Latest / Rounding Up / See It! Say It! Symbolize It! with Dr. Patrick Sullivan

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See It! Say It! Symbolize It! with Dr. Patrick Sullivan · Rounding Up

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Representations, language, and symbols are all important ingredients
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in building students' conceptual understanding and fluency.
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But how can we understand the relationship between those three ingredients?
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And what happens if we leave any one of them out of the recipe?
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We'll explore these questions today with Patrick Sullivan,
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a professor of mathematics at Missouri State University.
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Welcome to the podcast, Patrick.
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It's so great to have you joining us today.
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I'm excited to be here. Thank you for this opportunity.
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So you've written a book, it's called See It,
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Say It, Symbolize It. And early on in the book,
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you tell a story about your Grandma Edna,
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and I love this story. You talk about the role that she played in your math journey.
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So I wonder if you could share that story with our listeners.
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Yeah, my grandmother played an instrumental role in my life.
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She passed away about four years ago,
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about nine, at 99 [years] and 20 days.
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So very good full life. But ~ a little bit about her.
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She grew up in a small town in Missouri,
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was a valedictorian to that school in class of 1939,
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and didn't have an opportunity to go to college,
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so she ended up going to hairdressing school.
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So part of the story was to honor her life in sharing that story,
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but there's a poignant moment of my life,
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where
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I'm in elementary school and I'm this kid that,
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with allergies, so I'm carrying around a handkerchief.
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You know, my two favorite subjects are recess and lunch.
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~ I stuttered, and I I didn't read very well either.
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So There was not a whole lot where I felt like school was a winning place for me,
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but that one thing that I felt like I was good at was math.
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Things were going OK. I wasn't going home in tears.
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And along came 2-digit multiplication.
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And I remember vividly that day,
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just feeling devastated, like,
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"OK, this is it." And the worst part of it was—again,
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my two favorite subjects are lunch and recess.
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They didn't take lunch away—but we had to stay in for recess,
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which was crushing, 'cause I loved recess.
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So I was pretty distraught when I got home and [it was] one of those moments
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in life where things just were not going well.
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My, both my parents were were looking for different jobs,
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so they weren't there, but I was
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I was blessed with having my grandparents next door.
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So I go over that night, and my grandmother could tell something was wrong.
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And so she asked, and I said, "Well,
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I just cant figure this 2-digit multiplication business out." She
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got out these 3-by-5 cards and this ~ mechanical pencil,
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and she just wrote down at the top,
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"35 times 24" at the top of this card.
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And she said to me, "Do you know what 4 times 5 is?" I said,
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"Yeah, 4 times 5 is 20, 4 times 30 is 120,
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20 times 5 is 100, and 20 times 30 is 600." And she just explained that to me.
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She wrote the header above the number,
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she wrote the place value, and she talked about how the 20 was really 2 tens
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and the 30 was 3 tens. But it,
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it just all clicked. I mean, it was amazing.
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We spent the next hour and a half doing that.
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And the crazy part of it is, the next day in class,
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I felt like
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I was on cloud nine. I got it.
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And not only did I get it, I could explain it.
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And so the next day, there were about five of us in what I call,
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lovingly, math jail. And so I explained how I did it.
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The teacher gave me the space to do that.
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And I and I'm forever thankful for that.
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She could have just could have said,
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you know, "No, that's not the way I do it.
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So we're not gonna do it that way." But she let me go up to the board and show
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it to my peers. And then as my peers got it,
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they were like, "Oh, yeah, that's makes a lot of sense."
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And so that again, really elevated my stock.
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You know, I really, I felt like a million bucks there,
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like, "I'm the math guy now." And after that day we all were out of math jail.
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And so we all got to leave. And I I felt empowered.
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And that one day changed everything for me.
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And still, to this day, I'm 50 years removed from it,
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and it still brings back the emotions.
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There is so much in that story,
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Patrick. So I think I wanna, I wanna toss you another question.
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There are a lot of things in that story that spoke to me that validated some
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of the experiences I had as a kid.
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There was wisdom ~ both in how Grandma Edna helped you and also what your teacher
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did in response. But I wonder if you could be transparent with folks and
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say some of the meanings of that story that you think are important.
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You know, I want to make a point about how one moment in math around a
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big idea can change a trajectory for life.
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And it truly did. I mean that,
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to me, I—I go back to it and I remember it so vividly.
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You know, and it doesn't take much to get out of what I call "the math game."
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You know, the top three areas I often hear are fractions,
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multiplication, long division.
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You know, even when I ask adults when they say that they're not good
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at math and, and for me, this book is about redemption.
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It it's about a chance for people who said
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I've never really been good at math.
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Well, here's a different take on it.
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This is where you can get back in the math game and see math as a different
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in a connected way to think about.
