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A panel addressed the following questions related to the mathematical knowledge of teachers: What are the mathematical resources that teachers need to teach well? How can teachers learn the mathematics they need to teach well? The papers that follow are edited transcripts of their remarks. Moderator: Deborah Loewenherg Ball, University of Michigan Zalman ~ Usiskin, Professor, University of Chicago Deborah Schifter, Senior Scientist, Education Development Center Marco Ishigaki, Professor, Waseda University Miho Ueno, Mathematics Teacher, Tokyo Gak~gei University Senior High School

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WHAT MATHEMATICS DO TEACHERS NEED THAT THEY ARE LIKELY NOT TO ENCOUNTER IN THEIR MATHEMATICS COURSES? Zalman P. Usiskin, University of Chicago It is a truism. A teacher of mathematics should know a great deal of mathematics. The higher the level taught, the more the teacher needs to know. For a teacher of high school mathematics, this means knowing a good deal of number theory, algebra, geometry, analysis, statistics, computer science, mathematical model- ing, and history of mathematics. This is what we might view as the traditional background of a teacher who is consid- ered to be well prepared mathematically. Even though it is good to take more and more mathematics, there is a problem that taking more mathematics creates. Often the more mathematics courses a prospective teacher takes the wider the gap between the courses taken and the courses the teacher will teach. The gap is both in the mathematical content and the ways that content is approached. An entire body of mathematical knowledge is ignored. L~ ~ t.,'d'` ~ There is a substantial body of math- ematics that arises from teaching situa- tions in much the same way that statistics arises from data and applied mathematics arises from real situations, and that deserves to be viewed as a branch of mathematics in its own right. ~ call this "teachers' mathematics." A project currently underway entitle(1 "High School Mathematics from an A(lvance(1 Stan(l- point" is (leveloping a first course in teachers' mathematics for high school teachers, and second and third courses are being planned. ~ will attempt to describe the motivation and content of these courses. THE PROBLEM Every teacher of mathematics nee(ls 1. to see alternate (definitions and their consequences; 2. to know why concepts arose anti how they have changed over time; 3. to know the wi(le range of applications of the mathematical i(leas being taught; 4. to discuss alternate ways of approach- ing problems, inclu(ling ways with anti without calculator anti computer technology;

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5. to see how problems and proofs can be extended and generalized; and 6. to realize how ideas studied in school relate to ideas students may encounter in later mathematics study. The result of the lack of teaching these ideas to prospective teachers is that teachers are often no better prepared in the content they will teach than when they were students taking that content. For instance, they may know no more about logarithms or factoring trinomials or congruent triangles or volumes of cones than is found in a good high school text. THREE KINDS OF MATHEMATICS FOR TEACHERS Three kinds of mathematics content are particularly needed by teachers. Each might be said to consitute a facet of looking at school mathematics content at a deeper level than is possible for high school students. One focus is on mathematics particu- larly useful to high school teachers that might not normally be encountered in the standard courses taken by mathematics majors. Box ~ contains an example. There is an analogous theorem for inequalities, which ~ do not have the time You can add the same number to both sides of an equation, or multiply both sides of an equation by the same nonzero number, and the resulting equation is equivalent to the given one. But if you square both sides of an equation, you may gain solutions. And cubing both sides of an equation Joes not affect the solutions. What about taking the log of both sinless Or taking the sine of both sinless How can one tell, in general, whether an operation on both sides of an equation will change the solutions to the equations Here we are concerned with real-number solutions. Then an equation in one variable can be thought of being of the form fix) = Ax), where x is real. Applying an operation to both sides is like applying a function h to both sides. This results in the equation htf~x)) = h~gfx)~. There is a very nice theorem: The two equations fix) = gtx) and htf~x)) = h~gtx)) are equivalent if and only if h is a one-to-one function on the ranges of fix) and Ax). Examining this theorem and its special cases unifies the solving of equations and gives the teacher new insight into the process of equation solving. MATHEMATICAL KNOWLEDGE OF TEACHERS

