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r. rN ~4 1 - 1 ~ ~ ~ The opening session of the workshop included presentations designed to give partici- pants some sense of mathematics education and teacher preparation and development programs in the two countries. Deborah Schifter and Zalman Usiskin gave a broad overview of elementary and secondary teacher preparation and development programs in the United States. Keiichi Shigematsu and Keiko Hino described a case study of a middle-grades teacher who had taken a part in a graduate teacher education program in one Japanese university, while Toru Handa and Mamoru Takezawa spoke of other examples of professional development efforts in Japan at the secondary level. Elementary Mathematics Education in the United States Deborah Schifter, Senior Scientist, Education Development Center Mathematics Teacher Education in Grades 7-12 in the United States Zalman ~ Usiskin, Professor of Education and Director of the University of Chicago School Mathematics Project, University of Chicago A Study of Teacher Change Through Inservice Mathematics Education Programs in Graduate School Keiko Hino, Associate Professor, Nara University of Education Keiichi Shigematsu, Professor, Nara University of Education Recurrent Education in Japan: Waseda University Education Research and Development Center Toru Handa, Mathematics Teacher, Waseda University Honjo Senior High School Recurrent Education in Japan: Kanagawa Prefectural Education Center Mamoru Takezawa, Mathematics Educator, Kanagawa Prefectural Education Center

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:~ ~~ - Deborah Schifter, Education Development Center ~ have been asked to provide some background for this workshop on elementary-level mathematics instruction in the United States. ~ wall be touching on four main themes: (~) the tradition of local control of education policy in the United States, (2) the fragmentation of teacher education, (3) current efforts to improve education, and (4) challenges to this improvement. WHO CONTROLS U.S. EDUCATION? First, and this is not specific to elemen- tary education, is the question of who controls U.S. education policy. The federal government has limited power over our schools; it does not establish a national curriculum. Nor is teacher education or assessment a matter of national policy. Instead, each of the 50 states has its own policies in such matters as teacher certification and curriculum, and in some states, curriculum decisions are actually left to local school districts or even individual schools. In spite of the extreme fragmentation of authority over U.S. education, there is considerable uniformity in what is taught largely attributable to the way textbooks are produced, marketed, and adopted. A second mechanism for ensuring a degree of uniformity is the concern that students are prepared for college entrance exams. Although this (Erectly affects only secondary school instruction, this concern exerts indirect pressure on instruction in the lower grades. With such decentral- ized control over education policy, along with the tendency of all institutions to maintain the status quo, it is (difficult to make fundamental changes in our schools. However, two arms of the federal govern- ment, the National Science Foundation and the U.S. Department of Education, do provide grants for projects that appear to promise improvement in e(lucational practice. Through this mechanism, the federal government can exert some influence on policy (recisions. FRAGMENTATION OF TEACHER EDUCATION ~ now consi(ler fragmentation of teacher education and, specifically, implications at the elementary level. In the Unite(1 States, it is wi(lely believe(1 that a stable school environment better serves young children. This is accomplishe(1 by keeping

