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Participants were divided into five small groups for discussion throughout the work- shop. These groups were charged with responding to two of four questions using the experiences of the workshop to ground their thinking and discussion and to draw on the activities done together as a way to give meaning to their responses. Each group had assigned coordinators from the United States and Japan as well as a translator. Each U.S. coordinator was responsible for a summary of the discussion of their group. These small groups provided the opportunity for the participants from the countries to talk informally about teaching and professional development. This section contains summa- ries of the discussions, which varied depending on the focus of the groups, and an introduction to video clips of highlights of some of the small group work. Questions acIcIressec! in small group discussion: I. Teaching mathematics requires that teachers understand mathematics well them- selves. In particular, what mathematics do teachers need to know beyond the content of the curriculum that they teach to their students? 2. How do teachers use mathematics together with other kinds of knowledge and skill in order to connect students with mathematics? 3. How do skilled teachers learn about and make use of their students' knowledge and capabilities to help them learn mathematics? What are different approaches to helping teachers develop the mathematical under- standing they need to teach? What are the advantages and potential problems of different approaches? Group ~ Report Coordinator: Michelle Manes, Project Director, Education Development Center Group IT Report Coordinator: Denisse Thompson, Associate Professor, University of South Florida

Group Ill Report Coordinator: Susan Beal, Professor, St. Xavier University Group {V Report Coordinator: Ramesh Gangolli, Professor, University of Washington Group V Report Coordinator: Susan Wood, Professor, I. Sargeant Reynolds Community College Video Small Group Discussion Highlights SMALL GROUP DISCUSSION

Group Members Jack Burrill, Cindy Connell, Daniel Goroff, Toru Handa *, Keiko Hino ~ *, Jackie Hurd ~ ~ *, Shunji Kurosawa, Michelle Manes *, Toshio Sawada, Deborah Schifter, Lee Stiff ~ ~ *, Mamoru Takezawa, Hajime Yamashita Report Coordinator; ~ ~ Translator; ~ ~ HIS. International Congress on Mathematics Education (ICME) Travel Group Teaching mathematics requires that teachers understand mathematics well themselves. In particular, what mathemat- ics do teachers need to know beyond the content of the curriculum that they teach to their students? RESPONSE The question of what mathematics teachers need to know was a challenge for the group. One view is that perhaps that question is easier to answer for Japanese teachers than for teachers from the United States the national curriculum in Japan defines the content teachers need to teach and therefore defines at least the minimal level of content they need to understand. One example of this from the workshop discussions: It was striking to participants from the United States that all of the Japanese teachers in the room were familiar with Deborah Ball's "coin prob- lems," and that there was general agree- ment that these were "sixth grade prob- lems." That degree of agreement about content at each grade level simply does not exist in the United States. It would be surprising if a group of teachers were all familiar with the same problem, much less agreeing on the curricular purpose for the problem and at which grade level it belonged. Perhaps teachers in the United States get bogged down in particular pieces of content (both for themselves and their students) because they lack this consensus about what happens when in a child's mathematics education. However, we agreed that there are some things teachers need to know, or be able to do, that transcend any particular curriculum. For example, it is important that teachers · recognize the value of persistence · understand mathematics as a science, something you explore · recognize patterns · understand mathematics as a system · have the ability to follow a child's mathematical ideas and determine if there are problems with the logic · have a positive attitude about math- ematics and the expectation that it will make sense There was also some discussion about the content teachers needed to know. One point of agreement was that elemen-

tar y school teachers probably do not need to study calculus and it seems, in both countries, it is unlikely that they do. There seemed to be consensus that teachers should know, at a minimum, the content of the courses that they teach in addition to the courses before and after theirs. As we dug into this idea, problems soon arose, however. In a district where each school has a different curriculum, what does it mean to know the content of the courses before and after? If a teacher teaches different grade levels, how does the minimal requirement change? Some people felt that simply one year on either side was not enough, and it should be broadened. However, a teacher in the group reminded us that elementary school teachers are generalists, saying, "You can't make every elementary teacher into a math professor and a historian and a literature professor. Elementary teachers have to teach everything." The closest we could come to consen- sus was to agree on the requirement of knowing the content of their own course and one course on either side, along with some particular pieces of mathematics- like the base 10 system in deep ways. An interesting fact was that in both countries, there are concerns about what mathematics teachers need to know and about the weak backgrounds of elemen- tary school teachers. In Japan, two recent books have been popular: The University Student Who Can't Do Fractions (Okabe et al., 1999) and The University Student Who Can't Do Decimals (Okabe et al., 2000~. In Japan, the population (lecrease is making it easier to get into college, and there is concern that the elementary teachers are among the least prepared college students in terms of mathematical backgrounds. In the United States, the situation is similar, with schools of education having propor- tionally more of the students with the lowest entrance test scores (IJ.S. Depart- ment of Education, 2000~. In both countries, there are claims about what elementary teachers do not know. In the United States, there is a strong research base that shows weak- ness of mathematical knowle(lge among prospective and practicing elementary teachers (Ball, 1990, 1991; Post et al., 1991; Ma, 1999~. It is important, however, to talk about this research with respect for teachers, rather than disparage them. How do teachers use mathematics together with other kinds of knowledge and skill in order to connect students with mathematics? RESPONSE The question of how mathematical knowledge comes together with pedagogi- cal knowledge as teachers do their jobs was in a way easier to a(l(lress. . . It is clear from the Japanese lessons that great care is given to planning things like blackboar(1 use, order of problems presented, anti thinking about what might happen (luring a lesson. All of the plans for "if students do this, I'll react this way..." anti even knowing what students are likely to say and do with particular problems bring together both mathematical knowle(lge anti the un(lerstan(ling of what students know anti how they think about math- ematics. Helping to connect students with content requires that teachers un(ler- stan(1 the content in (jeep ways anti that they can think flexibly about it. Examples given include making topics like simultaneous linear equations more SMALL GROUP DISCUSSION

