I’m Paul Cobb, Vanderbilt University.
Uh, when I think about evidence for mathematical development, what I’m particularly interested in is something called quantitative reasoning, which is different—the emphasis typically is on how do we calculate to get answers. Whereas for me, a quantity is a measurable aspect of the situation and we typically think of measuring, we think of feet and inches and miles and so on, but we think of anything in mathematics in terms of quantity. So, for example, we can think when kids counting that in enumerating, they’re measuring, if you like, how many or the muchness of a pile of candies or whatever it is. So we’re interested in relations between quantities. So, for example, you take a simple missing addend problem where you’ve got 5 and how many more is 12. It’s the conception of 5 as the part of the 12. So I’m very interested in how kids think about the relations between quantities. And that’s really important to us because it gets not just at how they calculate but the reasons why they calculate in particular ways.
What is the te—when I—in evidence, some of the things to attend to are the quality of these explanations kids give. So I think typically the emphasis is on the answer. And also we try to encourage in the classroom where we work, a culture where kids are encouraged to draw pictures or diagrams or sketches that will help them think through and reason to an answer. So also very much interested in that. In fact what we often do is have items or questions or tasks that are from our point of view uh for assessment that from the kids point of view is just another task. And they’re de—designed specifically so we get a written trace of their thinking. So we can go away and figure out or get some evidence on how the kids are thinking. And that really helps us plan our next lessons.
The social context—I guess when I talk about the social context of the classroom; I guess it’s important. Maybe I should tell you a story about why this became important to us. This was the first we—what we do in our research of classroom teaching experiments where we’re responsible for those kids instruction. An entire class for say a year or it might be 12 weeks or whatever. The first time we did this, we steamed in there with the intention of just focusing on individual kid’s reasoning. These were 2nd graders. They only had one year of formal schooling and here’s what happened. About the second day, the teacher—the kids worked in groups to solve a particular task, a mathematical task. And it was something like the equivalent of uh 6 and how many more make 14. And there’s a whole class discussion. The first kid that gets up is a boy and he goes up and he’s an answer of 6. And he starts to explain his reasoning. Halfway through he realizes he’s got it wrong and he sort of puts his head down like this and turns away and won’t look at the teacher. And the teacher said, “You know, come on, explain your answer.” And he says, “I can’t. It’s wrong,” and turns his back to the whole class. He’s completely embarrassed. At that point the teacher we worked with—we were very lucky she was incredibly competent—stopped the conversation about the arithmetical problem and immediately said—had a discussion with the kids about making mistakes in her classroom. “It’s OK to make a mistake. We make mistakes all the time.” I can’t remember the kids name—let’s say it was Steve, did exactly the right thing. He figured out that he made a mistake and he thought it through and corrected it himself. And that’s what we like to see. In other words, what we learned from that incident and became the focus of our research—we hadn’t planned it—was to focus on how the teachers guide what we call “The Classroom Microculture.” It may be more just the social atmosphere of the classroom. And we found unless we understood that, everything else we intended to do in terms of problem solving was irrevalent. And so it profoundly influences what kids think a problem is, what counts as an explanation, what mistakes means and all that kind of stuff. So that’s why it’s crucial to us.
When we think of kids and self, there’s a notion that’s becoming quite popular called that of identity and we’re just toying with this idea now. When—a part of what we do is instructional design where we develop instructional sequences that last, you know, for 12 weeks, 14 weeks, or whatever. And one of these we’ve been working on recently is neuro statistics with middle schools. And I guess when we think of the notion of self or identity, we’re thinking of what sort of people are the kids becoming as doers of mathematics in school. That’s the essence of it. And I think the bigger extension to that is we sometimes just think of schools and mathematics as self-contained entities. When we want to justify our goals, we’re also thinking about what kids are going to do beyond school. How are they going to use what they learn in the classroom as it relates to what goes on beyond. So for example, very consciously in the stuff we do with the middle school students, our role was for them to develop an identity as a data analyst. And so we actually ended up having exclusive conversations with the kids about this and about the significance of this in wider society. So that would be one example.
OK. This idea of identity is something—is quite new in our work and I—I would say my thinking is still pretty fuzzy in this area. But I think one of the things I do at my university is I teach uh classes for respective elementary school teachers and it’s very interesting, these are successful people in many walks of life that are coming back to get a masters degree. They’ve already got an undergraduate degree. And yet when we start off and we have conversations about their own mathematical experiences, about how they feel about mathematics, the majority of those experiences should be negative. Some of them would be fear of mathematics and so on. So they have developed identities and it’s not a natural phenomenon. They have developed those identities as a consequence of instruction received. So a big part of my focus with those respective teachers at the beginning of the semester is to help them understand that it’s a consequence of what was done to them and also by extrapolation that their kids will be developing identities as doers of mathematics. And those have consequences not just on how well they do in their class, but consequences in the choices they make later. And I might add, I think this is really important with the issues of equity. If we’re thinking—if suppose I’m teaching elementary arrhythmic school, down the road these kids are going to choose whether to go on—on in mathematics, whether to go on, to choose, advanced mathematics or not. What sort of experiences are they having? How do they feel about mathematics? These things are very important in influencing whether they are going to go on to advanced mathematics. And I think that becomes at the fault particularly, if we look at gender, when we know that girls are opting out and we know that many of them are opting out not because they’re not competent, but because of how they feel of their relation to mathematics or their mathematical identities. And I would guess, although I haven’t seen research on this, exactly the same occurs with African-American students, with Latino students and so on. So that’s why I think it’s a notion that has considerable repercussions.