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one teacher with the same class for the full school day. In most schools, teachers are responsible for teaching all subjects to their students mathematics, reading and writing, science, and social studies. From the background reading for this workshop (Appendix L), ~ understand this situation is the same as in Japan. In U.S. un(ler- graduate teacher preparation and inservice professional development programs, there is pressure to learn about instruction in all of these subjects. As a consequence, most teacher preparation programs determined by state certifica- tion requirements generally offer one or two courses in the teaching of mathematics (National Center for Education Statistics, 1995; America Federation of Teachers, 1997; Ingersoll, 19981. Some also require that prospective elementary teachers take at least one mathematics course, while others have no such requirement. On the other hand, there are some states like Georgia that now require undergraduate students to take several mathematics courses to become certified as elementary teachers. However, it is unusual for the mathematics courses to be coordinated with the education programs at the ~ university. In many states, teachers must spen(1 a certain number of hours in professional development settings to maintain their certification. However, once again, there is no uniform system of professional development. Some workshops are organized by the school or the school system, and these may be led by district personnel or by consultants who are hired from outside to work with teachers for a single (lay. The content might involve a new teaching technique or assessment procedure, student behavior or some such social issue, classroom (liscipline, or time management. They rarely (leal with mathematics or how chil(lren learn (Garet et al., 1999; Shields et al., 1999~. It is unusual for teachers to come together regularly to (liscuss substantive or prob- lematic issues in their practice although there are exceptions. Some teachers on their own initiative seek out courses at universities or summer programs, which again vary greatly from one another (McLaughlin, 20001. And there are some schools or school districts that have organized and coordinated substantive, professional development programs. CURRENT EFFORT TO IMPROVE ELEMENTARY EDUCATION Perhaps the most significant point in our discussion at this workshop is that until recently, the principal goal of elemen- tary mathematics instruction in the United States has been computational proficiency (National Council of Teachers of Mathematics, [NCI~M] 19891. The emphasis has been on remembering facts and algorithms and being able to produce correct answers with speed. The curricu- lum also inclu(le(1 learning the names of particular shapes, the use of a ruler, the formulas for the area anti perimeter of a rectangle, and sometimes the volume of a rectangular solid. Like the rest of the mathematics content, word problems have been treated mechanically. This emphasis on memorization rather than sense-making activity is evident in such errors commonly seen in the elementary classrooms as shown in Box I. In these examples, the chil(lren are applying single-digit math facts but are not remembering the computational procedures. Because they are not think- ing about the size of the numbers they start with or what the operations (lo, they form no reasonable estimate of the outcomes. If neither the chil(lren nor BACKGROUND CONTEXT FOR TEACHER PREPARATION

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26 +58 714 42 - 29 27 54 x23 162 +103 270 their teachers have learned to approach such problems with the expectation that they should make sense, it is difficult to correct the misconceptions at the base of these errors. This brings us to efforts to improve elementary mathematics in the United States. For many years, going back at least as far as the work of John Dewey, there have been educators devoted to rethinking K-12 mathematics, designing materials, working with teachers, and producing policy statements. The most dramatic recent change came just a decade or so ago when some of these people came together under the auspices of the NCrM to produce a set of stan- dards documents. NCrM is a profes- sional organization of mathematics educators whose work is mainly conducted by volunteers and supported by a full-time staff. The first three NCrM documents set out standards for curriculum, teaching, and assessment respectively: Curriculum and Evaluation Standards for School Mathematics,1989; Professional Standards for Teaching Mathematics, 1991; Assess- ment Standards for School Mathematics, 1995. The fourth, Principles and Stan- dards for School Mathematics, 2000, was E LEME NTARY MATH EMATICS E DUCATION intended to update, refine, and elaborate on the issues presented in the first three. The main principles guiding these docu- ments are that mathematics is about reasoning; that children be recognized as mathematical thinkers; and that eliciting, assessing, and buil(ling upon their think- ing should be at the heart of instructional practice. In particular, the documents argued that children should be encour- aged to use a variety of methods for representing and solving problems and then present their work to their ciass- mates for further analysis. These docu- meets also emphasize that mathematics is more than arithmetic. Geometry and data should be made significant components of the curriculum beginning in kindergarten. The fe(leral government (foes not set a national curriculum. These stan(lar(ls are offered as recommendations, without any requirement that people should follow them. However, at the federal level, the National Science Foundation and the U.S. Department of Education have supported these reforms. They have funded research centers anti professional (1evelopment programs anti provi(le(1 grants to produce curriculum materials at the elementary, middle, and secondary levels. These materials started to become available around 1996 anti 1997. To provide an idea of what the stan- dards support and the kinds of activities inclu(le(1 in the new curricula, ~ offer two examples taken from the newest NCrM document (2000~. A vignette is presented of a fifth-gra(le class that ha(1 been given the homework problem 728 divided by 34. One chil(l, Henry, presente(1 his solution metho(1 (Box 2~. Henry explaine(1 to the class, "twenty 34s plus one more is 21. ~ knew ~ was pretty close. ~ (li(ln't think coul(1 a(l(1 anymore 34s, so I subtracted 714 from 72S, and got 14. Then I had 21 remainder 14."

