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The first full day of the workshop began with a presentation by Deborah Ball designed to set the stage for the use of teaching practice as a medium for professional develop- ment. She framed a central question for workshop participants to consider: "How do teachers in Japan and the United States use practice to work on their teaching?" The rest of the workshop was spent exploring three different examples of the use of teaching practice as a medium for professional development: lesson study, video records of a class, and two cases describing mathematics classes. The sessions that focused on lesson study included general descriptions of the design and enactment of a lesson and postlesson rejection by the instructor and observers. Yoshinori Shimizu provided background on the nature of lesson study, and Makoto Yoshida extended that background and described one project on the implementation of lesson study in a U.S. school. Participants viewed two classroom lessons and the follow- up postlesson discussions. Hiroshi Nakano, the teacher in the fourth-grade lesson, gave a brief description of how lesson study was carried out at his school and his thoughts about the value of lesson study for his own professional growth. Video excerpts from the lesson and a printed version of the postlesson discussion are included in the following section as part of the proceedings. The day ended with three panelists reflecting on lesson study, the lessons they had observed, and how the day's experiences related to their own backgrounds as mathematics educators. Setting the Stage Deborah Loewenherg Ball, Professor, University of Michigan What can be learned from using practice as a means of (developing teachers' knowledge of mathematical content and how to teach that mathematics? What questions should frame our thinking?

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Lesson Study: What, Why, and How? Yoshinori Shimiz~, Associate Professor, Tokyo Gak~gei University How does lesson study work and what is its role in developing teachers' content knowledge and understanding of how to teach? Framing Lesson Study for U.S. Participants Makoto Yoshida, Professor, Columbia University Teachers College Lesson Study from the Perspective of a Fourth-Grade Teacher Hiroshi Nakano, Elementary Teacher, Setagaya Elementary School and Tokyo Gak~gei University LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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Deborah Loewenberg Ball, University of Michigan The question we want to work on together is, "How do teachers in Japan and the United States use practice to work on their teaching?" Why are we so interested in learning how teachers in both countries use the practice of teach- ing itself to work on their teaching? Let us think about other practices for a moment, for example, singing opera, writing poetry, playing soccer, cooking, or the practice that we are interested in here, teaching. It is important to notice three things about learning practices. First, you do not learn a practice simply by doing it. For example, poets do not become good at poetry simply by sitting with paper anti pencil and writing. You do not learn to cook simply by taking pans out of the cupboard and putting a pan on the stove with some food. Practices also are not learne(1 simply by acquiring knowle(lge. No one becomes a good soccer player by reading books about soccer. Finally, practices are not learned only by watching experts do them. If you attend a concert and listen to an opera singer perform opera, it is not likely that you wall be able to perform opera yourself. Each of these can help. It can help to watch experts engage in a practice. It can help to acquire knowledge about the practice. And it can help to do the practice. But none of these is enough. LEARNING PRACTICES Let me make some points about how practices are learne(l. First, learning a practice requires stu(ly. It requires trying things, anti it requires analyzing how the things that you tried work. Such analyses enable improvements. You (levelop new i(leas of things to try. This is true for playing soccer. It is true for writing poetry. It is true for many practices. We are interested in how the practice of teaching is learned, and all of these things are important to learning the practice of teaching. Second, there are practices involved in learning a practice; for example, watching teaching is not common sense. If you bring someone into a classroom anti ask them to watch a lesson, they may not know what to watch. They may not notice what the chil(lren are (loin". They may not know how to listen to the very specific way a teacher asks a question. There are things to learn about how to watch teaching carefully. There are things to learn about how to (liscuss teaching with colleagues.