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Because that story for me is a lot about seeing ideas that were disconnected.
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And in one stroke, in one conversation,
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she connected a lot of ideas. It's not only from the perspective,
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as I said, from what I learned,
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but I give it a lot of credit—I talked a lot about my Grandma Edna,
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but my teacher also gave me a platform to share my thinking with others and,
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again, in the 1970s, that was probably unheard of.
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That was the [era of] "Sit down and do your work and don't say anything".
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So for the teacher to do that,
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and not just reject what I wanted to share,
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was empowering.
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You know, when I first saw the title of the book,
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I really imagined that See It,
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Say It, Symbolize It might be a set of steps in a sequence.
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But after reading the book, I think I'd describe See It,
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Say It, Symbolize It as ingredients in a recipe.
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And I wanted to ask you: How do you think about the relationship between
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the three phrases in the title—See it,
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say it, and symbolize it?
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OK, first all, I love the analogy.
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I'm stealing it. I'm gonna use that from now on.
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It's a combination of what makes the recipe work.
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And as I've done many times in my ~ cooking experiences,
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if you leave one of them out, the results are not the same.
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You know, and so the "see it" part is the main ingredient.
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It's the base. Maybe that's like flour or something.
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But anyway, it's it's the base.
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For me, it's the foundation. And the "see it" part is
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the representation activities that are used to make the mathematical idea most
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visible.
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So for example, let's consider 2-digit addition.
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And we can think about either a horizontally written problem or
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a vertically written problem, a 2-digit problem.
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But many teachers utilize base ten blocks or bundled popsicle sticks.
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They're great manipulatives to teach addition.
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But then what type of activities do we engage students in with them?
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So it's not just the representations that matter,
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but it's how we utilize them and the activities we engage students.
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So
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I like base ten blocks, for example.
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They make the idea of equal exchanges explicit because you can hand me
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10 ones and I'm going to give you 1 ten.
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There's this switching out that happens.
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There's an exchange. But with bundling popsicles—the sticks—what's,
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what's most visible is that when you bundle those,
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you have the 1 ten as a bundle,
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but you still have the 10 ones.
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It's there. So it's still visible.
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So it's really tied to
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our learning goals and and being very intentional with the representations
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we use and the activities that we engage students as we're doing that.
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I think the "say it" part is about students communicating their understanding.
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You know, I think, just very basic when we're thinking about addition,
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you know—old school the way I learned it,
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you know—you carry the 1 and we regroup.
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But the idea is, and I'll talk more about this later,
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but this idea of equal exchange:
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What are we exchanging? And by doing that,
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we're taking 10 ones and replacing it with 1 ten.
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And that happens throughout mathematics.
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And I feel like with the "say it" part is,
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to me, like the egg of a recipe.
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It binds everything together. And I feel like that,
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many times, is the missing piece there that,
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you know, you've done this great job with these physical or visual representations,
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and then you jump right into the symbolic and there's no connection between the two.
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It's just one more thing to learn.
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And so there's a disconnect between what you've done with the representations
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or the models
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and then what you're doing symbolically.
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So I think language is the binding part.
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And then the symbolizing kind of works two ways.
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The first is: Obviously, when you take a number—like 74,
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for example—that knowledge has to be shared and imparted on somebody else.
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It's not something they're just gonna know that the 7 is 7 tens and the 4 is 4 ones,
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that we're operating in a base ten number system.
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That just doesn't happen from a constructive standpoint.
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You know, there has to be some judicious telling.
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But then there's also this element of symbolizing where when
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you give students something like 7 times 3,
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how are they making sense of it?
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How are they symbolizing that now that you've shown them something symbolically?
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How are they symbolizing?
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So those three things bind together.
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And oftentimes I see we rush the symbolizing and hide the seeing and the saying,
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and then we wonder why students struggle with mathematics.
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I really love how you unpacked the relationship between those,
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Patrick. And I do want to go back and talk about "see it" because— —one
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of the terms that you used in the book that I thought was really interesting
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is this notion of a concept image.
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And I wonder if you could share a little bit about what a concept image
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is and the theory behind what a concept image offers to students.
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I'm using Vinner & Tall's definition from the early 1980s,
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but the concept image—you we have a concept definition of what something is,
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whether it be a triangle or a fraction.
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But then we also have the concept image.
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What do we as the individuals,
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that total cognitive structure that we associate with the concept?
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And so we want those things to align at some level,
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but there's a lot of things, different associations that go into it.
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So
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let me give you example of a fraction.
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The concept definition of a fraction,
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for students based on a third grade Common Core standard,
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is the fraction "m over n" is m copies or iterations,
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a quantity that can be counted,
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how many of a unit fraction 1 over n,
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a unit that can be measured, like how much.