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34 X 10 = 340 34 X 20 = 680 680 +34 714 728 -714 14 SOURCE: Aclaptecl from NCTM, 2000, p. 153. Another child, Michaela, presented her solution (Box 3~. Michaela described the steps of the conventional division aigo- rithm. "34 goes into 72 two times, and that's 68. You've got to minus that, bring down the 8, and then 34 goes into 48 one time." Apparently, their teacher had not shown the conventional division algorithm to her students, and Michaela's classmates said they did not understand her method. Asked to explain, Michaela took the class through the steps again but with the same response. Then the teacher asked the class to identify the similarities in the two procedures and assisted them by insert- ing a 0 so that the chil(lren coul(1 more easily see where Henry's 680 shows up in Michaela's process. Through the discus- sion that followed, using Henry's solution as a point of reference, some of Michaela's classmates could begin to see the logic of the steps she hall taken. Another example is also taken from Principles and Standards for School Mathematics. In this fifth-gra(le class, the students had been given a word problem that involved adding 1.14, 0.089 and 0.3. They were asked to work in groups to come up with an answer. Although they had done preparatory work on decimals, a(l(ling them was a new topic. One group presented this solution (Box 4~. They explaine(1 that you change all the (lecimals into regular numbers. Then you a(l(1 them all up anti get 206. When challenged by their teacher to consider whether a number that size makes sense, they restored the decimal point. And their final answer was 2.06. The second group, which had done the calculation correctly, explaine(l, '~he reason we (li(ln't line up all the numbers was because we had to line up tenths with tenths, and the hundredths with the 21 34 - -68 48 -34 14 SOURCE: Aclaptecl from NCTM, 2000, p. 15~. 114 89 + 3 206 SOURCE: Adapted from NCTM, 2000, p. 195. BACKGROUND CONTEXT FOR TEACHER PREPARATION

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hundredths to make it come out right, and," they added, "the thousandths with the thousandths." However, when the class came together to discuss their solution methods, neither group was convinced by the other's explanation. Both answers could not be correct, but as a class, they could not resolve the stale- mate before the lesson ended that day. The next day, the teacher suggested that some of the students work on this problem with base 10 blocks. Their representation is shown in Box 5. The children in the class were convinced by this demonstration. Not only did it help them determine that 1.529 is indeed the correct answer, but it also helped them interpret the argument the second group had given you add tenths with tenths, etc. These examples illustrate the spirit of the mathematics education reforms in the United States. In the past, accepted practice would have required the children to memorize the steps of the division algorithm as presented by Michaela, with no attempt to explore the reasoning behind it; or in the past, children would have had to have memorized two different rules for adding numbers: For whole numbers, line them up at the right, but for decimals, line them up at the decimal point. In the vignettes, we see children sorting out the logic of the calculation procedures. It is noteworthy that in both examples, the children have been given responsibility for explaining their reason- ing to themselves and to their classmates. Their teacher posed a question or offered a suggestion that helped them find the sense in the procedures they were learning. CHALLENGES TO THE IMPROVEMENT OF MATHEMATICS EDUCATION One of the most difficult issues that the United States faces in improving elemen- tary mathematics education is that most ~ ~ ~ ~~ \ 1.14 0.089 0.3 Total SOURCE: Schifter et al., 1999, p. 1 18. E LEME NTARY MATH EMATICS E DUCATION 1.529

