The Case for Choral Counting with Fractions with Christy Pettis & Terry Wyberg

How can educators help students recognize similarities in the way whole numbers and fractions behave? And are there ways educators can build on students' understanding of whole numbers to support their understanding of fractions?

The answer from today's guests is an emphatic yes. Today we're talking with Terry Wyberg and Christy Pettis about the ways choral counting can support students' understanding of fractions.

Terry Wyberg is a senior lecturer in the Department of Curriculum and Instruction at the University of Minnesota. His interests include teacher education and development, exploring how teachers' content knowledge is related to their teaching approaches.

Christy Pettis is an assistant professor of teacher education at the University of Wisconsin-River Falls.

Choral Counting & Counting Collections: Transforming the PreK-5 Math Classroom by Megan L. Franke, Elham Kazemi, and Angela Chan Turrou

Teacher Education by Design

Number Chart app by The Math Learning Center

Mike Wallus: Welcome to the podcast, Terry and Christy. I'm excited to talk with you both today.

Christy Pettis: Thanks for having us.

Terry Wyberg: Thank you.

Mike: So, for listeners who don't have prior knowledge, I'm wondering if we could just offer them some background. I'm wondering if one of you could briefly describe the choral counting routine. So, how does it work? How would you describe the roles of the teacher and the students when they're engaging with this routine?

Christy: Yeah, so I can describe it. The way that we usually would say is that it's a whole-class routine for, often done in kind of the middle grades. The teachers and the students are going to count aloud by a particular number. So maybe you're going to start at 5 and skip-count by 10s or start at 24 and skip-count by 100 or start at two-thirds and skip-count by two-thirds.

So you're going to start at some number, and you're going to skip-count by some number. And the students are all saying those numbers aloud. And while the students are saying them, the teacher is writing those numbers on the board, creating essentially what looks like an array of numbers. And then at certain points along with that talk, the teacher will stop and ask students to look at the numbers and talk about things they're noticing. And they'll kind of unpack some of that. Often they'll make predictions about things. They'll come next, continue the count to see where those go.

Mike: So you already pivoted to my next question, which was to ask if you could share an example of a choral count with the audience. And I'm happy to play the part of a student if you'd like me to.

Christy: So I think it helps a little bit to hear what it would sound like. So let's start at 3 and skip-count by 3s. The way that I would usually tell my teachers to start this out is I like to call it the runway. So usually I would write the first three numbers. So I would write "3, 6, 9" on the board, and then I would say, "OK, so today we're going to start at 3 and we're going to skip-count by 3s. Give me a thumbs-up or give me the number 2 when you know the next two numbers in that count." So I'm just giving students a little time to kind of think about what those next two things are before we start the count together. And then when I see most people kind of have those next two numbers, then we're going to start at that 3 and we're going to skip-count together.

Are you ready?

Mike: I am.

Christy: OK. So we're going to go 3…

Mike & Christy: 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36.

Christy: Keep going.

Mike & Christy: 39, 42, 45, 48, 51.

Christy: Let's stop there.

So we would go for a while like that until we have an array of numbers on the board. In this case, I might've been recording them, like where there were five in each row. So it would be 3, 6, 9, 12, 15 would be the first row, and the second row would say 18, 21, 24, 27, 30, and so on. So we would go that far and then I would stop and I would say to the class, "OK, take a minute, let your brains take it in. Give me a number 1 when your brain notices one thing. Show me 2 if your brain notices two things, 3 if your brain notices three things." And just let students have a moment to just take it in and think about what they notice.

And once we've seen them have some time, then I would say, "Turn and talk to your neighbor, and tell them some things that you notice." So they would do that. They would talk back and forth. And then I would usually warm-call someone from that and say something like, "Terry, why don't you tell me what you and Mike talked about?" So Terry, do you have something that you would notice?

Terry: Yeah, I noticed that the last column goes up by 15,

Christy: The last column goes up by 15. OK, so you're saying that you see this 15, 30, 45?

