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

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How can educators help students recognize similarities in the way whole numbers
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and fractions behave?
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And are there ways educators can build on students' understanding of whole
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numbers to support their understanding of fractions?
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The answer from today's guests is an emphatic yes.
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Today we're talking with Terry Wyberg and Christy Pettis about the ways coral
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counting can support students' understanding of fractions.
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Welcome to the podcast, Terry and Christy.
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I'm excited to talk with you both today.
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Thanks for having us.
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Thank you.
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So for listeners who don't have prior knowledge,
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I'm wondering if we could just offer them some background.
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I'm wondering if one of you could briefly describe the choral counting routine.
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So how does it work?
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How would you describe the roles of the teacher and the students when they're
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engaging with this routine?
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Yeah, so I can describe it.
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The way that we usually would say is that it's a whole class routine for
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often done in kind of the middle grades.
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The teachers and the students are going to count aloud by a particular number.
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So maybe you're going to start at five and skip count by tens or start
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at 24 and skip count by a hundred or start at two thirds and skip count by two
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thirds.
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So you're going to start at some number and you're going to skip count by some
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number. And the students are all saying those numbers aloud.
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And while the students are saying them,
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the teacher is writing those numbers on the board,
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creating essentially what looks like an array of numbers.
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And then at certain points along with that talk,
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the teacher will stop and ask students to look at the numbers and talk about
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things they're noticing. And they'll kind of unpack some of that.
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Often they'll make predictions about things. They'll come next,
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continue the count to see where those go.
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So you already pivoted to my next question,
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which was to ask if you could share an example of a choral count with the
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audience. And I'm happy to play the part of a student if you'd like me to.
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So I think it helps a little bit to hear what it would sound like.
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So let's start at three and skip count by threes.
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The way that I would usually tell my teachers to start this out,
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as I like to call it the runway.
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So usually I would write the first three numbers. So I would write 3, 6,
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9 on the board, and then I would say, okay,
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so today we're going to start at three and we're going to skip count by threes,
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give me a thumbs up or give me the number two when you know the next two numbers
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in that count.
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So I'm just giving students a little time to kind of think about what those next
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two things are before we start the count together.
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And then when I see most people, I have those next two numbers.
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Then we're going to start at that three and we're going to skip count together.
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Are you ready?
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I am.
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Okay. So we're going to go
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3,
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6,
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6,
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9,
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12,
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12,
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15,
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18,
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21,
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24,
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27,
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30,
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33,
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36,
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keep going.
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30,
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39,
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40,
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42,
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40,
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45,
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48,
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51.
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Let's stop there.
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So we would go for a while like that until we have an array of numbers on the
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board. In this case,
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I might've been recording them like where there were five in each row.
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So it would be 3, 6, 9, 12, 15 would be the first row,
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and the second row would say 18, 21, 24, 27, 30, and so on.
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So we would go that far and then I would stop and I would say to the class,
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okay, take a minute, let your brains take it in. Give me a number one.
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When your brain notices one thing, show me two if your brain notices two things,
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three, if your brain notices three things.
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And just let students have a moment to just take it in and think about what they
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notice. And once we've seen them have some time, then I would say,
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turn and talk to your neighbor and tell them some things that you notice.
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So they would do that. They would talk back and forth.
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And then I would usually warm call someone from that and say something like,
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Terry, why don't you tell me what you and Mike talked about. So Terry,
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do you have something that you would notice?
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Yeah, I noticed that the last column goes up by 15.
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The last column goes up by 15. Okay,
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so you're saying that you see this 15, 30, 45?
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Yes.
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In that last column.
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And you're thinking that 15 plus 15 is 30 and 30 plus 15 is 45.
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Is that right?
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Yes.
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Yeah. And so then usually what I would say to the students is say, okay,
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so if you also noticed that last column is increasing by 15,
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give me a me too sign. And if you didn't notice it, show an open mind sign.
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So I like to give everybody something they can do and then we'd say,
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let's hear from somebody else. So how about you, Mike?
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What's something that you would notice?
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So one of the things that I was noticing is that there's patterns in the digits
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that are in the one's place.
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And I can definitely see that because the first number three in the first
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row in the next row,
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the first number is 18 and the eight is in the one's place.
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And then when I look at the next row, 33 is the first number in that row,
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and there's a three again.
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So I see this column pattern of three in the one's place,
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eight in the one's place, three in the one's place, eight in the one's place.
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And it looks like that same kind of a number, a different number.
