**DEBORAH BALL**

I’m Deborah Ball um University of Michigan.

Um, so libratory pedagogy is a term that many people use I think with a variety of meanings. I don’t actually use that term particularly in my work but the way I think about it does relate to my work, which is um—it’s about an approach to thinking about teaching that enables kids to um learn things that might go beyond what they might otherwise do. That gives them options that opens up horizons and paths that they might not otherwise have that, I guess you could think of, that liberates them from um a path they might be on if one didn’t think about that more carefully. So it’s a more of an opening.

Well, I think in general instruction—good instruction is about teachers um learning how to hear what their students say, watch what their students do, and keep in mind as they’re doing that um—let’s say in mathematics, try to think about um what kind of work kids need to be doing in mathematics, what it is that there is to be learned, and that make very um wise matches between what it is that mathematics entails and what it is that kids should be learning to develop in making good matches between them and what kids bring. And that’s just that much more challenging when teachers are faced with trying to hear and make sense of students who are less familiar to them. Teachers also aren’t that great at it sometimes when they’re dealing with students who are more familiar to them because the task of understanding what kids are saying, what they’re thinking about, what they’re trying to do isn’t an easy task. But it’s complicated when you’re dealing with students who come from cultural backgrounds which are different from the teacher’s, linguistic backgrounds that the teacher may have less preparation in, hearing, making sense of, and there’s some tendency for teachers to think of their students as somehow not capable of doing things when the students face difficulty. So again, I think this matching between what there is to be learned and what it takes to learn it and what students are currently trying to do is a very complicated piece of work.

Um, so you’re asking about how taking students needs seriously can contribute to student’s development. (interruption) To teacher’s development. Um, well, certainly the more that teachers pay close attention to what their students are saying and doing um the kinds of strengths that they bring, the kind of difficulties that they face, and pay close, close attention to learning to hear and see them, teachers are at least in the position to have the opportunity to learn more about students. The more one makes as one practice the study of students in the course of the work, the wider one’s experience becomes about he range of things that students of a particular age or a—around a particular topic might use. So, for example, I’ve taught for, you know, 25 years and by now the ages um and levels of students and range of students that I’ve taught have contributed to my having a rather large um base of knowledge about what I might um be prepared to hear, 3rd graders or 4th graders or 5th graders say or do. That doesn’t mean a student couldn’t surprise me or say something I’ve never heard. That still happens all the time. But I have a greater wealth then a beginning teacher has about what I might expect a student to say. Um, and I guess the tricky part about that is that one needs to keep expanding that because although there’s certainly patterns of what students um say and do, they’re also slightly less patterned kinds of responses or kinds of things kids do. And if you’re too quick to think that something is just like something a student may have said some other time or some things students typically do, then you may actually lose the opportunity to learn something new. Um, so I’m impressed that even after as long as I’ve taught, that there’s still a lot more to learn about students. But I think that this attention to student’s needs, knowledge, beliefs, practices, and so on gives teachers kind of in the course of their practice an opportunity to learn about students.