Terry: Yes.

Christy: In that last column. And you're thinking that 15 plus 15 is 30 and 30 plus 15 is 45. Is that right?

Terry: Yes.

Christy: Yeah. And so then usually what I would say to the students is say, "OK, so if you also noticed that last column is increasing by 15, give me a 'me too' sign. And if you didn't notice it, show an 'open mind' sign." So I like to give everybody something they can do. And then we'd say, "Let's hear from somebody else. So how about you, Mike? What's something that you would notice?"

Mike: So one of the things that I was noticing is that there's patterns in the digits that are in the ones place. And I can definitely see that because the first number 3 [is] in the first row. In the next row, the first number is 18 and the 8 is in the ones place. And then when I look at the next row, 33 is the first number in that row, and there's a 3 again. So I see this column pattern of 3 in the ones place, 8 in the ones place, 3 in the ones place, 8 in the ones place. And it looks like that same kind of a number, a different number. The same number is repeating again, where there's kind of like a number and then another number. And then it repeats in that kind of double, like two numbers and then it repeats the same two numbers.

Christy: So, what I would say in that one is try to revoice it, and I'd probably be gesturing, where I'd do this. But I'd say, "OK, so Mike's noticing in this ones place, in this first column, he's saying he notices it's '3, 8, 3, 8.' And then in other columns he's noticing that they do something similar. So the next column, or whatever, is like '6, 1, 6, 1' in the ones place. Why don't you give, again, give me a 'me too' [sign] if you also noticed that pattern or an 'open mind' [sign] if you didn't."

So, that's what we would do. So, we would let people share some things. We would get a bunch of noticings while students are noticing those things. I would be, like I said, revoicing and annotating on the board. So typically I would revoice it and point it out with gestures, and then I would annotate that to take a record of this thing that they've noticed on the board.

Once we've gotten several students' noticings on the board, then we're going to stop and we're going to unpack some of those. So I might do something like, "Oh, so Terry noticed this really interesting thing where he said that the last column increases by 15 because he saw 15, 30, 45, and he recognized that. I'm wondering if the other columns do something like that too. Do they also increase by the same kind of number? Hmm, why don't you take a minute and look at it and then turn and talk to your neighbor and see what you notice." And we're going to get them to notice then that these other ones also increase by 15. So if that hadn't already come out, I could use it as a press move to go in and unpack that one further.

And then we would ask the question, in this case, "Why do they always increase by 15?" And we might then use that question and that conversation to go and talk about Mike's observation, and to say, like, "Huh, I wonder if we could use what we just noticed here to figure out about why this idea that [the numbers in the] ones places are going back and forth between 3, 8, 3, 8. I wonder if that has something to do with this." Right? So we might use them to unpack it. They'll notice these patterns. And while the students were talking about these things, I'd be taking opportunities to both orient them to each other with linking moves to say, "Hey, what do you notice? What can you add on to what Mike said, or could you revoice it?" And also to annotate those things to make them available for conversation.

Mike: There was a lot in your description, Christy, and I think that provides a useful way to understand what's happening because there's the choice of numbers, there's the choice of how big the array is when you're recording initially, there are the moves that the teacher's making. What you've set up is a really cool conversation that comes forward. We did this with whole numbers just now, and I'm wondering if we could take a step forward and think about, OK, if we're imagining a choral count with fractions, what would that look and sound like?

Christy: Yeah, so one of the ones I really like to do is to do these ones that are just straight multiples, like start at 3 and skip-count by 3s. And then to either that same day or the very next day—so very, very close in time in proximity—do one where we're going to do something similar but with fractions. So one of my favorites is for the parallel of the whole number of skip-counting by 3s is we'll start at 3 fourths and we'll skip-count by 3 fourths. And when we write those numbers, we're not going to put them in simplest form; we're just going to write 3 fourths, 6 fourths, 9 fourths. So in this case, I would probably set it up in the exact same very parallel structure to that other one that we just did with the whole numbers. And I would put the numbers 3 fourths, 6 fourths, 9 fourths on the board.