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There are things to learn about how to make records of one's teaching so that later one can examine one's own work and the work of the students, show these records to other people, and discuss them. None of these things are automati- cally known. These are the things ~ mean by practices important to learning a practice; and if we talked about opera or cooking or soccer, we could make a list of practices important to learning those practices as well. In the workshop, we want to learn what practices teachers in Japan and the United States use to learn the practice of teach- ing. What do teachers in these two countries do that enables them to develop their teaching practice? MATHEMATICS TEACHERS' PRACTICE Keiko Hino and Keiichi Shigematsu gave a (definition of teaching practice in their paper. They said that mathematics teachers' practice can be construed as a set of abilities grouped into three parts: prelesson, midlesson, and postlesson. Prelesson. The mathematics teachers' practice is construed as the ability to first, in the prelesson, organize and transform mathematical knowledge according to the goals and purposes of the lesson. Second, the prelesson requires teachers to have what they call the "eyes" to evaluate how that organization of the mathematics works for the students. Midlesson. Next, they talked about the midlesson. They mentioned several things that teachers must do while enacting the lesson in class. First, teachers must execute the plan they have made. They must move from the piece of paper with the design and use it with the students in their class. This means they must create activities that lead to their goal, and they must take notice of students' current situations all the time. In the sixth-grade classroom we visited at Tokyo Gakugei University Elementary School the other day, we noticed that the teacher was constantly looking at the students, watch- ing, trying to figure out how the lesson was being received and experienced. Do the students understand? Do they know what I am asking them to do? Do they understand each other? Sometimes teachers apply routines they already know, but they must make judgments; they must decide that this moment is the moment for that routine. Sometimes situations arise in teaching for which teachers have no routine, and at those times, as Professors Hino and Shigematsu suggest, teachers must promptly invent new actions to manage what they see happening in their class. This is very complicated work. Postlesson. In the postlesson, teachers reflect on what happened in the lesson. They analyze how the design worked with their students, and they develop concrete plans for the next lesson on the basis of what happened in that lesson. Summary. This is a cycle of design: generating designs, using the design with students, analyzing how it works, revising the design for the next step. How do teachers learn to do this well? What activities and practices do teachers in Japan and the United States use to develop their abilities to carry out this practice of teach- ing? Keiko Hino and Keiichi Shigematsu indicated that there are several abilities that are important in a mathematics teacher's practice. One is the ability to design lessons, to organize and transform math- ematical knowledge and have eyes to evaluate the results of the design. Another is the ability to enact lessons, to carry them out in class. This involves creating activi- ties, taking notice of students, applying routines, and inventing actions depending on how students understand the content. LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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Finally, this indicates the need for yet another ability: the ability to analyze lessons. Through such analyses, teachers develop knowledge about teaching and work out concrete plans for next lessons. How do teachers develop knowledge about teaching so that each day, each year, with each class of students they become more skillful, more able to do these things? What practices enable them to become increasingly more adept at designing lessons, carrying out those lessons with their students, and all the time observing how the lesson is working with their students? WHAT DO TEACHERS NEED TO LEARN? So ~ offer a short but difficult list. What do teachers need to learn to engage in this practice of mathematics teaching? One thing that repeatedly came through in their discussion of practice was how to pay attention to and teach every student in the class. We were impressed that with a class of 33 students the teacher was watching and looking around, trying to understand what the students were learning. This is not an easy thing to do. How do you figure out what the students un(lerstan(l, whether they are paying attention, whether they are following, if the examples make sense, if they are interested, and if they are learning? Students are different from one another. This requires great skill to do what sounds like a very simple thing. Second, teachers need to know how to know mathematics and to use it to help their students learn. This is not the same thing as knowing mathematics to do mathematics by yourself. How do teachers learn to know mathematics in ways that enable them to organize the content, create activities, and adjust the activities to address the goals of the lesson as well SETTING THE STAGE as particular students' interests, needs, problems, difficulties, and so on? This is another big area. In the presentations from the United States, we heard that we face serious problems of teachers not knowing mathematics well enough to help each of their students learn mathematics. Teachers' mathematical knowledge equips them to teach all students, so the first and second points are very related. Third, anti perhaps a little (lifferent, is that teachers need to learn how to work with others on developing knowledge for teaching. Some of us watched a group of Japanese teachers discuss a lesson last week, and today we will learn about lesson stu(ly. One important practice for developing teaching is to work with others on teaching, to learn to do the kinds of things that ~ have been talking about. One interesting point in Professors Hino and Shigematsu's study was how fre- quently a note was made that Mr. A engaged in discussions with others about his teaching. We are interested in what Mr. A (li(1 in these (liscussions, what he talke(1 about, what he learne(l. How (lo teachers learn to work together with others to develop their teaching? QUESTIONS FOR CONSIDERATION ~ want to end by suggesting some simple questions that we should ask ourselves as we learn about what Japanese anti U.S. teachers are doing to work on their practice. As we learn about lesson study, for example, we want to know what the teachers actually (lo as they engage in the practices of lesson stu(ly. What (lo they work on? What (lo they use to work on this? Do they look at students' work? Do they look at mathematics books? Do they rea(1 articles? Do they bring in other people with whom to talk? What (lo they

