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Understanding the Role of Language in Math Classrooms with William Zahner · Rounding Up

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How can educators understand the way that language interacts with the
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mathematical concepts and skills their students are learning?
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And how can educators focus on the mathematics of a task without losing sight of
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its language demands as their planning for instruction?
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We'll examine these topics with our guest, bill Zaner,
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director of the Center for Research in Mathematics and Science Education at San
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Diego State University. Welcome to the podcast, bill.
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Thank you for joining us today.
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Oh, thanks. I'm glad to be here.
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So I'd like to start by asking you to address a few ideas that often surface in
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conversations around multilingual learners and mathematics.
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The first is the notion that math is universal and it's detached from
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language. What, if anything,
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is wrong with this idea and what impact might an idea like that have on the
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ways that we try to support multilingual learners?
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Yeah, thanks for that.
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That's a great question because I think we have a common sense and
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strongly held idea that math is math no matter where you are and who
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you are. And of course,
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the example that's always given is something like two plus two equals four,
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no matter who you are or where you are. And that is true,
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I guess on the sense that two plus two is four unless you're in base three or
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something.
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But that is not necessarily what mathematics in
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its fullness is.
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And when we think about what mathematics broadly is,
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mathematics is a way of thinking and a way of reasoning and a way
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of using various tools to make sense of the
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world or to engage with those tools on their own
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right.
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And oftentimes that is deeply embedded with language.
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Probably the most straightforward example is anytime I ask someone to
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justify or explain what they're thinking in mathematics,
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I'm immediately bringing in language into that case.
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And we all know the old funny examples where a kid is asked to show their
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thinking and they draw a diagram of themselves with a thought bubble on a math
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problem. And that's a really good case where I think a teacher can say, okay,
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clearly that was not what I had in mind when I said show your thinking.
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And instead,
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the demand or the request was for a student to show their reasoning or their
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thought process typically in words or in a combination of words and
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pictures and equations.
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And so there's where I see this idea that math is detached from
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language is something of a myth that there's actually a lot of
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mathematics.
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And the interesting part of mathematics is often deeply entwined with language.
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So that's my first response and thought about that.
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And if you look at our common core state standards for mathematics,
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especially those standards for mathematical practice,
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you see all sorts of connections to communication and to language
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interspersed throughout those standards.
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So create viable arguments,
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that's a language practice and even attend to precision,
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which most of us tend to think of as round appropriately.
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But when you actually read the standard itself,
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it's really about mathematical communication and definitions and using those
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definitions with precision. Again, that's an example,
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bringing it right back into the school mathematics domain where language and
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mathematics are somewhat inseparable from my perspective here.
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That's really helpful.
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So the second idea that I often hear is the best way to support multilingual
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learners is by focusing on facts or procedures.
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And that language comes later, for lack of a better way of saying it.
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And it seems like this is connected to that first notion,
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but I wanted to ask the question again,
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what if anything is wrong with this idea that a focus on facts or procedures
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with language coming after the fact,
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what impact do you suspect that that would have on the way that we support
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multilingual learners?
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So that's a great question too, because there's a grain of truth, right?
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Both of these questions have simultaneously a grain of truth and simultaneously
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a fundamental problem in them.
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So the grain of truth and an experience that I've heard from many folks who
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learned mathematics in a second language was that they felt more
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competent in mathematics than they did in say,
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a literature class where the only activity was engaging with texts
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or engaging with words because there was a connection to the
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numbers and to symbols that we're familiar.
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So on one level,
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I think that this idea of focusing on facts or procedures comes out of this
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observation that sometimes an emergent multilingual student feels most
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comfortable in that context, in that setting.
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But then the second part of the answer goes back to this first idea that really
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what we're trying to teach students in school mathematics now is not simply
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or only how to apply procedures to really big
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numbers or to know your timestables fast. I think we have a much more
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ambitious goal when it comes to teaching and learning mathematics.
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That includes explaining, justifying, modeling,
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using mathematics to analyze the world and so on.
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And so those practices are deeply tied with language and
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deeply tied with using communication.
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And so if we want to develop those, well,
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the best way to do that is to develop them to think about what are the
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scaffolds,
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what are the supports that we need to integrate into our lessons or into our
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designs to make that possible?
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And so that might be the takeaway there,
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is that if you simply look at mathematics as
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calculations, then this could be true.
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But I think our vision of mathematics is much broader than that,
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and that's where I see this potential.