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So with that in mind, we as teachers are challenged to provide students with
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experiences that align students' concept image with that concept definition.
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And that's often dependent
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on students' first experiences with fractions and the representations that
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are utilized.
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So the problem is, as I review many US curricula,
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what they see due to an emphasis on—I call a missing piece area models.
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Like, you have this area model that's been separated into eight pieces
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and seven of those are shaded,
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is a concept image more aligned with m pieces shaded out of n total pieces,
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which to me is the underlings of a ratio,
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but it fits more with that interpretation than seeing fractions as numbers.
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So
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those experiences, at least in the beginning,
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need to initially engage in activities with models that make
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the unit fraction explicit. For example,
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given a length of one unit and a length of part of that unit that represents
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a particular unit fraction, how can we describe the length that lies somewhere
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between 0 and 1 while engaging them in the activity of—if we're going
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to talk about iterating, then are we in engaging them in iterating or making copies?
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So I think if the iterated length is the same length of the whole after three
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iterations,
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we have a name for that, and that's one-third.
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And the reason why it's one-third is because three copies of that unit length
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of one-third is the same length of the whole.
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So thinking about the language that I want that to align with the experience,
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we really want them to see and develop a concept image of fraction
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as a measurement that involves a unit,
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and it's derived from a different unit,
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which is the whole. So we've got that units of units perspective.
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And so if that's the goal and the concept definition is applied to that idea of
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m over n as being iterations of that unit fraction,
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that we want the experiences students have have to align with that.
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You know, something that you said,
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I wanna just hold on it for a second.
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I really appreciated that part of what we're trying to do when we're in
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the "see it" phase is think about what explicit features we really want
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to be thinking about. Whether it's a visual model or a manipulative,
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there's a need for like a close analysis of:
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"What does this model
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help them see?"
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Yes.
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I think that calls for us as teachers to really interrogate the models that
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we are putting in front of kids and then look back and say,
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"To what extent do these actually reflect what I want my kids
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to understand?" Does that make sense?
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Ab, absolutely. You know, I had teacher ~ experience where she asked us
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to to come in her classroom, and we did some interviews and a lot
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of students were saying that the fractions five-,
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six-, and seven-eighths were equal.
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And she was shocked because she'd never told them anything to think that five-,
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six-, and seven-eighths were equal.
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But then we said, "OK, so let's look at your instruction,
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and let's talk about it. And so one of the activities that
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she did with students—it's a great activity,
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and and I I wanted to honor the teacher in that idea—but we started
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to think about, "Well, what could lead to this?" And so,
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what she did is she had a fraction tower that had a sheet that had halves,
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thirds, fourths, fifths, sixths,
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eighths, tenths, twelfths. And so when she was doing fractions,
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she was having students for like the fraction three-fourths put a jelly belly
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on each of the one-fourths. So,
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with five-, six-, and seven-eighths,
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guess what? There was one jelly belly missing.
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Yep.
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So we're not attending to the size of the unit.
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So there was a misalignment between the understanding that she desired
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and the activity that she was engaging students in.
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And so just by having that conversation,
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we were able to shift the focus a little bit.
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And again, this teacher was doing the best they could.
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I, I'm not faulting the teacher at all,
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but just really being reflective on,
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as you said, those representations or models that we use and is there alignment with
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the learning goals that we have.
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Gotcha. Yeah. That's a great example.
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So I would love to shift and talk about some of your ideas about language.
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So this is the "say it" part of the book.
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OK.
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Let's start with a big broad question.
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You talked about the role that language plays.
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You said that it's the stuff that binds.
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I wonder if you could talk a little bit more about what you mean
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by language playing a role of binding,
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especially when it comes to elementary math classrooms.
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OK.
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So there's two main reasons that I think it's important.
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And, you know, again, using your analogy,
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you know, I feel like it's underutilized,
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and it is the egg that binds everything together.
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And obviously, if things aren't binding together,
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let's say you're making cookies,
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and you don't put an egg in it,
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it's not going to bind together.
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It's just going to be this chunk,
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you know, you're essentially making,
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probably granola or something.
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I don't know.
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But we have models but no way to bind them together.
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And we often wonder why students easily use base ten blocks to solve
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an addition problem, but then struggle to connect those actions
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with what they do symbolically.
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I had this conversation with a group of teachers yesterday.
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Think about the problems—I'll give the,
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you know, 34 plus 58, and then just for semantics,
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just say 0.34 plus 0.58. And so part of that binding language together
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in those ideas, many students—again,
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my adult learners—see those as two different ideas.
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There's this conceptual divide between addition with whole numbers,
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addition with the decimals, and addition with fractions,
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for example.
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And my argument is: it is
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the same thing. All we are doing is combining quantities of the same size a unit.
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And so we can use that common language to condense what we're talking about.