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use to enable this work? Who works with teachers? Do teachers do this alone? Do others work with them? What do teachers seem to learn and how do they learn these things? We want to ask these same questions tomorrow when we examine some of the practices of teachers in the United States who also work on practice to develop their teaching. We hope to leave here with more knowledge about what it takes to use practice as the site for working on something that cannot be learned only through study, only through watching experts, or only by working alone and just (loin" it. What (toes it take to use practice as a site for (leveloping practice? LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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`'A Yoshinori Shimiz~, Tokyo Gak~gei University Lesson study is a common element in Japanese educational practices. At the outset, however, there are differences in cultural background that should be considered as we discuss lesson study at the elementarylevel. Professor Usiskin indicated, for example, that student teachers in the United States usually have eight to ten weeks for training in the classroom, but the Japanese student teachers spend three or four weeks or in some cases just two weeks in classroom work. Also, in Japan in elementary schools and junior high schools there is a very large teachers' room where every teacher has his or her own desk. Those teachers who are not giving lessons spend their time in this room. Most U.S. teachers have their own room where they spend their time when they are not teaching. Consider one other example. About ten years ago, ~ visited a middle school in San Francisco. When the lesson started, a boy began to eat an apple during the lesson. "Why was this boy eating?" ~ asked the teacher after the lesson. She said, "He must have been hungry." What ~ wanted to ask was why he had to eat an apple in his class during a lesson, because it would never happen in a Japanese classroom. So things that we take for granted in our own culture may be some things that are not natural at all on the other si(le of the ocean. We have to keep that in mind when we consider any cultural activity like teaching. What follows is a brief outline of lesson study with a special focus on the role of lesson plans. Sometimes this is called the agenda or schedule, but whatever its name, for Japanese teachers it is something that is taken for granted, although they do not always prepare the lesson plan. Lesson studies are held at (1ifferent levels, and there are different types as well. Lesson stu(lies are con(lucte(1 as part of the preservice teacher training programs for student teachers. There is another type, calle(1 intraschool lesson studies, where maybe three times a year lesson studies are held within a particular school. Lesson studies are also held on a prefectural level, city level, or a school (listrict level, anti consequently organiza- tions anti programs vary, which is an important consideration to remember. Finally, lesson studies are held at the national level, open to outsiders. ~ just listed four (1ifferent types, but this, of course, is not an exhaustive list. There may be some other types as well. Generally a lesson study consists of the following: the actual classes taught to

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Before Deciding a "theme" (and organizing a team) Selecting a particular topic for the study Writing a lesson plan (analyzing the topic to be taught, assessing students' learning, examining the task to be posed, thinking teacher's roles, etc.) Discussing and revising the lesson plants) Tried by other teachers, or in another class Reflecting on the lesson and re-revising the plan pupils, observation by others, immediately followed by intensive discussion called the study discussion. Designing, enacting, and analyzing are the three stages that evolve before, during, and after the lesson, in other words preparation, implementation, and analysis (Box 11. There is extensive preparation made before the class, and there wall be exten- sive work done after the lesson study as well, which wall be used as a follow-up and as a preparation for the next lesson studied. These events form a cycle. Lesson studies also have different objectives and aims. One is to educate student teachers, and a second is to monitor and instruct novice teachers. In the late 1980s, a new system of teacher education programs was introduced in Japan for new or novice teachers. Newly hired teachers are closely supervised for a one-year period by the (1eputy school a=,' .c.~.~.l,.~er.tr=~_~er~el`~.~.l,~.~ml~e During Teaching/observing the lesson Recording what the teacher and students said, how students worked on the task during their seat work, and what was written on the chalkboard Extensive discussion on the lesson A se~f-ref~ection by the teacher Discussion on the task, students' responses, teacher's roles, and so on Comments and suggestions by a mathematics educator or an experienced teacher After Ideas are used in the following lessons Next theme may be identified A report of the lesson is sometimes shared with outside people LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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principal or by well-experienced teachers. As part of the program, novice teachers have to take various classes, and the contents of these courses are well defined, and often lesson study is a part of these courses. Another objective for all teachers is to improve teaching skills, which is what counts most of all. All teachers also need to learn the roles of maintaining and managing the school, and therefore collaborations among teachers are needed. Lesson studies are conducted to maintain such collaborations. Lesson study can be used to improve teacher content knowledge. In addition, the national-level study programs that are organized by a group of teachers or by a school are sometimes used to share new ideas, new methods, test new materials, or new approaches, as well as to demonstrate those new approaches. Usually lesson studies begin by choos- ing a specific theme. For example, a focus in the current movement of educational reform is on helping children