Oh, well um one of the things that had troubled me in recent years was the ways in which people who worried about ethical questions or moral questions in teaching um seemed to think of those as somehow a part from questions of knowledge. And Suzanne Wilson and I wrote a paper in which we tried to show by looking inside of the practices of teaching and learning how almost everything teachers do entails a kind of simultaneous management of both moral and intellectual issues. For example, if a student um brings up uh a solution to a mathematics problem in class, one issue for the teacher to be concerned with is um “Is this mathematically viable? Is this worth taking up with the whole class? How might I um try to use what this student is doing?” But the minute you start to think about, “Should I bring this up in class? Should I have the other students discuss it?” you’re outside of merely intellectual questions. Or at least your—you’ve expanded your realm of concern beyond intellectual matters because it’s not just mathematical questions that—that you have to be thinking about, but also “Who is this child? How do other chil—children regard that child? Um, what are the consequences of asking a student to present an idea in front of the rest of the class at a particular time?” Those are questions that are not just intellectual but have ethical and moral dimensions about whose getting to talk and whose ideas are getting play and so on. And that’s just one illustration where someone might think of that as either a moral question like, “Who gets to talk?” Or they might think of it as an intellectual question like, “Is this an important mathematical idea that the whole class ought to work on?” Or “Is this a learning issue that this child needs help with?” But, in fact, if you trace it inside of the decision a teacher would be making, it would be both moral and intellectual. And that would be true about a number of things I could give you examples of. It would be true is we discussed homework, which seems like the most sort of routine piece of teacher work. But a question, you know, that teachers face as they design homework if they really face it closely is, you know, what sort of homework ought to get sent home when you are honest about the fact that parents will differ in their capacity to help students with their homework, the time they might have available. If you send home very challenging, intellectual work, perhaps the consequence of that is that some kids will get to do homework and some won’t. And then in some sense you’re by—you’re very active trying to engage kids in complex work outside of school, perhaps contributing to disadvantage in certain ways. On the other hand, if you give homework that’s very intellectually kind of simple and routine, then in some ways you’re perhaps denying the need to push kids intellectually. But the whole question about the interplay between home and school over something as mundane as homework would be another very nice place to consider intellectual and moral questions.

Um, I guess that that’s a phrase that um Maggie Lampert and I use. That is, what it means for teachers to investigate their practice. But, I guess the way I would think about that in a way to say it more clearly is that it’s—you asked me um at one point about what teachers can—how teachers can develop from paying close attention to their students. Um, and like it or not, practice is the main place that teachers learn about their teaching. Teachers have said that for years, that the professional development they had didn’t contribute nearly as much as anything that they learned just from doing it. And I think teacher educators have sometimes been skeptical because they’ve seen teachers not grow over time as they um gained an experience. That is, years of experience wasn’t necessarily um correlated with increases in knowledge or skill by the teacher. But if we understand that practice is where teachers spend their time, then there’s a great deal to learn about how practice can be um a source for teacher’s to learn. So I think that might reorient the way that we might work on um what we ask teachers to do, what we equip them with in pre-service preparation. But what I mean by it is the very act of designing a problem carefully with one’s students, watching carefully what happens, and then using the way that problem plays out in class as an opportunity to consider “What was it about the design of this math problem?” What did I learn about how students treated it? What did I see in my own efforts to try and make it work? How would I redesign that problem if I used it again?” And teachers can certainly investigate practice or study their practice alone, but um we also have a lot to learn in this country about how to structure teacher’s opportunities to work on study or investigate their practice with other teachers. So I’ve become interested recently in the structures of the ways in which Japanese teachers learn practices of studying their teaching. And I don’t think it’s about something one might call lesson study or anything formal like that. I think it’s about developing um a knew set of questions and habits about the way one goes about looking carefully at what one does with students and preferably having colleagues with whom one can then talk about those same questions. Um, but simply asking teachers to study their practice or conduct research on their practice I don’t think would do it. I think it’s about thinking very carefully about what is it one can do around one’s teaching that can be productive for learning. That might involve designing lessons with other teachers. It might mean examining student work in a very particular kind of way. It might mean looking at videotapes of one’s teaching trying to track the role that one plays in the course of a lesson. There are number of critical questions that one might choose to study over time, but I think simply sitting and reflecting wouldn’t be what I would call investigating practice.

So it’s a very interesting question what tools might help teachers um make profitable use of their own work as an opportunity to learn. One thing that I’ve been thinking about recently is that um having a—a frame of questions that one might typically discuss with other teachers could be a set of tools like that. Um, I had the experience recently of showing a videotape of my teaching to a group of Japanese teachers and I was very struck that the questions that they were all inclined to ask me about my lesson were pretty different from ones that I’ve typically been asked in this country. And it made me think a little bit about how those questions were a very profitable for what we all were that were able to talk about. They asked me very specific things about how I had designed the particular problem I was using and why I had set it up the way I had. And then they asked questions about the way I had used time in a lesson like why had I explored at the beginning of class the les—they knew exactly how many minutes I had spent on that and they expected me to have some rational for why I had done it for that long and then moved on. Why I had pulled the lesson to closure in the way that I did. I mean, it’s not that American teachers have never asked me questions like this, but the detail with which they were talking to me and the questions that they all seemed to share that they expected to discuss about a lesson struck me as very useful. So I guess questions could be a tool. Another kind of tool could be material things like being able to videotape one’s own teaching, um being able to have good records of kids work. And in general making records—I think that it’s unrealistic to imagine that much will happen without creating reference of teaching because it—so much happens so quickly that without having things that to which one can return and look whether alone or whether they’re teachers, it seems to me that it’s less likely that—you’re more likely to be talking from impression and what you think happened rather then being able to look and re-look at either what kids produced or what you did or how a problem turned—you know, evolved across a lesson.

