Nancy Anderson, Keep Calm and Press for Reasoning
ROUNDING UP: SEASON 1 | EPISODE 13
We often ask students to share their strategies. But what does it look like to uncover and highlight the reasoning that informs that strategy? Today on the podcast, we’ll talk with Nancy Anderson, a classroom teacher and professional learning developer about strategies to elicit the reasoning at the heart of a student’s thinking.
Mike Wallace: We often ask students to share their strategies. But what does it look like to uncover and highlight the reasoning that informs that strategy? Today on the podcast, we'll talk with Nancy Anderson, a classroom teacher and professional learning developer, about strategies to elicit the reasoning at the heart of the student's thinking.
Welcome to the podcast, Nancy. I am so excited to talk to you today.
Nancy Anderson: Thank you. Likewise, Mike.
Mike: I'd like to begin with a quote from your article, “Keep Calm and Press for Reasoning.” In it, you state: “Mathematical reasoning describes the process and tools that we use to determine which ideas are true and which are false.” And then you go on to say that “in the context of a class discussion, reasoning includes addressing the strategy’s most important ideas and highlighting how those ideas are related.” So, what I'm wondering is, can you talk a little bit about how eliciting a strategy and eliciting reasoning may or may not be different from one another?
Nancy: So, when we elicit a strategy, we're largely focused on what the student did to solve the problem. For example, what operations and equations they might have used, what were the steps, and even what tools they might have used. For example, might they have used concrete tools or a number line? Whereas eliciting reasoning focuses on the why behind what they did. Why did they choose a particular strategy or equation? What was it in the problem that signaled that particular equation or that particular operation made sense? And if the strategy included several steps, what told them to go from one step to the next? How did they know that? And then similarly for the tools, what is it in the problem that suggested to them a number line might be an effective strategy to use? And lastly, eliciting reasoning sort of focuses on putting all those different pieces together so that you talk about those different elements and the rationale behind them in such a way that the people listening are convinced that the strategy is sound.
Mike: That's actually really helpful. I found myself thinking about two scenarios that used to play out when I was teaching first grade. One was I had a group of children who were really engaging with the number line to help them think about difference unknown problems. And what it's making me think is, the focus of the conversation wasn't necessarily that they used the number line. And it's like, “Why did this particular jump that you're articulating via number line—what is it about the number line that helped you model this big idea or can help make this idea clearer for the other students in the class?”
Nancy: Exactly, yes. So, when I think about reasoning, I think about different pieces coming together to form a cohesive explanation that also serves as a bridge to using a particular strategy for one particular problem, [and] as a tool for solving something similar in the future.
Mike: So, I have a follow-up question. When teachers are pressing students for their reasoning, what counts as reasoning? What should teachers be listening for?
Nancy: Broadly, mathematical reasoning describes the processes and tools that we use to determine which ideas are true and which are false. Because mathematics is based upon logic and reasoning—not a matter of who says it or how loudly they say it or how convincingly they say it—but rather, what are the mathematical truths that undergird what they're saying? That's sort of a broad definition of mathematical reasoning, which I think certainly has its merits. But then I think about the work of teaching, particularly at the elementary level. I think it's helpful to get much more specific. So, when we think about elementary arithmetic, reasoning really focuses on connecting computational strategies to the operations and the principles that lie underneath. So, in the context of a class discussion, when we have a student explain their reasoning, we're really trying to highlight a particular strategy's most important ideas and how those ideas are related, but in such a way that others can listen and say, “Oh, I get it. If I were to try the problem again, I do believe that's going to lead to the correct answer.” Or if it was this problem, which is similar, “I think I can see how it might make sense for me to use this approach here with these slight adjustments.” So, do you want to take an example?
Mike: Yeah, I'd love to.