develop their own thinking ability. This focus may guide the selection of the theme. More general themes may also be chosen, such as "teaching pupils how to live." A team of teachers is organized as part of the process. When the size of the school is small, the entire school wall often be involved in the team. A particular topic is selected for the study, and the lesson plan is written by the team. One thing to be emphasized here is that by writing and revising a lesson plan, we work on the lesson plan, refining it in an iterative manner. The content knowledge, the pupiT's learning level, the specific tasks to be presented to that pupil are part of the lesson plan. "Are you going to use the number 10 or 12 for this particular task of multiplication?" It can make a big differ- ence, and such minute details are well planned before the class is given. Usually LESSON STUDY: WHAT HOW AND WHY? the class duration is 45 minutes long, but hours of preparations are made before- hand. Sometimes the same topic is taught by other teachers in other classrooms for trial purposes. Then you revisit and reject on the lesson to rerevise the lesson plan. The activities above occur before the class. During the study lesson in class, the observers will take very detailed notes. What are the responses of the pupils to the given task and what did the teacher say? What were the questions raised by the pupils? What the teacher wrote on the blackboard is recorded as well. In other words, many things happen during the observation phase. During the postlesson (liscussion that follows the study lesson, the teacher who taught the class would share his or her own impression or reflection about the class with the observers. This is followed by intensive discussions on the tasks, students' response, teacher's role, anti on anti on. An invite(1 principal, mathematics educator, or experienced teacher may give comments and suggestions about the class as well. After the stu(ly lesson, the feedback from this class woul(1 be use(1 for the next class, and the theme for the next class will be identified as well. Sometimes a report is put together, the ideas from a lesson stu(ly are presente(1 in journals, or the materials are (listribute(1 within the school or within the school (listrict to be share by fellow teachers. That is the basic outline of lesson study, but ~ woul(1 like to say a few wor(ls about the lesson plan. Throughout the lesson studies, the lesson plan serves as a medium for communication among teachers (Box 2~. Lesson plans have various purposes or objectives as well. Box 3 shows the common framework for lesson plans. The matrix shows the steps that shoul(1 be followe(1 (luring a 45-

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Sharing ideas Discussing various aspects of the lesson Used as a frame of reference for the lessons Shaping the lesson flow (script) Steps Posing a problem Students' problem solving on their own Whole-class discussion Summing up ~ Exe rc ice/Extension Main Learning Activities Anticipated Students' Responses Remarks on Teaching minute class. These give the teacher a certain image of how the class is going to move forward. Sometimes an exercise or an extension will be provided as the final step. Keeping the common framework for lesson study in mind, we can interpret some of the fin(lings of the Thir(1 Inter- national Mathematics anti Science Stu(ly videotape classroom study conducted by Stigler anti his colleagues that compare eighth-gra(le mathematics lessons in Germany, the Unite(1 States, an(l Japan. One of the biggest (lifferences among the three countries was, for example, the alternative solutions presente(1 by the teachers anti by the students (luring one class (Figure I). As Figure ~ indicates, more alternative solutions are presente(1 by students in Japan than in Germany anti the Unite States. This is naturally interprete(1 as closely relate(1 to the lesson plan because it is reflected in considering the antici- LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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FIGURE 1 Lessons that included alternative solution methods. Teacher Presented 45 ~ 40 ~ 35 - ~n o In o a) <~ 20- a) c' a) 15 - Student Presented 42 30 - 25 - ~9 O- 12 = 14 Germany Japan United States SOURCE: Data adapted from Stigler, 1999. pated response of the children and in the course of discussing and revisiting the lesson plan. In fact, the anticipated students' responses make up a large part of the lesson plan Appendix D). Finally, throughout the discussion on lesson study, the teacher's content knowI- edge and understanding of teaching practice wail improve, and through the entire lesson stu(ly this will be further refined (Box 4~. As was mentioned above, lesson study is a common element in Japanese educa- tional practices. Also it is a necessary element for improving teachers' content knowledge and understanding of how to teach. _=R ~er.~.~.~i.~.~=r~c~.~~l~r~i~~.~.~ MAR:_ Interwoven in a certain way Reflected on his/her anticipation of students' response to the task to be posted Developed through examining and discussing lesson plans and by observing and reflecting on the lesson Elaboratec~ in the process of lesson stuc~ies LESSON STUDY: WHAT HOW AND WHY?

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on the school. However, in interschool study, each school is required to conduct in-schoollesson study. During the prelesson study, the teachers teaching first and second grade form one group, third- and fourth-grade teachers form another group, and the fifth- and sixth- grade teachers form a third group for the discussion. The research promotion committee chairperson sometimes joins those groups, and sometimes we would call in an external adviser to have the discussions. Then we conduct the class and have a discussion afterwards, in which the teachers within the school participate. ~ was primarily engaged in war(l- initiated lesson studies and school- initiated lesson studies. For me, what is lesson study? It is an indispensable field of training. When ~ was in the university, did practice teaching just like lesson study. The teacher responsible for my teaching training told me that it is very important to have lesson study, and that idea has not changed since then. ~ would like to make my class be enjoyable for children's thinking. ~ want the class to operate so that the children's thinking can be recognized by others and also by the teachers. ~ also like to make the class feel that they can find out about the similarities and differences of their ideas in relation to others. To realize these wishes means training. That is how ~ see lesson study. Lesson study is where