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The same number is repeating again,
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where there's kind of like a number and then another number.
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And then it repeats in that kind of double like two numbers and then it repeats
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the same two numbers.
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So what I would say in that one is try to revoice it,
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and I'd probably be gesturing, I said, where do I do this? But I'd say, okay,
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so Mike's noticing in this one's place,
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in this first column he's saying he notices it's
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3 8, 3 8.
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And then in other columns he's noticing that they do something similar.
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So the next column, however, is like 6 1, 6 1 in the one place.
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Why don't you give again,
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give me a me too if you also noticed that pattern or an open mind if you didn't.
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So that's what we would do. So we would let people share some things.
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We would get a bunch of noticings while students are noticing those things.
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I would be, like I said, revoicing and annotating on the board.
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So typically I would revoice it and point it out with gestures,
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and then I would annotate that to take a record of this thing that they've
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noticed on the board. Once we've gotten several students noticings on the board,
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then we're going to stop and we're going to unpack some of those.
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So I might do something like, oh,
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so Terry noticed this really interesting thing where he said that the
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last column increases by 15 because he saw 15, 30, 45, and he recognized that.
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I'm wondering if the other columns do something like that too.
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Do they also increase by the same kind of number?
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Why don't you take a minute and look at it and then turn and talk to your
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neighbor and see what you notice.
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And we're going to get them to notice then that these other ones also increase
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by 15. So if that hadn't already come out,
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I could use it as a press move to go in and unpack that one further.
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And then we would ask the question, in this case,
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why do they always increase by 15? And we might then use that question and that
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conversation to go and talk about Mike's observation then to say like, huh,
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I wonder if we could use what we just noticed here to figure out about why this
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idea that these ones places are going back and forth between 3, 8, 8,
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wonder if that has something to do with this, right?
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So we might use them to unpack it. They'll notice these patterns,
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and while the students were talking about these things,
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I'd be taking opportunities to both orient them to each other with linking moves
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to say, Hey, what do you notice?
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What can you add on to what Mike said or could you revoice it?
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And also to annotate those things to make them available for conversation.
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There was a lot in your description, Christy,
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and I think that provides a useful way to understand what's happening because
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there's the choice of numbers,
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there's the choice of how big the array is when you're recording
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initially, there are the moves that the teacher's making.
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What you've set up is a really cool conversation that comes forward.
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We did this with whole numbers just now,
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and I'm wondering if we could take a step forward and think about, okay,
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if we're imagining a choral count with fractions,
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what would that look and sound like?
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Yeah, so one of the ones I really like to do is to do these ones that are just
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straight multiples, like start at three and skip count by threes.
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And then to either that same day or the very next day. So very,
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very close in time in proximity,
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do one where we're going to do something similar but with fractions.
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So one of my favorites is for the parallel of the whole number of skip counting
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by threes is we'll start at three fourths and we'll skip count by three-fourths.
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And when we write those numbers, we're not going to put them in simplest form,
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we're just going to write three-fourths, six fourths, nine fourths.
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So in this case,
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I would probably set it up in the exact same very parallel structure to that
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other one that we just did with the whole numbers.
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And I would put the numbers 3, 4, 6, 4, 9, fours on the board. I would say,
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okay, here's our first numbers. We're going to start starting at three-fourths.
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We're going to skip count by three-fourths and give me a thumbs up or the
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show me a two when you know the next two numbers.
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And then we would skip count them together and we would write them on the board.
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And so we'd end up,
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and in this case I would probably arrange them again in five columns just to
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have them and be a parallel structure to that one that we did before with the
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whole numbers. So it would look like three fourths, six fours, nine fourths,
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12 fourths, 15 fourths on the first row. And then the next row,
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I would say 18 fourths, 21 fourths, 24 fourths, 27 fourths, 30 fourths.
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And again,
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I'd probably go all the way up until I got to 51 fourths before we'd stop and
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we'd look for patterns.
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So I think what's cool about that, it was unsaid,
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but it kind of implied is that you're making a choice there.
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So that students had just had this experience where they were counting in
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increments of three and 3, 6, 9, 12, 15,
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and then you start another row and you get to 30 and in this case,
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three-fourths, six fourths, nine fourths, 12 fourths, 15 fourths.
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So they are likely to notice that there's something similar that's going on
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here. And I suspect that's on purpose.
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Right?
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That's precisely the thing that we want right here is to be able to say that
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fractions aren't something entirely new,
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something that you just very different than anything that you've ever seen
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before in numbers.