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That's really clarifying.
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I think the way that you unpack that is if you view mathematics as simply a set
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of procedures or calculations maybe. But I would agree with you.
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What we want for students is actually so much more than that.
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One of the things that I heard you say when we were preparing for this interview
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is that at the elementary level,
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learning mathematics is a deeply social endeavor.
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Tell us a little bit about what you mean by that bill.
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Sure. So mathematics itself,
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maybe as a premise is a social activity.
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It's created by humans as a way of engaging with the world and a way of
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reasoning.
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So the learning of mathematics is also social
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in the sense that we're giving students an introduction to this way of
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engaging in the world using numbers and quantities and shapes
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in order to make sense of our environment.
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And when I think about learning mathematics,
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I think that we are not simply downloading knowledge and sticking it into our
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heads.
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And in the modern day where artificial intelligence and computers can
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do almost every calculation that we can imagine,
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although your AI may do it incorrectly just as a fair warning,
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but in the modern day,
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the actual answer is not what we're so focused on.
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It's actually the process and the reasoning and the
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modeling and justification of those choices. And so when I think
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about learning mathematics as learning to use these language tools,
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learning to use these ways of communication, how do we learn to communicate?
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We learn to communicate by engaging with other people,
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by engaging with the ideas and the minds and the feelings and so on
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of the folks around us, whether it's the teacher and the student,
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the student and the student, the whole class and the teacher.
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That's where I really see the power. And most of us who have learned,
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I think can attest to the fact that even when we're engaging with a text,
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really fundamentally we're engaging with something that was created by somebody
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else. So fundamentally,
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even when you're sitting by yourself doing a math word problem or doing
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calculations,
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someone has given that to you and you think that that's important enough to do,
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right? So from that stance,
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I see all of teaching and learning. Mathematics is social,
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and maybe one of our goals in mathematics classrooms is beyond
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memorizing the timestables, is learning to communicate with other people,
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learning to be in this activity with other folks.
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One of the things that strikes me about what you were saying, bill,
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is there's this kind of virtuous cycle, right?
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That by engaging with language and having the social aspect of it,
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you're actually also deepening the opportunity for students to make sense of the
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math.
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You're building the scaffolds that help kids communicate their ideas as
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opposed to removing or stripping out the language that's the context in some
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ways that helps them filter and make sense.
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You could either be in a vicious cycle,
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which comes from removing the language or a virtuous cycle.
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And it seems a little counterintuitive because I think people perceive language
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as the thing that is holding kids back as opposed to the thing that might
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actually help them move forward and make sense.
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And actually that's one of the really interesting pieces that we've looked at in
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my research and the broader research is this question of
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what makes mathematics linguistically complex
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is a complicated question.
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And so sometimes we think of things like looking at the word count as a way to
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say, if there are fewer words, it's less complex. And if there are more words,
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it's more complex. But that's not totally true.
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And similarly, if there's no context, it's easier or more accessible.
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And if there is a context, then it's less accessible.
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And I don't see these as binary choices.
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I see these as happening on a somewhat complicated
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terrain where we want to think about how do these words
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or these contexts add to student understanding
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or potentially impede?
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And that's where I think this social aspect of learning mathematics,
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as you described,
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it could be a virtuous cycle so that we can use language in order to
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engage in the process of learning language.
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Or the vicious cycle is you withhold all language and then
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get frustrated when students can't apply their mathematics.
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Maybe the most stereotypical answer, my kids can do this,
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but as soon as they get a word problem, they can't do it. And it's like, well,
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did you give them opportunities to learn how to do this?
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Or is this the first time? That would explain a lot.
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Well, it's an interesting question too,
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because I think what sits behind that in some ways is the idea that you're kind
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of going to reach a point,
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or students might reach a point where they're ready for word problems.
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And I think what we're really saying is it's actually through engaging with word
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problems that you build your proficiency,
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your skillset that actually allows you to become a stronger mathematician.
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Right? Exactly. And it's a daily practice, right?
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It's not something that you just hold off to the end of the unit,
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and then you have the word problems,
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but it's part of the process of learning and thinking about how you integrate
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and support that. That's the key question that I really wrestle with.
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Not trivial, but I think that's the key and the most important part of this.
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Well,
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I think that's actually a really good segue because I wanted to shift and talk
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about some of the concrete or productive ways that educators can support
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multilingual learners. And in preparing for this conversation,
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one of the things that I've heard you stress is this notion of a consistent
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context.