you can express your pleas and also you can improve your position and status. Through these lesson studies you can make presentations about your teaching within the mathematical education com- munity and establish your own ideas. What we learn from lesson stu(ly changes as we accumulate experiences. When you have little experience, you learn methodology and how to run the class. You are taught by many people through the prelesson studies. As you gain experience, rather than learning how to conduct the class, in prelesson stu(ly you can get to know the value of math- ematics anti the value of the materials. Accumulating such experiences was a great asset for me. Through the prelesson study ~ am taught by others. Having others point out my weaknesses, un(lerstan(1 what they are. This leads to motivation to improve for the next occa- sion. However, lesson stu(ly is rather (difficult even if you plan ahead very well. Even though you think you did well, others might point out what went wrong. So lesson stu(ly is both (difficult anti rewarding. LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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l l ~ l a-l ~ ~ 1 - LESSON STUDY Workshop participants viewed two elementary classes and the follow-up or postlesson discussion for each lesson. As part of the Ninth International Congress on Mathematics Education CCME-9) that preceded the workshop, workshop participants had the opportunity to visit a s~xth-grade class during an introductory lesson on functions. Makoto Yoshida translated the lesson and the postlesson discussion for those in attendance. The lesson plan and transcripts of the selected portions of the lesson and postlesson discussion are available in Appendix E. During the workshop itself, participants viewed a video of a fourth-grade lesson on place value along with a video of the postlesson discussion by the observers. The lesson plan, a description of the content, and a summary of the postlesson discussion held by the Japanese, both '1~-1 translated by Makoto Yoshida, and a transcript of the selected portions from the actual lesson and discussion can be foun(1 in Appen(lix F. VIDEO SELECTIONS A Demonstration Lesson: Function Thinking at Sixth Gra(le taught by Shunji Kurosawa, Tokyo Gakugei University Setagaya Elementary School Lesson anti Postlesson Discussion among S~xth-Grade Lesson Observers A Study Lesson: Large Numbers at Fourth Gra(le taught by Hiroshi Nakano, Tokyo Gakugei University Setagaya Elementary School

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A panel addressed the following questions related to the fourth- and s~xth-grade lesson: How do the two lessons compare? What are the differences and the similarities? What was the mathematical content and how did the lessons develop student under- stan(ling? Panel Moderator: Keiichi Shigematsu, Nara University of Education Jacqueline Goodioe, Resource Teacher, Burrville Elementary School, Washington, DC Jerry Becker, Professor, Southern Illinois University Ichiei Hirabayashi, Professor Emeritus, Hiroshima University

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COMPARISON OF THE TWO LESSONS "lacqueline GoocIloe, Burrville Elementary School First ~ would like to say that this has been a very rewarding experience, both the Ninth International Congress on Mathematics Education (ICME-9) and this workshop. ~ am an elementary mathematics resource teacher. A math- ematics resource teacher in the District of Columbia public schools does not have responsibility for one classroom but services all the teachers in that building in mathematics. Our school has about 360 students, a relatively small enrollment with classes from all day prekindergarten through sixth grade. ~ want to point out some of the similari- ties that ~ saw in the two lessons. Both lessons addressed important mathematical ideas, getting a sense of large numbers, posing questions, finding functional relationships. Most elementary ciass- room teachers do not often discuss these mathematical ideas. The teachers in the videos often related the task to concepts with which the students were familiar. Time was allowed for students to talk and explain their thinking. In both examples, class size was large, between 30 and 40. That is large for some U.S. classrooms. The planning was evident. Another similarity that is familiar to some of us was the teachers' genuine delight in student learning and student understanding. The strength ~ see in lesson study is the growth that develops through the collabo- ration and discussions with other teachers. This is not always evident in U.S. schools- thinking about "why we do what we do." Reflecting on practice is an area in which would like to see some improvement by teachers in the United States. REFLECTIONS ON VIDEOS: PANEL In both videos we saw the teachers moving around in the classes. However, we did not discuss a lot today about what specifically the teachers were looking for that was going to help them assess the student learning. They were moving throughout getting an un(lerstan(ling of what the students were thinking, but what were the teachers thinking as they watched the students? Much of the lesson study in the preplanning deals with anticipated student responses, the student results. ~ would like to know more about how much time is spent developing that part of the lesson anti the source of that information. Is it all from the teacher's experiences? Is it all from the study group's experiences? Where do we get the notions of how students will respond and how can that benefit teachers? As we look at the videotapes anti sit in (1iscussion groups, something else about lesson study emerges. There are implications about what "teacher talk" is about. The last (liscussion about new teachers, preservice teachers, and those that are new to lesson study and how they must know the curriculum, the textbook, and the math- ematics was important. When does this happen anti over how long a perio(1 of time are we talking about? If lesson stu(ly is such a powerful tool, ~ am wondering if videotaping of the sessions shoul(1 be done so that other teachers can get the benefit of the experience. There are other questions as well. For example, what does lesson study look like at the high school level? Do skille(1 teachers, inexpe- rience(1 teachers, anti preservice teachers all have (lifferent perspectives on the lesson study? This past week ~ have heard more and more about the Japanese intent to engage students in interesting anti motivating activities and, in fact, one of the speakers