I would say, "OK, here's our first numbers. We're going to start starting at 4 fourths. We're going to skip-count by 3 fourths. And give me a thumbs-up or the show me a 2 when you know the next two numbers." And then we would skip-count them together, and we would write them on the board. And so we'd end up—and in this case I would probably arrange them again in five columns just to have them and be a parallel structure to that one that we did before with the whole numbers. So it would look like 3 fourths, 6 fourths, 9 fourths, 12 fourths, 15 fourths on the first row. And then the next row, I would say 18 fourths, 21 fourths, 24 fourths, 27 fourths, 30 fourths. And again, I'd probably go all the way up until I got to 51 fourths before we'd stop and we'd look for patterns.

Mike: So I think what's cool about that—it was unsaid, but it kind of implied—is that you're making a choice there. So that students had just had this experience where they were counting in increments of 3, and 3, 6, 9, 12, 15, and then you start another row and you get to 30, and in this case, 3 fourths, 6 fourths, 9 fourths, 12 fourths, 15 fourths. So they are likely to notice that there's something similar that's going on here. And I suspect that's on purpose.

Christy: Right, that's precisely the thing that we want right here is to be able to say that fractions aren't something entirely new, something that you—just very different than anything that you've ever seen before in numbers. But to allow them to have an opportunity to really see the ways that numerators enumerate, they act like the counting numbers that they've always known, and the denominator names, and tells you what you're counting. And so it's just a nice space where, when they can see these in these parallel ways and experience counting with fractions, they have this opportunity to see some of the ways that both fraction notation works, what it's talking about, and also how the different parts of the fraction relate to things they already know with whole numbers.

Mike: Well, let's dig into that a little bit more. So the question I was going to ask Terry was: Can we talk a bit more about the ways the choral counting routine can help students make sense of the mathematics of fractions? So what are some of the ideas or the features of fractions that you found choral counting really allows you to draw out and make sense of with students?

Terry: Well, we know from our work with the rational number project how important language is when kids are developing an understanding of the role of the numerator and the denominator. And the choral counts really just show, like what Christy was just saying, how the numerator just enumerates and changes just like whole numbers. And then the denominator stays the same and names something. And so it's been a really good opportunity to develop language together as a class.

Christy: Yeah. I think that something that's really important in these ones that you get to see when you have them. So when they're doing that language, they're also—a really important part of a choral count is that it's not just that they're hearing those things, they're also seeing the notation on the board. And because of the way that we're both making this choice to repeatedly add the same amount, right? So we're creating something that's going to have a pattern that's going to have some mathematical relationships we can really unpack. But they're also seeing the notation on there that's arranged in a very intentional way to allow them to see those patterns in rows and columns as they get to talk about them.

So because those things are there, we're creating this chance now, right? So they see both the numerator and denominator. If we're doing them in parallel to things with whole numbers, they can see how both fractions are alike, things that they know with whole numbers, but also how some things are different. And instead of it being something that we're just telling them as rules, it invites them to make these observations.

So in the example that I just gave you of the skip-counting, starting at 3 fourths and skip-counting by 3 fourths, every time I have done this, someone always observes that the right-hand column, they will always say it goes up by 15. And what they're observing right there is they're paying attention to the numerator and thinking, "Well, I don't really need to talk about the denominator," and it buys me this opportunity as a teacher to say, "Yes, I see that too. I see that these 15 fourths and then you get another, then you get 30 fourths and you get 45 fourths. And I see in those numerators that 15, 30, 45—just like we had with the whole numbers—and here's how I would write that as a mathematician: I would write 15 fourths plus 15 fourths equals 30 fourths." Because I'm trying to be clear about what I'm counting right now. So instead of telling it like it's a rule that you have to remember, you have to keep the same denominators when you're going to add, it instead becomes something where we get to talk about it. It's just something that we get to be clear about. And that in fractions, we also do this other piece where we both enumerate and we name, and we keep track of that when we write things down to be clear. And so it usually invites this very nice parallel conversation and opportunity just to set up the idea that when we're doing things like adding and thinking about them, that we're trying to be clear and we're trying to communicate something in the same way that we always have been.

