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ROUNDING UP: How You Say It Matters: Teacher Language Choices That Support Number Sense with Dr. James Brickwedde Carry the 1. Add a 0. Cross multiply. All of these are phrases that educators heard when they were growing up. This language is so ingrained that many educators use it without even thinking. But what's the long-term impact of language like this on the development of our students' number sense? Today, we're talking with Dr. James Brickwedde about the impact of language and the ways educators can use it to cultivate their students' number sense. BIOGRAPHIES James Brickwedde is the director of the Project for Elementary Mathematics. He served on the faculty of Hamline University's School of Education & Leadership from 2011–2021, supporting teacher candidates in their content and pedagogy coursework in elementary mathematics. RESOURCES The Project for Elementary Mathematics TRANSCRIPT Mike Wallus: Carry the 1, add a 0, cross multiply. All of these are phrases that educators heard when they were growing up. This language is so ingrained, we often use it without even thinking. But what's the long-term impact of language like this on our students' number sense? Today we're talking with Dr. James Brickwedde about the impact of language and the ways educators can use it to cultivate their students' number sense. Welcome to the podcast, James. I'm excited to be talking with you today. James Brickwedde: Glad to be here. Mike: Well, I want to start with something that you said as we were preparing for this podcast. You described how an educator's language can play a critical role in helping students think in value rather than digits. And I'm wondering if you can start by explaining what you mean when you say that. James: Well, thinking first of primary students—so, kindergarten, second grade, that age bracket—kindergartners, in particular, come to school thinking that numbers are just piles of ones. They're trying to figure out the standard order. They're trying to figure out cardinality. There are a lot of those initial counting principles that lead to strong number sense that they are trying to integrate neurologically. And so, one of the goals of kindergarten, first grade, and above is to build the solid quantity sense—number sense—of how one number is relative to the next number in terms of its size, magnitude, et cetera. And then as you get beyond 10 and you start dealing with the place value components that are inherent behind our multidigit numbers, it's important for teachers to really think carefully of the language that they're using so that, neurologically, students are connecting the value that goes with the quantities that they're after. So, helping the brain to understand that 23 can be thought of not only as that pile of ones, but I can decompose it into a pile of 20 ones and three ones, and eventually that 20 can be organized into two groups of 10. And so, using manipulatives, tracking your language so that when somebody asks, "How do I write 23?" it's not a 2 and a 3 that you put together, which is what a lot of young children think is happening. But rather, they realize that there's the 20 and the 3. Mike: So, you're making me think about the words in the number sequence that we use to describe quantities. And I wonder about the types of tasks or the language that can help children build a meaningful understanding of whole numbers, like say, 11 or 23. James: The English language is not as kind to our learners [laughs] as other languages around the world are when it comes to multidigit numbers. We have in English 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And when we get beyond 10, we have this unique word called "eleven" and another unique word called "twelve." And so, they really are words capturing collections of ones really then capturing any sort of tens and ones relationship. There's been a lot of wonderful documentation around the Chinese-based languages. So, that would be Chinese, Japanese, Korean, Vietnamese, Hmong follows the similar language patterns, where when they get after 10, it literally translates as "10, 1," "10, 2." When they get to 20, it's "2, 10"—"2, 10, 1," "2, 10, 2." And so, the place value language is inherent in the words that they are saying to describe the quantities. The teen numbers, when you get to 13, a lot of young children try to write 13 as "3, 1" because they're trying to follow the language patterns of other numbers where you start left to right. And so, they're bringing meaning to something, which of course is not the social convention. So, the teens are all screwed up in terms of English. Spanish does begin to do some regularizing when they get to 16 because of the name "diez y seis," so "ten, six." But prior to that you have, again, sort of more unique names that either don't follow the order of how you write the number or they're unique like 11 and 12 is. Somali is another interesting language in that—and I apologize to anybody who is fluent in that language because I'm hoping I'm going to