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My students throw things at me after a while 'cause I have tons of dad jokes,
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but again, this goes back to my grandmother,
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but: "How do you catch a, a unique rabbit?" "Unique up on it." OK,
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that was the line. "And how do you catch a tame—
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That was an impressive dad joke,
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by the way.
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I know, I know. ~ And you know,
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"How do you catch a tame rabbit?" You know,
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"It's the tame way."
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And so my argument is adding whole numbers,
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adding decimals, and adding fractions,
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it's "the tame" way. We're just combining quantities of the same size of unit.
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And so we want to condense that language.
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And part of that language is: There are a couple terms I use
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are "equal exchanges," which is,
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to me, just equivalent expressions,
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but it, it permeates throughout.
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It's an easy way to think about anybody that's exchanged money
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can, you know, [exchange] 10 ones for 1 ten-dollar bill,
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you know, we, we can associate with that.
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And then there's also the idea later on of what I call "analogous problems."
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And that is: problems that involve the same quantities,
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the same actions, but different sizes of units.
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And again, they permeate through all four operations.
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So I, I see that throughout.
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And so, we talk a lot about that communication about discourse,
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like,
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again, fractions. You know, students can tell me what the numerator
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and denominator of a fraction is,
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but what does it mean? Well, numbers are just stressing this idea that numbers
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are just a multiplicative tethering of quantities and sizes of units.
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And once we started talking about that,
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it just connected for these adult learners who are like,
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"Oh, my gosh, why didn't nobody ever tell me that?" You know,
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and so I suppose even with fractions—I,
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I know we talked about this a little bit—but,
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like, a fraction like five-eighths.
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Just to
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put emphasis on the size of the unit,
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we talk about five groups of one-eighth.
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We really want to talk about the quantity,
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which is how many, and the size of the unit,
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which is how much.
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And so that is just a couple examples of where I use that language.
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And so, for me, you know, in summary,
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it it's a connector between critical ideas.
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So I want to back up because I think the addition problem that you,
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you were talking about with that group of teachers was 34 plus 58.
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Yeah.
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And then you were also talking about,
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as you said, like, 0.34 plus 0.58.
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Uh-huh.
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And in my head, I started to think about:
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Well, what we're really talking about in the first case is:
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3 tens and 4 ones, 5 tens and 8 ones.
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And so we're [combining]
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the 3 tens and the 5 tens and the 4 ones and the 8 ones.
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And then to follow the process through,
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it's three-tenths and five-tenths,
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and it's four-hundredths and eight-hundredths.
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And really, the process we're going through is the same.
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And there's an equal exchange that happens along the way in both of those processes.
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Yeah.
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But the processes themselves stay the same.
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Yes. Yeah, like with 34 plus 58,
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you're adding 4 ones and 8 ones together.
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So you get 12 ones. So you do the ex—equal exchange:
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10 of those ones is 1 ten. And then with 30 point—I would,
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I say 0.34 plus 0.58—
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—you got twelve-hundredths. And so equal exchanging 10 of those hundredths
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for one-tenth. And so, you know,
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the equal exchange is happening,
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but with different sizes of units.
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Yeah, and I think you made a reference to eighths,
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and I wanna just name this for listeners in case they missed it.
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The difference between describing something as five-eighths and—how
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did you say it?
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Five groups of one-eighths, one-eighth,
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I really want to put an emphasis on:
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We're working with children. They are [having] a hard time distinguishing—you
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know, that's another thing with the language,
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you know, the, the, the suffix t-h-s—or the morpheme t-h-s—is you know,
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it's easy with obviously halves,
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and thirds is a little problematic.
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But fourths, fifth, sixth, and sevenths,
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we can associate that with the partitioning action—
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you know, fourths is partitioning that unit,
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whatever that unit is, into four equal pieces;
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fifths, five equal pieces, ~ or equal lengths.
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So it's embedded in there. I just don't think we rely enough
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on it because students—many times,
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I, I see this happening again and again—students don't hear
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the difference between, let's see,
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5 eights, as in five groups of 8,
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and five-eighths, five groups of one-eighth.
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And so they don't make that distinction many times,
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but also the emphasis is on: When you say "five-eighths," you're
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not really—many times we're finding students aren't attending to the size
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of the unit. But if you describe it as five groups of one-eighth,
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we've got our unit fraction, which again,
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everything else builds from that unit fraction.
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So it's much more explicit and meaningful.
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And then when we talk about addition of fractions,
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we need a common denominator because again,
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addition is about combining quantities of the same size of unit.
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So,
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five groups of one-eighth and four groups of one-eighth is nine groups
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of one-eighth. You know, it just kind of fits.