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But to allow them to have an opportunity to really see the ways that numerator
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enumerate,
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they act like the counting numbers that they've always known and a denominator
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names and tells you what you're counting.
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And so it's just a nice space where when they can see these in this parallel
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ways and experience counting with fractions,
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they have this opportunity to see some of the ways that both fraction notation
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works, what it's talking about,
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and also how the different parts of the fraction relate to things they already
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know with whole numbers.
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Well, let's dig into that a little bit more.
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So the question I was going to ask Terry was can we talk a bit more
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about the ways the coral counting routine can help students make sense of the
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mathematics of fractions?
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So what are some of the ideas or the features of fractions that you found coral
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counting really allows you to draw out and make sense of with students?
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Well,
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we know from our work with the rational number project how important language is
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when kids are developing an understanding of the role of the numerator and the
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denominator and the coral counts really just showed what Christy was just
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saying,
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how the numerator just nuMe rates and changes just like whole numbers and then
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the denominator stays the same and name something.
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And so it's been a really good opportunity to develop language together as a
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class.
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I think that something that's really important in these ones that you get to see
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when you have 'em. So when they're doing that language,
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they're also a really important part of a choral count is that it's not just
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that they're hearing those things,
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they're also seeing the notation on the board.
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And because of the way that we're both making this choice to repeatedly add the
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same amount,
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so we're creating something that's going to have a pattern that's going to have
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some mathematical relationships we can really unpack,
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but they're also seeing the notation on there that's arranged in a very
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intentional way to allow them to see those patterns in rows and columns as they
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get to talk about them. So because those things are there,
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we're creating this chance now, so they see both the new denominator.
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If we're doing them in to things with whole numbers,
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they can see how both fractions are alike,
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things that they know with whole numbers,
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but also how some things are different. And instead of it being something that
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we're just telling them as rules, it invites them to make these observations.
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So in the example that I just gave you of the skip counting,
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starting at three-fourths and skip counting by three-fourths,
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every time I have done this,
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someone always observes that the right hand column that will always say goes up
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by 15. And what they're observing right there is they're paying attention to the
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numerator and thinking, well,
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I don't really need to talk about the denominator and it buys me this
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opportunity as a teacher to say, yes, I see that too.
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I see that these 15 fourths and then you get another,
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then you get 30 fourths and you get 45 fourths.
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And I see in those numerator that 15, 30,
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45 just like we had with the whole numbers,
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and here's how I would write that as a mathematician,
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I would write 15 fourths plus 15 fourths equals 30 fourths. I'm
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trying to be clear about what I'm counting right now.
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So instead of telling it like it's a rule that you have to remember,
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you have to keep the same denominators when you're going to add instead becomes
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something where we get to talk about it.
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It's just something that we get to be clear about. And that infractions,
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we also do this other piece where we both enumerate and we name and we keep
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track of that when we write things down to be clear.
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And so it usually invites this very nice parallel conversation and opportunity
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just to set up the idea that when we're doing things like adding and thinking
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about them,
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that we're trying to be clear and we're trying to communicate something in the
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same way that we always have been.
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Well Terry,
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it strikes me that this does set the foundation for some important things,
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correct?
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Yeah, it sets the foundation for adding and subtracting fractions and how that
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numerator counts things. And the denominator tells you the size of the pieces.
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It also sets up multiplication. The last column,
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we can think of it as five groups of three-fourths.
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And the next number underneath there might be 10 groups of three-fourths.
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And as we start to describe or record what students'
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noticings are, we get a chance to highlight those features of adding fractions,
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subtracting fractions, multiplying fractions.
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We've played around the edges of a big idea here.
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And one of the things that I want to bring back is something we talked about
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when we were preparing for the interview. This idea that learners of any age,
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generally speaking,
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they want to make use of their understanding of the way that whole numbers work
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as they're learning about fractions.
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And I'm wondering if one or both of you want to say a little bit more about
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this.
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I think a mistake that we made previously in fraction teaching is we stayed
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under one.
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We just stayed and worked within zero and one and we didn't go past it.
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And if you're going to make one a benchmark or to a benchmark or any whole
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number a benchmark,
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when you're counting by three-fourths or two thirds or whatever,
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you have to go past it.
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So what Coral County has allowed us to do is to really get past these benchmarks
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and kids saw patterns around those benchmarks and they see 'em.
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And then I think we also saw a whole number thinking get in way.