Mike: Well, Terry, it strikes me that this does set the foundation for some important things, correct?

Terry: Yeah, it sets the foundation for adding and subtracting fractions and how that numerator counts things and the denominator tells you the size of the pieces.

It also sets up multiplication. The last column, we can think of it as 5 groups of 3 fourths. And the next number underneath there might be 10 groups of 3 fourths. And as we start to describe or record what students' noticings are, we get a chance to highlight those features of adding fractions, subtracting fractions, multiplying fractions.

Mike: We've played around the edges of a big idea here. And one of the things that I want to bring back is something we talked about when we were preparing for the interview. This idea that learners of any age, generally speaking, they want to make use of their understanding of the way that whole numbers work as they're learning about fractions. And I'm wondering if one or both of you want to say a little bit more about this.

Terry: I think a mistake that we made previously in fraction teaching is we kind of stayed under 1. We just stayed and worked within 0 and 1 and we didn't go past it. And if you're going to make 1 a benchmark or 2 a benchmark or any whole number a benchmark, when you're counting by 3 fourths or 2 thirds or whatever, you have to go past it. So what choral counting has allowed us to do is to really get past these benchmarks, and kids saw patterns around those benchmarks, and they see them.

And then I think we also saw a whole-number thinking get in the way. So if you ask, for example, somebody to compare 3 seventeenths and 3 twenty-thirds, they might say that 3 twenty-thirds are bigger because 23 is bigger than 17. And instead of embracing their whole-number knowledge, we kind of moved away from it. And so I think now with the choral counting, they're seeing that fractions behave like whole numbers. They can leverage that knowledge, and instead of trying to make it go away, they're using it as an asset.

Mike: So the parallel that I'm drawing is, if you're trying to teach kids about the structure of numbers in whole number, if you can yourself to thinking about the whole numbers between 0 and 10, and you never worked in the teens or larger numbers, that structure's really hard to see. Am I thinking about that properly?

Terry: Yes, you are.

Christy: I think there's two things here to highlight.

So one of them that I think Terry would say more about here is just the idea that, around the idea of benchmarks. So you're right that there's things that come out as the patterns and notation that happen because of how we write them. And when we're talking about place value notation, we really need to get into tens and really into hundreds before a lot of those things become really available to us as something we talk about, that structure of how 10 plays a special role.

In fractions, a very parallel idea of these things that become friendly to us because of the notation and things we know, whole numbers act very much like that. When we're talking about rational numbers, right? So they become these nice benchmarks because they're really friendly to us, there's things that we know about them, so when we can get to them, they help us.

And the choral count that we were just talking about, there's something that's a little bit different that's happening though because we're not highlighting the whole numbers in the way that we're choosing to count right there. So we're not—we're using those, I guess, improper fractions. In that case, what we're doing is we're allowing students to have an opportunity to play with this idea, the numerator and denominator or the numerator is the piece that's acting like whole numbers that they know. So when Terry was first talking about how oftentimes when we first teach fractions and we were thinking about them, we were think a lot about the denominator. The denominator is something that's new that we're putting in with fractions that we weren't ever doing before with whole numbers. And we have that denominator. We focus a lot on like, "Look, you could take a unit and you can cut it up and you can cut it up in eight pieces, and those are called eighths, or you could cut it up in 10 pieces, and those are called tenths."

