A university professor and education researcher who also teaches 5th graders, Magdalene Lampert has compiled a library of videotapes that bring images of thoughtful practice right into the teacher education classroom.

*Many of those working to improve schooling these days talk about the need to teach for understanding. Most teachers probably think they're already doing that. What's your idea of teaching for understanding, and how does it differ from what generally goes on in classrooms?*

To me as a teacher, understanding means sense-making. It means the conversation in the classroom is dominated by the questions, “Does this make sense to you?” and “Why or why not?”

And it involves communication—being able to explain your thinking so that someone else grasps your ideas. So teaching for understanding has two components: the individual component and the shared component. From that standpoint, I see my responsibility as a teacher as both helping individuals to understand and helping them to express their understanding so that other people in the group understand. It's all related—because we know from the work of psychologists that explaining your ideas is a pretty good way to improve your own understanding.

**But isn't that what goes on in classrooms all the time?**Not really. People talk a lot in classrooms, but they don't always pay careful attention to whether the talk is being understood.

At the most basic level, we might compare teaching with telling and learning with listening. But teaching for understanding is much more than telling, and learning with understanding involves more than just listening. In my classroom, I use the word “communication” to replace the word “telling.” What I mean by communication is that not only are you responsible for expressing your ideas; you're also responsible for finding out whether the person you're talking to understands what you're talking about. That notion applies to both the teacher and the students.

**Much of the talk in typical classrooms could be classified as either lecture or recitation. In recitation, the questions have a predetermined answer.**Yes, and the exchange often stops with the teacher's judgment of the answer to the question. The interaction is question, answer, judgment. I'm trying to add something to that pattern. The next step is for the teacher to ask, “Why do you think that?” In other words, when a student gives an answer, he or she has only begun to participate in the conversation. The teacher needs to work to understand the student's thinking before passing judgment on the answer.

**You've been referring to your students in your classes. You do things very differently from most professors of education; you actually teach mathematics to a class of elementary school youngsters in a public school.**I have usually done that, but I'm not doing it this year. I've changed jobs, going from Michigan State University to the University of Michigan, and I'm taking some time to settle in and find a place where I can continue my school teaching.

**In Lansing, you taught the same class of students day after day?**Everyday from the first day of school until the last, for about an hour a day.

**Why did you do that?**For several reasons. One of them has to do with establishing a culture in the classroom. A big piece of teaching for understanding is setting up social norms that promote respect for other people's ideas. You don't get that to happen by telling. You have to change the social norms—which takes time and consistency.

Another reason for teaching a class every day has to do with building a coherent curriculum that lets students examine mathematical ideas in considerable depth. Instead of doing lots of little problems each day, I try to spread a big multifaceted problem over time. We keep getting back to it, seeing different ways of looking at it.

The third reason for teaching every day is that, as a researcher and a teacher educator, I feel it's important for me to know those aspects of teaching that you might not notice if you just taught demonstration lessons now and then.

**You use examples from your own teaching in your university teacher education classes, right?**Yes. At first I invited my methods students to come to visit my 5th grade classroom. Sometimes I had them come in groups of three, observe the lesson, and then meet with me for an hour afterward. Sometimes I had them interview the 5th graders—I put them in a sort of one-on-one relationship—and then the education students came back later to watch lessons on the topics of the interview.

In 1987, I started making some videotapes of my lessons. The tapes were used not only by me in my courses but by other faculty members in the school of education in order to bring an image of practice right into the classroom. Based on this experience, I developed a project with my colleague, Deborah Ball, to collect an entire year's worth of documentation of our lessons.

**An entire year of classes on video?**Yes, and the video is supplemented by lots of other materials. Deborah teaches 3rd grade in the same school where I taught 5th grade. And for a year, we videotaped our math lessons almost every day, starting on the first day of school and lasting until the end of the year. One reason for doing that was that often people would look at a videotape of a lesson or visit a class in, say, January, and the situation would look very unusual to them. They would be curious about how we organized things so that the students would behave in the ways they were observing. So we thought it would be useful to show how that happened over time.

**It's very unusual to have a visual record of a whole year of teaching a single class.**And it's very complete, because in addition to the videotapes we have all the children's written work for every day. We have the teacher's journal, which is a written record of both plans for each lesson and reflections on how the lesson went. We also had observers there every day with a set of five or six questions that they answered about every lesson. We have that material from every day of the year, plus quizzes, interviews with the children across the year, and an interview with each one a year later.

**I believe I recall that you also have videos of the chalkboard, showing how it changed in the course of some of the lessons.**Yes, that's right. Not all classroom communication is verbal; some of it is written. For example, we use drawings to build mathematical models and illustrate mathematical ideas.