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So if you ask for example,
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somebody to compare three seventeens and three 20 thirds,
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they might say that three 20 thirds are bigger because 23 is bigger than 17.
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And instead of embracing their whole number knowledge,
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we moved away from it. And so I think now with the Coral County,
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they're seeing that fractions behave like whole numbers.
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They can leverage that knowledge and instead of trying to make it go away,
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they're using it as an asset.
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So the parallel that I'm drawing is if you're trying to teach kids
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about the structure of numbers in whole numbers,
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if you can yourself to thinking about the whole numbers between zero and 10 and
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you never worked in the teens or larger numbers,
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that structure's really hard to see. Am I thinking about that properly?
15:39
Yes, you are.
15:41
I think there's two things here to highlight.
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So one of 'em that I think Terry would say more about here is just the idea that
15:47
around the idea of benchmarks.
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So you're right that there's things that come out as the patterns and notation
15:52
that happen because of how we write them.
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And when we're talking about place value notation,
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we really need to get into tens and really into hundreds before a lot of those
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things become really available to us as something we talk about that structure
16:05
of how 10 plays a special role in fractions,
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a very parallel idea of these things that become friendly to us because of the
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notation and things we know whole numbers act very much like that. And
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when we're talking about rational numbers,
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so they become these nice benchmarks because they're really friendly to us,
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there's things that we know about them, so when we can get to them,
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they help us. And the coral count that we were just talking about,
16:28
there's something that's a little bit different that's happening though because
16:32
we're not highlighting the whole numbers in the way that we're choosing to count
16:36
right there. So we're not, we're using those, I guess,
16:39
improper fractions in that case.
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What we're doing is we're allowing students to have an opportunity to play with
16:46
this idea the numerator and denominator or the numerator is the piece that's
16:50
acting like whole numbers that they know.
16:52
So when Terry was first talking about how oftentimes when we first teach
16:56
fractions and we were thinking about them,
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we were think a lot about the denominator.
16:59
The denominator is something that's new that we're putting in with fractions
17:02
that we weren't ever doing before with whole numbers.
17:05
And we have that denominator we focus a lot on like look,
17:08
you could take a unit and you can cut it up and you can cut it up in eight
17:10
pieces and those are called eight,
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or you could cut up in 10 pieces. And those are called tents and we focus a lot
17:15
on that because it's something that's new.
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But the thing that allows 'em to bridge from whole numbers is the thing that's
17:20
the same as whole numbers. That's the numerator.
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And so when we want them to have chances to be able to make those connections
17:28
back to the things they know and see that yes,
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there is something here that's new, it's the denominator,
17:34
but connecting back to the things they know from whole numbers,
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we really do need to focus some on the numerator and letting them have a chance
17:41
to play with what the numerator is to see how it's acting and to do things.
17:45
It's not very interesting to say to look at a bunch of things,
17:49
say like two thirds plus four thirds equals six thirds, right?
17:52
Because they'll just start to say, well, you can ignore the denominator,
17:56
but when you play with it and counting and doing things like we was talking
17:59
about setting up a whole number count and a fraction count in parallel to each
18:03
other, now they get to notice things like that invites them to say things like,
18:07
oh,
18:08
so adding 15 in the whole numbers is kind of adding 15 fourths in the
18:13
fourths so they get to say this because you've kind of set it up as low hanging
18:17
fruit for them,
18:18
but it's allowing 'em really to play with that notion of the numerator and a
18:21
common denominator setting. And then later we can do other kind of things that
18:25
let them play with the denominator and what that means in those kinds of pieces.
18:29
So one of the things I really like about coral counts and coral counts with
18:31
fractions is it's setting up this space where the numerator becomes something
18:35
that's interesting and something worth talking about in some way to be able to
18:39
draw parallels and allow them to see it. And then of course,
18:42
equivalency starts to come into play too.
18:44
We can talk about how things like 12 fourths is equivalent to three holes,
18:47
and then we get to see where those play their role inside of this count too.
18:51
But it's just something that I really like about coral counting with fractions
18:53
that I think comes out here. And it's not quite the idea of benchmarks,
18:58
but it is important.
19:00
Well, let's talk a little bit about equivalency then. Terry.
19:03
I'm wondering if you could say a little bit about how this routine can
19:07
potentially set up a conversation around ideas related to equivalency.
19:12
We could do this choral count instead of just writing improper fractions all the
19:16
way through. We could write 'em with mixed numbers.