Nancy: So, for example, in a first grade class, there might be a class discussion about different strategies for adding seven plus eight. And I think in a lot of classes at one point, the teacher would likely want to highlight the fact that you can find that sum using doubles plus one. So, in this particular instance, if a student were to talk about their reasoning, we'd want to encourage that student and certainly help that student talk about the following ideas: the connection between seven plus eight and seven plus seven, and the connection between their answers, namely because the second addend has changed from seven to eight, and noting the connections between the second addend and the answers, namely, if the second addend increases by one, so, too does the sum. And finally, we'd want to emphasize what it is we're doing here. Namely, we are using sums that we know to find sums we don't know.
Nancy: So, that's an effective example of what reasoning sounds like in the elementary grades. It's very specific. So even though reasoning is the thing that allows us to move from specific examples to generalizations in elementary mathematics, it's oftentimes by really focusing on what's going on with specific examples…
Nancy: …that students can begin to make those leaps forward. Some of my thinking lately about what I do in the classroom comes from the book Make It Stick, which talks a lot about learning processes and principles in general. And one of the points that the authors make in the book is that effective learners see important connections, for whatever reasons, sometimes more readily or more quickly than others.
So, what I try to do with my teaching then is to say, “OK, well, how can I help all learners see those relevant and important connections as well?”
Mike: Absolutely. So, it really does strike me that there are planning practices that educators could use that might make a press for reasoning more effective. I'm wondering if you could talk about how might an educator plan for pressing for reasoning?
Nancy: One thing that I think teachers can do is anticipate, in a very literal sense, what is it that they want students to say as a result of participating in the lesson? So, I think oftentimes we, as classroom teachers, focus on what we want students to learn (i.e., the lesson objective or the essential aim). But that can be a big jump from thinking about that to thinking about the words we literally want to hear come out of students’ mouths. So, I think that that's one shift teachers can make to thinking not just about the lesson objective as you'd write on the board, but literally what you want students to say, such that when you walk around and you sort of listen in on small groups, those moments where you say like, “Oh yeah, they're on the right track.” And then I think another key shift is thinking more towards specific examples rather than generalizations.
Nancy: So, as an example, suppose that in a third or fourth or fifth grade classroom, students were talking about fraction comparison strategies, and the teacher had planned for a lesson where the objective was to determine if a fraction was more or less than a half by using the generalization about all fractions equal to a half. Namely, that the numerator is always half of the denominator. So, that certainly could be something that we might see in a, you know, teacher's guide or perhaps in a teacher's planning book. But that's different than what we'd want to hear from students as the lesson progressed. For example, I think the first thing that we'd want to hear as the students were talking, is a lot of examples, right? The kinds of examples that are going to lead to that key generalization. Like if a student was talking about nine-sixteenths, I think we'd want to hear that student reason that nine-sixteenths is more than half because half of 16 is 8, and nine-sixteenths is a little bit more than eight-sixteenths.
Nancy: And so, what's effective about that kind of planning is that it alerts you to those ideas when you hear them in the room. And it can then help you think about, “What are the pieces of the explanation that you want to press on.” So, in this case, the key ideas are finding half of the denominator, connecting that value to the fraction that is equivalent to one-half, and then comparing that fraction to the actual fraction we're looking at so that we can bring those key ideas to the fore, and the ideas become a strategy for students to use moving forward.
Mike: You're making me think about two things kind of simultaneously. The first is, I'm reflecting back on my own practice as a teacher. And at that time, my grade-level team and I, we tried to really enact the whole idea of anticipating student strategies that comes out in the Five Practices book. But what you're making me wonder about is, we went through, and we said, “Here are some of the ways that children might solve this. This is some of the strategies.” The step we didn't take is to say, “We know that there are multiple ways that children could attack this or could think about this, but what's the nugget of reasoning?” What would we want them to say in conjunction with the strategy that they had so that we were really clear on if a student is counting on to solve this problem, what's the nugget of reasoning that we want to either press on or encourage. If the’re direct modeling, again, what's the nugget of reasoning that we want to press on. If they're decomposing numbers? Same thing. So, really it makes me think that it's helpful to anticipate what kids might do. But the place that really, like, supercharges that is that thing that you're talking about, is—what's the thing that we want them to say that will let us know that they're onto the reasoning behind it?