Um, I think that in the past we’ve tended to think about teaching and learning as occurring like within a classroom so that you can pay attention to kids and teachers and the mathematics. And somehow later other people will worry about what’s the political context in which the school system exists? What’s the policy context in the state? What does this district or community care about? What are the pressures on teachers? And I’m increasingly convinced um, along with my colleague, David Coen, that it’s not actually the way things look when you look at—when you look at it from inside the classroom. That those things we think of as environmental factors of teaching enter right inside of instruction and if they don’t they probably don’t matter for it. So, for example, you can look from the outside and you can observe that a group of teachers works in a district that has a very high stakes new assessment in place. But suppose you went into that classroom of one of the teachers and you noticed that that teacher was designing a lot of her teaching now to try to match up with that assessment. And then you went down the hall and you went into another classroom and that teacher seemed relatively unconcerned about the new assessment. You’d have to understand that it’s impossible to understand environments just from the outside. It’s really how each of those teachers perceives um the pressures or lack of pressures in her environment and what that means for the way that she constructs um teaching with her class. And you could name a lot of examples like that. Like parents—parent pressure, for example, about reform curriculum programs. Um, those aren’t things that exist at school board meetings. Those exist in the ways that kids come to school saying, “My dad said that this is a stupid problem and this isn’t math and when are we going to start doing math?” That’s how the environment enters inside of what teachers do with kids in classroom and it seems to us increasingly that that’s very much part of teacher’s work so that if we have a conception of teaching as somehow in a vacuum, we’re probably missing a way to understand a lot of what it means to actually make teaching and learning work in—in real schools.

Um, typically the way people have thought about trying to improve um instruction in classrooms is they pick one element of instruction and try to intervene on that. For example, a most typical in math education has been to improve curriculum materials. And I think most curriculum developers have gradually come to realize that um curriculum materials work in—in the ways that teachers understand what the problem is, how teachers reinterpret and make choices about them, and even at the micro level as they present certain problems to their class, what those—their kids then ask about and how they then helped them with it so that curriculum materials work as an ingredient in the interactive kind of um interactions within teaching and learning. So that if you think you’re going to change instruction simply by changing curriculum materials, you’ll be humbled pretty quickly. And somehow—I mean, I guess that’s the kind of theory behind systemic reform. It’s just that often systemic reform hasn’t really thought a lot about teaching and learning inside of classrooms. It’s thought more about these larger environmental factors. So the points of intervention—I mean, my bets would be that interventions that take seriously that kids have things to learn in order to engage in math differently. That teachers have things to learn in order to help kids learn to learn math differently, that there are things for teachers and kids to learn about a new curriculum. That there are things that leaders and others in the environment would have to learn to see it as a problem of everyone’s learning and that those things interplay with one another would be a more profitable way to intervene then to imagine—you know, simply training teachers differently would make a difference. If teachers learn more math but don’t learn how to use that mathematical knowledge to hear their students or to use their curriculum materials differently probably won’t matter that teachers know more math all by itself, Just as having more curriculum—better curriculum materials won’t matter at all by itself. So I guess my sense about intervention has to do with taking more seriously that all of these elements interact. And that doesn’t mean there’s any single way to intervene, it just means that interventions that take those interactions more seriously and design to enhance the ways in which teachers, kids, and mathematics, and environments come together, those interventions seem to me more likely to be fruitful then one that think you can intervene on just one element.

