My name is Jeremy Kilpatrick. I’m at the University of Georgia.

Well, I—I think what we try to—what we try to accomplish in schooling is that—that students would leave from their mathematics education with confidence that they can approach any kind of situation that is potentially mathematical. That they can think about that situation and use the ideas of mathematics that they’ve learned in a school and that—that they’re comfortable in doing tasks in their daily life or—or solving problems that emerge in their daily life. Uh using any mathematical skills they’ve acquired and—and it’s always a big question how—how—what level do—do we want of that. Uh but it’s—I guess it’s like general literacy. Uh it’s—it’s being able to function. Uh and I think the big problem in mathematics is the fear that so many people have or the—or the uh—the—the experiences that they’ve had in school that block them from actually using the mathematics that they should know. So I think one of the—one of the real challenges for math teachers is to try—try to uh make the mathematics something that—that’s not only understood but that students can actually put to prac—put into practice. So most peop—most people are trying to do that in the classroom more then they have in the past and I think that’s raising the level of literacy. I hope it is.

Uh I think the reform—to talk about changes that the reform movement has—has—has stimulated in—in—as far a pedagogy goes, uh first of all I don’t think its been as dramatic as—as some people have portrayed it. I think actually many of these elements of pedagogy that—that are being advocated today as part of the reform movement have been present for at least a hundred years. And people have been talking about doing this kind of thing. I think a lot of—a lot of what we’re doing is returning to some of the ideas that people like John Dooey had. But basically the idea is that the student should be more a—a constructor and discoverer of mathematical knowledge and the student should be more active in—in responding. Uh I think there have always been teachers who—who try—try to conduct classes where students were—were um more active and more engaged with a mathematical ideas. I think what the reform has done for many teachers is to give them permission to do that. Uh I—I think, for example, that many elementary teachers no how to do that in—in reading, in—in other subject fields. But they—they felt that they needed to operate differently in mathematics. Uh and I think this—this has sort of told them that they could—they could uh do the same kinds of things. Have small groups, have kids working on projects, have this kind of thing going on. Uh and for secondary teachers I think it’s in some ways been a more radical change because secondary mathematics teachers are, in many cases, borrowing their pedagogy from the college mathematics that they know and uh—and from the high school that they experienced. So uh I think for them its been a much more difficult thing to try to change their pedagogy. But I guess uh the main uh contribution of their reform has been to—to try to put our uh—to give it—to—to get teachers to attend more to the discourse that’s going on in their mathematics classroom and to value more what the students have to contribute to that. Uh I can’t really characterize what pedagogy was like before uh because I think there always has been and there probably always will be a lot of mathematics teachers who are expositors of mathematics and there’s a place for that. But I think what the reform has done is opened up some space for—for students to be more active uh and do more of the work uh together and in—in a collaboration with their teacher rather then opposition or in some other form.

I—I—I’m—I’m pleased to talk about the—the communication idea because that’s actually uh I think one of the reasons that this been—has been foregrounded in the—the mathematics reform. In some earlier work that was done at the College Board where they were looking across the curriculum at how students might be better prepared in all subject fields. And this was with the focus on preparation for college but it—it gradually came to apply to everything that we—people began to realize that in many mathematics classes there’s so little communication, real communication, between teachers and students. The questions that the teacher was asking were not real questions. And it uh—and students knew that the teacher knew the answer. So this was not any—any genuine communication and as far as asking students to speak about mathematics to each other uh or to the teacher, there was really very little of that except at a low level of answering very specific questions. So I think the emphasis on—on communication in—in the mathematics classroom has opened—opened a lot of doors for students to learn in very different ways. They learn to speak comfortably with each other and with the teacher about mathematics. They learn to write about it, they learn to read each other’s work, and to talk about that. And so we have a much higher level of—of—of uh discourse in the mathematics classroom. I—I think during—I don’t want—again, I don’t want to characterize to uh any decade or too—too strongly, but I think during the 70’s there was a lot of emphasis on individualized instruction in mathematics where students went off and did mathematics on their own without talking to anybody really about it unless the teacher came by and would ask them how they were doing. And I think this deprived them of the opportunity to express um what they knew. And in—in—I guess in language uh we talk about the difference between uh your reception vocabulary, the kinds of things that—that you can um understand when you see them or here them and your production vocabulary which are the—which is the kind of thing that you can actually produce and talk about. And I think what was happening in mathematics class is—and it probably still is happening in some mathematics classes is that kids can see—can know that mathematics when they see it but they can’t produce it because they haven’t been asked to do this in a communication context. So I think what communication—the emphasis on communication has provided students with a context for becoming literate in a production sense uh in mathematics.