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teachers do not understand the mathe- matics they teach very well (Ball and Wilson, 1990; Ma, 1999; National Center for Education Statistics, 1995; National Research Council, 2001~. In the past 15 years, there have been many studies that have documented this problem, which has become an increasing concern to many U.S. educators and policy makers. In my first example about division, the teacher herself understood the comparable reason- ing behind the two methods presented by her students, and so she could draw their attention to it. In the second, the teacher understood the principle that underlies addition of whole numbers and decimals and so could suggest a representation that would help her students see it too. How- ever, such teachers are, at this time, more the exception than the rule in the United States. Most, having been educated in mathematics restricted to memorization of procedures, have not learned how to make sense of mathematics. In general, U.S. teachers have not had opportunities to learn mathematics from a conceptual perspective. The examples ~ have given come from the most recent Standards document published in spring 2000. When the first Standards document appeared in 1989, there were few instances of curriculum materials that laid out a coherent math- ematics program. By the early 199Os, it became apparent there were serious problems with the way teachers were interpreting the Standards documents. These were largely rooted in a failure to appreciate that the core idea of the reform was making sense of mathematics. For example, one interpretation held that the standards were about teaching essentially the same mathematics content, but now the children were sometimes to work in small groups, or were to use manipulatives, as well as paper and pencil (Weiss et al., 1994~. So there were many professional development workshops devoted to these teaching strategies. However, when the children were put into groups, they were not necessarily given problems that required real thinking. And when they were given blocks to use, they were merely shown yet another set of steps to remember. Another interpretation was that the new practice was to emphasize problem solving. So some teachers committed one day each week to it. Although many of these problems did require some hard thinking, they were not necessarily related to a coherent conception of the content that the children needed to learn. In one very disturbing interpretation of the Standards, some teachers agreed on the importance of eliciting ideas from their students but did not understand that they had a further responsibility to critically analyze those ideas for math- ematical soundness. Indeed, having themselves been taught that mathematics is memorization, many teachers have never developed the skills required for assessing the logic of a mathematical argument. Nor do they even realize that this is something that they, much less their students, should be doing. There was some research in the early 199Os that alerted us to these problems of practice in the name of the Standards (NCrM, 19911. And there were some programs for college students preparing to become teachers, and for teachers already practicing, working to address them. As a result of these programs, and with support from new curriculum materials, there are now a greater number ~ . . . . . .. ot classrooms in WhlCh the new vision of mathematics teaching practice has successfully taken hold (Briars and Resnick, 20001. BACKGROUND CONTEXT FOR TEACHER PREPARATION

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However, since 1995, a movement has emerged to put a stop to these reform efforts. This movement has arisen out of several concerns. Some people became justifiably alarmed when they saw some of the problematic practices ~ have just described. And even where the mathe- matics was being taught wed, some parents were made uneasy by the unfamiliar look of the mathematics their children were bringing home. Others felt that the new practices would interfere with children learning basic arithmetic and algebraic skills, without which they believe math- ematical understanding cannot develop. Still others were distrustful of teachers who now aspire to a more complex, intellectually, and demanding practice. In any case, some reformers now confront groups of parents supported by some professional mathematicians and backed by wealthy funding sources who have succeeded in blocking local and state reform efforts (Battista, 1999; Becker and Jacobs, 2000; Grossman and Stoduisky, 1995~. Although this conflict has made its way into the media, society at large- including the journalists covering the conflict is unaware that the questions concerning how learning takes place or how children put mathematics concepts together are complex and profound. The one mathematical specific that seems to come up regularly in the debate is the role of conventional computation algorithms. In the public arena it is posed baldly: Should teachers encourage children to devise procedures that make sense to them for example, to add 45 + 36, children will often find the sum of 40 and 30 first, then add 5 and 6, and then total the partial sums: 70 + Il = 81 or should teachers demonstrate conventional algorithms? As my earlier example suggested, the dichotomy is a false one. E LEME NTARY MATH EMATICS E DUCATION Nonstandard procedures can render the meanings of the operations transparent and the reasoning behind the standard algorithms accessible. However, as the debates get played out in public, sensible and deep discussion of such issues becomes increasingly (lifficult. To summarize the context in which we are working to educate teachers and support their professional development, many teachers have yet to be introduced to the idea that mathematics is about reasoning, that children, anti they them- selves, have mathematical ideas and the capacity to explore and critically assess them. Once they come to appreciate that doing mathematics is a sense-making activity, there is much mathematics content for them to learn. Then they must create a teaching practice that is (lifferent from the practice they know, that helps their students understand the mathematics too. Finally, they must do all this while keeping up with comparable (levelop- ments in the teaching of literacy, science, and social studies. And yet there exists no coherent teacher education and professional development infrastructure to support these efforts. Furthermore, we are trying to do this work in a period in which basic directions in education policy are the subject of contentious political (rebate. Wanting to (lo the best for their students, teachers are caught in the middle, attacked for their students' failures, hel(1 accountable for an impossibly difficult task. However, as ~ anti my colleagues have (liscovere(l, once teachers begin to learn mathematics in settings that support the (levelopment of their own powers of reasoning, they become eager to learn more mathematics for themselves and to provide their students with opportunities they missed out on as children.