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to mention here. But the more important point is that there is a lot of content of this type: theorems that integrate content that might be taught in different units or different years; theorems that shed light on formulas, figures, or functions; and so on. The second focus is on the extended analysis of problems. Recall the four problem-solving steps of Polya (19521: understanding the problem, devising a plan, carrying out the plan, and looking back. Most analyses of problem solving devote their time more to devising a plan than any other step. This is important, but it is also quite important to examine the last step: looking back. This means looking at a problem after it has been solved and examining what has been done. Will the method of solution work for other problems? Here is an example from Dick Stanley of Berkeley (Box 2), who is one of the main authors of the materials we are . ~ clevlslng. The third type of mathematics is the explication and examination of concepts (Box 3~. Recall the well-known problem in which a rectangular sheet of cardboard is folded into a box by cutting out four congruent squares from each corner. What is the maxi- mum volume of the box that can be created If the cardboard is 12" by 18" and each square has side x, then the box has height x and length and width 12 - 2x and 18 - 2x. So the problem is to maximize xt12 - 2x)18 - 2x) over the range of possible values of x. The problem can be Jone these Jays by graphing the function fix) = xt12 - 2x)18 - 2x), or it can be Jone with calculus, or in a numerical way by appropriate substitution. It happens that the volume is maximized when x = 5 + ~17 . But that tells us very little- it does not give us intuition into the problem. Why are there two solutions Are they related in some ways If we leave the problem without examining such questions, then we have gone no farther than the typical class. We can gain more intuition by letting the length of the rectangular sheet be 1 (say 1 foot) and the other dimension be w. If w is near zero, then the rectangle is long and thin. If w is near 1, then the rectangle is near a square. If w is large, then the rectangle again becomes long and thin. In our example, since 18 is 1 .5 times 12, w is 1 .5. How is the value that maximizes volume related to we The relationship turns out to be inter- esting and gives insight into the problem that was not obvious. MATHEMATICAL KNOWLEDGE OF TEACHERS PANEL

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Consider the idea of parallel lines. (1 ) Parallel lines are lines that are equidistant from one another. In this conception, parallel lines are an instance of parallel curves. This conception explains why train tracks are called parallel. (Tracks are parallel even when they curve.) (2) Parallel lines are lines that do not intersect. This conception places parallel lines as an example of disjoint sets. This is the usual definition of parallel lines. (3) Parallel lines are lines that go in the same direction. A~aebraica~Y. this means lines ... .. . . . ... .. . . v ,, wits the same slope ano so under thus conception unless an exception is mane a fine is parallel to itself. Sameness of direction intuitively underlies why, when parallel lines are cut by a transversal, corresponding angles have the same measure. Virtually every mathematics concept and all the important ones can be examined in a variety of ways. When we give a definition for an idea, we almost immediately put blinders on the other ways of looking at the idea. By reexamining the variety of ways from a broader perspective, we can appreciate why students may have difficulty con- necting various aspects of the same idea. SUMMARY '~eacher's mathematics" is a field of applied mathematics that deserves its own place in the curriculum. There is a huge amount of material that falls under this heading. However, this material is usually picked up by teachers only haphazardly through occasional articles in journals, or by attending conferences like this one, or by reading through teachers' notes found in their textbooks, or by examining research in history and conceptual foundations of school mathematics. This mathematics is often not known to profes- sionalmathematicians. It covers both pure anti applie(1 mathematics, algorithms and proof, concepts and representation. Teachers' mathematics is not merely a bunch of mathematical topics that might be of interest to teachers but a coherent held of stu(ly, (listinguishe(1 by its own important i(leas: the phenomenology of mathematical concepts, the exten(le(1 analyses of related problems, and the connections and generalizations within anti among the (1iverse branches of mathematics. The importance of teachers' mathematics thus goes well beyond the nee(ls of teachers to inclu(le all those who study the learning of mathematics and the mathematics curriculum. Deborah Schifter, Education Development Center When ~ spoke on Sunday, ~ mentioned that one problem with the implementation of the National Council of Teachers of Mathematics Standards is that many MATHEMATICAL KNOWLEDGE OF TEACHERS