Well, I—I think—I think the reform effort has—has—has produced some real challenges for teacher education. I think we have to reconceptualize what we’re doing when we prepare teachers to teach. Uh we’ve tended to dichotomize their preparation and say, “OK, first we’ll teach you all the mathematics that you need to know and then we will try to give you some ideas on methodologies for teaching and then we’ll—and then we’ll let you practice all of that.” And I think what’s needed is a much more integrated way of, first of all, teaching—teaching the mathematics and having the teachers look at the mathematics that they will be teaching in a—in a much more uh creative way and a deeper way and to do that in the context of thinking about how children learn this and how teachers can help children learn this. So this separation between content and pedagogy I think in teacher preparation has to begin—we have to begin to put that aside in favor of a more integrated approach to—to looking at math—the deep ideas of elementary school mathematics in the context of how children come to understand those and how teachers can help those children. So what we need in teacher preparation to give teachers many more opportunities to look at what children are doing, what children can do, and to work with children as they think about the mathematics that they are going to be teaching.

Well, I would like teachers uh to think of—of the standards as uh—the various standards documents actually, as documents that they need to be responding to as uh—as critical readers and ideally uh I would hope that groups of teachers might come together and have conversations about the standards. I think there’s a danger that people will take these as sort of uh some sort of doctrine or gospel (clears throat) and would—excuse me. There’s a danger that—that teachers might—might uh, uh take these as something to be followed. And I would rather they—that they took them as something to be challenged and thought about because we recognize that there’s nothing perfect about them. Uh they are meant to be stimu—stimuli for conversations. And I think they give teachers a framework to think about their teaching. I hope they give teachers some ideas for what they might do and do differently. But I also hope that teachers will be critical of them and will say, “Well this—I don’t really understand this or I don’t agree with this or whatever” and begin to talk with other teachers about how they see it because uh I think it’s in that process that the teachers uh will become uh more aware of what they’re doing and more professional in the way they approach their—their teaching.

Well, I wasn’t involved with—with the production of the first standards document and—and—or the decision of why to take evaluation. I—I think—I think it’s clear that they took curriculum first because this instance in some ways the easiest thing to talk about, much easier then talking about teaching. I think we can all in mathematics talk about the kind of things we want taught in mathematics but to talk about how to teach it is a much more challenging thing. And I think probably just to guess that the decision was made to—to include evaluation because there was a certain amount of—of um concern about how students were being evaluated in mathematics. What happened—and I think people—some teachers have wondered why then was there a need for another document, “The Assessment Standards.” And I think over the time from 1989 to 1995 there—a lot of people’s thinking changed with respect to assessment and a lot of uh—I think people realized that what had been said in the earlier assessment—the earlier evaluation standards wasn’t wrong, but it certainly needed to be supplemented. And I think some of the best new ideas—probably—probably the single best new idea in the—in the uh assessment standards was this notion that when you—that assessment in mathematics should be an opportunity not just for the teacher to learn what the student knows and can do, but for the student himself or herself to actually be learning some mathematics. In other words, in so much mathematic assessment uh the assessment is a time away from learning and you—you stop learning and you are assessed. And I think the—the whole idea of using the assessment experience as an opportunity for the student as well as the teacher to learn mathematics uh I think that’s—that’s a very important uh idea that came out in the assessment standards.

I don’t know that I could boil it down to any one thing. I—I do th—I do hope that as—as teachers look at the more recent standards document, the principles and standards document that came out in April 2000, I do hope that teachers will refer back to the earlier assessment document from ’95 which is not so well-known. Uh because I think the—while there is a principle on—on assessment, uh I don’t think it—is they’re able to go into as much depth and detail as was done in the ’95 document. So uh whereas in some sense the 2000 document superceded the document that was first produced in 1989, I there is a recognition that the teaching standards from ’91 and the assessment standards from ’95 are still very much alive and valuable for teachers to keep looking at. And so those assessment principles, the—the—excuse me—the uh assessment standards, the original six in ’95, uh are still very important for teachers to be considering. I think the standard having to do with openness, for example, is extremely important when teachers think about assessment.