and what they can skip. No course covers this. CONCLUSIONS . . interesting by setting the problems in a social science or physics context and connecting solving equations to a broader notion of "doing and undoing" which students understand. This goes beyond contextualizing problems to answer the question of "What is this good for? When wall ~ use it?" to helping students better understand the content by showing the bigger picture and connections to things they already understand. In planning a lesson, teachers need to move beyond what they know as teachers and think about what the students know and understand. There is a difference between the way a teacher understands a piece of content and the appropriate level for students. Understanding a piece of content as a teacher means being able to talk about it appropriately for different levels of students. When an experienced teacher listens to a student's explanation, they are able to repeat back what the student said (an example of careful listening) and pick out the essential mathematics (bringing the careful listening together with the leacher's consent knowledge). Inexpe- rienced teachers are unable even to repeat what the students have said. Even if they know the mathematics, they are unable to really listen to their students. If they can not understand what their students are saying, they can not present the mathematics appropri- ately or address misunderstandings. · In reading students' journal responses and problem solutions, experienced teachers are able to pick out the good ideas and provide guidance to students, whereas less experienced teachers may have difficulty making sense of what students write. Teachers need to know enough to make decisions about what is useful to pursue . G R O U P One major concern, discussed during two of the three meetings, was the fact that lesson study is used primarily at the elementary level in Japan. According to the Japanese participants in the group, lesson study exists but is less common in middle school and not used at all in high school and college. The Japanese teachers explain this by saying that the high school teachers graduate from the mathematics department; they like math and have an easier time teaching it. Yet they also said that exploration of mathematics content by the teachers may be part of lesson study but is not always. Another explana- tion is the tracked high school system in Japan. In high school, the focus is prepa- ration for university and entrance exams; this is much less of a worry at the elemen- tary level. It seems that in Japan, elemen- tary teachers take to lesson study because of an opportunity to learn content. It is not the only purpose, but it is a big attraction. To the U.S. participants, it was clear that a lot of pedagogical learning went on in lesson study: how to plan a lesson, how to anticipate student responses, how to use the blackboard, and so on. The purpose of lesson study as described by one Japanese participant was to "look at the delta the difference between what was planned and what actually happened in a class." That seems like a valuable experience for a teacher at any level, and in fact Daniel Goroff described a lesson study like program at Harvard University. The program is for mathematics professors, and the goal is to improve teaching, not content knowledge.

Another issue was around the evalua- tive nature of classroom observation in the United States. Teachers wall still be observed as part of their evaluation, so how is this formal observation different from a lesson study experience, and how can it be made clearly different for the teachers? This brought up issues around evaluation of mathematics teachers: Some members of the group felt strongly that in evaluating teaching of mathematics, you have to have mathematics experts (and not just administrators) involved. The Education Development Center has a project called Lenses on Learning focused on designing materials and videos to help administrators look at teaching practice and understand what they should look for in a mathematics classroom. In discussing the two main types of teacher professional development pre- sented in the workshop, participants identified clear positives and negatives in both approaches. One Japanese participant explaine(1 that one negative aspect of lesson study is that teachers try to create lessons that no one would criticize. That kind of pressure can keep a teacher from taking risks and from exercising creativity and originality. In addition, the point was made that when watching a real lesson with real kids, the issues that come up may not be about the lesson. Whatever happens happens, anti you can not always predict or arrange to discuss particular ideas based on study lessons. Another point about lesson study, or at least its potential implementa- tion in the United States, is that teachers see being observe(1 as a "showcase." They (lo something they (lo not usually (lo with their students. The feedback received on a study lesson may not be useful if it does not apply to daily practice. In thinking about records of practice, one participant compare(1 that form of professional development as training people to be opera critics rather than singers; he claimed that watching a video of teaching does not help teachers with their teaching practice. Defenders of the mo(lel, however, claim that it's not about being a critic, but rather to learn skills that are important in teaching. It is true that records of practice are removed from actual practice, but there are things you can learn: You can learn to examine students' talk, for example. What are they really saying? Do their mathematical icleas make sense? Also, records of practice are less threatening than having · ~ someone come in ant examine your own practice. Using vicleoancicases might be a way to cultivate the kind of atmosphere that wouic! allow lesson stucly to happen. Deborah Schifter clescribec! a professional clevelopment exercise she uses: Teachers write a narrative of a classroom conversa- tion. They react each others' narratives, en c! professional clevelopment staff react en c! comment on them. Some teachers say that the exercise makes them hear their students' mathematics clifferently. They need to attend clifferently to recorc! it. This kind of activity can serve as a kind of bridge between examining videos en c! cases to examining one's own practice. In comparing the two methocis, there are some interesting points of contrast. In using records of practice for professional clevelopment, you can specifically select records for particular learning goals. In essence, you can build a curriculum around these records. However, it takes an outside expert to collect, examine, en c! catalog the records, and possibly to cleliver the professional clevelopment. With lesson stucly, you cannot select for particular learning goals. Whatever happens is what there is to discuss. But lesson stucly can be clone in a school without the participation of outside SMALL GROUP DISCUSSION

professional developers. Lesson study groups require leadership and organiza- tion but that can come from among the teachers in a school. RESEARCH ISSUES In looking at the issues raised in our discussion, some interesting areas for research emerged: . What is the potential for implementing a lesson study or similar professional development structure at the high G R O U P school level? Is there potential benefit for teachers at this level? Is there an impact of participation in lesson study on daily practice? One point made in the workshop is that Japanese teachers do not spend the same amount of time on a regular lesson as on a study lesson. So is it true that study lessons are "showcases" with little relation to a regular class, or does going through the process of carefully planning some lessons affect how a teacher presents all lessons? If there is an impact, what is it?

Group Members Hyman Bass, Phyllis Caruth ~ ~ *, Frances Curcio ~ ~ *, Alice Gill, Ichiei Hirabayashi, Tadayuki Ishida, Carole Lacampagne, Carol Malloy, Nob~hiko Nohda, Semi Soga *, Akihiko Takahashi, Denisse Thompson *, Hiroko Uchino, Tad Watanabe Report Coordinator; ~ ~ Translator; ~ ~ HIS. ICME Travel Group The group did not make any specific assumptions in order to answer the assigned questions. However, some misunderstandings among the American educators that were not clarified until Tuesday may have impacted the discus- sions. In particular, there was confusion about what was meant by lesson study. Not until Tuesday did it become clear that Japanese teachers might participate in several lesson study groups throughout the year on a regular basis to discuss a variety of instructional materials and content; participation in a long term lesson study occurred much less frequently. From the discussions and from addi- tional information provided by Tad Watanabe, it appears that lesson study is a practice that occurs typically at K-6 schools. Some of these may focus on developing a lesson over a period of time, and others may focus on providing mutual support for (laity problems or for more broad-based issues, such as how to implement a new national course of study. At the lesson study sessions focusing on the development of particular lessons, different individuals may bring plans and have others critique these plans. Lesson stu(ly affords teachers the opportunity to talk about different aspects of lessons or to explain what happened when they tried a lesson in the classroom. Formal lesson stu(ly with open-house stu(ly lessons occurs only a limited number of times during the school year and is for the explicit purpose of research- ing the lesson. These lessons often take place at elementary schools attache(1 to universities; education researchers at the universities might serve as outside observers anti reactors to the lesson study. The availability of lesson study at the elementary schools affiliate(1 with universities seems to fulfill one of the missions of these schools, namely, to provi(le national lea(lership in improving teaching in all subjects inclu(ling math- ematics. It appears that not all Japanese teachers regularly participate in a formal lesson study as a teacher being observed teaching a lesson to children. A(l(litionally, lesson studies appear to be practices that occur primarily at the K-6 levels. Very little (liscussion occurre(1 about professional (levelopment practices for teachers at gra(les 7-12. In responding to the charge to (liscuss the two questions liste(1 below, the group focused on both questions simultaneously.