And we focus a lot on that because it's something that's new. But the thing that allows them to bridge from whole numbers is the thing that's the same as whole numbers. That's the numerator. And so when we want them to have chances to be able to make those connections back to the things they know and see that yes, there is something here that's new, it's the denominator, but connecting back to the things they know from whole numbers, we really do need to focus some on the numerator and letting them have a chance to play with what the numerator is, to see how it's acting, and to do things. It's not very interesting to say—to look at a bunch of things and say, like, "2 thirds plus 4 thirds equals 6 thirds," right? Because they'll just start to say, "Well, you can ignore the denominator." But when you play with it and counting and doing things like we was talking about—setting up a whole-number count and a fraction count in parallel to each other—now they get to notice things like that. [It] invites them to say things like, "Oh, so adding 15 in the whole numbers is kind of adding 15 fourths in the fourths." So they get to say this because you've kind of set it up as low-hanging fruit for them, but it's allowing them really to play with that notion of the numerator and a common denominator setting.

And then later we can do other kinds of things that let them play with the denominator and what that means in those kinds of pieces. So one of the things I really like about choral counts and choral counts with fractions is it's setting up this space where the numerator becomes something that's interesting and something worth talking about in some way to be able to draw parallels and allow them to see it. And then of course, equivalency starts to come into play too. We can talk about how things like 12 fourths is equivalent to 3 wholes, and then we get to see where those play their role inside of this count too. But it's just something that I really like about choral counting with fractions that I think comes out here. And it's not quite the idea of benchmarks, but it is important.

Mike: Well, let's talk a little bit about equivalency then. Terry. I'm wondering if you could say a little bit about how this routine can potentially set up a conversation around ideas related to equivalency.

Terry: We could do this choral count—instead of just writing improper fractions all the way through, we could write them with mixed numbers. And as you start writing mixed numbers, the pattern becomes "3 fourths, 1 and a half, 2 and a quarter," and we can start bringing in equivalent fractions. And you still do the same five columns and make parallel connections between the whole numbers, the fractions that are written as improper fractions and the fractions with mixed numbers. And so you get many conversations about equivalencies. And this has happened almost every time I do a choral count with fractions is, the kids will comment that they stop thinking. They go, "I'm just writing these numbers down." Part of it is they're seeing equivalency, but they're also seeing patterns and letting the patterns take over for them. And we think that's a good thing rather than a bad thing. It's not that they're stopped thinking, they're just, they're just—

Christy: They're experiencing the moment that patterns start to help, that pattern recognition starts to become an aid in their ability to make predictions. All of a sudden you can feel it kick online.

So if you said it in the context, then what happens is even in the mixed-number version or in the improper-number version, that students will then have a way of talking about that 12 fourths is equivalent to 3, and then you're going to see that whole-number diagonal sort of pop in, and then you'll see those other ones, even in the original version of it.

Terry: Yeah, as we started to play around with this and talk with people, we started using the context of sandwiches, fourths of sandwiches. And so when they would start looking at that, the sandwiches gave them language around wholes. So the equivalence that they saw, they had language to talk about. That's 12 fourths of a sandwich, which would be 3 full sandwiches. And then we started using paper strips with the choral counts and putting paper strips on each piece so kids could see that when it fills up they can see a full sandwich. And so we get both equivalencies, we get language, we get connections between images, symbols, and context.

Mike: One of the questions that I've been asking folks is: At the broadest level, regardless of the number being counted or whether it's a whole number or a rational number, what do you think the choral counting routine is good for?

Christy: So I would say that I think of these routines, like a choral count or a number talk or other routines like that that you would be doing frequently in a classroom, they really serve as a way of building mathematical language. So they serve as a language routine. And then one of the things that's really important about it is that it's not just that there's skip-counting, but that count. So you're hearing the way that patterns happen in language, but they're seeing it at the same time. And then they're having chances, once that static set of representations on the board, those visuals of the numbers has been created and set up in this structured way, it's allowing them to unpack those things. So they get to first engage in language and hearing it in this multimodal way. So they hear it and they see it, but then they get to unpack it and they get to engage in language in this other way where they get to say, "Well, here's things that stand out to me."