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teachers and professional developers have emphasized new teaching strategies at the expense of what the reforms were actually about, i.e., mathematical understanding. If this is indicative of a deep-seated tendency in the United States to adopt superficial strategies to get at deep problems we must be aware of this in the context of lesson study too. ~ am concerned that people wail get very enthusiastic about the strategy of lesson study and lose the essence of what it is about. One question ~ have is whether there are shared ways of thinking about math- ematics, learning, teaching, and ciass- rooms implicit in the practice of lesson study in Japan, ways of thinking that would need to be cultivated among teachers in the United States to make lesson study profitable. For example, when we viewed the video of Mr. Nakano's fourth-grade lesson, we watched one student explain the reasoning behind his incorrect answer reasoning that was quite easy for us to follow but which bypassed the mathematics of the problem. In our discussion of his lesson, Mr. Nakano commented that when students make an error or have difficulty with an idea, as this child did, this is when "the fun begins." My interpretation of his remark is that this is when his work begins the teacher becoming aware of difficulties his students are having, figuring out what it is they do understand in relation to the learning objectives he has for his students, and then developing a path to reach these objectives. My question is, is this understanding shared among Japanese teachers who engage in lesson study? And since it is not shared among teachers in the United States, is lesson study an appropriate context for developing it, or are there other, more propitious settings in which teachers might better develop this disposi- tion toward their work? Similarly, do Japanese teachers who engage in lesson study share an understanding of the mathematics of the curriculum they teach? Given the mathematical needs of many U.S. elementary teachers, is lesson study the appropriate context to a(l(lress these? There is evidence that many teachers, anti Americans generally, lose touch with their capacity to think mathematically as early as in the primary gra(les. It's at this point that they start to learn that math- ematics is memorization, in the process losing touch with their own powers of reasoning about mathematics. Anti so when we look at the work that teachers need to do, we must keep this in mind, un(lerstan(ling at the same time that this (foes not reflect on teachers' intelligence but is the result of their own schooling. As we discuss their serious needs in math- ematics, it is very important to maintain a spirit of respect for the teachers who still have so much to learn. In order to convey some of the issues raise(1 by elementary teachers' math- ematical deficits, ~ will describe three (lifferent reactions to one set of activities frequently (lo. In these activities, we look at some very common strategies children devise for solving multidigit calculations and then ask the teachers to apply the chil(lren's methods to other pairs of numbers. When ~ begin a course this way, ~ consistently provoke several (lifferent reactions from teachers. Some actually get quite agitate anti argue that all the children (li(1 the calculations the wrong way. Apparently, these teachers believe there is only one way to solve a given problem and that is to apply the algorithm they were taught in school. This points to a very important learning need for teachers: They must come to see that understanding the MATHEMATICAL KNOWLEDGE OF TEACHERS PANEL

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mathematics knowing operations and calculations involves more than being able to apply a single algorithm. This must be one of the goals for these teachers, and it is the work of their instructors to help teachers recognize that there is a larger world of mathematics they can enter. A second common response is one of relief: teachers recognizing that the children's procedures are ones they themselves have always employed. But under the impression that there was something wrong with their work, they had always kept it secret and felt some- what ashamed. So it comes as a relief to have their own strategies for calculating acknowledged as valid. However, having engaged in such "freelance" mathematical reasoning in secret, this capacity to think on their own remained underdeveloped. It is important now to encourage these teachers to move forward, to develop their powers of mathematical thought. In many cases, they learn, to their surprise, that they are strong mathematical thinkers. It is worth adding that many such teachers, once they discover that their ways of thinking were mathematically valid, go through a period of sadness or anger over lost opportunities, over the many years they could have been doing satisfying mathematics hall they hall the right encouragement. A third common response is illustrated by what happened the first time ~ did an exercise like this with teachers. We had been working for some time, when one of the teachers blurted out, "I can follow the student's procedure. ~ can apply the method to a different set of numbers. But don't understand why it works. This is another meaningless algorithm to me." An(1 many of the other teachers in the class agree(1 with her. At that time, ~ was quite surprised, but since then ~ have come to understand what was going on here. That is, again, the teachers have learned the mathematics by rote. But unlike those who react in the first way (lescribe(l, these teachers (lo un(lerstan that reasoning must play some role in mathematics. However, they never developed models or representations, no sense of what the operations actually do, to call upon to make sense of mathematical procedures. To illustrate the kinds of mo(lels or representations, the kind of mathematical imagination, teachers need to develop, consider Mr. Kurosawa's students. They were working with a sequence of images of (lots with an accompanying story starting with one virus, the viruses grow by adding four each minute. Some students represented the number of viruses after three minutes as 4 x 3 + I, others as 3 x 4 + 1. To explain these different arithmetic representations, students groupe(1 the (lots in (1ifferent ways (Figure 11. This is precisely the sort of mathemati- cal imagination teachers need to (levelop. Given the mathematical nee(ls of so many elementary teachers in the U.S., our first priority must be to help them reconnect with their own capacities for mathematical thought, to help them (levelop meanings for the symbols and objects of mathemati- cal study. But how is this to be done? It certainly isn't happening in most math- ematics courses offered at colleges anti universities. One possibility is to work from records of practice, perhaps like those we have seen today, that highlight chil(lren'smathematicalthinking. Such records, which reveal children's math- ematical i(leas in process, coul(1 provi(le access to those same ideas for teachers who (li(1 not have opportunities to (1evelop these ways of thinking when they, them- selves, were children. MATHEMATICAL KNOWLEDGE OF TEACHERS