Nancy: Exactly. And I think the conversations you're having or have had with your colleagues reflects where we are with the field generally. I think that the field of mathematics education is at a place where, for the most part, we're on board with the use of discussion as a pedagogy. I don't think that it's a tough sell to convince a lot of folks that students should be spending some amount of time talking. But I don't think that we as a field are nearly as clear on what to do next. And again, as you alluded to with the Five Practices book, and while I would certainly agree that all of these are important aspects of classroom talk, I think that they skip over this essential idea of pressing for reasoning. Namely, staying with the student beyond just their initial explanation so that their ideas become clear, not just to others, but also clear to them.
Mike: I love that. I want to go in a direction that you started to allude to, but you really got to in, in your article. This idea that there's a certain number of questions for follow-up that can really have a tremendous impact on kids. I'm wondering if you could talk a little bit about that.
Nancy: My article and more broadly, my interest in press for reasoning, is motivated in large parts by my professional interest in figuring out, you know, what it is about discussion that makes it such a powerful tool for learning. So, although we have enough empirical evidence to support discussion as an effective pedagogy in math class, we as a field are much less clear in knowing which of the aspects of discussion are most efficacious for learning. What are the mechanisms of student talk that help students learn math more deeply? I had the good fortune many years ago to find some compelling research by Megan Franke and Noreen Webb and their colleagues at UCLA who did some digging into press for reasoning. And through their studies, they have shown that follow-up questions, questions that press students to clarify and strengthen their initial explanation, are associated with students giving more robust and more accurate explanations.
Nancy: What their research revealed is that it takes two to three specific follow-up questions in order to either have the student say, more math and more accurate mathematics. So, I think about that so often in my work in the classroom because so often I'll ask a student to explain their reasoning and because they're learning, the explanation comes out either partially correct or partially complete, and I need them to say more. And I might ask them the first follow-up question and either they or I suddenly start to worry. The student might think, “Am I saying something wrong? Am I totally off track here? Uh, I'm not really sure why I did what I did.” And then I, of course, as the teacher, I'm so worried about, “Am I putting the student on the spot? Am I losing the rest of the class?” And in those moments, I hear myself say, “Two to three follow-up questions, two to three follow-up questions,” as a way to remind myself to stay with the student. That if we really do believe that students learn by talking, then it only makes sense that we should expect them to need more than just one turn to get their ideas out in such a way that are clear and accurate to them as well as to the listeners.
Mike: So, that's fascinating, Nancy. I think there's two things that stood out from what you said. One is, as a classroom teacher, I appreciate the fact that you acknowledge that feeling of, “Am I losing the class?” [It] is something that always exists when you're trying to question and support. But I think the thing that really jumps out is, we have research that says that this actually does have a tremendous impact on kiddos. So even though it might feel counterintuitive, staying with the press for those two to three questions really does have a tremendous impact. I'm wondering what it might sound like to take a student's initial response and then follow up in a way that presses for reasoning.
Nancy: So, suppose a fourth grade class is working on strategies for multi-digit multiplication, and one particular strategy that the teacher would like to emphasize, or showcase, is compensation. Namely, how we can change one or both factors in a multiplication to create an easier computation and then make an adjustment accordingly. For example, we can multiply 19 times 40 by thinking about 20 times 40, and then subtracting 40. Let's suppose that students are working in groups and—on this computation—and the teacher overhears a student talking to their partner about how they use this exact strategy, and briefly checks in with the student and asks, you know, if they'd be willing to share their strategy with the whole class. And the student agrees. So, the teacher calls on the student to tell us, “How did you compute 19 times 40?” And the student says, “Well, I did 20 times 40 minus 40, and I did that because 20 times 40 is easier.”