**How do you expect the videos to be used?**The videos and their accompanying materials are being developed under a teacher preparation grant from the National Science Foundation. We hope to begin sharing them in two ways. Here at the University of Michigan, I'm working with a team to develop what we're calling the Instructor's Working Environment—an electronic support system for using multimedia cases in teacher education.

Then for the next couple of summers, Deborah Ball and I will be running institutes for teacher educators who want to come and work with some of the materials and take them back and try them out in other settings.

**What makes your research most unusual, it seems to me, is that you focus on your own teaching rather than on the teaching of others.**Yes, I collect a lot of information on my own teaching. Having six or seven years of detailed information gives me an opportunity to stand back from the situation and look at it with a somewhat less passionate eye. I think there are definite trade-offs. I'm invested in my work as a teacher, but I think that when writing about myself I can deal with things that would be hard to handle if I were writing about someone else.

**Some researchers would undoubtedly criticize your approach as too idiosyncratic. You're studying the teaching of one individual—yourself. How can you generalize about that to apply to others?**Well, I was talking about that with a friend recently. I think that the model that best fits my way of thinking about this is the narrative. When a person writes a story, the issue is not generalizability; it's whether the story has enough information to let the reader judge whether the story is believable.

**What else can you say about what you have learned from the kind of research you do?**One thing I've learned is how hard it is to teach for understanding. I think I've learned something about how one makes strategic decisions in a context, about how one brings knowledge and understanding to bear on the day-to-day, minute-by-minute activities in the classroom.

But I'm also learning what the practice of teaching mathematics for understanding can be—what's possible. In medical research, before a drug or treatment is widely used, it's tested under ideal circumstances to see what its powers are. Once it's established that it is possible to do a heart transplant, for example, doctors can begin to do it more commonly. That's what I see myself doing: learning what's really possible—what teaching for understanding can look like in a classroom that has 29 children of different socioeconomic, racial, and cultural backgrounds and different ability levels. And what it takes to maintain that kind of teaching on a day-to-day basis—recognizing, by the way, that I only teach mathematics and not all the other things that most teachers do.

**Does your way of teaching apply to other subjects as well, in your opinion?**I think so. We began by talking about sense-making. The expectation that people should do things and say things that make sense applies across subject matters. But the way in which we answer the question, “Why does that make sense?” is different in different subject matters. Mathematical evidence is different from the evidence one would use to support a claim in history or literary criticism or economics.

**It fascinates me that you work at so many different levels. For example, in the teacher education classes in which you use videos of yourself teaching, you try to create the same kind of conditions for learning that you do for your elementary students.**Yes. In both I try to establish a culture of respect for other people's ideas. And respect for your own ideas. It's quite remarkable that many people get to college without having a whole lot of respect for their own ideas. They expect the professor to do the thinking in the class, and then they judge it to be relevant or irrelevant. And trying to turn that around and make it a situation in which the learner also does the thinking is as difficult in undergraduate teacher ed. courses as it is in the 5th grade.

**Can you give an example? Let's start with the 5th grade. What does it look like when you're trying to get 5th graders to understand something in math?**All right, here's an example. We were working on rate and ratio in my class, so I gave my students this problem: if a car is going at a constant speed of 50 miles per hour, how far will it go in 10 minutes? Now, if you're familiar with elementary school students, you know that often they'll look at a problem and ask, “What should I do with the numbers?”

Well, in this problem, there's a 50 and there's a 10. You could add them, subtract them, multiply, or divide them, right? Well, there's some sense that in time, speed, and distance problems, you're supposed to either multiply or divide. If you multiply, you get 500. That doesn't seem sensible, because if a car is going 50 miles an hour, in 10 minutes it's not going to go 500 miles.

So the first thing I would say is, “What do you think about this? How far do you think the car would go?” And let's just say that some students say 500 miles, and some say 5 miles. Then I would say, “Let's look at these two ideas. Let's look first at the 500 miles. Does that make sense? Have you ever been in a car that could go 500 miles in 10 minutes?” I wouldn't say, “That's wrong,” or “That's ridiculous”; I'd say, “Let's think about it.”

Now, that one is pretty straightforward, but let's look at five miles. When I gave this problem, that was the answer that most of my 5th graders originally asserted made sense. And if you think about riding in a car, it seems reasonable that you might go 5 miles in 10 minutes if you were going 50 miles an hour. But one of the students said, “I don't think that makes sense, because the car's supposed to be going 50 miles in an hour, and if it only goes five miles in 10 minutes, then in 20 minutes it'll go 10 miles, in 30 minutes it'll go 15 miles, and in an hour it'll only go 30 miles.” So I asked, “What do the rest of you think about that?” and everyone went busily back to work, trying to figure out, “Well, gee, we thought it was 5. Ten into 50 is 5. Why doesn't that work?” The burden was on their shoulders to figure out why it didn't work.