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~ ~ ~ ~ ~ 1 ~ Irk row 51e' 1~.1 r ram I ~T~ ~ Zalman I? Usiskin, University of Chicago Good afternoon. It is a pleasure and an honor to present this overview of second- ary mathematics teacher education in the United States. At least since the 1960s, when the results of the International Study of Mathematics were announced and Japan scored at the top and the United States at the bottom, Japan has been seen as a place to which we might turn to improve mathematics education in the United States. In the early l980s, at the University of Chicago, we ha(1 the opportunity to translate Soviet and Japanese high school texts, and we were able to see the algebra and geometry taught in junior high school and how virtually an entire population can be brought to higher mathematical performance. The recent third Inter- national Mathematics and Sciences Study suggests that we may be able to learn not only how to raise student performance but also instructional techniques from Japan (National Center for Education Statistics [NCES], 1996). In turn, the size of the United States and the length of time we have been working to improve mathematics education has led to numerous studies and projects in the United States. We hope that some of the ideas we have been working on wail be informative and stimulating, and we look forward to sharing them. In the workshop materials, we presented data regarding mathematics teacher education in the Unite(1 States. These are pages 31-34 of a longer (1ocument that was distributed at the Ninth International Congress on Mathematics Education (Dosseyan(lUsiskin, 20001. Rather than repeating what is there, ~ will elaborate on the picture the (lata present with regar(1 to mathematics teacher education for those who teach mathematics in grades 7-12. With respect to the first two questions consi(lere(1 in Deborah Schifter's remarks, the situation is the same at gra(les 7-12 as at grades K-6. Decisions are left to states anti schools, anti education is fragmented. To un(lerstan(1 the picture of mathe- matics teacher education, it helps to have knowle(lge of the mathematics curriculum at the secondary school level. There has never been a national curriculum in the United States. But at a given time, most schools follow much the same curriculum. This is because, among the most-use(1 textbooks, there is a strong tendency to have the same content anti approach. ~ begin with a typical curriculum of 50 years ago for high school anti the first year of college for well-prepare(1 students

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(Table I). At that time, algebra and geometry were avoided almost entirely until ninth and tenth grades. Only a minority of students took mathematics beyond tenth grade (National Advisory Committee on Mathematical Education [NACOME], 1975~. The most-used and most influential of the new math curricula, the School Mathematics Study Group (SMSG) curriculum, and Me demands for increased performance in mathematics and science modified the 1950s curriculum in a number of ways. Some content was moved into lower grades, and other content was compressed and integrated. Also, many schools maintained a second curriculum for slower students. As a result, by 1975 a typical curriculum was bifurcated as shown in Table 2 (NACOME, 1975~. In this 1970s curriculum, there was some algebra and geometry in eighth- TABLE 1 Typical Curriculum for High School anc! First Year of College, 1950 Grade Content 7th 8th 9th 1 0th 11th 12th College Arithmetic Arithmetic First-year algebra Geometry Seconcl-year algebra Solicl geometry; trigonometry with logarithms Analytic geometry; first-semester calculus SOURCE: NACOME, 1 975. TABLE 2 Typical Curriculum, 1975 Grade Most Students Slower Students 7th 8th 9th 1 0th 11th 12th College 7th-gracle mathematics 8th-gracle mathematics First-year algebra Geometry Seconcl-year algebra with functions Functions and trigonometry Calculus 7th-gracle mathematics 8th-gracle mathematics General mathematics (arithmetic) Consumer mathematics (financial arithmetic) SOURCE: NACOME, 1 975. M AT H E M AT I C S T E A C H E R E D U C AT I O N I N G R A D E S 7- 1 2