19:19
And as you start writing mixed numbers, the pattern becomes three-fourths,
19:23
one and a half, two and a quarter,
19:25
and we can start bringing in equivalent fractions and you still do the same five
19:29
columns and make parallel connections between the whole numbers,
19:33
the fractions that are written as improper fractions and the fractions with
19:37
mixed numbers. And so you get many conversations about equivalencies,
19:41
and this has happened almost every time I do a choral count with fractions is
19:45
the kids will comment that they stop thinking, they go,
19:49
I'm just writing these numbers down. Part of it is they're seeing equivalency,
19:54
but they're also seeing patterns and letting the patterns take over for 'em.
19:58
And we think that's a good thing rather than a bad thing.
20:01
It's not that they're stopped thinking, they're just.
20:05
They're experience in the moment that patterns start to help that pattern
20:09
recognition starts to become an aid in their ability to make predictions.
20:12
All of a sudden you can feel it kick online. So if you said it in the context,
20:17
then what happens is even in the mixed number version or in the improper number
20:20
version,
20:21
that students will then have a way of talking about that 12 force is equivalent
20:25
to three,
20:25
and then you're going to see that whole number diagonals sort of pop in,
20:29
and then you'll see those other ones, even in the original version of it.
20:33
As we started to play around with this and talk with people,
20:36
we started using the context of sandwiches, fourths of sandwiches.
20:40
And so when they would start looking at that,
20:43
the sandwiches gave them language around holes.
20:46
So the equivalence that they saw, they had language to talk about.
20:50
That's 12 fourths of a sandwich, which would be three full sandwiches.
20:54
And then we started using paper strips with the coral counts and putting paper
20:58
strips on each piece so kids could see that when it fills up they can see a full
21:03
sandwich. And so we get both equivalencies, we get language,
21:07
we get connections between images, symbols, and context.
21:12
One of the questions that I've been asking folks is, at the broadest level,
21:17
regardless of the number being counted or whether it's a whole number or a
21:20
rational number, what do you think the choral counting routine is good for?
21:25
So I would say that I think of these routines like a choral count or a number
21:30
talk or other routines like that that you would be doing frequently in a
21:33
classroom.
21:34
They really serve as a way of building mathematical language.
21:38
So they serve as a language routine.
21:40
And then one of the things that's really important about it is that it's not
21:42
just that there's skip counting, but that count.
21:45
So you're hearing the way that patterns happen in language,
21:49
but they're seeing it at the same time and then they're having chances once
21:54
that static set of representations on the board,
21:56
those visuals of the numbers has been created and set up in this structured
22:01
way, it's allowing them to unpack those things.
22:04
So they get to first engage in language and hearing it in this multimodal way.
22:07
So they hear it and they see it,
22:10
but then they get to unpack it and they get to engage in language in this other
22:14
way where they get to say, well,
22:15
here's things that stand out to me. So they make these observations and they
22:19
will do it using informal language.
22:21
And then it's buying the teacher an opportunity then to not only highlight that,
22:25
but then to also help formalize that language. So they might say, oh,
22:28
I saw a column goes up by five. And I would get to say, oh,
22:31
so you're seeing that you add each time to this column and here's how a
22:34
mathematician would write that. And we would write that with those symbols.
22:37
And so now they're getting chances to see how their ideas are mathematical ideas
22:42
and they're being expressed using the language and tools of math.
22:45
Here's the way you said it,
22:46
here's what your brain was thinking about and here's what that looks like when a
22:50
mathematician writes it.
22:51
So they're getting this chance to see this very deeply authentic way and just
22:56
also buying this opportunity not only to do it for yourself,
22:58
but then to take up ideas of others. Oh, who else saw this column?
23:02
Or do you think that we could extend that? Do you think it's anywhere else?
23:05
And they get to then immediately pick up that language and practice it and try
23:10
it. So I look at these as a really important opportunity,
23:13
not just for building curiosity around mathematics, but for building language.
23:17
Let's shift a little bit to teacher moves to teacher practice,
23:20
which I think y'all were kind of already doing there when you were talking about
23:23
opportunities.
23:24
What are some of the teacher moves that you think are really critical to
23:29
bringing choral accounting with fractions particularly to life?
23:33
I think just using the strips to help them visualize it and it gave 'em some
23:37
language. I think the context of sandwiches or whatever it happens to be,
23:41
gives them some ways to name what the unit is.