Well, I think most of us believe that teacher knowledge has a lot to do with student performance. We feel it in our own teaching when we don’t know something well. We notice it, we feel it, we can’t help kids learn. We’ve seen teachers who make gross errors in the course of lessons because they don’t know the content. Um, so we believe it but most people I think are either surprised to learn or already know that we’ve not been very successful in documenting that um empirically. So depends whether you’re interested—which part of that you’d be interested in knowing. Um, empirically I think we haven’t been able to establish a good relationship between teacher—what teachers know about math and what their students learn because we’ve had too limited an idea about both of those. That is, too limited an idea about what mathematical knowledge matters for teaching and too limited an idea about how to trace its effects into kids learning. So if you wanted to be clever about it, you’d probably study something like, a student presents um a somewhat un—unanticipated response during class to a math problem and then what you studied was different teachers capacities to muster their mathematical knowledge to figure out what the kids might have meant, and to ask a good question or to point something out, or to clarify something. That’s kind of a point at which I think mathematical knowledge makes huge amounts of difference. One can’t understand what kids are saying without understanding quite a lot about the math. But we haven’t studied it that way. We tend to look at teacher knowledge as a function of how many courses teachers have had or whether they have good grades in math. And that’s not getting close enough to what it means to really make use of mathematics, to do the work of teaching. And if we could find ways to study that, my bet would be strong that we would see a very high relationship. Now the reason that that matters is it means that the way we help teachers learn mathematics ought to be focused on that use. Ought to be focused on what you have to know about math and how you have to use it so that you can do teaching, not mathematics, not biology, not something else but do teaching, which is intensively mathematical work. But we’ve tended to give it—we’ve tended to short change how mathematical—the teaching of mathematics is. And as a function of that I’ve think we’ve pretty much failed often to equip teachers with the mathematical knowledge that could make a huge difference in their work.

Um, I think the—there are a lot of challenges that teachers face where trying to teach mathematics more effectively. And it probably varies—you know, it varies for different people um, but there’s some common threads one could—could certainly identify. One would be it’s not at all easy to take seriously what kids think because (clears throat) first of all it’s very hard to understand what students say. It’s very easy to think that you know what students are saying and be wrong. It’s very easy to be so worried about coverage that you don’t slow down enough to hear whether students in fact are getting what you’re talking about or what the class is doing. It’s very easy to think that if a couple of kids give smart answers that everyone’s with you. So the very act of trying to understand what 30 or 40 or 20 students are learning mathematically is a very challenging piece of work. And I think that a chal—a problem we faced isn’t—is over romanticizing that, making it look as though—if you just cared about that you could do it. I think we need very much closer discussion on the identification of what are strategies for trying to do that in the complex environment of a classroom. And that takes me back to something that I wanted to talk about which was how important mathematical knowledge is to that listening and to that hearing. Um, teachers—it’s a vacuous statement to urge teachers to pay attention to their kids thinking without realizing how mathematical that work is. Um, and it’s interesting because it’s not the kind of mathematics necessarily that a mathematician would know. I’ve had a lot of experience showing um videotapes of classroom lessons to mathematicians who are much more puzzled about what the kids are saying then uh an experienced teacher might be. But it works both ways because mathematicians also sometimes hear um deeply mathematical issues arising out of kids’ talk that someone with knowing less math might not realize. Or they might get very excited about something a student brings up and you sort of wonder why they’re getting so excited about this. And if you pay closer attention, it grows out of a kind of um esthetic or appreciation of mathematics then someone who’s very highly uh educated in a—in a domain like mathematics has. So it really works both ways. I think there are things experienced can hear in kids thinking and I think there are things mathematicians can think. But if I had to pick a challenge for teachers, it’s that um kind of taking the mathematics seriously and at the same time taking kids thinking and knowledge seriously and managing to find the ways to stitch those together. Um, and some people talk about one and not the other, but teaching is about both.