Although responses do not fit neatly with only one of the two questions, this sum- mary attempts to sort responses to provide some coherent comments to each question. How do teachers use mathematics together with other kinds of knowledge and skill in order to connect students with mathematics? RESPONSE In the course of discussing this ques- tion, it became clear that the mathematics knowledge of the teacher is crucial at all points in the instructional process. Teachers design lessons, enact lessons, respond to students' ideas and make use of those ideas in furthering the lesson, analyze lessons, and study student work. All of these points require considerable mathematical knowledge. For instance, in the sixth-grade lesson the teacher explic- itly wanted the children to construct the task of the lesson the teacher's knowI- edge guided the lesson by guiding the children until the task for the day emerged. That is, students raised ideas related to area models; the teacher guided the children to consider other ideas until the linear growth model emerged. Clearly the teacher had to know what mathematical structure was inherent in his goal. Other issues that arose in relation to the question are discussed below. Teachers use their knowledge of mathematics as they develop appropriate tasks to use with their children. The teacher's mathematics knowledge is used in conjunction with his or her knowledge of children to develop tasks that will be of interest to the children and motivate them to learn. This intersection of the knowI- edge of children and mathematics was GROUP 11 evident in the fourth-grade lesson in which the teacher used a quiz show format in an attempt to motivate children to deal with issues of large numbers. Teachers use their knowledge of mathematics to consider real-life prob- lems and examples that people need for everyday life. Japanese K-6 mathematics education seems to focus on "mathematical literacy," so that Japanese teachers attempt to begin a lesson with a context that children might encounter; this was evident in the fourth-grade lesson in which the teacher dealt with large num- bers related to size and costs. It was also evident in the Japanese teachers' concerns about the U.S. third-grade lesson dealing with the number of pennies, nickels, and dimes to equal a given amount of money. Japanese teachers thought the problem woul(1 be more compelling if the teacher had the coins in her pocket to show to students or to have students consi(ler what the coins might purchase. Teachers use their knowledge of mathematics to un(lerstan(1 the (1ifferent aspects of the mathematics in a problem that may impede the work of the students on the given task. Teachers need to be aware of the cultural dimensions that provide gates that may have an effect on learning, either positively or negatively. For instance, in the fourth-gra(le lesson on large numbers, the numeration system involves grouping by ten thousands in Japanese, rather than by thousands as in English. The different mental structure require(1 to comprehen(1 the numbers in this system was a source of difficulty for some of the American educators watching the lesson; focusing so har(1 on compre- hen(ling the numbers represented in an unfamiliar way ma(le it more (difficult to focus on the mathematics embe(l(le(1 in the lesson. Although this (lifference in grouping was not a problem for the

Japanese children, the cultural differences caused a problem for the American educators. Given the cultural diversity of children in U.S. classrooms, it serves as a potent reminder of the need to be sensi- tive to children's perspectives. When cultural backgrounds cause children to think differently, it is often easy to dismiss their thinking as incorrect; however, student explanations and good question- ing on the part of the teacher can help both students and teachers to bridge the cultural divide and enhance learning. Teachers use their knowledge of mathematics when they take a simple problem from their textbooks and con- sider how the problem might be adapted or extended to explore the mathematics more deeply. This was evident in the sixth-grade lesson as the teacher took a basic problem dealing with growth by four per minute and had children look for patterns and attempt to express those patterns by means of functions and variables. As part of this work with functions, the teacher had children considering the different interpretations of 4 x 48 + ~ and 48 x 4 + ~ in the context of the situation. This functional work foresha(lowe(1 anti lai(1 the foundation for more advanced work in later grades. Teachers use their knowledge of mathematics to anticipate various student solutions that may arise during classroom discussions about a task. Anticipating student solutions helps teachers deter- mine where the discussion may go and enables them to make appropriate deci- sions on how to procee(1 with the lesson. For instance, in the sixth-grade lesson, children's ideas about area were leading to a quadratic relationship. By anticipat- ing students' solutions, the teacher was able to steer the discussions until the linear relationship that he wanted the chil(lren to investigate was generated. Teachers use their knowledge of mathematics as they make connections between mathematics and other disci- plines. What are different approaches to helping teachers develop the mathematical a~nder- standing they need to teach? What are the advantages and potential prohiems of different approaches? RESPONSE A number of issues arose regarding the various professional development ap- proaches presente(1 at the seminar- research study lesson, video case record, written cases, and professional develop- ment in general. The discussions before and after the stu(ly lesson help teachers know anti un(lerstan(1 the mathematics embe(l(le(1 in the lesson. The (liscussions are important because they help teachers reflect on how to fix structural errors in the lesson. For instance, in the fourth-gra(le lesson, the fact that each successive question in the task increase(1 by a factor of ten cause some students to focus on the pattern being generate(1 rather than the un(lerly- ing mathematical relationships, thereby obtaining an incorrect result for the comic book problem. Through (liscussions about the lesson after its presentation, teachers are able to think about this structural problem anti how it might be resolved prior to teaching the lesson again. One approach that is particularly helpful for teachers is to use vertical articulation of topics. That is, teachers shoul(1 stu(ly topics anti consi(ler how they change anti grow across the various gra(les anti levels. If elementary anti lower secondary teachers understand SMALL GROUP DISCUSSION

where topics eventually lead in high school, then they might teach those topics differently. For instance, the s~xth-grade teacher taught his lesson to lay founda- tions for more formal work with functions in a later grade. It appears that in Japan, teachers are encouraged to participate in lesson study at various levels so that they have a good sense of the mathematics that is taught prior to and following the level at which they teach. The same is not necessarily true in the United States. Although the recently released Principles and Standards for School Mathematics (National Council of Teachers of Math- ematics, 2000) encourages this vertical articulation through the content stan- dards, it is not clear that such articulation typically occurs in practice. U.S. teachers have little opportunity to interact with teachers at levels other than their own to gain a better understanding of the math- ematics taught at the different levels. The national course of study in Japan helps teachers make assumptions related to the issue. This course of study helps teachers make assumptions about the mathematics that children should know, both in terms of instruction at their grade level and in terms of prerequisite knowI- edge. U.S. teachers are not able to make such assumptions quite so readily. Materials need to be available in teachers' editions that show how ideas are related across levels. The materials in the Japanese teachers' editions are not the same as in the pupils' editions. The teachers' editions contain supplementary problems and discuss how a topic relates to previously taught ideas and to ideas that wall appear in later grade levels. The Japanese teacher materials provide support to teachers, pointing out the important ideas of the lesson and where students might have problems; in addi- tion, the teachers' edition may provi(le GROUP 11 different solutions or ways that children think about a problem. Some of these same instructional approaches are present in the teachers' editions of newer U.S. textbooks that provide additional prob- lems, ideas for review, reteaching ideas if students have difficulties, or extensions for more advanced work. A big difference may be that the course of study in Japanese education helps focus on connections to earlier and later courses that are not possible in U.S. textbooks; hence, Japanese materials often seem more focused than corresponding U.S. teacher materials. Research is needed to evaluate the relation between school mathematics that teachers are expected to teach and college mathematics that students are expected to learn. University mathematics does not necessarily help in teaching school mathematics. The mathematics teachers stu(ly shoul(1 begin with the mathematics in the textbooks of the curriculum. This perspective closely relates to the work- shop paper delivered by Zalman Usiskin in which he raised the issue that school mathematics is a rich source of study in its own right. Japanese anti U.S. e(luca- tors agree(1 that there was a need to stu(ly mathematics for teaching and not just mathematics for mathematicians. The three different professional devel- opment approaches seen at this workshop facilitate the integration of content with methods. This integration is important because teachers' pedagogical content knowle(lge comes into play in un(lerstan(l- ing how students (levelop their math- ematical knowle(lge. The purpose of the formal stu(ly lesson is to provide research on the lesson and its effectiveness. However, written cases anti videos could be use(1 in lesson stu(ly groups anti coul(1 help in the (development of a research lesson. Published study lessons coul(1 also be use(1 in stu(ly