23:44
We found starting with that runway,
23:46
it really helps to have something that they can start to kind of take off and
23:50
start the counting routine.
23:51
We also found that the move where you ask 'em what do you notice?
23:55
What patterns do you notice? We really reserve for three and a half rows,
23:59
so we try to go three full rows and a half and it gives everybody a chance to
24:03
see something. If I go and do it too quick,
24:06
I find that I don't get everybody participating in that noticing as well as
24:11
doing three and a half rows.
24:12
It just seems to be a magic part of the array is about three and a half rows in.
24:18
I want to restate and mark a couple things that you said, Terry.
24:21
One is this notion of a runway that you want to give kids and that functions as
24:25
a way to help them start to think about again, what might come next.
24:29
And then I really wanted to pause and talk about this idea of you want to go at
24:34
least three rows or at least is it three or three and a half? Three and a half.
24:39
When you have three of something, then you can start to use patterns.
24:42
You need at least those three for even to think there could be a pattern.
24:45
So when you get those, at least three of them and they have that pattern to do,
24:49
and like Tara is saying, when you have a partial row,
24:51
then what happens is those predictions can come from two directions.
24:54
You could keep going in the row,
24:55
so you could keep going horizontally or you could come down a column.
24:58
And so now it kind of invites people to do things in more than one way when you
25:02
stop mid row.
25:04
So let me ask a follow-up question.
25:07
When a teacher stops or pauses the count,
25:11
what are some of the first things you'd love to see them do to spark
25:16
some of the pattern recognition or the pattern seeking that you just talked
25:20
about?
25:22
Teacher moves?
25:23
Yeah.
25:24
Okay. So we do get to work with pre-service teachers all the time.
25:28
So this is one of my favorite parts of this piece of it.
25:30
So what do you do as a teacher that you want?
25:32
So we're going to want an array up there that has enough,
25:35
at least three of things in some different ways people can start to see some
25:39
patterns. You can also, when you do one of these counts,
25:42
you'll hear the moment what Terry described earlier, stop thinking.
25:46
You can hear a moment where people,
25:47
it just gets easier to start the pattern starts to help you find what comes
25:52
next, and you'll hear it.
25:53
The voices will get louder and more confident as you do it.
25:56
So you want a little of that. Once you're into that kind of space,
25:59
then you can stop because you've just heard them get a little more confident
26:03
that their brains are going. So you're kind of looking for that moment.
26:07
Then you're going to stop in there again partway through a row so that you've
26:10
got a little bit of runway in both directions.
26:11
So they can keep going horizontally, they can come down vertically and you say,
26:14
okay, and you're going to give them now a moment to think.
26:18
And so that stopping for a second before they just talk,
26:21
creating space for people to formulate some language,
26:24
to notice some things is really, really important.
26:26
So we're going to create some thinking space,
26:28
but we know there's some thinking happening,
26:30
so you just give them a way to do it.
26:32
Our favorite way to do it is to instead of just doing a thumbs up and thumbs
26:35
down in the chest,
26:36
we just do a silent count at the chest rather than hands going up.
26:40
We just keep those hands out of the air and I say,
26:42
give me a one at your chest. So a silent like number one right at your chest
26:47
when you've noticed one thing. And if you notice two things,
26:50
give me a two and if you notice three things, give me a three.
26:53
They will absolutely extrapolate from there.
26:55
And you'll definitely see some very anxious person who definitely wants to say
26:58
something with a 10 at their chest.
27:00
But what you're doing at that moment is you're buying people time to think and
27:04
you're buying yourself as a teacher some insight into where they are.
27:09
So you now get to look out and you can see who's kind of taking a while for that
27:14
one to come up and who has immediately five things and other things.
27:18
And you can use that along with your knowledge of the students now to think
27:21
about how you want to bring people into that discussion.
27:24
Somebody with 10 things,
27:25
they do not need to be the first person you call on. They are desperate to share
27:29
something and they will share something no matter when you call on them.
27:32
So you want to use this information now to be able to get yourself some
27:37
ideas of like, okay,
27:38
I want to make sure that I'm creating equitable experiences that I want to bring
27:41
a lot of voices in.
27:43
And so the first thing we do is we have now a sense of that we just watched,
27:47
we gave ourselves away into some of the thinking that's happening,
27:51
and then we're going to partner that immediately with a turn and talk.
27:54
So first they're going to think and then they're going to have a chance to
27:56
practice that language in a partnership. And then again,
28:00
you're buying yourself a chance to listen into those conversations and to know
28:04
that they have something to share and to bring it in.