articulate it correctly—I believe that there, when they get into the teens, it's "1 and 10," "2 and 10," is the literal translation. So, while it may not be the "10, 1" sort of order, it still is giving … the fact that there's ten-ness there as you go. So, for the classrooms that I have been in and out of—both [in] my own classroom years ago as well as the ones I still go in and out of now—I try to encourage teachers to tap the language assets that are among their students so that they can use them to think about the English numbers, the English language, that can help them wire that brain so that the various representations—the manipulatives, expanded notation cards or dice, the numbers that I write, how I break the numbers apart, say that 23 is equal to 20 plus 3—all of those models that you're using, and the language that you use to back it up with, is consistent so that, neurologically, those pathways are deeply organized. Piaget, in his learning theory, talks about young children—this is sort of the 10 years and younger—can only really think about one attribute at a time. So that if you start operating on multidigit numbers, and I'm using digitized language, I'm asking that kindergartner, first [grader], second grader to think of two things at the same time. I'm, say, moving a 1 while I also mean 10. What you find, therefore, is when I start scratching the surface of kids who were really procedural-bound, that they really are not reflecting on the values of how they've decomposed the numbers or are reconfiguring the numbers. They're just doing digit manipulation. They may be getting a correct answer, they may be very fast with it, but they've lost track of what values they're tracking. There's been a lot of research on kids' development of multidigit operations, and it's inherent in that research about students following—the students who are more fluid with it talk in values rather than in digits. And that's the piece that has always caught my attention as a teacher and helped transform how I talked with kids with it. And now as a professional development supporter of teachers, I'm trying to encourage them to incorporate in their practice. Mike: So, I want to hang on to this theme that we're starting to talk about. I'm thinking a lot about the very digit-based language that as a child I learned for adding and subtracting multidigit numbers. So, phrases like, "Carry the 1" or "Borrow something from the 6." Those were really commonplace. And in many ways, they were tied to this standard algorithm, where a number was stacked on top of another number. And they really obscured the meaning of addition and subtraction. I wonder if we can walk through what it might sound like or what other models might draw out some of the value-based language that we want to model for kids and also that we want kids to eventually adopt when they're operating on numbers. James: A task that I give adults, whether they are parents that I'm out doing a family math night with or my teacher candidates that I have worked with, I have them just build 54 and 38, say, with base ten blocks. And then I say, "How would you quickly add them?" And invariably everybody grabs the tens before they move to the ones. Now your upbringing, my upbringing is the same and still in many classrooms: Students are directed only to start with the ones place. And if you get a new 10, you have to borrow and you have to do all of this exchange kinds of things. But the research shows when school gets out of the way [chuckles] and students and adults are operating on more of their natural number sense, people start with the larger and then move to the smaller. And this has been found around the world. This is not just unique to US classrooms that have been working this way. If, in the standard algorithms—which really grew out of accounting procedures that needed to save space in ledger books out of the 18th, 19th centuries—they are efficient, space-saving means to be able to accurately compute. But in today's world, technology takes over a lot of that bookkeeping type of thing. An analogy I like to make is, in today's world, Bob Cratchit out of [A] Christmas Carol, Charles Dickens's character, doesn't have a job because technology has taken over everything that he was in charge of. So, in order for Bob Cratchit to have a job, [laughs] he does need to know how to compute. But he really needs to think in values. So, what I try to encourage educators to loosen up their practice is to say, "If I'm adding 54 plus 38, so if you keep those two numbers in your mind, [chuckles] if I start with the ones and I add 4 and 8, I can get 12." There's no reason, if I'm working in a vertical format, to not put 12 fully under the line down below, particularly when kids are first learning how to add. But then language-wise, when they go to the tens place, they're adding 50 and 30 to get 80, and the 80 goes under the 12. Now, many teachers will know that's partial sums. That's not the standard algorithm. That is the standard algorithm. The difference between the shortcut of carrying digits is