So they make these observations and they will do it using informal language. And then it's buying the teacher an opportunity then to not only highlight that, but then to also help formalize that language. So they might say, "Oh, I saw a column goes up by 5." And I would get to say, "Oh, so you're saying that you add each time to this column, and here's how a mathematician would write that." And we would write that with those symbols. And so now they're getting chances to see how their ideas are mathematical ideas and they're being expressed using the language and tools of math. "Here's the way you said it; here's what your brain was thinking about. And here's what that looks like when a mathematician writes it." So they're getting this chance to see this very deeply authentic way and just also buying this opportunity not only to do it for yourself, but then to take up ideas of others. "Oh, who else saw this column?" Or, "Do you think that we could extend that? Do you think it's anywhere else?" And they get to then immediately pick up that language and practice it and try it. So I look at these as a really important opportunity, not just for building curiosity around mathematics, but for building language.

Mike: Let's shift a little bit to teacher moves, to teacher practice, which I think y'all were kind of already doing there when you were talking about opportunities. What are some of the teacher moves that you think are really critical to bringing choral counting with fractions particularly to life?

Terry: I think just using the strips to help them visualize it, and it gave them some language. I think the context of sandwiches, or whatever it happens to be, gives them some ways to name what the unit is. We found starting with that runway, it really helps to have something that they can start to kind of take off and start the counting routine. We also found that the move where you ask them, "What do you notice? What patterns do you notice?," we really reserve for three and a half rows. So we try to go three full rows and a half and it gives everybody a chance to see something. If I go and do it too quick, I find that I don't get everybody participating in that, noticing as well, as doing three and a half rows. It just seems to be a magic part of the array is about three and a half rows in.

Mike: I want to restate and mark a couple things that you said, Terry. One is this notion of a runway that you want to give kids. And that functions as a way to help them start to think about, again, "What might come next?" And then I really wanted to pause and talk about this idea of, you want to go at least three rows, or at least—is it three or three and a half?

Terry: Three and a half.

Christy: When you have three of something, then you can start to use patterns. You need at least those three for even to think there could be a pattern. So when you get those, at least three of them, and they have that pattern to do—and like Terry was saying, when you have a partial row, then what happens is those predictions can come from two directions. You could keep going in the row, so you could keep going horizontally, or you could come down a column. And so now it kind of invites people to do things in more than one way when you stop mid-row.

Mike: So let me ask a follow-up question. When a teacher stops or pauses the count, what are some of the first things you'd love to see them do to spark some of the pattern recognition or the pattern seeking that you just talked about?

Christy: Teacher moves?

Mike: Yeah.

Christy: OK. So we do get to work with preservice teachers all the time. So this is one of my favorite parts of this piece of it. So what do you do as a teacher that you want? So we're going to want an array up there that has enough, at least three of things in some different ways people can start to see some patterns.

You can also, when you do one of these counts, you'll hear the moment—what Terry described earlier as "stop thinking." You can hear a moment where people, it just gets easier to start, the pattern starts to help you find what comes next, and you'll hear it. The voices will get louder and more confident as you do it. So you want a little of that. Once you're into that kind of space, then you can stop. You know because you've just heard them get a little more confident that their brains are going. So you're kind of looking for that moment. Then you're going to stop in there again partway through a row so that you've got a little bit of runway in both directions. So they can keep going horizontally, they can come down vertically. And you say, "OK," and you're going to give them now a moment to think. And so that stopping for a second before they just talk, creating space for people to formulate some language, to notice some things is really, really important.

So we're going to create some thinking space, but we know there's some thinking happening, so you just give them a way to do it. Our favorite way to do it is to, instead of just doing a thumbs-up and thumbs-down in front of the chest, we just do a silent count at the chest rather than hands going up. We just keep those hands out of the air, and I say, "Give me a 1 at your chest"—so a silent number 1 right at your chest—"when you've noticed one thing. And if you notice two things, give me a 2. And if you notice three things, give me a 3." They will absolutely extrapolate from there. And you'll definitely see some very anxious person who definitely wants to say something with a 10 at their chest. But what you're doing at that moment is you're buying people time to think, and you're buying yourself as a teacher some insight into where they are. So you now get to look out and you can see who's kind of taking a while for that 1 to come up and who has immediately five things, and other things.