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talked about developing a sense of wonder and a new approach for mathematical thinking. ~ would like to know, if lesson study is so powerful a resource for professional (levelopment, why aren't more lesson studies done in all the public schools as much as is done in schools attached to a university? It would seem that the public school teachers could benefit from this powerful resource as well. The workshop has opened my eyes to some wonderful and powerful ways of looking at mathematics and of looking at mathematics teaching. Teaching is hard, and thinking about teaching is even harder. FOURTH GRADE AND SIXTH GRADE: OBSERVATIONS OF THE TWO LESSONS "ferry Becker, Southern Illinois University ~ feel very fortunate to have had the opportunity to see Mr. Kurosawa's lesson, the film of Mr. Nakano's lesson, and to have had the benefit of the discussion of these lessons. If ~ put both of these lessons together for a moment, that is, if just speak in aggregate of some of the observations ~ made, the first thing ~ would say is this, Teaching mathematics is really a big job! Overall ~ thought the tasks in both of the lessons were very good. To begin with ~ was not so sure about the task around which Mr. Nakano (levelope(1 his lesson plan, but eventually through the discussion ~ came to see that it was also a very good task. One of the things ~ noticed is that the productions and the observations of the students during the lesson are written on the board for everyone to see. They are left there so they can be referred to and so students have enough time to write them down in their notebooks and to refer to them. We saw in each of the lessons that the teachers wrote down a number of important observations made by the students. In the s~xth-grade lesson ~ think the number of observations on the part of the students was nine or ten, and none were suggested by the teacher. The teacher rather skillfully set the learning situation and then provided time for the students to use their own natural thinking abilities. So the lesson proceeded on the basis of the productions of the students, which ~ think is significant. Another thing ~ noticed is that in both of the lessons not only (li(1 the students seem to be enjoying them anti sometimes getting excited about what was going on, but they felt perfectly comfort- able in sharing their observations. The significance of this to me is that the class- room situation has all the makings of a community of scholars. These young people feel perfectly comfortable once the learning task is set, and they are given time to make their observations, to share them, and to get reactions from the other students. It was also very clear to me that the teachers were experts on the problem task anti in han(lling anti managing the lesson so that when various responses were given by the students the teacher knew how to respon(1 to those anti how to (leal with them. Clearly they were very knowle(lgeable about what was going on. ~ think the students hall time to work on the tasks, to use their own natural thinking abilities. The teacher took time to view the work the students were doing so that the responses that fit the objective of the teacher coul(1 be written up on the boar(1 for everyone to see, where it coul form the basis for a discussion. There was also encouragement to the students to write (town their observations, writing out their observations in wor(ls as well as LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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writing down what they were thinking in mathematical symbols. ~ saw a very nice contrast with what we commonly see in the United States. There is a difference between verbalizing observations and verbalizations. Verbaliza- tions come as a consequence of learning mechanical procedures, but here in the lessons we saw the students commenting on and explaining the reasons for the observations that they made. The teacher asked for clarifications. Students were asked to verbalize their thinking. To me this forms the basis for developing the language of mathematics in the class- room, and ~ think this is very important. At the end of the lessons, the teacher asked students, for example, to think of a general expression for a given number of minutes later. The common approach is to go right to that expression, but here the focus was on all of the processes that lead up to that. Also, we found the teachers making a conscious attempt to put the problem situation into the reality of the students, and ~ thought that was very goo(l. In discussing the lessons, it occurred to me that what we are talking about here is the assessment of lessons. During {CME-9 ~ heard several talks in which the points were made that good problems are crucial in assessment, the assessment of lessons as well as the assessment of individual learning. It is the case that the teacher can get whatever the teacher asks for, and that means the richer and the more open the problems, then the more potential they have for revealing students' under- standings and abilities. For example, at the beginning of Mr. Nakano's lesson, ~ thought he was going to engage in direct teaching, and ~ wondered why he was going to do that. It took me a while to realize that was not the case. In fact, he was seeking to find out what the students' REFLECTIONS ON VIDEOS: PANEL understandings and abilities were, and he had a systematic way of approaching the situation so he could get some insight. We saw that insight also in the discussion of the lesson. So there was time given to the students to show what they could do with these learning tasks, and in so doing we could get some insight into the difficul- ties they had, the ability that they seemed to be able to demonstrate, and so on. Since my first visit to Japan many years ago when ~ talked about the things ~ thought ~ had learned from observing numerous class lessons, my colleagues back in the United States commented, "But, Jerry, you have to remember that is Japan. The culture is very (lifferent from ours. You cannot import what goes on in another culture." ~ (li(ln't believe that then, anti ~ (lon't believe it now. ~ think we can look at the tasks, for example, what was on the blackboard (Appendix F). We can have very good insight, regar(lless of culture, about the problem the students are given and the different ways the students approach it. While we watch the students and listen to them develop their own ways of (leafing with the problem situation, we can see that they exhibit responses that are qualitatively different. The students generate a situation where they can (liscuss, in a mathematical way, many responses anti perhaps even come to a consensus about which are the better ones from a mathematical point of view. don't think that is specific to any one culture. What that is specific to in my judgment is mathematics, anti teaching anti learning mathematics. Finally, in our small group (liscussion we were considering the question: How (lo skille(1 teachers learn about anti make use of their students' knowle(lge anti capabilities to help them to learn math- ematics? We have seen (luring the discussions here that the skilled teacher