groups. In the study groups, experienced teachers mentor novice teachers. A question that arises with all of these approaches is how much mathematics knowledge a procedurally oriented teacher gains from studying such cases. CONCLUSIONS The group considered similarities in the ways that teachers in the two cultures look at the different models of profes- sional development: lesson study, written case study, and video case analysis. In discussing the various approaches, Japanese teachers focused on the math- ematics in the model, whether a written or video case, to determine whether or not the mathematics was appropriate for the level of the student. What was the math- ematics that the teacher was trying to help students understand? With the lesson study approach, teachers have thought about why they are doing some- thing, what mathematics has come before, and what mathematics they may want to foreshadow. That is, the Japanese teachers are interested in how the mathematics of a particular lesson relates to the curriculum for the whole year; the course of study provides a useful perspective on the content that should be taught in the classroom. In contrast, U.S. teachers often consider pedagogical issues first. Paying serious attention to the mathematics in the lesson development would likely be beneficial to U.S. teachers. One issue that arose is that all the various approaches to professional development depend on the teachers' knowledge of mathematics and how this knowledge can be used to help students. If teachers do not know enough math- ematics, they cannot take the ideas very far. A valuable aspect of lesson study is to help the teacher understand the intrica- cies and details of a lesson choice of numbers for problems, selection of manipulative materials to enhance learn- ing, and so on. The detail in such discus- sion would be exceedingly valuable in the first few years of teaching. Video and written cases provide an opportunity for teachers to learn the mathematics that is situated in their daily practice. Teachers analyze mathematical tasks, think about how the tasks might be selected, reflect on the mathematics the tasks elicit, and consider how the given task is embedded in a family of problems that can be scaled up or down as needed. Discussions begin with how the ideas play out in the given class being studied, and then teachers begin to think about exten- sions. The task is the seed for a lesson that ultimately can be generalized math- ematically. If teachers are able to see how ideas grow from practice, they are not as likely to question the generalization of problems. It seems important for teachers to develop the mathematics that is related to the school mathematics they must teach. Industrial mathematics, commercial mathematics, and economic mathematics all have different perspectives and emphases. At present, much of the mathematics taught in K-12 schools is based on the mathematics of mathemati- cians. When students understand all the mathematics in the K-12 schools, they are able to enter a collegiate mathematics program designed for future mathemati- cians, scientists, or others needing to use higher-level mathematics. The emphasis on the mathematics of mathematicians does not always seem relevant to prospec- tive teachers. If the mathematics teachers learned was situated more in practice, such as in the video cases discussed by Hyman Bass and Deborah Ball, then SMALL GROUP DISCUSSION

perhaps teachers would focus more on the content of the mathematics and be more comfortable doing so. The professional development dis- cussed in the workshop requires a longi- tudinal commitment. Intensive work with cases or videos is the type of professional development we need to consider more often rather than the short variety that is common in the United States. At present, there is often no framework to put the different professional development experiences into a curriculum or into the larger mathematical picture. In the hands of a skilled facilitator, the approaches discussed in the workshop can provide a framework for professional development because of their focus on how children learn mathematics. The facilitator needs to be explicit in helping teachers make connections to the larger K-12 curriculum in order to understand how the math- ematical ideas fit together and to provide the forward and backward look into curriculum that appears to be a hallmark of the Japanese materials. RESEARCH ISSUES Several questions arose: · Why is the study lesson conducted only at elementary school and not also at high school if it is such an important part of teachers' professional develop- ment? · Teachers in the Japanese research study lessons are both instructors and researchers. Do American teachers have this (lual perspective? If not, how could we begin to foster this dual perspective? MATHEMATICAL KNOWLEDGE OF TEACHERS PANEL

n Group Members Susan Beal *, Jacqueline Goodioe ~ ~ *, Fargo Ishigaki, Yoshihiro Hobo, Kouichi K~magai, Carol Midgett ~ ~ *, Tatsuo Moroz~mi ~ (second days, Keiichi Nishimura *first days, Takaski Nakamura, Nanette Seago, Yoshishige Sugiyama, Zalman Usiskin, Makoto Yoshida Report Coordinator; ~ ~ Translator; ~ ~ HIS. ICME Travel Group The group spent time un(lerstan(ling the three methods or approaches to professional development in Japan and the United States that were presented and discussed in the workshop: lesson study, video records, and written cases studies. Teaching mathematics requires that teachers understand mathematics well themselves. In particular, what mathematics do teachers need to know beyond the content of the curriculum that they teach to their students? RESPONSE In summarizing the response to the first question, the group felt that the teachers nee(le(1 to un(lerstan(1 the mathematics they were teaching but did not discuss what teachers needed to know beyond the content of the curriculum. In two of the examples used during the workshop (sixth-grade lesson, 4n + I, and the hexagon example, 4n + 2, in the case study), students could use a recursive formula to describe a pattern: add four each time. How (lo we know? What constitutes proof at these levels? The teacher is looking for an explicit (definition of sequence, i.e., a close(1 formula for the nth term. In the United States studying the relation between recursion anti a close(1 formula is not in the curriculum, and it was believed by the U.S. partici- pants that U.S. teachers do not have much experience in working with this relation- ship. A(l(ling four is relate(1 to the slope of the line. Do teachers know this? Development of variables as a concept, although it is a major part of Japanese curriculum, is not explicit in Japan. Boxes represent x. The letter x is use(1 in the seventh gra(le, anti Mom there polynomials are intro(luce(l. The use of symbols in the linear equation, y = m x + b, where m is constant and x is a variable, is currently being researche(1 in Japan; what should m anti x be called? Moving from particulars to a more general mathematical idea is what teachers nee(1 to learn. How do skilled teachers (master teachers, experienced teachers) learn about and make use of students' knowledge and capabilities to help them (the students) learn mathematics?