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FIGURE ~ Students show different ways to group the Jots. Haruo Ishigaki, WasecIa University What do teachers learn and how do they learn about mathematics? When it comes to that issue, then it is basically the same as how students learn and what students learn. Therefore, ~ would like to share with you the following four points. First of all, the knowledge that any excellent teacher has is not a perfect one. Teachers need to be aware that their knowledge always has to be updated, and it has to be reconstructed. They should encourage students, but they should also encourage themselves. Good teaching is 80 percent confidence and 20 percent doubt. Teachers need to have both of these. When ~ just entered the university, ~ listened to the lecture of a Nobel laureate in quantum mechanics. And suddenly in the middle of the lecture, he stopped and started thinking. He had forgotten absolute zero. Was it minus 273 degrees or minus 237 degrees? He wasn't sure. If he was a Nobel laureate and forgot, it is okay to have doubt, ~ thought. That was a very big motivation for my work. Second, students say extraordinary things and commit extraordinary mis- takes. And they give explanations which are not understandable to us. There are many situations, however, when there is some very valuable information in those little things that students or pupils say. slid not let a particular student take my graduate course, so he went to another university. After two years, he contacted me. He was going to talk at an academic meeting and wanted me to come and listen to his presentation. ~ realize(1 that had lost a very big treasure. ~ should have listened to his mathematics. This is true even when students are still children. Anti third, in Japan the teachings of Confucius were common in education. One of his teachings is that you shoul(1 always correct your mistakes. So at an appropriate place, you shoul(1 recognize that you have already committed a mis- take anti acknowle(lge it. Often if some- bo(ly asks a student questions, the student thinks he is being criticized. ~ listened to one of the lectures by a famous mathema- tician. He ma(le a mistake, anti the au(lience began to mumble anti give him a(lvice. But ~ was very much impresse(1 by the professor's attitude. He aske(1 the MATHEMATICAL KNOWLEDGE OF TEACHERS PANEL