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poses a problem situation and lets the students show what they can do. That is a good way to get insight. We also dis- cussed or had described to us the impor- tance of and the meaning of a detailed lesson plan. So before the lesson is ever taught, the teachers sit down and discuss the problem situation and try to list all of the anticipated responses of the students. If the three of us are the team that is getting ready to develop a lesson, then am going to learn quite a lot about that problem situation from my colleagues, and perhaps they will learn something from my insights on the problem too. While the lesson is in progress, the teacher has the option of purposefully scanning the work of the students. By looking at their work, the teacher can decide which responses everyone should see. The teacher knows which responses should be shared on the board and why they should be discussed. The reason, of course, is because the discussion wall tie in with the objective of the lesson that the teacher has in mind. LESSON AS A DRAMA AND LESSON AS ANOTHER FORM OF THESIS PRESENTATION Ichioi Hirabayashi, Hiroshima University Prelucle ~ remember it was around 1975 that ~ happened to be visited by two American professors: one of them was in science education whose name ~ have forgotten, and the other was in mathematics educa- tion named M. Vere Devault. ~ hall seen his name in the 32nd Yearbook of the National Council of Teachers of Mathe- matics anti other publications, but it was my first time to meet him. At that time was a young professor at Hiroshima University, and it was very impressive to hear from him that there were two features in the research on curriculum in mathematics education: technology and humanity. Ithink the same two features would be in lesson stu(ly. Since then, in Japan, and perhaps in America, the research on technological features has developed remarkably, but not much attention has been paid to the humanistic feature. This feature is far more important than the technological one because of its external effect on children's future life. There also are the same two features in the lesson, and would guess in a workshop on lesson study as a method of research about mathematics teaching, the technological feature would also be emphasized far more than the other. In this atmosphere of research, it is important to stress the humanistic feature of the lesson, anti this is the very reason that ~ wrote this short paper. A Reminiscence: Complexity of a Lesson ~ wish to start with a reminiscence of my own when ~ was mathematics teacher in the lower secondary school soon after the graduation from university. In this school ~ ha(1 the same two courses for two classes in the same gra(le. ~ taught them the same topics almost every time in the same way in each class. But in the final examination, ~ was surprise(1 to fin(1 that the achievement of the class taught in the second lesson was far better than the first, although each class was equally divided according to their ability at the start of the term. The reason seemed evident: The teaching ability of the teacher (me) ha progressed from the first lesson to the second. But at that time it was difficult, anti still is, for me to analyze this reason clearly anti persuasively. The lesson is a LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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very complicated phenomenon with many subtle factors interrelating with each other: the teacher's ways of speaking, asking, responding, and time, place, gestures, and so on. Among these factors, there may be some vital importance in deciding the success or failure of the teaching. Lesson is a complex phenomenon in a "classroom culture," as pointed out by Heinrich Bauersfeld (Bauersfeld, 19961. The advantage of lesson study as a method of research in mathematics education is that it is the way to grasp the true state of affairs of the problem in its whole and to bring a synthetic, totally recognized interpretation to it, being aware of many factors that are subtly interacting with each other as if it were in one organism. The lesson, as ~ mentioned above, is a very complicated process, and its effect on students' learning is too subtle to express in a simple literal thesis. It can be evaluated only through close observation about what is actually going on; a mere written report is not only unable to describe the actual state of affairs but often overlooks the kernel and essential points of the lesson. Lesson Demonstration in Conference Before War World IT, we had two or three normal schools for the education of primary school teachers in each 47 prefectures of the country and several higher normal schools for the education of the secondary school teachers in some districts of the country. One or two primary or secondary schools were attached to these teacher training schools for the training of student teachers and the practical research of education. In each attached school it was a custom to have a conference usually once or twice a year to present educational ideas to the teachers of the (listrict. It also was a custom to have a session of lesson observ- REFLECTIONS ON VIDEOS: PANEL ing, where an expert teacher from this school showed a lesson, which was a model for the participating teachers. During a renowned teacher's lesson, the class might be surrounded by many attending teachers occasionally number- ing more than the pupils in the class. There was one thing to be noticed here: The observing teachers seemed to believe that a unique best teaching method was embodied in this expert teacher's demon- stration of teaching and could be learned only through observing this model lesson (Erectly. At that time we had a national curriculum (as is still the case), and in primary school, textbooks were exclusively edited by the Ministry of Education. Un(ler these e(lucational-political circumstances, it was natural that teachers at that time believe(1 in the existence of a best metho of teaching and seriously wished to acquire it. Certainly it would be an obvious falsehood that there is a unique best teaching method. However, there is a profound implication in believing that a teaching method, whether goo(1 or ba(l, is embo(lie(1 in a teacher's performances or even embe(l(le(1 (1eeply in the teacher's character. The only way to know is to see the lesson. This tra(lition of lesson observing (luring a conference has been maintaine(1 over a long perio(1 of time in every school attache(1 to a university or its (1epartment of education. An(1 even more than this, custom has spread to almost every regional educational conference of teachers. Most teachers in Japan, in transmitting their educational ideas to others, seem to prefer (demonstrating an actual lesson to a formal oral or written presentation. To show their beliefs through an actual lesson is far easier than to express them on paper. ~ think this may be the reason we see the "(lemonstra-