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I really appreciate this, Patrick,
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because this seems so simple a move on the part of the teacher,
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in the sense that I can control the language choices I'm using when
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I'm describing quantities or the ways that we're acting on quantities.
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And that can have a really profound effect on kids' understanding.
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Absolutely.
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I think what I'm trying to say is it's very much within my control as a teacher
21:18
to make those kinds of shifts,
21:19
but the impact on the other side could really be profound if I can build
21:25
the habit of using language the ways that you're describing.
21:28
Yes. Yes indeed. And like you said,
21:31
I like the idea of not, this is not a monumental shift.
21:34
I mean, you may start to see mathematics differently as you read the book
21:38
and some of the ideas—but the shift,
21:40
these are easy shifts to make.
21:42
This just more of a little bit of a nudge towards adjustment in language.
21:46
Before we move from language and talk about "symbolize it," I just want
21:50
to go back to the phrase "equal exchange." And I'm really taken with this.
21:56
I heard you on another podcast talk about equal exchange.
21:59
And I think it also is connected to this idea that you have some pretty meaningful
22:06
feedback around the kinds of language that we might use with kids.
22:10
But I wonder if you'd talk about equal exchange and then also how
22:13
you think about the kind of language we use
22:17
in math classrooms.
22:18
I'm gonna put you on the spot here for a second,
22:19
Michael. I if I have you write down the symbol string 3,
22:23
6, 0, 0, 0, 0,
22:26
what are you gonna say back to me?
22:29
So, I want
22:29
to clarify to make sure I've got it right.
22:31
If I wrote down 3, 6,
22:34
and then four 0s afterwards—
22:37
Yes. You got it. You got it.
22:39
—what will I say?
22:40
Yeah.
22:40
What's occurring to me is: It will take me a moment to think about the magnitude,
22:45
and I would probably place a comma to help me clarify ~
22:51
how big this number is. So after doing that,
22:55
I would say three hundred and sixty thousand.
22:59
Dead on. So we can all agree that's 360,000,
23:02
right?
23:02
We're good with that. OK. What is it 36 of?
23:06
That same amount would be 36 what?
23:09
It's 36 ten thousands.
23:12
OK. Or 3.6 what?
23:15
Oh, it's 3.6 one hundred thousands.
23:19
Exactly. And so, again, this is kind of to show the trajectory of where this could
23:24
go. You have just learned scientific notation.
23:27
You know, this idea that you've taken 360 and you said "thousand,"
23:33
so that's 360 times 10 to the third,
23:35
which is 36 times 10 to the fourth, [or] 3.6 times 10 to the fifth.
23:42
So as the quantity decreased by a factor of 10,
23:47
the size of the unit increased by a factor of 10.
23:50
And so this is an idea, again,
23:52
of an equal exchange. And so we're,
23:54
when we're dealing with 10 units of ten—so,
23:56
for example, 1 ten for 10 ones—I have 10 ones and I'm giving
24:02
you 1 ten. So the quantity decreased by a factor of 10,
24:06
but the size of the unit increased by a factor of 10.
24:10
So there's that reciprocal relationship that happens as we do equal exchanges.
24:17
And so that happens throughout mathematics.
24:19
And what's interesting is I was just playing Monopoly one day and I thought,
24:22
"Oh, my gosh, it's there. It's always been there." You know,
24:25
where you give the banker 2 ten-dollar bills and they give you 1 twenty-dollar bill.
24:31
That back-and-forth relationship,
24:33
that reciprocal relationship between the quantity and size of unit
24:36
has always been there, which is why they're equal exchanges because
24:39
the product of that is always going to be one-tenth times 10 is 1,
24:44
one-half times 2 is, is 1. It's always been there.
24:46
And so it bridges all the way up through mathematics,
24:49
starting with elementary school.
24:51
You know, what's cool about this is:
24:52
I think initially when I read the term,
24:55
what I was thinking about was fairly procedural,
24:58
which is we use language like "carry the 1" or "borrow from the
25:02
6," and that obscures place value.
25:05
And an equal exchange I think makes what's happening clearer in
25:10
the form of language. But what I'm really taking away from this now is that part
25:15
of the
25:17
outcome is that if you engaged in the kind of tasks or questions that
25:21
you were describing, you also build a much clearer sense of what "equal" means
25:26
mathematically, which is a huge benefit for children,
25:30
particularly as they're moving through elementary school and starting
25:34
to move into secondary math. And, and so,
25:37
it's a bit of a two-for-one in the sense that the language clarifies the arithmetic,
25:42
but it also sets up this understanding of
25:44
equalness, which really, really pays dividends in secondary.
25:48
Yeah, 'cause we're we're going to talk about equivalent expressions,
25:51
numerical, algebraic, equivalent equations,
25:53
equivalent systems. So there's all this equivalence that happens.
25:56
I feel—to me this is one of the binding elements.