OK, I’ll give an answer and tell me if it makes sense. If not, we can redo it. One of the—before—no. We’ve been talking a lot about the classroom ‘Microculture,’ the classroom atmosphere and one of the things we’re trying to do—at least how I think about it when we’re actually responsible for these kids learning over a period of time. We’re focusing on the learning of the group or the learning of the classroom community, even their mathematical learning, how ways of reasoning and thinking and arguing develop. And if—and that might seem to imply that everybody thinks the same, that they all progress in a lock stead fashion. And in fact at any point in time as we will know you’ve got huge, qualitative differences in the kids reasoning. That’s really important to us pragmatically to focus on as teachers. For example, we have whole class discussions at the end of each lesson typically. Those are sort of the capstone. That’s where every book we so draw on, the kids thinking is a resource. And to plan for those discussions, we spend our time when they’re working in groups or they’re working individually. Not running around to try and put out fires or to try and work with kids individually. We spend the time going around the classroom to understand the different ways that kids are thinking and reasoning because that’s the primary material, the primary resource we’re going to draw on in the whole class discussion. So for example, we might conjecture if we have this solution from this student compared with that solution from that student. This really important mathematical idea could emerge from their thinking as a topic of conversation for everybody. So that diversity for us isn’t an obstacle, it’s a resource. We would be in trouble if everybody was thinking the same way. It’s something to be capitalized on.
Well, it’s very interesting—over the years my thinking about classroom discourse has evolved and shifted. In the first teaching experience we did—and this was longer ago then I’d like to admit, 14 years ago—uh, we were just very happy if we had kids in sustained conversations that appeared to be mathematical. We were thrilled to bits. And I guess over the years we’ve focused—and others as well—teased our aspects or characteristics of conversations that we think are particularly crucial in supporting kids learning. One is, of course, that they’re talking about substan—a substantial idea that advances our agenda. Uh one aspect—let me try and articulate it. It’s quite subtle and it’s a distinction between what we would call conceptual and calculational discourse. Sometimes people translate that immediately into conceptual versus procedural meanings. So—but that’s not what it means. Let me try and give an example and I’ll give a very elementary one. Uh, if again we go back to simple arithmetic and think about missing add-in problems and the problem is something like, let’s say again, 6 and how many more makes 14. We know—it’s pretty predictable if you’re doing that in 1st or 2nd grade, you’re going to have a bunch of kids who are going to add. They’re going to solve it not as 6 and how many more make 14, they’ll take the 14 and they’ll add the 6 to it and get 20. There’ll be other kids who have done it as a missing add-in, have some very sophisticated solutions. Now suppose I’m a kid who’s just added, that’s how I understood the problem. If I’m listening just to how the other kids calculated, I have no idea what’s going on. For me to change my thinking, I’ve got to figure out how they understood the problem all on my own. If on the other hand, we can get out into the conversation how they understood the problem—suppose it was a—and let’s just say it was a silly little problem like “Two kids went fishing. One kid—let’s say Juan caught 6 fish and between them they get—caught 14, how many—how many did Mike get—how many fish did Mike get?” We might push or the teacher’s role would be to have them—each child talk about what the 6 meant. Whose fish are those? What do the 14 mean? And then those differences in their thinking would become explicit. Now we’re back to talking about how they interpreted or understood the problem. That would be an aspect of the conceptual conversation whereas if we had just focused on, “Well, this kid did 6 and 6 is 12 and 2 more is 14, so they said it was 8.” Even though it was completely meaningful, that would be a calculation conversation. So that would just be one characteristic, for example. And I think if you look in the literature, a lot of which is with the teachers, you’ll find other aspects now. We’re getting to the point where we can begin to pinpoint aspects of conversations if they’re important and the teacher’s role in bringing them about.
I think sort of a standard question is, “Why did you calculate in that way?” Now initially that’s how—I’ve asked that question and the kid said, “To get the answer.” In other words, we talk past each other famously. Uh, I think its encouraging kids to develop inscriptions or representations or drawings of their thinking to help communicate. Initially there’s a role for the teacher to do that. To maybe play a role in a whole class discussion in developing drawings for everybody to see. I think its things like—in the missing add-in problem—what does the 6 mean or whose fishes are those? It’s pushing the conversation to that level.