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tion lessons" in many conferences of teachers. But recently ~ began to feel some anxieties: Has "lesson observing" become something like a mere ritual of the confer- ence without any reflections on its func- tion as a method of studying the problem of mathematics education? Functions of Lesson Stuck ~ think we can notice two functions for lessons open to other teachers in an educational meeting: as a method of research and as a place for presenting new ideas or findings. In the first, the lesson is open to colleague-teachers in a school or in a regional conference, and the analysis is divided according to its aim: to ask fellow teachers to see a typical lesson to become aware of the teaching problems through the many eyes of the participants and to find causes of failure or success of the lesson or to focus attention on some particular problematic factors. In both cases it would be better to prepare a written lesson plan for the participants' reference. The second function shoul(1 be a carefully plannedlesson. Observers should be informed about its aim, topic, teaching process, suppose(1 pupils' activities, and intended results before- han(l. Traditionally in research in mathematics education, we present the results of our research in a paper, but ~ think there is another way of presentation. The demon- stration lesson as in the second function, would like to say, is a form of research report that can be compared to the thesis in a paper. Our findings during lesson studies could be ones that are (difficult to inform others about using the usual form of thesis in a paper. They may be trans- mitted best through the demonstration lesson. Such a lesson, ~ think, should qualify as a written paper. These lessons are different only in form and have the same values as the written thesis. They have the same quality as the performance of a musician or the masterwork of an artist, which should be considered as a whole, not broken into pieces. Conclusion and Aciclitional Comments Lesson study is a synthetic method of research in teaching anti learning. We should regard this method as legitimate as the usual analytic method of science. As mentioned above, lesson stu(ly is a very complicated phenomenon and not easily studied by a strict analytic method of investigation. The natural complexity of this originates in our thinking or learning activities. An analytic method is difficult to use in treating such a complex phenomenon and often overlooks some subtle anti essential factors. The (demonstration lesson is another form of thesis presentation, and we should consider how such a lesson should be undertaken. Among many things to be considered, the most important is how to make a lesson plan. Here ~ give some of my ideas about how to write lesson plans as ~ am not satisfie(1 with the current ways of (loin" so: . The anticipated process of teaching may be a mere outline of the lesson, but the teacher's intention should be clearly (lescribe(l. In Japan we often see a written lesson plan with a very (letaile (lescription of the process, including the teacher's behavior anti pupils' LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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. . activities. But actually the lesson wall not go so well as in the planning. ~ was often bored reading lesson plans and thought ~ would enjoy them if they were written more like a work of literature. A lesson plan may be like the playbook or the scenario of a drama which permits the actor's timely digressions or ad fibs in a large scale. In fact, a teacher may be an actor or actress, as Polya (1963) said: The classroom may be the stage of a theatre, and a good lesson may be a good performance of a drama. But in many schools in Japan, a lesson may be seen as the time to cram knowledge into the brains of pupils to prepare for the entrance examination to the upper school. The lesson plan may necessarily turn to a recipe for cram- ming mathematics effectively. PostIucle In Japan, we have traditional arts called raka~go, which is something like an entertainment held in a small theatre telling a short comic story. Some years before, we invited to the annual confer- ence of the Japan Society of Mathematics Education the famous rakugo teller named Katsura Beicho; he had a high reputation for his refined way of talking. The subject of the story was tsa~ho-zan (bottle math), with a very simple and very stupid plot. REFLECTIONS ON VIDEOS: PANEL A man went to the bottle shop and bought a bottle for 100 yen. But on the way home he changed his mind and wanted a larger one. He returned to the shop and changed to the larger one for 200 yen. When he was going out the shop, of course the master of the shop asked him to pay 100 more yen. But the man said to him, "I had already paid 100 yen when ~ bought a smaller one and just now ~ paid with another bottle worth 100 yen. In total, ~ paid to you 200 yen, and ~ can have a large bottle for 200 yen." The master was very confused and could say nothing. That's the full story, and to tell the story takes only a few minutes, but this rakugo teller attracted the entire audience throughout one solid hour. It is the art not the technology that attracts the attention of the audiences. Mathematics teachers should have such art in the classroom. For instance, the first-gra(le teacher has to teach 2 + 3 = 5 to pupils taking at least one hour without boring them. Or, to tell the answer to the problem "how many remain in a box of 10 candies if 2 are eaten" needs only a few seconds. But it would be a marvelous thing for chil(lren to know eight candies remain in the box without opening the box and counting them. To make them understand this marvelous thing as such may take more time, and it is not only the technology of teaching but also the art of the teacher to do so.

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