28:06
I will pretty much always make that a warm call. I won't say who wants to share,
28:10
I will say Terry or Mike, let's hear.
28:14
And then I won't just say, Terry, what was your idea? I would say, Terry,
28:18
tell me something that either you or Mike shared that you noticed.
28:23
So we'll give a choice. So now they've got a couple ways in.
28:26
They just said something.
28:28
So you're creating this space where you're really lowering the temperature of
28:32
how nerve wracking it is to share something.
28:34
They have something to say and they have something to do.
28:37
So I want all of those moves.
28:39
And then I kind of alluded to it when we were doing the practice one,
28:43
but the other one I really like is to have all class gestures so that everyone
28:47
constantly has a way they need to engage and listen.
28:50
And so I like to use ones not just the me too gesture,
28:53
but we do the open mind gesture as well so that everyone has one of the two.
28:58
Either it's something that you were thinking or they've just opened your mind to
29:03
a new idea. And it looks,
29:04
we use it kind of like an open book at your forehead. So if's,
29:09
best way I can describe you put both hands at your forehead and you touch them.
29:12
They're opening up opening doors, and so everyone does one of those, right?
29:16
And then as a teacher, you now have some more information. You could say, oh,
29:20
Terry, you just said that was open mind. You hadn't noticed it. Well,
29:23
tell us something different you noticed.
29:25
So you get that choice of what you're doing.
29:27
So you're going to use these things as a teacher to not just get ideas out,
29:32
but to really be able to pull people in ways they've sort of communicated
29:37
something to you that they have something to share.
29:39
So I love it for all the ways we get to these teacher moves that don't just then
29:44
work in just this choral count,
29:46
but that do a really great job in all these other spaces that we want to work on
29:49
with students too,
29:50
in terms of equitably and creating talk orienting students to one
29:55
another, asking them to listen to and build on each other's ideas.
29:59
When you first start doing this, you want to just stop and listen.
30:02
So I think some of my mistakes early on was trying to annotate too quickly,
30:07
but I found that really good teacher move is just to listen and I got to listen
30:11
when they're think pair sharing,
30:13
I get a chance to listen when they're just thinking together,
30:16
I get a chance to listen when they describe it to the whole class.
30:19
And then I get to think about how I'm going to write and record what they said
30:24
so that it amplifies what they're saying to the whole class.
30:28
And that's the annotation piece.
30:29
And getting better at annotating is practicing what you're going to write first
30:34
and then they always say something a little different than what you anticipate,
30:38
but you've already practiced. So you can get your colors down,
30:41
you can get how you're going to write it without overlapping too much with your
30:45
annotations.
30:47
I think that feels like a really important point for someone who is listening to
30:50
the podcast and thinking about their own practice.
30:53
Because if I examine my own places where I sometimes jump before I need
30:57
to,
30:58
it often is to take in some ideas but maybe not enough and
31:03
then start to immediately annotate.
31:05
And I'm really drawn to this idea that there's something to,
31:09
I want to listen enough to kind of hear the body of ideas that are
31:14
coming out of the group before I get to annotation.
31:17
Is that a fair kind of summary of the piece that you think is really important
31:21
about that?
31:23
Yes. And as I'm getting better with it,
31:25
I'm listening more and then writing after.
31:28
I think I know what they're saying and I check with them as I'm writing.
31:33
So you started to already go to my next question, which is about annotation.
31:38
I heard you mention color, so I'm curious,
31:41
what are some of the ideas about annotation that you think are particularly
31:44
important when you are doing it in the context of a coral count?
31:48
Well, yeah, I think a coral count.
31:50
So color helps just to distinguish different ideas.
31:54
So that's a useful tool for that piece of it. What we typically want,
31:58
people will notice patterns usually in lines.
32:01
And so you're going to get vertical lines and horizontal lines,
32:03
but you'll also get diagonals. That's usually where those will be.
32:06
And they will also notice things that are recognizable. So like the 15, 30,
32:10
45 being a number sequence that is a well-known one is typically wouldn't going
32:13
to be the first ones. We notice another one that happens along a diagonal.
32:17
And the examples we gave will be 12, 24, 36, it comes on a diagonal.
32:22
People will often notice it because it's there.
32:23
So then what you want is you're going to want to draw in those lines that help
32:26
draw students' eyes, other students' eyes, not the ones who are seeing it,
32:30
but the ones who weren't seeing it to that space. So they can start to see that
32:33
pattern too.