RESPONSE Three aspects emerged as important elements in response to the second question: self-reflection, attention to details, and summary of the lesson. Points of cultural differences between the Unite(1 States an(l Japan were also (lis- cussed: how teachers are known to be good teachers, direct comparison of teachers, working hard in mathematics verus "math people," and what it means to be a teacher. Self-Reflection The most important commonality among the methods and approaches expressed by the participants in our discussion group was the role of self- reflection. Each approach or method built on self-reflection is a vehicle to help teachers grow, mature, and learn more about pedagogy and mathematics. Partici- pants commented that it was difficult to reflect on thinking and teaching in the context of actually teaching. One of the roles discussion plays in lesson study is to help teachers reflect on their lesson, on students' learning, and on the questions asked by the teacher to develop the concept being taught in the lesson. One participant mentione(1 that "they help you see what you can't about yourself." Another commented that "self-reflection is most important, but you have to actually test (your ideas in the) classrooms." In the video records example presented by Deborah Ball, the teacher kept a journal each day, reflecting on the lesson, how the students were learning, and what they were doing during the lesson. The journal can be used by others to under- stand what the teacher was trying to accomplish in her lesson. The three phases of video records for professional development are design, enactment or GROUP 111 facilitation, anti analyzing or reflecting. Thus, in the construction and use of records, self-reflection plays an important role. In case study, self-reflection was indeed what Catherine and David related with respect to their lessons anti with the process of change in which they were involved. For example, from page 10 of the case stu(ly: A "few weeks later when he met with his colleagues and shared a ten-minute segment of a videotaped lesson, their reaction to the tape led David into reflecting on how leading he was with the students." Our discussion group felt that you have to look at your teaching and, in doing so, share with colleagues. Attention to Details Attention to (retails was particularly important in lesson study. Sensitivity to (retail was calle(1 the essence of lesson study, and a teacher's expertise in this area provides evidence of a skilled teacher. For example, if a teacher can explain clearly what a student said, the teacher has a "good eye" for teaching. Attention to (retail inclu(les such things as the importance place(1 on the example selected to introduce the lesson, and the order of examples used in the lesson, the language of questions and what it tells about the concept un(ler investigation. Lesson study also tries to anticipate responses students woul(1 or coul(1 make anti formulates plans to inclu(le accommo- (lations for the (lifferent nature of the responses. Furthermore, the benefits (lerive(1 from lesson stu(ly inclu(le paying attention to student (retail. Lesson stu(ly focuses on learning what the students are (loin". Another teacher could use the concept of lesson study and teach a lesson to fin(1 how his or her students use or benefit from the materials (action research). Attention to (retail for

daily lessons is important as is the need to plan for every lesson. Video records as an approach to professional development can focus on the details associated with the lesson. Materials available to be studied consist of detailed documentation of teaching and learning, although all of these are not necessarily analyzed. The materials are used to encourage teachers to learn how to observe, discuss, and make records- all important in learning a practice such as teaching as described by Deborah BaD in her opening remarks. Her study took place over an entire academic year in two classes. The richness of the detail allows explora- tions of what took place in the teaching- learning process over a period of time. Lesson Summary All three approaches for professional development have strengths and weak- nesses. Many study lessons are tran- scribed, which is very time-consuming. One participant noted that he has one of his students copy what is on the black- board and write about his or her thoughts on the lesson, rotating the task so that all students have the opportunity. In lesson study, the blackboard can be used as a record for research. Participants observed that it takes a long time in constrained circumstances to watch a video, while opportunities to read a case study were more flexible. All agreed that it is nice to have a summary of the lesson. However, whether it is video or written, a point of view is always present, depending on the person who videotaped the lesson or the one who summarized it. Since lesson stu(ly has many observers, there is not one point of view. Because each approach for summarizing or recording a lesson has strengths and weaknesses, the group thought that it would work well to com- bine the methods. One characteristic of lesson study is that the physical records of the lesson are on the blackboar(l, which can help as teachers analyze the lesson. In fact, when (liscussing the sixth-gra(le lesson, we ha(1 the physical "blackboar(l" to help us in our analysis and recollection. The blackboard is organize(1 to help students keep notes. Furthermore, we can categorize what the teacher said by using the notes from the blackboard. CONCLUSIONS Cultural Differences and Their Impact on Professional Development How Teachers Are Known to Be Good Teachers In Japan some teachers are known for their teaching and become known because they have been observed in the classroom, perhaps through lesson stu(ly. Many also write about their ideas in one of the many magazines for teachers. If the idea appears in a mathematics-specific issue, comments are ma(le by mathematics educators. Also, Japanese bookstores have many books written by teachers to share pleas. The U.S. participants felt that U.S. teachers become known by their profes- sional affiliations anti accomplishments anti not necessarily by their teaching. The professional journals, for example those publishe(1 by the National Council of Teachers of Mathematics, include articles by teachers that (lescribe their i(leas or classroom activities, but it is very (difficult to know from this how a teacher actually teaches. An(1 although there are magazines such as Teacher, which might mirror those in Japan, these magazines (lo not necessarily support the mathematics education community. SMALL GROUP DISCUSSION

Direct Comparison of Teachers Direct comparison of one teacher with another does not happen in Japan, as in the case study example with the compari- son of Catherine's and David's classes. In Japan, the lesson study consists of a set of hypotheses on how to conduct the lesson, which are then tested. It is these hypoth- eses that are critiqued during the discus- sion following the observation of the lesson. Furthermore, to prevent over- whelming the teacher with too many suggestions during the discussion of the lesson study, focal points are established and the participant observers categorize what the teacher said, using the black- board as a point of reference. Usually, the lesson has been planned with others, so the emphasis is on the work done by the group. The notion of "we will help you improve" in Japan starts as early as when you are a student and so naturally comes to the teaching process, but this is not so in the United States. Working Harc! in Mathematics Versus "Math People" The attitude that all students can learn mathematics seems to be more prevalent GROUP 111 in practice in Japan than in the United States, even though NCrM (and the Mathematical Association of America) specifically include this in the literature. Being a Teacher In the United States, teachers go back to college to get advanced degrees but not in Japan. Becoming a teacher is very difficult in Japan. Many teacher appli- cants fail the screening to become a teacher. The test to become a teacher is to teach in front of a set of examiners. As in the U.S., supply and demand has a lot to (lo with how rigorous the requirements are to become a teacher. In Japan there is also a first-year teacher training program in lieu of student teaching. The year includes 30 days in school with a mentor to help. Every five years teachers go back to the education center for continued professional develop- ment. In the United States, 33 states require a small amount (minimal) of professional development (Dossey and Usiskin, 2000~. In most states teachers have to do some form of professional (levelopment to retain certification.

