom. My roots run deep in early childhood. With that said, I think I stand on the back of giants, teacher practitioner researchers for early childhood who have spent decades observing children and recording their thinking. I mentioned cognitively guided instruction, which features the research of Thomas Carpenter and his team and their book. Children's Mathematics is a great guide for K five teachers. Another teacher researcher is Kathy Richardson, and some listeners may know her from her books, the Developing Number concept series or number talks in the primary classroom. She also wrote a book called How Children Learn Number Concepts, A Guide to the Critical Learning Phases, which Targets Pre-Kindergarten through grade four. And then I think also the work of Julie Sama and Doug Clement. They have a website that looks at learning progressions starting at birth all the way through grade three. And this website is learning trajectories.org, and it's one of those that is always evolving, so it not only explains learning trajectories for all early childhood math concepts, but there are literally thousands of videos and lessons for teaching math and new content is always being added. So any of those would really give teachers some good ideas on how children learn the progressions that they go for and really help them notice and put a reference to what they're seeing kids do.

Mike: You mentioned Giants, and those are some gigantic folks in the world of mathematics education. I had a really similar experience with both CGI and Kathy Richardson in that a lot of what they're describing are the things that I was seeing in classrooms. What it really helped me do is understand how to place that behavior and what the meaning of it was in terms of students' understanding of mathematics. And it also helped me think about that as an asset that then I could build on. Are there resources you would invite our listeners to engage with if they want to continue learning?

Shelly: I think if listeners are interested in learning more about developmental progressions in math, the resources I mentioned, children's Mathematics, cognitive guided instruction by Thomas Carpenter Al, the How Children Learn Number Concepts by Kathy Richardson or SAMA in Clements learning directories.org are good places to start. But honestly, Mike, it's about teachers making purposeful observations, understanding what they're seeing and hearing, and then knowing what to do with that information. The Latin root of the word assessment means to sit beside. I would like to invite our listeners to sit beside their students. Listen, watch, question, take note, because developing the capacity to observe children in action, listening to their thinking and then acting upon what they see in hear takes practice, takes effort, and once teachers become fascinated with children sitting in front of us, we can become students of our students. As Alan Fisher would say. That's when teachers really see the benefits. They'll recognize that all their students have math abilities, and that these math abilities are specific and actionable. And when we nurture our students with what they know and what they need to know, they will grow.

Mike: I think that's a great place to stop. Shelly Schaeffer, thank you so much for joining us.

Shelly: Thank you so much, Mike, for having me. It's been a pleasure talking with you.

Mike: This podcast is brought to you by the Math Learning Center and the Meyer 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