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So can you just talk a little bit more about what you mean by that and how
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educators can use that when they're looking at their lessons or when they're
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writing lessons or looking at the curriculum that they're using?
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Absolutely. In our past work,
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we engaged in some cycles of design research with
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teachers looking at their mathematics curriculum and
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opportunities to engage multilingual learners in communication and reasoning in
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the classroom.
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And one of the surprising things that we found just by looking
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at a couple of standard textbooks was a surprising number of
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contexts were introduced that are all related to the same
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concept.
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So the concept would be something like rate of change or ratio,
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and then the contexts,
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there would be a half dozen of them in the same section of the book. Now,
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this was, I should say at a secondary level,
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so not quite where most of the bridges work is happening,
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but I think it's an interesting lesson for us that we took away from this.
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Actually at the elementary level,
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Catherine Cheval has made the same observation.
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What we realized was that contexts are not good or bad by
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themselves. In fact,
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they can be highly supportive of student reasoning or they can get in the
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way, and it's how they are used and introduced.
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And so the other way we thought about this was when you introduce a
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context,
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you want to make sure that that context is one that you give sufficient time for
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the students to understand and to engage with that is relatable,
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that everyone has access to it,
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not something that's just completely unrelated to students' experiences.
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And then you can really leverage that relatable,
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understandable context for multiple problems and
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iterations and opportunities to go deeper and deeper.
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To give a concrete example of that,
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when we were looking at this ratio and rate of change,
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we went all the way back to one of the fundamental contexts that's been
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studied for a long time,
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which is motion and speed and distance and time. And that seemed like a
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really important topic because we know that that starts all the way back in
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elementary school and continues through college level physics and beyond.
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So it was a rich context.
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It was also something that was accessible in the sense that we could do things
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like act out story problems or reenact a race that's
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described in a story problem.
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And so the students themselves had access to the context in a deep way.
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And then last,
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that context was one that we could come back to again and again,
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so we could do variations on that context on that story.
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And I think there's lots of examples of materials out there that start off
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with a core context and build it out.
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And thinking of some of the bridges materials,
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even on the counting and the multiplication,
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I think there's stories of the insects and their legs and wings and counting and
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multiplying, and that's a really nice example of it's accessible.
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You can go find insects almost anywhere you are. Kids like it,
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they enjoy thinking about insects and other icky, creepy crawly things.
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And then you can take that and run with it in lots of different ways, right?
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Counting multiplication, division ratio, and so on.
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This last bit of our conversation has me thinking about what it might look like
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to plan a lesson for a class or a group of multilingual learners.
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And I know that it's important that I think about mathematical demands as well
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as the language demands of a given task.
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Can you unpack why it's important to set math and language development
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learning goals for a task or a set of tasks,
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and what are the opportunities that come along with that?
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If I'm thinking about both of those things during my planning?
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Yeah, that's a great question, and I want to mark the shift, right?
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We've gone from thinking about the demands to thinking about the goals and where
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we're going to go next.
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And so when I think about integrating mathematical goals,
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mathematical learning goals and language learning goals,
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I often go back to these ideas that we call like the practices or
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these standards that are about how you engage in mathematics.
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And then I think about linking those back to the content itself.
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And so there's kind of a two piece element to that.
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And so when we're setting our goals and lesson planning,
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at least here in the great state of California,
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sometimes we'll have these templates that have what standard are you addressing?
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What language standard are you addressing? What ELD standard are you addressing?
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What SEL standard are you addressing?
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And I've seen sometimes teachers approach that as a checkbox, right? Tick, tick,
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tick, tick, tick. But I see that as a missed opportunity.
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If you just look at this,
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you're plugging things in because as we started with talking about how learning
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mathematics is deeply social and integrated with language that we can
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integrate the mathematical goals and the language goals in a lesson.
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And I think really good materials should be suggesting that to the teacher,
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you shouldn't be doing this yourself every day from scratch.
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But I think really high quality materials will say,
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here's the mathematical goal and here's an associated language goal,
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whether it's productive or receptive functions of language.
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And here's how the language goal connects the mathematical goal.
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Now, just to get really concrete,
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if we're talking about an example of reasoning with ratios,
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so I was going back to that, then it might be generalized,
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the relationship between distance and time and that the ratio of distance
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and time gives you this quantity called speed,
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and that different combinations of distance and time can lead to the same speed.