And you can use that along with your knowledge of the students now to think about how you want to bring people into that discussion. Somebody with 10 things, they do not need to be the first person you call on. They are desperate to share something, and they will share something no matter when you call on them. So you want to use this information now to be able to get yourself some ideas of, like, "OK, I want to make sure that I'm creating equitable experiences, that I want to bring a lot of voices in." And so the first thing we do is we have now a sense of that because we just watched, we gave ourselves away into some of the thinking that's happening. And then we're going to partner that immediately with a turn and talk. So first they're going to think and then they're going to have a chance to practice that language in a partnership. And then, again, you're buying yourself a chance to listen into those conversations and to know that they have something to share. And to bring it in, I will pretty much always make that a warm call. I won't say, "Who wants to share?" I will say, "Terry or Mike, let's hear." And then I won't just say, "Terry, what was your idea?" I would say, "Terry, tell me something that either you or Mike shared that you noticed." So we'll give a choice. So now they've got a couple ways in. You know they just said something. So you're creating this space where you're really lowering the temperature of how nerve-racking it is to share something. They have something to say, and they have something to do. So I want all of those moves.

And then I kind of alluded to it when we were doing the practice one, but the other one I really like is to have all-class gestures so that everyone constantly has a way they need to engage and listen. And so I like to use ones not just the "me too" gesture, but we do the "open mind" gesture as well so that everyone has one of the two. Either it's something that you were thinking or they've just opened your mind to a new idea. And it looks, we use it kind of like an open book at your forehead. So, the best way I can describe it to you, you put both hands at your forehead and you touch them like they're opening up, opening doors. And so everyone does one of those, right? And then as a teacher, you now have some more information because you could say, "Oh, Terry, you just said that was open mind. You hadn't noticed it. Well, tell us something different you noticed." So you get that choice of what you're doing. So you're going to use these things as a teacher to not just get ideas out but to really be able to pull people in ways they've sort of communicated something to you that they have something to share.

So I love it for all the ways we get to practice these teacher moves that don't just then work in just this choral count, but that do a really great job in all these other spaces that we want to work on with students too, in terms of equitably and creating talk, orienting students to one another, asking them to listen to and build on each other's ideas.

Terry: When you first start doing this, you want to just stop and listen. So I think some of my mistakes early on was trying to annotate too quickly. But I found that a really good teacher move is just to listen. And I get to listen when they're think-pair-sharing, I get a chance to listen when they're just thinking together, I get a chance to listen when they describe it to the whole class. And then I get to think about how I'm going to write and record what they said so that it amplifies what they're saying to the whole class. And that's the annotation piece. And getting better at annotating is practicing what you're going to write first and then they always say something a little different than what you anticipate, but you've already practiced. So you can get your colors down, you can get how you're going to write it without overlapping too much with your annotations.

Mike: I think that feels like a really important point for someone who is listening to the podcast and thinking about their own practice. Because if I examine my own places where I sometimes jump before I need to, it often is to take in some ideas but maybe not enough and then start to immediately annotate. And I'm really drawn to this idea that there's something to, I want to listen enough to kind of hear the body of ideas that are coming out of the group before I get to annotation. Is that a fair kind of summary of the piece that you think is really important about that?

Terry: Yes. And as I'm getting better with it, I'm listening more and then writing after I think I know what they're saying. And I check with them as I'm writing.

Mike: So you started to already go to my next question, which is about annotation. I heard you mention color, so I'm curious: What are some of the ideas about annotation that you think are particularly important when you are doing it in the context of a choral count?