I don’t know what deep learning is. (laugh) Um, I mean I assume that um—when people say something like deep learning they mean um concerns for students—uh, learning things in ways they can actually make use of and remember as opposed to uh memorize for a test or learn in some rout way that they can’t mentally use. It’s both about remembering. That is, being able to hold on to something for more then a few days, or a few hours, or a few minutes, and it’s about um knowing it in a way that enables you to I guess use it on one hand and I would add to that maybe reconstructed if you do forget it. So I think if I had to pick things about deep learning, it would be um those features of having learned something. Um, but the problem with the work learning is we use it in so many ways. We use it to describe the accomplishment of learning, which is I think what you mean when you ask about deep learning. But you could also consider what does it mean to learn? That is, what is it that you have to do to learn mathematics? And we talk about that a lot. But when you ask me about the challenges teachers face, a challenge we face is we don’t—we don’t always know enough about um what you have to do to learn some mathematics, not just cognitively, but what the mathematical work of it is. And that would be another way to think about deep learning is what does it look like to work on mathematics um in ways that, you know, enable you to learn it?

I think to try to characterize standards-based mathematics instruction you have to do one of two things. You either have to decide that you’re going to accept that—what that refers to is a set of ideas that many people have been trying to promote over the last decade, or you can be kind of obstinate and say, “It doesn’t really mean anything at all.” Because literally what it refers to is an effort in the last decade to try to establish some common um understanding, something that kind of vision for improved mathematics instruction. But in a technical sense, they’re not really standards because their standards-based has meant an association with a set of ideas that were essentially visions more then standards. But if I want to kind of adopt that as a phrase then what I think it generally seems to refer to is more—a set of practices that people expect to see in classrooms, a commitment to um mathematics taught more for understanding and not uh—I would say for understanding and also for skill as opposed to simply for road um skill. But it’s a phrase that hasn’t served us particularly well because it’s a cover for a whole bunch of things I’m sort of clumsily list—listing. And it hasn’t—it hasn’t promoted the kind of discussion about what is it about the math that we’re hoping to accomplish? What is it we expect that kids have to be able to do in order to learn that math? What do classrooms need to be like? And the answers to those questions, which are the real questions are more complicated then a label um because there’s no single way a classroom should look in order to um promote a, let’s say, deeper—deeper learning of mathematics that’s of a very worthwhile kind. It could look a number of ways but by having a label and then listing things like open-ended problems, or real-world problems, or discussion-oriented classrooms, we don’t do ourselves very much good. So I would say that in general we’re better off trying to disaggregate that phrase “Standards-based instruction” and try to think instead what is it about math that we’re talking about? What is it about kids’ opportunities to work on math we’re talking about? What is it we’re talking about what teachers might do in classrooms to promote that? And I think that would get us further.

Um, sure. (interruption) I mean I think that we have a lot—we have some history of trying to talk about the mathematics we’re hoping kids might learn and that includes both an expansion of the topics we think school mathematics ought to cover and weakly or maybe strongly, depending on how you think about it, more articulation of what I would call practices of mathematics. Sometimes they’re referred to as processes. And in the least specific way we say things like problem solving or communication. But I think that we’re better off when we try to think more mathematically about the phrases we use. For example, mathematical reasoning is clearly something many people have been concerned with. But mathematical reasoning um entails a set of practices that one does in order to make a mathematical conjecture, in order to investigate it, in order to establish whether you think it’s viable, in order to prove it. And if you look carefully at what you have to do to do those different things, it opens up um a lot more detail about what teachers might try to design for kids to be doing in classrooms, which is a far sight from simply saying, “Tell me what you think” or “Why did you come up with that answer?” Um, the more detail you can put around an idea like mathematical reasoning and the more you can try to uncover what are some of the things one does to build a mathematical argument, the closer you are then to being able to start considering what is it 1st graders might do to begin developing that kind of capacity? What would you hope the 5th graders would be doing? Um, and so one aspect would be that I think we need—we’ve done and need to do more of is what are the sort of content areas and practices of mathematics that we think are crucial for kids to develop and why? And then we have a host of questions about um how classrooms could be organized and designed? How learning experiences could be designed that teachers and kids could do together to promote that kind of mathematics? But I don’t believe there’s any single way to do that. I don’t think it’s about whether kids are in small groups or um, you know, what the format of the class is. I think instructional formats have kind of distracted us from the work of teaching learning. I think it has much more to do with um what kinds of questions teachers ask kids to work on? What kinds of interplay there is between what kids produce and what teachers do with it? So you can imagine entirely whole group instruction that would be very sensitively uh tuned to taking up what kids produce, using it to pro—move the lesson along and to tying it up at the end. And you can also imagine a teacher doing—working on similar goals but having much more opportunity for kids to talk with each other, much more chance for um there to be small group work. In either case the issue has to do with what the kids engagement with the mathematics is and whether and how the teacher can make sure that mathematics stays in focus and that things don’t kind of veer off to become social or about the context of the problem or something else. We would do ourselves again, I think, more good to elaborate the forms that teaching could take and keep more focal that the whole thing has to do with what kids are actually doing mathematically in classrooms. And that I don’t think can have any single form. (interruption) I mean to be pre—(interruption) Yeah, to be perversed you can—I can imagine extremely good mathematics lectures. I’m not advocating that for 1st graders but I think it’s silly to say lecturing can’t be sensitive to kids thinking and that small group work always is. It’s clearly not the case. It’s much more subtle then that.