25:59
Equivalence is one of those binding concepts that under,
26:01
underpins mathematics.
26:02
So I feel like we probably need to move to "symbolize it." And part
26:07
of what really struck me in the phrase "symbolize it" was "it," meaning you're
26:12
symbolizing something. It could,
26:13
I suppose, be an action, a relationship,
26:16
a visual model, maybe an arrangement of physical materials,
26:20
even a context. But that symbolizing to me felt like an act of translation.
26:26
Am I understanding what you're getting at properly?
26:29
Or is there a different interpretation that I should be thinking about,
26:32
Patrick?
26:33
You know, there's a lot of, it's—symbolizing [is] a very broad term,
26:36
and you know, just by you asking that question has got me reflecting on it a lot.
26:40
But I think about you know, we write a number,
26:42
like I said earlier, like 74. We have to share that information.
26:46
We use the Hindu Arabic description of it to write those numerals down
26:50
and we talk about it: 7 tens and 4 ones,
26:52
if that's the context. And so there's always going to be
26:55
a translation involved because,
26:57
again, just me communicating what I just said to you is a translation
27:01
of what was written symbolically.
27:03
You know, I had to interpret that.
27:04
And so, to me, symbolizing is always about interpretation.
27:08
Just last week, I was interviewing a group of fifth graders
27:12
on the multiplication problem 3 times 7.
27:14
And what was fascinating to me—again,
27:17
writing those symbols that we all kind of had shared understanding
27:20
and what that means. But what they did to communicate their understanding
27:24
to me was so radical, so many different possibilities.
27:27
You know, one student drew 3 groups of 7 dots and counted them individually.
27:32
Another student wrote 14 and 14,
27:34
and then subtracted 7. And so I had to ask [about it]. [They said,] "Well,
27:38
that was 2 times 7 and 2 times 7.
27:40
And then I subtracted one group of 7." And that was just a couple ways that were
27:44
done. But there were all these ways that when I just flashed this commonly
27:48
understood symbol string in front of students,
27:50
it cued something—like I said earlier,
27:52
that concept image—it just spewed things out of each mind,
27:56
and they were radically different depending on the student.
27:58
And so that aspect
28:00
is the dynamic part of symbolizing to me is that students can come at
28:05
it from so many different perspectives.
28:06
And the beauty of teaching is,
28:07
first of all, I've got to communicate clearly what 7 and 4 actually represents when
28:13
it's a string, and talk about the meaning of that and why.
28:16
But also, as a teacher, when I let my students loose and they start interpreting
28:21
the symbols I use, I have to navigate the terrain of trying to put
28:25
all those pieces together. To me,
28:27
that's the beauty of teaching.
28:28
I love taking
28:29
what my students say and try to mold something beautiful with it.
28:32
That's when it's fun for me.
28:34
I 100% agree with you. It's interesting,
28:36
the way you describe that as a symbol string (chuckles)—7 times 3 is
28:40
a string of symbols put together—and understanding what the meaning of that is,
28:46
that's a special sauce in—
28:48
—in teaching and also in learning.
28:49
It can also be the thing that scares the heck out of folks because—
28:53
—there's maybe a loss of control or creating space for kids
28:57
to share their thinking. And I think you put your finger on something that
29:01
can either be incredibly exciting
29:03
or nerve-racking for a lot of folks.
29:06
I think a lot about this where we pose a symbol string in front of students:
29:09
"3 X 7" or whatever, "3 dot 7." And we kind of impose that meaning on them,
29:14
but we we don't give them a language,
29:16
and just say, "OK, this is multiplication." And depending on which country
29:20
you come from, this is interpreted differently.
29:21
But the first numeral in a multiplication problem I interpret as a quantity.
29:25
And the second is the 7—like,
29:28
in 3 times 7—is a multiplicative unit.
29:31
So it is a unit. It's like, "We have 3 sevens,"
29:35
you know; we commonly use the term.
29:36
So there's a lot that goes into that that when I'm teaching that,
29:40
I want to make clear how I want the students to interpret that symbol string,
29:44
but the way they communicate their understanding of that symbol string
29:48
can be very diverse depending on their own understandings.
29:51
I think part of what jumps out for me too is by offering the chance
29:54
to communicate their understanding of that symbol string,
29:57
it really is a formative opportunity for you because you get a sense
30:01
of what's going on in the room,
30:03
like how are they thinking? And then doing that allows you to connect where
30:08
you want them to get to with where they are right now.
30:10
Whereas if you forego the opportunity to offer them the opportunity to make sense
30:15
of it, you don't know if you're tethering to something they understand
30:19
and working from there,
30:21
or if it just absolutely went over their heads because it wasn't
30:25
in any way connected to their understanding.