Now, in this case, another student started making a diagram. She drew a line, and next to 10 minutes she put 5 miles, next to 20 minutes she put 10 miles, and so on. And using the diagram she figured out that we needed a number that, when multiplied by 6, would be close to 50. Eight was pretty close, but wasn't quite big enough, so eventually they decided that the car goes somewhere between 8 miles and 9 miles.

Now, it sounds like the kids are doing all the work. What's my job? Well, first of all, I asked questions that would lead them to question their assumptions. I monitored the discussion so that students could challenge one another in ways that were civil and relatively safe. And when their thinking came close to a big mathematical idea, I helped them to see those connections.

**But in doing that, you always ask rather than tell.**Well, I wouldn't say I never tell anybody anything. But whenever possible I try to figure out a question to ask that will lead them to develop their own understanding. Or I'll “tell” them something that will provoke them to revise their own ideas.

**Now, how does this same idea apply in undergraduate teacher education?**What I want to do is make it possible for preservice teachers to formulate the kinds of questions about teaching and learning that researchers have studied, and explore them in real teaching situations. One way of doing that is show them a piece of videotape that will cause them to make a lot of conjectures like, “If the teacher only calls on students whose hands are raised, a lot of students will not be engaged in the problem,” or something like that. When that happens I say, “Why do you think that?” “Well, such and such happened on the videotape.” “Okay,” I say, “Let's look at another videotape. Let's also look at the notebooks of the students who were raising their hands. And let's refine that conjecture.”

**How did you decide to begin using videos for these purposes?**I began to use video when I wanted to communicate with other people about this kind of teaching. How do you explain to a beginning teacher or a school superintendent or a policymaker or another researcher what this kind of teaching is like? I was drawn into multimedia by an idea from Wittgenstein that understanding is enhanced by traversing the same territory in lots of different directions. That's something people can't do when they observe a live lesson. For one thing, they can't stop and ask questions while the lesson is actually going on; that would be too disruptive. They can only look at it with one lens at a time—or maybe two or three at most—whereas, if it's captured on tape, they can go over it as many times as they want. And it's remarkable how much new stuff you see every time you go over something, especially if you can do it with other people and talk about different perspectives. Using interactive multimedia enables us to do a more thoughtful kind of clinical education.

**That's especially important, because you're demonstrating a kind of teaching that in many cases your teacher education students haven't experienced much themselves.**That's right.

**As we've said, teaching for understanding is not common in schools; it's very demanding. Can we really expect large numbers of teachers to learn to teach that way?**Yes. I'm very optimistic about that—but it's not going to happen quickly. It's not going to happen by fiat. It's only going to happen if we recognize that understanding teaching is a problem similar to understanding mathematics or understanding history or understanding science. It's that kind of parallel that makes me use the same pedagogy with my teacher ed. students that I use with my 5th graders. Active, engaged, constructive learning is as important for college students learning to be teachers as it is for children learning to do math.

**What is your view of what schools need to be like?**One thing I feel strongly about is that schools need to make available lots of opportunities for teachers to work together on the problems of their practice. For example, groups of teachers ought to get together on a regular basis to look at children's work and say, “Do you think this student understands this idea? How would you work with this student?” Teachers of mathematics can get together to talk about what good problems are like: “What happened when you tried that problem in your class?” If teachers had more opportunity to work with one another on their practice, they could develop some mutual understanding about what's important to learn and how to teach it.

But teachers, on the whole, have not been treated as being capable of doing that sort of thing. They've not been treated as people who can and should think about curriculum and instruction. They've been treated as people who have to be told what to do, who can't think for themselves. And that seems particularly ironic when we hear so much about trying to get

*children*to think for themselves.One thing I've learned is that teaching is not about solving problems; it's about managing complexity—because every time I set out to solve a problem, another problem crops up. It was a big insight for me to say that's a characteristic of this practice, rather than my failure; that teaching necessarily involves trying to accomplish a lot of conflicting goals.

**I once heard you say that when your students watch a video of you teaching, they ask how you decide which student to call on, and they think there's a correct answer.**Yes, that's a good example. Their assumption that there

*is*a correct answer comes not only from their need for security but from the idea implied in a lot of research articles that, if we just do enough research, we'll come up with a perfect formula for whom to call on, when, and how often. That's just not going to happen. We might come up with such a formula under certain very carefully controlled conditions, but every classroom is different, and every child is different, and every moment of teaching is different.