Well, a complication of attending to the standards movement in developing teachers is that teachers go out to work in a wide variety of contexts and um unless you’re doing entirely site-based work with teachers—if you work in a teacher education institution or you do professional development with teachers across a range of context, um you have to think a bit responsibly about the varying environments in which teachers might have to do their work. So if you become heavily involved in promoting a particular approach or a particular form of form of teaching, I think you could end up serving teachers less well. Similarly, if you end up not doing the unpacking that—um, it seems to really require to consider um mathematics teaching. If you don’t talk about the mathematics, if you don’t talk about what it is teachers might be doing with the math, you cover up a lot of the work. So this—again, the standards-based movement has sometimes not promoted the kind of detailed investigation of the work of teaching in ways that really serve teachers well.

Um, I think three things. One thing that makes um learning to teach mathematics for understanding difficult is that most people, including people you would not expect to have this problem, have not learned mathematics in a way that equips them to do this work with students. Um, people may have been very successful at mathematics themselves. That’s not always a very good ticket to becoming a good math teacher. Um, someone might have struggled with mathematics and also not be really well prepared to teach mathematics. So most people haven’t had the kind of mathematical experience and education that they might need and most people are involved in relearning, unlearning, unpacking, or some form of mathematics learning that would be critical in order to teach for understanding. A second is that we have very few images about what teaching—that looks like teaching for understanding would look like and even less in the way of language to try to unpack it when we do see it. So you can watch classrooms where it looks like very different forms of work are happening by kids around mathematics but quite often when people watch examples of that they say things like, “The teacher doesn’t seem to be doing anything.” And as long as that’s where the conversation stays, then we actually don’t have very much developed knowledge about what the practices are for the teacher. And I can tell you from having worked on this for, you know, over 25 years myself, there’s an enormous amount of work for the teachers. It’s a very structured approach to teaching to design for kids to engage in serious mathematical work in school. So without kind of images and language to start studying what the practices are for the teacher, that’s very challenging in trying to learn. One ends up trying to invent a lot by oneself, so we don’t make the kind of progress that we deserve to make as a—as a community of practitioners, teacher educators, researchers.

I think um using more records of practice would be a sensible way to pursue it. That is, having more opportunities whether it’s um through research or in professional development with teachers to study records of practice, to learn to bear down more carefully on what teachers are doing, what kids are doing, to invent language if it’s necessary to do so. Um, there are undoubtedly also useful things to read. There have been people writing about practice. There might be ways of leveraging reading in order to learn to look differently. But I think records is at the heart of it. I think we haven’t made the kind of progress that we’ve needed to make around our knowledge of teaching largely because we haven’t had the ability to make records of teaching practice. And teaching practice as a result of that is neither public nor visible. Um, and if you think about the fact that the work of either learning to teach or educating teachers is about developing people’s capacity to engage in a very complex practice. If it’s invisible when it’s not public, you’re left with something very peculiar in—in the education of teachers or in one’s own learning. So it seems like a first step that we’re pretty well poised to uh use now is to create records that are possible to be used. But that’s only a small part of the step because records alone can’t do very much. It’s learning—developing ways of using those records whether for research purposes, or in teacher education courses, or um in professional development work that begin to take advantage of what’s afforded by having practice available to study.