30:27
And so I think there's a sense of maybe loss of control that comes along with giving
30:32
kids that opportunity, but it also pays dividends because once you know
30:36
how they're thinking, you can game-plan much more effectively as a teacher.
30:40
The risk is giving up control,
30:42
but the reward is what you gain because now you understand your kids better
30:46
and you can make moves more effectively.
30:48
Yeah, I tell people all the time:
30:50
I've been in education now for 34 years,
30:52
and for the first 20 I felt like I was pretty stagnant in my teaching.
30:56
I'd learn a few things, get a few things better,
30:58
but I felt like my teaching improved exponentially when I just started listening
31:02
to my students. And I'm like, ~ "Oh, they don't really understand that
31:06
the way I thought they would. And OK,
31:07
so now I can modify and I can latch on to what they're telling me as opposed
31:11
to just me thinking, "This is what's going to be best."
31:14
You know, that's I think a really great segue because we've been dancing around
31:18
the edges of my next question,
31:20
which is: How do you think about the role of an elementary teacher in symbolizing?
31:25
Like, is their role to show or to tell?
31:28
Is their role to pull students' ideas and then offer a symbolic model for that?
31:35
Are those two things in relationship with one another?
31:39
Or is there another way to think about the teacher's job when it comes to
31:43
symbolizing either the things that kids are saying or the things they're describing
31:47
or the things that they're seeing?
31:49
It's twofold. It's using precise language to communicate their understanding
31:52
of ideas. You know, a lot of times there's hidden mathematics between what we're
31:57
doing. As I've said with the number 74:
31:59
What base am I operating in? But that's kind of hidden.
32:03
And then it it really is this is the secret sauce of teaching.
32:07
When you set students loose and you give them a problem—I know we've talked
32:10
a lot about contextualized problems;
32:12
I'm a believer in contextualized problems,
32:14
but—that's where you kind of start to see
32:17
the different ways that our students are understanding ideas.
32:19
You know, that student, for example,
32:21
I said earlier, who drew out an array of 7 by 3 to communicate
32:25
her understanding multiplication,
32:26
that's great that she could create the represent the model,
32:28
but then she counted by ones. And so that tells me we're probably not ready
32:34
to be thinking about multiplication.
32:35
We need to start thinking about grouping at a,
32:38
at a, a little bit higher level.
32:39
So we need to start building that understanding.
32:42
So it really ~ provides teachers a lens
32:46
with scaffolding their instruction.
32:47
And without that, we're just kinda throwing darts hoping something hits.
32:51
I wanna touch on one other piece before we close,
32:54
Patrick. Another piece that I really liked about the book is:
32:57
You were really taking pains to help teachers see that there's connections between
33:02
what's happening at different grade levels.
33:04
And an equal exchange is an example of that.
33:07
One of the things that I've heard you say is:
33:09
The action remains the same; it's the size of the unit that changes.
33:12
And I wonder if you could be a little bit explicit about,
33:15
like, where are some examples of that in elementary mathematics that
33:20
would help someone who is a lower-grade kindergarten,
33:23
first, second grade teacher recognize how what they're doing continues;
33:28
it's just continuing with a different unit size.
33:31
~ Or vice versa, thinking about,
33:33
like, if I'm a third or fourth grade teacher,
33:36
where's the connection to what's happening in younger grades?
33:39
Great question. I love it. Thank you for asking it.
33:42
Let's start out with just 4 plus 8.
33:44
You know, there's something monumental that happens at 10.
33:47
That's a new unit. So again, even though it's easy to count 8,
33:52
9, 10, 11, 12, there's that idea of an equal exchange that happens right at 10.
33:57
That's the first one. So I love the base ten blocks a little bit later,
34:00
but the 10-frame. [The] 10-frame is such a neat tool to start to develop that idea
34:05
of making a ten.
34:06
So we've got, you know, kindergarten,
34:08
first grade, we're in that. And then we go to two digits.
34:10
And again, problems like—I, I've been using,
34:13
I think 34 plus 58 quite a few times here,
34:16
so I'll stick with it. But 34 plus 58,
34:19
340 plus 580, 3.4 plus 5.8,
34:24
we could go on and on and on, but those are what I call analogous problems.
34:28
And so that's a connector, you know,
34:30
because again, addition about combining quantities the same size a unit.
34:34
So the goal is to be consistent with that language all the way through.
34:37
And so we can make those connections between what we do with single-digit addition
34:42
to what we do with 2-digit addition to moving into decimals and fractions
34:46
all the way up the frame.
34:47
And the same is true with multiplication.
34:49
You know, 6 times 8 is 48. And we don't really think about it [in this way],
34:52
but it's 48 ones. That's the unit that's associated with it.
34:55
But we could also think of that as 4 tens and 8 ones.
