To culminate the Workshop activities that addressed the question, “What mathematical knowledge does a teacher need to teach well?” a panel reflected on their own experiences and how those experiences contributed to their perspectives on the relation between mathematical understanding and teaching.

Perspectives from a Mathematician

*Alan Tucker, State University of New York-Stony Brook*

Perspectives from a Mathematics Educator

*Deborah Schifter, Education Development Center, Inc.*

Perspectives from the Community College

*Gladys Whitehead, Prince George's County, Maryland, Public Schools*

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Knowing and Learning Mathematics for Teaching
What Kinds of Mathematical Knowledge Matter in Teaching?
To culminate the Workshop activities that addressed the question, “What mathematical knowledge does a teacher need to teach well?” a panel reflected on their own experiences and how those experiences contributed to their perspectives on the relation between mathematical understanding and teaching.
PANELISTS
Perspectives from a Mathematician
Alan Tucker, State University of New York-Stony Brook
Perspectives from a Mathematics Educator
Deborah Schifter, Education Development Center, Inc.
Perspectives from the Community College
Gladys Whitehead, Prince George's County, Maryland, Public Schools

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Knowing and Learning Mathematics for Teaching
Perspectives from a Mathematician
Alan Tucker
I am certainly humbled by coming here, and while I think I have some mathematical knowledge, I would never dream of thinking I could come close to teaching in an elementary classroom. So, I find the word knowledge a little difficult to process, and I think, perhaps, a lot of the issues are more about reasoning. Moreover, there's too much knowledge. Every single instance we've seen today required a lot of insight and knowledge, and it doesn't seem like a typical preservice teacher in three or six credit hours is going to pick up all this knowledge.
Teaching is as much an art as a science. I think, rightly or wrongly, that being a school teacher is like being a doctor. Doctors have two years of academic training, two more years of quick rotation in different settings, a residency, then two to six or seven or eight years of internship. With only a couple of years in the classroom, teachers are learning as they go along, and they continue to learn.
The mathematical knowledge teachers need is the foundation for their apprenticeship and their lifelong learning. They need critical-thinking skills. I like to think of keeping the principles and the knowledge in mathematics as simple and clean as possible and emphasize building on reasoning skills. As a simple example, if I'm asked a question about division, I'm going to think about division as repeated subtraction. For a lot of the theoretical questions, that gets me a long way. Simplifying the knowledge and beginning from a safe base is one way to get going. From another perspective, I'm running a “math across the curriculum ” project, and all the other disciplines seem to have the same problem we have. It's not a matter of facts; it's a matter of what my friends in other disciplines are calling critical-thinking skills. The focus is not mathematical knowledge or mathematical reasoning; I believe there is a more generic way of reasoning in which we're interested here.
Just a quick aside, at the University of Maine, the math department and the English department are in the same building, and one is at the north wing and the other is at the south end, on every floor. This is the critical-thinking skills building. Many of you may know that, before computer science was a well-defined discipline, math majors and English majors were the people that a lot of individuals thought made the best programmers. The point here is that many problems we have transcend