OK. Another aspect of classroom discourse we focus on is what we call reflective discourse. Uh and this is very simply uh where there’s a raising of the level. And let me think of an example. This is from a 1st grade classroom and this particular example. Uh the teacher and the kids—the teacher started by posing a simple task. She used an overhead projector and there’s a big tree here and there’s a little tree here. And let’s say there’s 8 monkeys who are playing the tricks. And the task was for the kids, “How many monkey’s could be in each tree?” And as the kids gave the combinations—you know, somebody might say there’s 5 here and there’s 3 there. Or there’s none here and there’s 8 there. The teacher sort of recorded—made a little table uh in the middle. At that point what the activity was about was simply finding the combinations. The teacher then asked a question that pushed the discourse up to another level. It was reflective shift. And that—this question was this, “How will we know—can we know for sure if we’ve got more?” And kids started checking and somebody suggested we haven’t got three and five in the tree and they checked, “Yes, they did.” And then the teacher then asked another question, “Can we prove it?” Now the conversation became very different. That record she’d made, that table became the focus in which kids began to organize the combinations. “Well, we’ve got 0 and 8 and we’ve got 8 and 0. And we’ve got 7 and 1 and 1 and 7,” and so on. So that was a reflective shift in the—what they’ve done before, itself became the object to think about. Another way of putting it is the process of mathematizing the situation that the kids and the teacher collectively went through. And so that is what we’d refer to as reflective discourse.
In terms of how we think about it, if you want to focus then—I’ve just spoken about how the discourse shifted, the bottom line is also “What are the kids learning—those individual kids learning?” One way to think about it is that the discourse in affect almost causes their thinking to evolve. Now in our experience, we find it useful to focus much more on the individual kids learning and to view it—we would use the word, a reflexive relationship between the shifts and the discourse in learning of the kids. The teacher asked the question, “Can we prove it?” But it was the kids who had to make the contributions that would push the discourse up to another level. The teacher can’t do it by his or her self. The kids have to play their part as well. The teacher can create the situations for the kids to make those shifts, but the—but it’s the actually kids who do the shifting. I guess that would be the best way we would think about he relationship between the two. Does that get it what you wanted?
What we think about—I think about cognitive theory, this sort of metaphor that comes to mind, is basically goes back to the idea of you acquire knowledge and then you apply it. So the metaphor to me is that it’s transporting or transferring knowledge from one situation for another. A lot of the school curricular is based on this. So, for example, we’re going to learn those general algorithms for adding and subtracting three digit numbers with the idea that we can then apply them towards situations. We’re going to transfer what we learned in the classroom to another situation. So that would be the basic metaphor. In contrast, if you take a situated view, you have a very different sort of metaphor that still—base is still rooted in that idea. But rather then focusing on knowledge transferred from one physical location to another, we would always look at people, including kids in a classroom as participating in what we call practices. So, for example, it would be a particular established way of reasoning and thinking about arithmetic or a data analysis would be a practice. And it’s that participation that constitutes the context in which their thinking occurs.
I have—when I think about two—I have slowly over the years drifted to a situated view and more and more so. When we started working, I took a very strong cognitive point of view. That was 14 years ago. And the reason for that relates to my goals as a math educator. We’re in the business of trying to find ways to support the improvement of kids thinking. And for me, given my purposes, taking a situated view I would argue is a major strength. It’s a good thing, not a negative thing. And here’s why. Suppose we go out and we work with these kids over a long period of time and we do these interviews and we get some achievement data at the end. If I just get the achievement data, all I know is some kids have learned something. I want to improve my instruction design. I want to improve so I can do a better job with the next group of kids. What I want to be able to do is to relate how they learn to the means by which it’s supported. Which includes the tasks, the computer tools, the classroom discourse, all that kind of stuff. In a situated analysis that we can now do, I can tie—or I and my colleagues can tie the development to the kids reasoning, to the means by which we supported it. So we can immediately go back and figure out how to improve those means of support. And so it’s seeing the kids reasoning is very much related to the classroom’s social context in which it occurs.
OK. In terms of what—I would say that teaching “in a way compatible with reform recommendations, the standards” then you stand as 2000 for ? is incredibly challenging as a form of practice. And the more I’ve worked in classroom’s and worked with teachers, the more I’ve come to admire them in terms of the skills and the knowledge and the dispositions that are involved. A lot’s been written about this—you know, I think it’s clear that it involves managing the classroom’s social climate. It involves having some relatively deep understanding of the mathematical ideas that are involved and of kids thinking. But there’s one aspect I want to focus on that typically isn’t talked about but in our experience it’s really important. And it relates also to the resources I think teachers need. We focus a lot more on what they need to know but not on what resources they need to help them. And I would say what’s needed, a relatively detailed instructional sequences. But what’s important is not just a stack of tasks or a materials, it’s the rational that underlies them. And for us that rational takes the form of a learning trajectory, which is again, is a sequence or learning root or whatever together with the means of supporting kids development. The idea is not the teacher’s root—march kids down this trajectory or that it’s just the different textbook. The hope is that the teachers—because they understand the trajectory—and it’s our job to figure out ways to help teachers reinvent that trajectory—and then are in a position to adapt and to adjust the materials to their situation. So it’s a view of teaching this idea-driven adaptation which I think relates very much the professionalization of teaching.