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And so explain and justify and show using words,
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pictures, diagrams. So that would be a language goal,
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but it's also very much a mathematical goal.
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And I guess I see the mathematical content, the practices,
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and the language really braided together in these goals.
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And that I think is the ideal, and at least from our work,
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has been most powerful and productive for students.
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This is off script, but I'm going to ask it and you can pass if you want to.
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I wonder if you could just share a little bit about what the impact of those
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kind of practices that you described, have you seen what that impact looks like?
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Either for an educator who has made the step and is doing that integration or
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for students who are in a classroom where an educator is purposely thinking
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about that level of integration?
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Yeah, I can talk a little bit about that. In our research,
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we have tried to measure the effects of some of these efforts.
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It is a difficult thing to measure because it's not just a simple true false
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test question type of thing that you can give a multiple choice test for.
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But one of the ways that we've looked for the impact that these types of
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intentional designs is by looking at patterns of student
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participation in classroom discussions and seeing who is
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accessing the floor of the discussion and then
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looking at other results like giving an assessment,
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but deeper than looking at the outcome,
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the binary correct versus incorrect,
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also looking at the quality of the explanation that's provided.
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So how you justify an answer.
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Does the student provide a deeper or a more mathematically
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complete explanation?
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That is an area where I think more investigation is needed,
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and it's also very hard to vary systematically. So from a research perspective,
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you may not want to put this into the final version,
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but from a research perspective,
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it's very hard to fix and isolate these things because they are
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integrated,
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because language and mathematics are so deeply integrated
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that trying to fix everything and do this,
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what caused this water to taste like water? Was it the hydrogen or the oxygen?
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Well, you can't really pull those apart, right?
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The water molecule is hydrogen and oxygen together.
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I think that's a lovely analogy for what we were talking about with mathematical
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goals and language goals.
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That I think is really a helpful way to think about the extent to which they're
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intertwined one another.
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Yeah, I need to give full credit to Vygotsky, I think,
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who said that something might be Vygotsky. I'll need to check my notes.
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I think you're in good company if you're quoting Vygotsky. Before we close,
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I'd love to just ask you a bit about resources. I say this often on the podcast.
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We have 20 to 25 minutes to dig deeply into an idea,
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and I know people who are listening often think about, where do I go from here?
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Are there any particular resources that you would suggest for someone who wanted
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to continue learning about what it is to support multilingual learners in a math
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classroom?
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Sure. Happy to share that.
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So I think on the individual and collective level,
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so say a group of teachers,
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there's a beautiful book by Catherine Cheval and her colleagues about supporting
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multilingual learners and mathematics,
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and I really see that as a valuable resource.
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I've used that in reading groups with teachers and use that in book studies,
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and it's been very productive and powerful for us. Beyond that,
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of course, I think the NCTM provides a number of really useful resources,
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and there are articles, for example,
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in the mathematics teacher learning and teaching PK to 12,
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that could make for a really wonderful study or opportunity to engage more
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deeply. And then I would say on a broader perspective,
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I've worked with organizations like the English Learner Success Forum and
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others.
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We've done some case studies and little kind of classroom studies that are
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accessible on my website,
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so you can go to that. But there's also from that organization,
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some really valuable insights if you're looking at adopting new
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materials or evaluating things,
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that gives you a principled set of guidelines to follow.
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And I think that's really helpful for educators because we don't have to do
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this all on our own.
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This is not a reinvent the wheel at every single site kind of situation.
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And so I always encourage people to look for those resources. And of course,
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I will say that the MLC materials, the bridges and mathematics,
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I think have been really beautifully designed with a lot of
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these principles right behind them. So for example,
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if you look through the teacher's guides on the bridges and mathematics,
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those integrated math and language and practice goals are a part of the design.
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Well, I think that's a great place to stop. Thank you so much for joining us,
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bill. This has been insightful and it's really been a pleasure talking with you.
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Oh, well thank you. I appreciate it.
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And that's a wrap for season three of Rounding Up.
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I want to thank all of our guests and the MLC staff who make these podcasts
23:30
possible, as well as all of our listeners for tuning in.
23:33
Have a great summer and we'll be back in September for season four.
23:39
This podcast is brought to you by the Math Learning Center and the Meyer Math
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Foundation dedicated to inspiring and enabling all individuals to discover
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and develop their mathematical confidence and ability.