Um, at times understanding a subject is similar to understanding it for teaching but often it’s not because understanding a subject for oneself means that you um see connections, that you can use mathematical—let’s say mathematics—you can you use mathematical knowledge (background noise), you can solve problems but teaching puts you in a domain where you’re using mathematics to solve a kind of problem that you don’t do if you’re not a teacher. For example, we’ve talked about um the problem of figuring out what kids know. A—a competent mathematician doesn’t have to spend his or her time studying what young children mean mathematically. That’s a kind of problem solving that a teacher has to do mathematically that a mathematician doesn’t have to do. So making up a quiz for 3rd graders to take on fractions is a mathematical problem not a cognitive or a pedagogical one. How do you decide which numbers to put, what questions to ask? Those are all mathematical questions that are examples of mathematical problem solving in practice that engage you in something that’s very different then simply knowing the math for yourself. And we’ve tended to think of those as pedagogy but if you look at them carefully, a lot of the thinking you’re doing is mathematical in order to do that sort of work. So for me, I guess, the difference between knowing a subject well for yourself and knowing it in a way that allows you to teach it is the capacity to consider questions like “What is there to know here? What does it take to do this particular thing?” You don’t have to have that kind of self-consciousness in order to perform competently on one’s own. But if you’re expected to help someone else learn mathematics, you have to be able to be much more self conscious about um the mathematical work involved so that you can solve the kind of mathematical problems that teaching requires.

Um, there are—I think some of the conditions that need to exist to help culturally diverse, linguistically diverse um—the kind of wide range of students that one um wants to help succeed in mathematics classrooms include um attention the kind of environment that the classroom is, the kind of norms that are established for a participation and engagement that make it possible for different—um students who bring different kinds of discursive habits to school to learn to engage in a broader range. You asked me earlier about libratory pedagogy. One element would be helping kids to develop discursive practices beyond the ones that they’ve learned at home. And that would be true for most students in the class but it’s particularly true for kids who are linguistically um diverse. Um, another might be to consider carefully the kinds of problems—mathematics problems that one uses in class. There’s been a great trend toward the use of contextualized real-world problems on the premise that those are somehow more interesting for students. But thinking carefully about that, you load onto the mathematics then—cultural baggage that may actually interfere with mathematics learning. And I think there’s quite a lot to be considered about whether in fact relatively abstract problems might engage a wider range of students better then our current tendency to use highly contextualized problems. Um, my experience has been that even young children get extremely fascinated with um highly abstract problems, almost puzzle-like problems that don’t require kids that on top of learning of mathematics to also negotiate cultural context and situations around the math that may not in fact be part of their own background. Um, and a final one that I think is worth considering is the range of participation structures that teachers provide for kids to engage in mathematically. For example, if a class is engaged in reasoning about a solution, there are many different kinds of turns that kids can take as they contribute to the reasoning about a solution. And I think that that’s an example of detailing teaching practice much more carefully. Some questions teachers pose can be ones that kids who are just learning English can answer easily and be part of the discursive stream and others require quite a lot more language competence. And a skillful teacher can dole out terms of that kind over time in ways that enable lots of kids to be in it. My experience has been that when kids take the more compressed, more um simple terms to answer and do so successfully, they also grab at the more complex ones over time. But that kind of technology around the questions that teachers pose and the kind of um opportunities to participate that they give kids is a very good example of without—if we don’t have that kind of detail, teachers will ask without thinking about it like I’ve done in my own work a lot of the same sort of questions and only some kids will jump on those. And then you start seeing classrooms where, you know, there are very uneven patterns of engagement with the mathematics and participation. So it takes me back again I guess to the study of the practice of teaching and learning to have a wide repatra of the kinds of questions that can be asked that are mathematical in nature but have different sorts of complexities to them and learning how to manage those to design a class discussion.