34:58
But then we have 6 times 80, 6 times eight-tenths,
35:00
6 times 8,000, 6 times eight-fifths.
35:04
We still have 48 of something.
35:06
Whether it's 48 tens, 48 hundreds,
35:09
48 thousandths, we still have 48 of something.
35:13
Forty-eight fifths, problems with fifths is the regrouping happens at five,
35:17
not at ten. So our equal exchange involves five-fifths for a whole unit.
35:22
So that's a distinction, but you're still taking six-eighth of a different unit.
35:27
So your result's gonna be 48 of something.
35:29
And that's the same with division.
35:31
You know, like I I think about,
35:33
for example, 24 divided by 2 and 8 divided by two-thirds.
35:37
When you first look at those problems,
35:39
you think, "There is no connection whatsoever." But when you put those
35:42
in context and students start modeling,
35:44
but let's say you have 24 peanut butter and jelly sandwiches and you're going
35:48
to give each person 2 peanut butter and jelly sandwiches.
35:50
Well, you're going to go through that whole process and you're going
35:53
to create 12 groups of 2. Well,
35:55
let's say it's 8 peanut butter and jelly sandwiches and you're going
35:58
to give two-thirds of a peanut butter-jelly sandwich to each person.
36:01
How many people can you give
36:03
two-thirds of a peanut butter-jelly sandwich to?
36:05
Again, when you create that 8 and you create the partitioning to make thirds,
36:09
you have done an equal exchange.
36:11
You've changed 8 ones to 24 thirds.
36:13
So now, how many groups of two-thirds are in twenty-four thirds?
36:17
Well, it's going to be 12. 12 of what?
36:19
12 groups of two-thirds. So again,
36:23
it's the same activities and the same quantities,
36:27
but on a different size of unit.
36:29
So it goes back to my silly joke about
36:32
the tame way again. But it it is.
36:33
There there's a structure that fits throughout mathematics
36:37
and it's not compartmentalized.
36:38
It fits together. And the key is to help our students understand where those places
36:43
are.
36:44
What you make me think is how much of elementary mathematics is about units
36:48
and unit exchange and recognizing equivalence between quantities of units
36:54
and other units. It's pretty powerful.
36:57
And I think what I love about it is:
36:58
It does help you see that there's a long journey that's going on,
37:03
but the themes of that journey continue to come back and show themselves
37:08
in slightly different ways.
37:10
I 100% agree. I think it's so amazing to me that to me units are a huge deal,
37:15
but with addition they're hidden behind place value.
37:19
With fractions they're in the denominator written as a numeral many times,
37:22
which students have already associated as being a quantity because everything
37:26
they've done with whole numbers has involved the numeral representing a quantity.
37:30
And the only time in mathematics [...] where a numeral represents a size
37:35
of unit is in, when it's a fraction.
37:37
And so that is a huge disruption
37:39
for me, and so, that needs to be,
37:41
you know—total, completely agree.
37:43
You know, I think this is actually a pretty good pivot point because we talked
37:48
a lot about, kind of, the big ideas in See It,
37:52
Say It, Symbolize It, I wonder if you could talk about how someone who's listening
37:57
to this and wants to keep learning,
37:59
~ how might they continue to dig in either by contacting you or
38:04
by engaging with content that you've put together?
38:07
See it, say it, symbolize it. It's on Amazon,
38:09
and it's also on Solution Tree.
38:11
And my website, it's still a work in progress,
38:15
but it's called: unleashingmathematicalbrilliance.com.
38:15
And I'm continuing to put ideas up there.
38:19
I have some graphic organizers related to what I call superpower understandings
38:24
on there, as well as a few other podcasts I've done and and a few blog entries
38:28
as well that I'm continuing to build on.
38:30
But just trying to build more resources to support teachers because what
38:34
I hear often is:
38:36
"I love the ideas, but how do I fit this into my curricula?" And
38:40
so many times there's nothing there.
38:42
Yeah. And I will say for listeners,
38:44
we will add links to all of the resources that Patrick talked about.
38:48
I could keep going, Patrick, but I think this is probably a really good place
38:51
to stop. I just want to thank you again for joining us and
38:55
for a lovely conversation. ~ Thanks so much for being a part of the podcast.
38:59
It was my pleasure. Like I said,
39:01
a huge shout-out to all the work that you guys do.
39:03
I fell into that work 25 years ago of all the work happening at
39:06
the Math Learning Center. And I've just been a huge fan of all the work that
39:10
you guys do there.
39:11
Ah, thank you so much.
39:13
And thank you, our audience, for listening.
39:16
We'd love to hear what you all think.
39:18
Is there a Grandma Edna in your life who helped you understand new math concepts?
39:22
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39:25
Say It, Symbolize It in your teaching practice?
39:28
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39:30
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