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mathematics. Critical-thinking skills run across the curriculum.
The reasoning skills teachers need are mind-boggling because there are many situations where we can use mathematics. There are endless numbers of applied situations. Sometimes, examples can't really connect at all, or only marginally, with drills about addition skills and subtraction skills. Sometimes there are natural connections; sometimes these skills seem unconnected.
I find that a common knowledge base can be both an asset and a liability. There are valid conclusions that people build and we use, but then there are lots of false conclusions that have to be broken down. Of course, this is the way the history of mathematics evolved, without a rigorous foundation for much of anything for years and years, until at some point or another, things became so complicated people had to restart. Somehow I think this evolution of our discipline is something we should try to model in the classroom. This approach has its pluses and minuses. Everything we do should have tremendous tension in it —that's they way I look at life—opportunities and things that can go wrong. Students have to be taking chances and welcome the opportunity to do so. The teacher has to be taking chances and try to build on something for which they were not prepared, but that's where the art comes in, as well as the experience. I think that there is also a tension between mathematical reasoning and very applied practical things. At the college level, we talk about teaching service courses in applicable mathematics. We have all sorts of clients telling us what to do and little time to do it. To explore a number pattern the way Deborah did with her class is unthinkable or pushing the envelope a long ways to do such things on a regular basis in calculus.
There are obviously valuable experiences in theory and in applications. I didn't see any really applied things during the workshop today. I didn't see examples of running a store. Many things like that can be done. My dad was a mathematician at Princeton, a very bright guy, and he said at Princeton the most important word in the mathematics department vocabulary was “taste.” There are all sorts of examples you can give. There are all sorts of theoretical examples. There are all sorts of applied examples. This is part of the art, to have good taste in the examples that bring things together. I don't know how to teach this to preservice teachers. When you see it, you feel really good about it. The taste in mixing these different approaches, in mixing pure things with applied things, is, I think, where the excitement lies.
Despite widespread concerns about the mathematical education of young people and about the general public's mathematical literacy, I'd like to strike a positive note in my conclusion. When I was in college in the ‘60s, I can remember headlines in the New York Times saying something like “63,255,000 People Employed Last Month.” Today we never hear how many people are employed. What we hear now is the increase, the delta. Not only that, the newspapers now talk about the change in the delta. They say that the growth in job creation slowed by 30 percent last month or that the number of new people applying for unemployment insurance dropped last month by 30 percent. Such sophisticated mathematical information was never printed thirty years ago. On Monday Night Football, I've heard an announcer say the receiver has practiced hundreds of times running downfield X yards and turning around to catch a pass. The idea of fixed yet unknown quantities is becoming part of the

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vernacular. I don't know what sophistication is going to exist in thirty or forty years when the teachers who are training now are still teaching.
While we lament students' skills, people have, at a common level, a knowledge base right now that is pretty high. One of the exciting challenges for teachers is to build on this knowledge. You can lament what people don't know, or you can take advantage of what they do know. What we saw today at every stage along the way is that young people know a lot, and the opportunities to build on and reinforce that knowledge seems to me very, very exciting.

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Perspectives from a Mathematics Educator
Deborah Schifter
Yesterday, Mark got us started making lists of what teachers need to know in order to teach mathematics effectively. In the time I have, I'd like to add a few items to those lists. Some of these items come out of the discussions we've been having; others are my own.
When we looked at Deborah Ball's video this morning, we paid particular attention to how she was helping her students articulate how they figured things out and how she used their own logic to show them the elements of a mathematical argument. This points to a first item that could be added to our list. Teachers need to have a strong sense of what constitutes a mathematical argument, mathematical justification. Furthermore, they must be able to draw out students' ideas to illustrate elements of mathematical justification.
In the last break-out session, we started talking about the importance of teachers being able to recognize valid mathematical argument. This is an issue that is more basic than those we identified in the morning. Many teachers haven't developed the skill of listening to the mathematical justifications or methods that students use to solve problems and determining whether they are mathematically valid. That is, some teachers have no way of assessing mathematical validity if a student presents a method or argument different from the one the teacher learned when he or she was in school. So this is another item to add to our list.
Even when a child presents a right answer, the teacher needs to go further, to look at the mathematical argument and determine whether the argument itself is valid. And when a mathematical argument is invalid (whether the child's answer is correct or not), the teacher must be able to examine the child's logic to determine what aspect of the child's thinking is valid. Is it on the order of a careless arithmetic error? Or is there something more substantial—an important idea that the child needs to work on? And if so, just what is that idea?
Another of the issues we touched on this morning is the importance of being attentive when a student offers an idea that broaches an important mathematical domain or “habit of mind.” We were thinking about the importance of generalization, and how in the video, Deborah Ball picked up on students' ideas to help the class think about a general claim. The role of generalization is critical in mathematics and teachers themselves need to