only a space-saving version of partial sums. Once you go to partial sums in a formatting piece, and you're having kids watch their language—and that's a phrase I use constantly in my classrooms—is, it's not a 5 and 3 that you are working with, it's a 50 and a 30. So when you move to the language of value, you allow kids to initially, at least, get well-grounded in the partial sums formatting of their work, the algebra of the connectivity property pops out, the number sense of how I am building the quantities, how I'm adding another 10 to the 80, and then the 2, all of that begins to more fully fall into place. There are some of the longitudinal studies that have come out that students who were using more of the partial sums approach for addition, their place value knowledge fell into place sooner than the students who only did the standard algorithm and used the digitized language. So, I don't mind if a student starts in the ones place, but I want them to watch their language. So, if they're going to put down a 2, they're not carrying a 1—because I'll challenge them on that—is "What did you do to the 12 to just isolate the 2? What's left?" "Oh, you have a 10 up there and the 10 plus the 50 plus the 30 gives me 90." So, the internal script that they are verbalizing is different than the internal digitized script that you and I and many students still learn today in classrooms around the country. So, that's where the language and the values and the number sense all begin to gel together. And when you get to subtraction, there's a whole other set of language things. So, when I taught first grade and a student would say, "Well, you can't take 8 from 4," if I still use that 54 and 38 numbers as a reference here, my challenge to them is, "Who said?" Now, my students are in Minnesota. So, Minnesota is at a cultural advantage of knowing what happens in wintertime when temperatures drop below 0. [laughs] And so, I usually have as a representation model in my room, a number line that's swept around the edges of the room, that started from negative 35 and went to 185. And so, there are kids who've been puzzling about those other numbers on the other side of 0. And so, somebody pops up and says, "Well, you'll get a negative number." "What do you mean?" And then they whip around and start pointing at that number line and being able to say, "Well, if you're at 4 and you count back 8, you'll be at negative 4." So, I am not expecting first graders to be able to master the idea of negative integers, but I want them to know the door is open. And there are some students in late first grade and certainly in second grade who start using partial differences where they begin to consciously use … the idea of negative integers. However, there [are] other students, given that same scenario, who think going into the negative numbers is too much of The Twilight Zone. [laughs] They'll say, "Well, I have 4 and I need 8. I don't have enough to take 8 from 4." And another phrase I ask them is, "Well, what are you short?" And that actually brings us back to the accounting reference point of sort of debit-credit language of, "I'm short 4." "Well, if you're short 4, we'll just write 'minus 4.'" But if they already have subtracted 30 from 50 and have 20, then the question becomes, "Where are you going to get that 4 from?" "Well, you have 20 cookies sitting on that plate there. I'm going to get that 4 out of the 20." So again, the language around some of these strategies in subtractions shifts kids to think with alternative strategies and algorithms compared to the American standard algorithm that predominates US education. Mike: I think what's interesting about what you just said too is you're making me think about an article. I believe it was "Rules That Expire." And what strikes me is that this whole notion that you can't take 8 away from 4 is actually a rule that expires once kids do begin to work in integers. And what you're suggesting about subtraction is, "Let's not do that. Let's use language to help them make meaning of, "Well, what if?" As a former Minnesotan, I can definitely validate that when it's 4 degrees outside and the temperature drops 8 degrees, kids can look at a thermometer and that context helps them understand. I suppose if you're a person listening to this in Southern California or Arizona, that might feel a little bit odd. But I would say that I have seen first graders do the same thing. James: And if you are more international travelers, as soon as, say, people in Southern California or southern Arizona step across into Mexico, everything is in Celsius. If those of us in the northern plains go into Canada, everything is in Celsius. And so, you see negative numbers sooner [laughs] than we do in Fahrenheit, but that's another story. Mike: This is a place where I want to talk a little bit about multiplication, particularly this idea of multiplying by 10. Because I personally learned a fairly procedural understanding of what it is to multiply by 10 or 100 or 1,000. And the la…
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