|1~. Le]ni.Ill ~ Group Members Jerry Becker, Shelley Ferguson ~ ~ *, Ramesh Gangolli *, Beth Lazerick ~ ~ *, Shinichiro Matsumoto, Judith Mumme, Hiroshi Nakano, Yasukiro Sekig~chi *, Kiyoaki ShodFa, Margaret Smith, Shinichi Suzuki, Daniel Teague ~ ~ *, Miho Ueno Report Coordinator; ~ ~ Translator; ~ ~ HIS. ICME Travel Group Although an attempt had been made to focus the group's discussion on the ques- tions suggested at the workshop, given the central position occupied by lesson study in the plenary sessions, it was perhaps inevitable that much of the small group discussion was related to assessing the role and effectiveness of lesson study as a tool of professional development. What follows is a summary of responses to the questions as they relate to Japanese teachers taken from the interaction at the workshop. It is based partly on what transpired in the small group sessions but also on information obtained through other opportunities (formal as well as informal) as they arose during the workshop. The remarks are directed toward mathematics (and not toward any other subject), and those that might seem to apply more generally to other subjects should be viewed as limited by this restriction. BACKGROUND Elsewhere in these proceedings one can find a full description of lesson study: what it is, how it is implemented, the contexts in which teachers encounter it, and the role it plays in their professional development. In this summary, it is not necessary to repeat that information. It will nevertheless be useful to mention here a few aspects of the process of lesson study that might run contrary to the conception of lesson study that some may have formed. These aspects will serve as a backdrop against which some of the remarks that follow may make more sense. . Lesson study sessions are not as pervasive a part of the daily life of teachers in Japan as one might sup- pose. The extent to which individual teachers engage in formal lesson study sessions is quite limited. Thus, many elementary teachers might do so only a few times a year at their school. They might also observe such sessions as a part of professional development activities at professional meetings, etc. Secondary school teachers engage in it to an even lesser extent. Schools attached to university departments that do research in mathematics education are exceptions. There, lesson study is a far more recurrent feature of life. Moreover, the lesson study sessions in which teachers participate might deal

only qualifying proviso . . with any subject (not necessarily mathematics). There is a tremendous range in the type of experience afforded by such lesson study sessions, depending on grade level, context, and purpose. Sessions might range Tom very informal ones in which a couple of colleagues might casually observe a small change of presentation and comment on it, to very formal heavily attended "study lessons" in which new investigations or major changes in existing lessons might be launched or dissected. The latter type of session is typically held in a university context. The end product of such sessions (after many iterations) might well end up as a published lesson made available to teachers. Great care is clearly taken in the preparation and implementation phases of a lesson study session. Thus, there is continual stress on the importance of (a) clearly identifying the conceptual and cognitive goals of a lesson, fib) designing interesting questions that will promote those goals, (c) weighing and selecting from different pedagogi- cal devices that might be useful in pursuing the problem, anti ((1) anticipat- ing a variety of student approaches and/or misconceptions. Notw~thstand- ing the scrupulous attention paid to these elements, the postdiscussion that follows a study lesson sometimes concentrates on conceptual content and at other times on individual pedagogical style. Discussions may range from polite noncommittal ones to long substantial debates. · In spite of these variations in form and ~ i! s that in order to be useful, this process must be undertaken on a continuing basis over a sufficiently long period. Bearing in mind these aspects, a summary of responses to four questions follows. What do teachers do? RESPONSE The simple answer is that they continu- ally engage in the process of trying to become more effective teachers, thinking about what they teach and how they teach it. But this is a glib summary that does scant justice to a number of institutional anti cultural forces that promote anti indeed ensure this behavior on a continu- ing basis. Notable among them are the following. Preservice Acculturation Although classroom-based preservice training (e.g., as a student teacher) is short typically just three or four weeks we were told by our Japanese colleagues that throughout the prepara- tory years certain expectations of what is regarded as professional behavior are built in as an integral part of preservice education. Among these are (a) the notion that as a teacher, one is expected to strive for "continuous improvement" throughout one's career and fib) that lesson study or other activities like it (involving reflection and analysis of one's teaching) will form the most important tool for achieving continuous improvement. substance, there is a universal accep- tance of the proposition that the pro- cess of reflection exemplified by the lesson study experiences is a powerful It seems to be a widespread practice tool for professional improvement. The that some structured mentoring is pro- GROUP TV Initial Mentoring

vided for the beginning teacher. This is especially the case for elementary teachers, who are usually not specialists in math- ematics.) It was not clear whether this mentoring is subject oriented (e.g., toward mathematics). The precise manner in which this is done was not clarified in our discussions, but it seems that it is usual to assign a fledgling teacher to a senior colleague who becomes responsible for mentoring that teacher for one or two years. A policy adopted in 1998 requires that such mentoring shall be provided as a rule to all beginning teachers. It is not clear whether this policy is fully imple- mented as of today, but it surely sends a strong message to the beginning teacher that her or his teaching effectiveness is of immediate concern to the school system. The system of mentoring seems to be an effective vehicle for transmission of the professional culture of the school and its cadre of teachers. Inservice Integration The mentor who is a part of the school's professional culture reiterates the expectations clearly conveye(1 (luring the preservice phase during the initiation phase. Simultaneously, the beginning teacher is also integrated as a part of a team of teachers in the school. Most of our Japanese colleagues seemed to indicate that this is the prevalent model. The integration is facilitated by the existence of a number of support struc- tures that promote the implementation of the model. Among these are a. Availability of time for teachers to prepare anti (liscuss lessons among themselves. b. Opportunities for getting expert advice; this may come via an ongoing or occasional liaison with a faculty at a university or by means of other specific opportunities (e.g., at lesson study sessions at the ward, prefectural, or national level). Stability of employment; after an initial probationary period, a teacher acquires tenure de facto. Implicitly, the other part of this social contract is the assumption that the teacher will fulfill the expectations of professional behavior described above. Fluctua- tions in the total number of teachers needed are handled by attenuating and adjusting replacements needed due to retirements or resignations, rather than by lay-offs. This encour- ages in(livi(lual teachers to view themselves as a part of a permanent value(1 structure, contributing signifi- cantly to a national need, rather than as functionaries who are hired and fire(1 base(1 on random (demographic · ~ exigencies. d. Standardized, unambiguous curricular materials, supplemente(1 by explicit implementation schedules and sup- porte(1 by (letaile(1 teachers' gui(les; the implementation schedules are expecte(1 to be observe(1 scrupulously, so much so that we were tol(1 that in any given week, essentially all stu- (lents at a particular gra(le level will be studying the same mathematical topic throughout the country. On the other han(l, the extent to which an in(livi(lual teacher woul(1 follow exactly the (letaile(1 lesson plan for teaching that topic as lai(1 out in the teachers' gui(le varies. The variation seems to depend on the experience anti initiative of the teacher, as well as the gra(le level. The adherence to a stan(lar(1 curriculum, both as to the sequencing of topics anti the time spent on them applies without excep- tion, so far we can tell, to elementary anti mi(l(lle school gra(les. There SMALL GROUP DISCUSSION