Nancy: Great. So, we've got some ideas on the table, and so now let's unpack. So, maybe the first question to ask the student is for them to interpret 19 times 40. What does that mean? Literally, it says 19 times 40, but can they give a context? Can they provide an interpretation of that expression with the hope of getting the idea out that we can think of 19 times 40 as 19 groups of 40. And similarly, 20 times 40 as 20 groups of 40. So, once we have the idea of groups of a number out there, can the student tell again why it made sense for them to think of 20 times 40? Why is that easier? Then another follow-up question to ask is, “Well, what's the connection between changing that first factor to 20 and subtracting 40?” Because if you think about it, if you're a listener who's unfamiliar with compensation, that's a pretty big leap to go from changing the first factor by one to a second step of subtracting 40. Huh?
Mike: It sure is.
Nancy: (laughs) Right? Like, how does changing it by one mean you subtract 40. And so, here the students can talk about the fact that we found 20 groups of 40, which is one too many groups. So, we compensate by subtracting 40. So, those are some follow-up questions that I think we'd want to ask.
Mike: This example just makes so many connections. I'm struck by the fact that, simultaneously, that press for reasoning is helping the child who came up with the idea really build a stronger vocabulary and a justification, and at the same time, it's actually providing access to that strategy for kids who didn't come up with it, who maybe kind of wondering, “What? Where did that come from?” So, really it's beneficial for the child who brought the reasoning to the table and to everybody else. The other thing that jumped out is, even in that question where you said, “Can you offer this in context?” That's kind of connecting representations, right? Like the child was articulating something that might show up in equation form and asking them to articulate that in a contextual form. [That] is actually a way of challenging their thinking as well.
Nancy: Exactly, yes. For many students—and, unfortunately, many more adults—symbols are just that, their symbols. Yet, we who engage in mathematics know that many times symbols are linked to not just one representation, but several, that there's certainly a literal interpretation of any kind of symbol string or numeric expression. But then we can interpret what those expressions mean by connecting back to the different meanings of the operation. So yeah, like you said, Mike, there's two things going on here at least: Helping the other students learn about this particular approach and trusting that it works, but also helping the original speakers see what it takes to convince others. And in this case, part of that includes the fact that, “Oh, when I talk about multiplication, it's helpful to remind people that multiplication refers to putting groups together. Or that it's helpful to think about multiplication in terms of putting equal groups together.”
Mike: Well, before we close the podcast, Nancy, I typically ask a question about resources because I suspect for some folks, this conversation is one that they've been thinking about for a while. And for other folks, this idea of thinking past strategies toward a reasoning might be a new idea. So, I'm wondering if you'd be willing to share resources that you think would help support people maybe taking this conversation we've had and deepening it.
Nancy: Sure. So, my work in this field rests upon the shoulders of many brilliant mathematics educators and some of whom are people I admire from afar, like Megan Franke and Noreen Webb and their team at UCLA. And still others who I've had the honor to work directly with and learn from over the past 20 years. And two educators, in particular, are Suzanne Chapin and Cathy O'Connor of Boston University, who are a mathematics educator and applied linguist, respectively.
Mike: I adore their work. I'm just going to cut in and say, I'm excited for the resource you're going to share because I've read some of their stuff, and it's phenomenal.
Nancy: They were kind enough and generous enough when I was very new in the field to invite me to collaborate with them on a book called Talk Moves, which is essentially a teacher's guide to facilitating productive math talk. Many years ago, Cathy, Suzanne and I worked together on a research project where we were using discussion in elementary math classes in the city of Chelsea, Massachusetts, and we realized that there really wasn't a how-to guide out there for doing this kind of thing. So, from our work together came the book Talk Moves, which is now in its third edition and includes written vignettes in the book showing composite examples of teachers and students using talk moves to learn more mathematics, but also includes a set of video clips that were filmed in actual math classes with real-life teachers and real-life students using productive talk moves—including press for reasoning—to help students talk about their reasoning and respond to the reasoning of others. It's a very user-friendly guide for people who want to dig more deeply and see what this thing called productive math talk looks like in action.
Mike: So, I'll add to your plug. I read that back when I was teaching kindergarten and first grade, and it actually had a huge impact on my practice and just understanding at a granular level what this could look like. Nancy, thank you so much for joining us. It really has been a pleasure talking with you today.
Nancy: Oh, it's been a real pleasure for me too, Mike. Thank you so much for having me. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.