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learn to develop the habit of asking such questions as, “Does this always work?” or “How do you know it will always work?”
I'd like to take this opportunity to point out a dangerous assumption we might make when we listen to students to see if they are getting close to an important mathematical issue. That is, it is easy to attribute too much to what the children are doing. Here is an example that comes from a second-grade teacher: She reported that, early in the year, when her students were working with sums up to 10, they noticed that four plus six equals 10 and that six plus four equals 10. They coined the term “turn-around” to indicate when you add two numbers together, you can reverse the order and they add up to the same sum.
Many of us, listening to the children's conversation, would say that they understand commutativity as a property of addition. But at a certain point in the year, the teacher decided to ask the class explicitly, Do turn-arounds always work? Even though it may have sounded to our ears all year long that the students had been talking about turn-arounds as a property of addition, when the question was posed, they weren't sure. Some of them thought the answer was yes. Many thought the answer was no. They all began to test it out with large numbers. Only after a period of exploration and discussion did several children consider what the operation of addition does in order to develop an explanation for why turn-arounds always work. This tendency of attributing too much understanding as we begin to listen to children's mathematical ideas could be an item on our list.
My discussion group was charged with the task of thinking about intuitions and dispositions, some of the things that might fall under the general umbrella of mathematical knowledge but don't generally get classified as mathematical content. In our discussions, we started to touch on some things that aren't likely to come up through the analysis of mathematical tasks we were doing in our break-out session. Before I name them, I want to say that, for some of us, these things are almost like the air we breathe. It's hard to see them because they come so naturally to us—but they do need to be stated explicitly.
The list generated by our thinking included: Teachers need to learn that mathematics makes sense and that they should approach mathematics with the expectation that they can make sense of it. It seems to me that many, perhaps most people—teachers among them—graduating from our high schools and colleges are separated very early on from their own mathematical sense-making abilities. It would be good for everyone, but it is particularly important that teachers be reconnected with these abilities. They need to learn that mathematics is about ideas. They need to come to see that they, themselves, have mathematical ideas. Teachers need to have the experience of having mathematical ideas and of making sense of them.
Through the experience of doing mathematics, teachers can become familiar with the pleasure of figuring things out, and seeing how things fall into place. But along with, or most often just prior to, the sense of pleasure come experiences of frustration and confusion. Teachers must learn to work through those uncomfortable feelings to come to the point where the problem they are working on becomes clear. Certainly, if the teachers themselves never have the experience of working through their frustration to this place where it all comes together, they're never going to be able to tolerate their students meeting frustra-

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tion. And if we want students to develop deeper understandings, they must be allowed to work through their frustrations and confusions.
Another item for the list is that teachers need to become curious about how mathematics works. They might become curious about the number system, for example, learn to formulate their own mathematical questions and learn how to pursue answers to those questions. Teachers have to become mathematical thinkers and questioners in their own right. It feels critical that these items become part of the explicit agenda in teacher education courses.
There's still another item I'd like to address. I have been having a hard time articulating this point, and when I talk to colleagues, they don't always see it as mathematical, but I think it is an important mathematical issue, so I'll try to say it here. Teachers need to learn to look at a classroom scene and discern the mathematics in it, to recognize what is mathematical in what a child is saying. Many of us talk about how, when we show a video, it is difficult to have a group focus on the mathematics. A group of teachers is likely, instead, to talk about whether the children in the class are paying attention, whether the child who speaks feels confident or not, and similar matters. Often, we attribute such behavior to avoidance and believe the group doesn't want to talk about mathematics. But I am coming to believe it's not a matter of avoidance. Instead, I wonder whether the mathematics in the video simply isn't seen; the mathematics is not recognizable. This then becomes part of the agenda in a course for teachers: to learn to attend to the mathematics in what children say and do.
As I'm talking about some of these very basic issues of what it means to do mathematics, to recognize mathematics, to be a mathematical thinker, I feel the need to mention how important it is for us to take an appropriate stance in our work with teachers. When we talk about what's missing in teachers' mathematical knowledge, about mathematical capacities that are lacking, it is often with a tone of disparagement. But if we want to encourage teachers to venture forward, to make public their mathematical ideas—which are often just baby steps and, in many ways, not so far ahead of the third-grade children we saw on videotape—as we're working to help teachers develop their mathematical capacities, we must act with respect and generosity.