seems to be much more variation at the high school level, both in regards to the lessons used as well as the pacing. However, the order of devel- opment of the topics and the inter- connections that are to be stressed seem to be governed by the collective wisdom of the national curriculum and is fairly uniformly observed. What do teachers work on? RESPONSE The honing of specific lessons in order to make them more effective is the motif that seemed to run through the efforts of the individual teachers. Observations of lesson study sessions indicate that teachers devote considerable attention to (a) clearly articulating the specific intellectual goal of the lesson (it is noteworthy that the goal is often phrased as an abstract goal of achieving understanding of a concept rather than the mastery of a skill or technique); (b) selecting a problem or an investigation that will help students to arrive at an understanding of that goal; (c) seeking the most effective way of posing the problem to the class; (~) anticipating student responses, including alternative approaches as well as misconceptions; and (e) encouraging or eliciting generalizations. Although these features are at the forefront in the lesson study sessions, one question that arose was to what extent does this type of thinking and reflection pervade the day-to-day activity of teachers? Necessarily, the role most often playe(1 woul(1 be that of an observer rather than a presenter, especially when the teacher is a novice, gaining experience. Presenters may be more experience(1 or repute(1 teachers. Thus, an important function GROUP TV served by lesson study sessions seems to be to provide each teacher with an oppor- tunity to experience and absorb the ingredients of a successful model as an observer but with the privilege of being an active participant as a friendly critic. Of course, soon a time comes for every teacher when that teacher needs to play the role of a presenter. By this time, however, the tension of being on the spot is dissipated by the previous experiences, in which they have observed that there is an impersonal protocol of criticism directed more at the content of the lesson (what works anti what (toes not) or suggestions about how one might more effectively introduce a specific detail of the lesson, rather than at the inadequacies of the teacher. Thus, lesson study sessions seem to provi(le a model for emulation rather than an active (lay-to-(lay operating procedure. Nevertheless, because the model is highly valued and the lessons produced and perfected by the model can be used on a national basis, it exerts a powerful exem- plary force. The idea of continuous improvement through reflection and analysis of specific pedagogic decisions is implicit in the model, and the general agreement seems to be that it is a tried- and-tested operational model worthy of adoption on a routine basis. What do teachers apse? RESPONSE At the elementary level, the standard curricular materials, containing specific lessons, together with the detailed teachers' guides seem to provide the foundation on which classroom lessons rest. For the novice teacher they provide a safe, secure, and acceptably effective map that can be

followed while that teacher is acquiring and honing professional skills. For the more experienced or enterprising teacher, they provide a template from which that teacher may (lepart by providing innova- tive variations of approach and treatment, while maintaining the conceptual goals of the lessons. A striking feature of the lesson study sessions that we observed was the use of the blackboard as a vehicle for recording the ideas generated by students, thereby acting as an archive of the ideas under discussion in class, rather than as a written repository of the ideas sought to be conveyed by the teacher. It was not clear what role manipulatives played in lessons at the elementary level. There was a short, oblique discussion of this at one point. The impression one gets is that they play a peripheral role, subject to the mathematical goal of the lesson. to works with teachers? RESPONSE Our Japanese colleagues said that during the preservice years interactions between faculty and students are formal for the most part. However, there seems to be some mechanism by which the student becomes aware quite early in the pre-service years that continuous improve- ment is the professional ideal, and that lesson study (or other similar reflective activity) is a powerful anti effective tool for continuous improvement. The impor- tance of the central role played by this cultural agreement about professional norms cannot tee overestimated. Japanese colleagues made statements such as, '~he importance of continuous improvement though lesson study is taken for granted in Japan," "One can always improve by studying one's own lessons with the help of colleagues", "How can you be a func- tioning teacher if you do not care to improve?," anti "Even excellent teachers can always improve. One must continu- ally seek to improve. Teaching is 80 percent confi(lence anti 20 percent (loubt." During the initiation/mentoring phase, we were told that experienced teachers work with new entrants to the profession. This seems to have been a common practice of the profession, now adopted as official policy by the Ministry of E(luca- tion since 1998. During this novitiate phase as well as later in the teacher's career, teachers seem to have some opportunities to get advice directly or indirectly from disciplin- ary experts such as mathematicians or mathematics educators in higher e(luca- tional institutions. In schools attached to university departments charged with the preparation of teachers, this happens quite regularly as a part of the ongoing implementation of lesson study in the research program of the university. In other schools it seems to happen less systematically. When it does happen, it would probably be in the context of lesson study events at the ward or prefectural level. On the whole, the development of deep conceptual understanding of the concepts that teachers teach seems to come from this continuing process of careful lesson planning and implementa- tion, followed by reflection about its effectiveness, informed by constant analysis of student ideas. Few research mathematicians seem to be involved in school mathematics education. CONCLUSIONS The process of lesson study seems to be a key ingredient in the professional SMALL GROUP DISCUSSION

development of Japanese teachers, especially at the elementary and middle school levels. However, the precise manner in which this process facilitates the acquisition of disciplinary as well as pedagogical content knowledge by teachers is subtler than one might assume at first sight. Lesson study sessions are not numerous enough to act as vehicles through which teachers can acquire deep content knowledge of the great many mathematical topics they need to teach- teachers must necessarily seek other ways in which this knowledge may be acquired. Likewise, lesson study sessions are not frequent enough to ensure that teachers will absorb their methodology through force of habit. However, lesson study sessions serve as exemplary models for several aspects of practice. The thoroughness of preparation and presentation, the stress laid on analysis and reflection, and above all the single- minded focus on the effectiveness of the lesson (rather than the effectiveness of the teacher) are ideals that the teacher is encouraged to emulate in daily classroom practice. By the avowed acceptance of lesson study as a long-term professional development tool, the individual teacher affirms a belief in the value of these practices and a commitment to put them into practice. The opportunity to engage in lesson study sessions on a regular basis has a practical as well as symbolic value. A lesson study session forces the partici- pating teacher to hone her or his skills to an edge sharp enough to withstand critical evaluation by colleagues and to continue to employ those skills in daily practice. On the other hand, due to the high status collectively accorded by the profession to the process of lesson study, the individual teacher can justly regard his or her successful participation in it as evidence of professional growth and competence. This periodic affirmation of exemplary GROUP TV professional values and practices greatly reinforces the teacher's image of himself or herself as a professional engaged in a process of continual improvement. Thus, in a roundabout way, the process of lesson study seems to have the effect of enabling the majority of teachers to arrive at and sustain a view of themselves as members of a professional community engaged in continualimprovement. On-thejob acquisition of deep understanding of content and pedagogy depends on this attitude more than any other single factor. In the final analysis, this effect may be as valuable as any other effect of the process of lesson study. RESEARCH ISSUES In the short time span of the workshop, it was (difficult to get an idea of the extent to which individual teachers implement on a day-to-day basis the practices exempli- fied in the lesson study sessions. A study of how lesson study effects day to day practice would be useful. Also in (liscussions the Japanese indicated that teachers were generally expecte(1 to have a "good un(lerstan(ling" of the mathematics they are expected to teach. An examination of various certifica- tion requirements coul(1 give us an idea of the formal (leman(ls that are ma(le on teachers in the way of technical content knowle(lge. How (lo teachers acquire the (leeper conceptual un(lerstan(ling (involv- ing major ideas anti their interconnections) as well as the pe(lagogical content knowI- e(lge (such as effective strategies for communicating those ideas, awareness of common misconceptions anti strategies of (leafing with them, etc.) ? Is it on the job, through a commitment to the i(leal of continuous improvement through reflec- tion anti analysis, as (lescribe(1 above?