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audience to wait for about ten minutes, as ~ remember. He went to the corner of the blackboard and worked through what he had explained. And then finally, he turned back to the audience and said, "you are right." And then he went on. And finally, let me go back to the mathematics, the topic of today. A math- ematical concept must be understood in context. While ~ was working as an editor of the academic society of mathematics, happened to encounter a contribution about rounding off calculations. Take a very easy example, 3.1 x 3.9. Let's make it 3 x 4. And the answer should be 12. The students, however, repeatedly made a calculation of 3.1 x 3.9, and rounded the results to get 12. The author wrote about how to correct the students' mistake. His intention was to teach the students that if you use substitutes which are very close to the exact numbers, the result will be very close to the real answer. In his explana- tion, however, he said that this is math- ematically wrong, but in order to be efficient, using the substitutes is a good way to do the problem. Many students are good students and try to get to the right solution, not the wrong solution. understood that this is why the teacher failed. ~ think, in this particular case, the teacher did not understand the real purpose of teaching this method to students. The approach should have been taught in a specific environment, where the teacher un(lerstoo(1 the real aim anti the real purpose of the strategy. Let me talk a little about concept building to conclude my presentation. The priority in concept buil(ling is to first define it as a collection of elements with something analogous and relate it to the external world. The contents, the com- mon features inside, should follow the (definition. That will come later. ~ think we can learn something from this about knowing mathematics for teaching. Miho Ueno, Tokyo Gakugei University I'm from the Oizumi Campus of Tokyo Gakugei University Senior High School. The Tokyo Gakugei University has teacher preparation and an education faculty. ~ work for the senior high school and teach at the university. That means every year we receive the students from the Tokyo Gakugei University teacher training program anti for lesson stu(ly. When students come to our school to have lesson study, ~ have a chance to see what level of knowledge they start out with in teaching mathematics. During the three weeks, ~ can also identify how they have developed their skills and their un(lerstan(ling of mathematics education. Student teachers come to our school anti try to remember what they (li(1 when they were high school students. This is their understanding of high school classroom activities. For example, one student sai(1 when he took mathematics classes in the past, students were always taking notes, and the teachers explained and let the students solve the problems. The next student said that when he was a high school student, he un(lerstoo(1 that mathematics was testing students' ability to memorize, anti they learne(1 mathemat- ics in that way. The thir(1 student sai(1 that about 40 students were in the same class. Mathematics class was always quiet. Children use(1 formulas anti wrote them down in notebooks. These students were typical. Every year when they conclu(le their three-week teacher training course, ~ ask them how their attitude about the classroom has change(l. If students share the same impression about what they did when they were high school students, then they have MATHEMATICAL KNOWLEDGE OF TEACHERS

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a misunderstanding about mathematics teaching. They only know how to help students solve the problems as they are told to do in their textbook. Prior to the training period, student teachers have certain anxieties about what they are supposed to do. But they are relatively confident about their abili- ties to teach mathematics. They believe that their ability is enough to attain the level required by the guidebook for their lessons. During the three-week period, student teachers have to compile a lesson plan prior to their classes. They have to describe the purpose of the class, what kind of teaching materials should be used, what they are supposed to say to the students, and what reactions they expect. The students understand that the defini- tion of the formula is not the purpose. They recognize that they have to learn how they can help students solve prob- lems and understand the mathematics. The student teachers realize that you cannot only rely on textbooks, but rather you have to convey the value and essence of the mathematics with your own wording and with your own understanding. You have to be very careful about how you lay out the mathematics on the blackboard. The student teacher, at the beginning of the training, knows that his role as a teacher is to convey information to his students. But gradually they begin to realize that the class is the place where students have to learn. You have to devise teaching materials. You have to think carefully how you can use the pace of the class and how you can use the blackboard to help students learn. Their view of students changes while they teach during the training period. They begin to learn that you have to find out the value of the teaching materials, and you have to go into them deeper yourself so you can leverage what is written in the materials. This is a capability teachers are required to have. Teachers must understand the variable aspect of mathematical concepts and how deep the mathematical formulas are. When they are trying to find out and explain these (leep aspects of the math- emetics, they always discover that what they have as an analogy is not sufficient. This applies not only to the student teachers but also to experienced teachers. Teachers have to use their existing knowledge as a basis, but they also have to keep in mind that they have to improve their knowledge. As teachers try to acquire new knowledge of mathematics, they have to decide how to learn. They also have to know how formulas anti algorithms came into existence. For example, you have to consider a particular mathematical task under certain condi- tions. But in other cases you have to change the conditions to the same task, or you may try to generalize the same task, anti by (1oing so you will be able to see a pattern. Acquiring this kind of compe- tence should be done not only by the student teachers but also experienced teachers as well as students in class. You have to encourage students to obtain and acquire this kind of thinking. Unless teachers have this competence, they won't be able to teach that concept to the students. The basic attitude teachers should have toward studies is to be modest and try to learn as much as possible. Teachers are researchers at the same time as they are teachers, but they cannot stay only in a very narrow scope of their research. If teachers want to expand their scope of knowle(lge, they have to cooperate with their colleagues, and they have to enjoy (liscussions with other mathematics teachers. With this attitude they can (leepen their knowle(lge of mathematics. MATHEMATICAL KNOWLEDGE OF TEACHERS PANEL

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