Christy: Well, yeah, I think a choral count. So color helps just to distinguish different ideas. So that's a useful tool for that piece of it. What we typically want, people will notice patterns usually in lines. And so you're going to get vertical lines and horizontal lines, but you'll also get diagonals. That's usually where those will be. And they will also notice things that are recognizable. So like the 15, 30, 45 being a number sequence that is a well-known one is typically wouldn't going to be the first one we notice. Another one that happens along a diagonal, and the examples we gave will be 12, 24, 36, it comes on a diagonal. People will often notice it because it's there. So then what you want is you're going to want to draw in those lines that help draw students' eyes, other students' eyes, not the ones who are seeing it, but the ones who weren't seeing it to that space so they can start to see that pattern too. So you're going to use a little bit of lines or underlining that sort of thing. These definitely do over time get messier and messier as you add more stuff to them. So color helps just distinguish some of those pieces.

And then what you want is to leave yourself some room to write things. So if you have fractions, for example, you're going to need some space between things because fractions take up a little bit more room to write. And you definitely want to be able to write "plus 15 fourths," not just, "plus 15." And so you need to make sure you're leaving yourself enough room and practicing and thinking. You also have to leave enough room for if you want to continue the count, because one of the beautiful things you get to do here is to make predictions once you've noticed patterns. And so you're going to probably want to ask at some point, "Well, what number do you think comes in some box further down the road?" So you need to leave yourself enough room then to continue that count to get there.

So it's just some of the things you have to kind of think about as a teacher as you do it, and then as you annotate, so you're kind of thinking about trying to keep [the numbers] pretty straight so that those lines are available to students and then maybe drawing them in so students can see them. And then probably off to the side writing things like, if there's addition or multiplication sentences that are coming out of it, you probably want to leave yourself some room to be able to sometimes write those. In a fraction one, which Terry talked about a little bit, because equivalency is something that's available now where we can talk about, for example, the really common one that would come out in our example would be that 12 fourths is equivalent to 3 wholes. Somehow you're going to have to ask this question of, "Well, why is that? Where could we see it?" And so in that case, usually we would draw the picture of the sandwiches, which will be rectangles all cut up in the same way. So not like grilled cheese sandwiches in fourth, but like a subway sandwich in fourths. And then you're going to need some space to be able to draw those above it and below it.

So again, you're kind of thinking about what's going to make this visible to students in a way that's meaningful to them. So you're going to need some space to be left for those things. What I find is that I typically end up having to write some things, and then sometimes after the new idea comes in, I might have to erase a little bit of what's there to make some more room for the writing. But I would say with fractions, it's going to be important to think about leaving enough space between, because you're probably going to need a little bit of pictures sometimes to help make sense of that equivalency. That's a really useful one. And leaving enough space for the notation itself, it takes a little bit of room.

Mike: Every time I do a podcast, I get to this point where I say to the guest or guests, "We could probably talk for an hour or more, and we're out of time." So I want to extend the offer that I often share with guests, which is if someone wanted to keep learning about choral counting or more generally about some of the ideas about fractions that we're talking about, are there any particular resources that the two of you would recommend?

Terry: We started our work with the Choral Counting & Counting Collections book by Megan Franke[, Elham Kazemi, and Angela Chan Turrou], and it really is transformational, both routines.

Christy: And it has fractions and decimals and ideas in it too. So you can see it across many things. Well, it's just, even just big numbers, small numbers, all kinds of different things. So teachers at different grade levels could use it.

The Teacher Education by Design [website], at tedd.org, has a beautiful unit on counting collections for teachers. So if you're interested in learning more about it, it has videos, it has planning guides, things like that to really help you get started.

Terry: And we found you just have to do them. And so as we just started to do them, writing it on paper was really helpful. And then The Math Learning Center has an app that you can use—the Number Chart app—and you can write [the choral counts] in so many different ways and check your timing out. And it's been a very helpful tool in preparing for quality choral counts with fractions and whole numbers.

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

Christy and Terry, I want to thank you both so much for joining us. It has really just absolutely been a pleasure chatting with you both.

Christy: So much fun getting to talk to you.

Terry: 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.

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