~;TTT" Group Members Angela Andrews ~ ~ *, Deborah Ball, Toshiakira Fujii, Henry Kepner, Jr., Jean Krusi, Marilyn Mays ~ ~ *, Keiichi Nishimura, Ottawa Sakai, Kayo Satou *, Mark Saul ~ ~ *, Keiichi Shigematsu, Yoshinori Shimiz~, Lucy West, Susan Wood Report Coordinator; ~ ~ Translator; ~ ~ HIS. ICME Travel Group The group asked and answered the questions listed below. Much of the discussion centered around the details of Japanese lesson study. The group discussed the three models of records of practice presented in the workshop: video (Ball), written cases (Smith), anti Japanese lesson stu(ly as seen through the video of a fourth-grade lesson and study group discussion and observation of a s~xth-grade lesson and study group discussion. How do teachers use mathematics together with other kinds of knowledge and skill in order to connect students with mathematics? How do skilled teachers learn about and make use of their students' knowledge and capabilities to help them learn mathematics? RESPONSES Both questions were addressed through the discussion of the record of practice feature(1 (luring the workshop. Both video anti written cases provi(le records of practice for long-term use. Video records of practice can recor(1 the learning of the same student over time. In one viewing, a spectator can see lessons from different parts of the year. The author of a written case study may affect how useful it is. Japanese colleagues found video an easier medium to work with than the case stu(ly. To them video is a familiar medium and easily allows forming of images. If a teacher poses a question and does not get any of the expected answers, live observation might provide insight into why the expected answers did not occur. In Japan, the lesson plan, student responses, and teacher's reflections form a sort of "case study," although not in written form. Overview of "lapanese Lessons In the lesson write-up, like a script for a play, the left column gives student activi- ties, the right side gives "cautious points" and "evaluation points." Usually, a middle column in the plan contains expected student responses. The evaluation of the lesson must be consistent with the lesson's aim. Stan-

cards for evaluation are in terms of students' · willingness, intent, and attitude toward learning; mathematical thinking displayed; representing and processing origi- nally called skills of content; and · understanding. Each lesson is like a mountain, with a slow climb to the peak. Each lesson has a rhythm. Teacher personal satisfaction is not enough. To help students gain mathematical satisfaction, the teacher must sum up the mathematical objective of the lesson in which the teacher moves from naive student solutions to those that are more sophisticated. Japanese teachers like whole group discussion, not the one- to-one discussion between student and teacher. Students often explain other students' ideas. The parts of the lesson are introduction, development, and summary. Lessons must be set in a clear context. The use of the board is carefully planned and makes a summary of the whole lesson. Students can see what was learned, even if they do not completely understand. Students' Notes Students' notes and errors are valued by teachers. Japanese students are taught to take notes in first grade. They copy the task and conclusion and record their own work in the middle. They are encouraged to write their own ideas. The levels of student note taking are · animpression "it was fun"; why ~ am interested in the lesson the math content "it was about x"; classmates' ideas; compare my ideas and friends' ideas see myself objec- tively; the student is the owner of an Plea; and GROUP V · self-reflection generalize the problem beyond the content of the lesson. Lesson Stucly arc! Teachers' Mathematical KnowlecIge In Japan, teachers' mathematical knowledge is important in enabling teachers to anticipate student responses which in turn strengthen that knowledge. Teachers also need mathematical knowI- edge to build on unanticipated responses and be ready to adapt to the students' responses and misunderstandings. Teachers select student examples that are close to their goal, then build on them to help accomplish the purpose of the lesson. Teachers must be alert to stu- dents' ideas that extend the lesson or for opportunities to probe for deeper under- standing of the content. Working on and revising the teaching plan is one way to buil(1 mathematical knowle(lge. The teacher must think mathematically while creating the plan. Much attention is given in lesson design to the specifics of the lesson, even to details such as using the number 12 instead of Il. Emphasis is placed on finding a suitable task, with the structure of the task being very impor- tant. Teachers must stay focused on the goal of the lesson. What is the nature of the postlesson discussion? RESPONSE The postlesson (liscussion focuses on · the gap between the plan anti the imple me nte (1 le s so n ; anti · the gap between lesson stu(ly anti the general sense of mathematics education. There can be confusion between these two aspects.

Teachers of the same grade cooperate- they are given the opportunity to look at existing lesson study plans and read related books. Japanese teachers have very different backgrounds, much the same as U.S. teachers. Lesson study focuses on the "whys" of lesson design. In lesson study, the conversation immediately following the lesson is crucial to understanding and improving the lesson. The Japanese believe that observing the class (seeing with your own eyes) is best for staff development. What is the role of the advisor in lesson study? RESPONSE The advisor's role in lesson study is to identify key points to improve the lesson and teaching, to identify the most valuable things mathematically and pedagogically even in a disastrous lesson. In lesson study, the quick feedback from and to the teacher is very powerful. Advisors can ask "Why (li(1 you write in that place on the blackboard?" A brief record of the entire lesson should be on the black- boar(l. The teacher evaluates the advisor and does not invite an advisor back if the comments are not deep. How are teachers taught to he good observers during lesson study? RESPONSE Observation skills for live observation are very important. Teachers need to know how to observe well. When the teachers are students and first visit classes, they cannot even take notes- they do not know what to look at. They are taught to look at the relationship between the class purpose for that day and what happens in the class. They study one student, sometimes standing beside them to observe all they do, to (letermine whether the stu(lent's actions reflect the purpose of the lesson. Obser- vation is from the teacher's side of the classroom, not the back of the classroom, in order to view the students' faces. Observers consi(ler what the teacher asks, discriminating one question from another, anti (1iscerning the teacher's moves. Itis(lifficultto teach teachers to observe well. A goo(1 advisor is nee(le(l. SMALL GROUP DISCUSSION