I guess on one of the things I’ve uh become increasingly curious about is when and how to use context around mathematics problems. Um, at times it seems that creating a context around a math problem helps to pull kids into something, helps to kind of uh engage them, causes a reason for the math to be learned, and so on. But at other times it can be very crowding. It can actually obscure some of what’s really exciting and interesting about the math. Furthermore, it can distract kids and kids can be very engaged into tales about the context and kind of lose sight of the mathematics. Now you could arguer, that’s the point, they should be learning to apply mathematics. So really answering that question requires thinking again about what the purposes of mathematics instruction are. But if you take seriously that one thing mathematics provides is the capacity to abstract and the capacity to manipulate very complex relationships simply and efficiently then designing ways for kids to engage in both contextualized and abstract problems and the movement back and forth between them seems very important. And I’ve experiment—I haven’t—wouldn’t say I’ve been a curriculum builder, but I’ve experimented a lot with how different kinds of tasks play out and I—I would say I’m still um learning as a teacher what kinds of tasks offer lots of opportunities for different kids to do different kinds of work that afford opportunities for abstraction and generalization as well as contextualization, and that I’m still very much exploring that. But I think what I have now that I didn’t have, you know, 15 years ago is more lenses to pay attention to what the problems are that I pose and better capacity to frame a problem and prove a problem, study how it works and prove it better for the next year or the next day. Um, but those issues about contextualization, about sort of layers of problems so that different kids can do different kinds of work with them, those are things that govern a lot of my work around um the curriculum that I use with kids.

Well, in my own experience—and this really is just for my own experience and not from some broader observation—I’ve taught in a school where over half the kids didn’t speak English as their primary language and many of them were very limited English speaking. And perversely what I saw over the years was that um having mathematics class be a place where there was a lot of talk and a lot of pictures drawing and a lot of reference to objects and pictures and symbols, enriched the language environment for the language development of the kids. Now some people questions that because they remembered a day in where in that school we had done largely paper/pencil work that was highly symbolic. And they use to say, “Well, when we got these kids and they didn’t speak English very well, that was a safe haven for them. They could at least manipulate the symbols.” But that wasn’t my experience. It was that over time being in an environment where there was so much talk and so much discussion and so much multiple representations of ideas seemed to speed the language development of the kids. Now I wouldn’t say that that’s research, it’s just that I became more—more convinced that—maybe I should say I became less concerned that there were such intense risks involved in using so much language around math. I thought there was a lot of care that one had to exert so that it didn’t just become an environment where only those who spoke English competently and fluently could participate. But if one exercised that care, it seemed that it contributed to the language development rather then um impeded it.

Um, I guess the two things that I’m most concerned with are—and they’re related—one is, making the practice of teaching a more visible and detailed object of study by teachers, researchers, teacher educators. Taking more seriously that it’s not enough to say this is complex work. That leaves us with nothing. It’s complex work but it can be studied and learn and it’s not something one is born to do, it’s not something you just figure out by chance. But we need a better kind of uh method by which we um develop a kind of focus and detail around the work without thinking that we’re trying to find a single method of teaching. And second, I guess, um my other personal soapbox is the crucial role played by mathematical knowledge in teaching well. I think that anytime we forget that—and it’s forgotten by two kinds of people. It’s forgotten by people who know a lot of math because they simply take for granted how much the math that they know is playing a role in what they’re doing and it’s also forgotten by the people who don’t know so much math because they somehow delude themselves into thinking that the concerns they’re bringing to mathematics instruction are sufficient to teaching well. And so I’m left sometimes feeling like I don’t have anybody to talk to about this because I’m a person who didn’t really have that much mathematical education but I became profoundly aware as a teacher how central that was to teaching well. And oddly its not been so easy to get either group, either those who are very concerned with math or those who aren’t to under—to take seriously enough, in my opinion, how crucial that—the role of mathematical knowledge is in teachers work.