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Knowing and Learning Mathematics for Teaching
Perspectives from the Community College
Gladys Whitehead
When I was first asked to sit on this panel and received the questions about mathematical content, I said, well, I can write my description of what the content should be on a piece of paper, but I would like to come and learn about the concerns of others. One of the things that impressed me is that you started with the tasks, and from the tasks, you decided to pull out what mathematical content would address those tasks. By the time we finish, we will be addressing another question, about teaching that content.
I've struggled with issues of teacher preparation from two perspectives. I have been at the community college, where we were training elementary education majors and constantly struggling with these concerns: (1) have we given them enough content, and (2) have we given them adequate teaching strategies. As the supervisor of mathematics in Prince George 's County, I have the opportunity to go into classrooms and observe teachers. There are two types of observations that are probably the nemesis of a supervisor. One is when you walk into a classroom and clearly there is no content, period; that's the easy one. The tough one is when you walk in and you know the teacher understands the content but is unable to help students get a grasp on the concept.
One question centers on what content we want teachers to have. A tougher question, and one that comes full circle in the question we consider for the next part of the Workshop, is how do we as college professors train our teachers to really teach. In other words, it is not enough to just know the mathematics. How do you give them the skill of pulling out the reasoning, of knowing how to question? We have learned our teaching strategies over the years. We are very experienced and have developed our strategies by trial and error. So, how do we capture all of this experience and hand it to a new teacher?
I watched an excellent new teacher miss what we call a teachable moment. I wondered, how would I have trained her so that she didn 't miss that opportunity? After the observation, I shared with her what should occur the next time this happens. The teacher introduced a problem where the students engaged in discussion, wrote in their journals, and shared their reasoning skills. “You have a dozen eggs, and then five of the eggs are broken. Write as a fraction how many are unbroken.” A little girl came up, drew a

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little egg carton and darkened in the broken eggs. She had turned the numbers around. She had come up with a right answer, but it was for a different question. The teacher made her sit down rather than build on why she had turned that problem around in her mind. The teacher sent another student up, who erased the board completely and started over. Teaching teachers how to teach is an abstract concept, but that is really what we are about here. I can make sure that teachers know the content, but if they are going to be teachers, how do I get them to use these questioning, teaching strategies?
At the community college, we decided that, as much as possible, we need to model what we want to happen. Many of us college professors are die-hard, “lecture-type” individuals. We do not always practice group learning. For example, some of us walk in, fill the board with “mathematics, ” turn around, ask if there are any questions, and we are finished. At the community college, we have been working hard to have labs with our students, hands-on activities, and to practice what it is we want to see happen in the classroom. With elementary education majors, more than that has to happen. You have to convey to them the sense of learning and the sense of being able to help students learn.
After I worked on teacher inservice for so long, I began to build on the advice of the teachers and try to give them some presentation experience in their first two years in our program. I asked my students to present a lesson in their second semester course. They had 10-15 minutes to explain a concept to their classmates. When we finished the exercise, some of my students were astonished to discover that they were really uncomfortable. They did not know how to ask their classmates questions to get at what they wanted. They had envisioned the presentation would be easy. After all, they would be teaching elementary students, and how hard can that be? This is really the question, “How do we get the universities to not only teach the content, the easy part for us, but how do we teach preservice teachers to really be teachers?”
I'm pleased that we are looking at the issue of teacher preparation. I think this implies a bigger question for colleges and for professors. We are the ones that need inservicing now. We need to know how to turn it around. The public schools have always been ahead of us. Prince George's is one of the most progressive community colleges in the state of Maryland. Working with our school system has helped us revamp our training of teachers. Thank you for the chance to share with you.

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