32:34
So you're going to use a little bit of lines or underlining that sort of thing.
32:39
These definitely do over time get messier and messier as you add more stuff to
32:42
them. So color helps just distinguish some of those pieces.
32:46
And then what you want is to leave yourself some room to write things.
32:49
So if you have fractions, for example,
32:51
you're going to need some space between things because fractions take up a
32:54
little bit more room to write,
32:56
and you definitely want to be able to write plus 15 four.
32:59
It's not just plus 15.
33:01
And so you need to make sure you're leaving yourself enough room and practicing
33:06
and thinking.
33:06
You also have to leave enough room for if you want to continue the count,
33:10
because one of the beautiful things you get to do here is to make predictions
33:13
once you've noticed patterns. And so you're going to probably want to ask at
33:16
some point, well,
33:17
what number do you think comes in some box further down the road?
33:20
So you need to leave yourself enough room then to continue that count to get
33:24
there.
33:25
So it's just some of the things you have to kind of think about as a teacher as
33:28
you do it, and then as you annotate,
33:29
so you're kind of thinking about trying to keep them pretty straight so that
33:33
those lines are available to students and then maybe drawing them in so students
33:38
can see them.
33:38
And then probably off to the side writing things like if there's addition
33:43
or multiplication sentences that are coming out of it,
33:45
you probably want to leave yourself some room to be able to sometimes write
33:48
those in a fraction one, which Terry talked about a little bit,
33:53
because equivalency is something that's available now where we can talk about,
33:57
for example,
33:58
the really common one that would come out in our example would be that 12
34:00
fourths is equivalent to three holes. Somehow you're going to have to ask this
34:04
question of, well, why is that? Where could we see it? And so in that case,
34:08
usually we would draw the picture of the sandwiches,
34:10
which will be rectangles all cut up in the same way.
34:13
So not like grilled cheese sandwiches in fourth,
34:15
but like a subway sandwich in fourth.
34:18
And then you're going to need some space to be able to draw those above it and
34:21
below it. So again,
34:22
you're kind of thinking about what's going to make this visible to students in a
34:26
way that's meaningful to them.
34:28
So you're going to need some space to be left for those things.
34:32
What I find is that I typically end up having to write some things,
34:36
and then sometimes after the new idea comes in,
34:38
I might have to erase a little bit of what's there to make some more room for
34:43
the writing. But I would say with fractions,
34:45
it's going to be important to think about leaving enough space between,
34:48
because you're probably going to need a little bit of pictures sometimes to help
34:52
make sense of that equivalency. That's a really useful one.
34:55
And leaving enough space for the notation itself, it takes a little bit of room.
35:01
Every time I do a podcast,
35:03
I get to this point where I say to the guest or guests,
35:06
we could probably talk for an hour or more,
35:10
and we're out of time.
35:13
So I want to extend the offer that I often share with guests,
35:16
which is if someone wanted to keep learning about coral counting or more
35:19
generally about some of the ideas about fractions that we're talking about,
35:23
are there any particular resources that the two of you would recommend?
35:27
We started our work with Coral County and County collection book by Megan
35:32
Frankie, and it really is transformational both routines.
35:37
And it has fractions and decimals and ideas in it too.
35:40
So you can see it across many things. Well, it's just even just big numbers,
35:44
small numbers, all kinds of different things.
35:46
So teachers at different grade levels could use it.
35:48
The teacher education by design,
35:50
the td.org has a beautiful unit on counting collections for teachers.
35:55
So if you're interested in learning more about it, it has videos,
35:57
it has planning guides, things like that to really help you get started.
36:02
And we found you just have to do 'em. And so as we just started to do 'em,
36:06
writing it on paper was really helpful.
36:08
And then the Math Learning Center has an app that you can use the number
36:13
chart app and you can write 'em in so many different ways and check your
36:17
timing out.
36:18
And it's been a very helpful tool in preparing for quality choral counts with
36:23
fractions and whole numbers.
36:25
I think that's a great place to stop. Christy and Terry,
36:28
I want to thank you both so much for joining us.
36:30
It is really just absolutely been a pleasure chatting with you both.
36:34
So much fun getting to talk to you.
36:36
Thank you.
36:40
This podcast is brought to you by the Math Learning Center and the Meyer Math
36:43
Foundation dedicated to inspiring and enabling all individuals to discover
36:48
and develop their mathematical confidence and ability.