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FIGURE 1 Framework of inservice mathematics education program at the graduate school of the Nara University of Education. Enrichment in Specialty Area ~ ~ Improvement of Practice Information Exchange | of Practice | Enrichment in | Improvement | Self-Expre';sion 1 Improvement T Informal Specialty Area of Practice of Practice Exchange Lectures and practices in Reflection Presentations at Reflection Attending study meetings graduate school seminars Alternative view Alternative view Visiting schools and The acquirement of requisite Presentations at study institutions knowledge for research Lesson analysis groups Lesson analysis Attending conferences u, Specifying the research Lesson Presentations at Lesson .~ theme, reviewing the organization conferences organ~zabon preceding research, planning, preliminary research Contribution to the university bulletin Plan and practice of research Reflection Research towards Reflection Attending study meetings master's thesis Getting results in shape Alternative view Alternative view Visiting schools and Writing thesis institutions Examining conclusions, Lesson analysis Lesson analysis o suggestions, and tasks for the Submitting thesis Attending conferences ~ future Lesson Lesson u, organization Presentation and organization contribution to the university bull Tin schools, not only in our prefecture but the neighboring prefectures and teachers in those prefectures. Through such ex- change and participation in academic meetings, teachers can further cultivate their professional skills. Figure ~ does not show lesson study, but teachers aggressively participate in a lesson stu(ly program. In the two-year program, the first year takes place at the graduate school level. However, in the second year teachers go back to their original school for further training or further education. With respect to the second purpose of our research characterizing the school teacher's change, we found that a teacher's growth in mathematics takes place in four phases. Phase one is consideration of their own teaching practice; teachers have to be aware of the framework for their own teaching practice. In other words, they have to be aware of the issues in their own view of their teaching. Unless they are aware of their problems or issues, it is very (difficult to motivate them to improve their teaching ability. Second, a clear un(lerstan(ling of their issues, BACKGROUND CONTEXT FOR TEACHER PREPARATION

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however, wall not be sufficient to move to the next step. They need to think about what they should do. So they must search for another implicit framework (phase two). In phase three, that framework is internalized, and finally through that process an improved framework should be developed (phase four). These four phases show that improvement in profes- sional ability, for example, having more students come to enjoy mathematics, happens only when teachers construct their own framework for mathematics teaching. An explicit diagram is advanta- geous (Box I). In our research we found that if a teacher can notice improvement, the teacher wail continue to further improve his or her professional ability. Now we go into more detail about our research concerning the teacher's r: =-1-1-1~i-~~ Based on the result, Mr. A's change was characterized by interviews conducted three times during and after the participation in the program. Activities Reviewing recorded materials on his lessons Taking lectures in graduate schools Having discussions with various people Phase 1 Confrontation with Mr. A's implicit view of mathematics teaching 1 Taking lectures in graduate school Having discussions with various people Teaching practice in the club activity in his school | Phase 2 Seeking alterrnative views of mathematics teaching l Lesson practices along with successive observation of experienced mathematics teacher's class Having discussions with various people Construction of the framework in his master's thesis study Presentation of the framework in meetings and conferences Phase 3 Concretizing the alternative view by querying the ideal of mathematics teaching and key words for approaching this . Master's thesis study (classroom teaching practice) Teaching practice in an "optional mathematics class" in this school Phase 4 Construction of Mr. A's own framework for mathematics teaching A STUDY OF TEACHER CHANGE

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change. Improvements in the teacher's teaching practice has been mentioned several times, but what do we mean by teaching practice? There are three perspectives from which we looked at the teaching practice (Box 2~. One is the class program itself. In other words, preparing for the class organizing the class is one practice. Another is during the class, the kind of practice the teacher carries out. And the third aspect is after the class, making improvements in the practice the teacher has just finished. In other words, prelesson, midlesson, and postlesson are the three different aspects pertaining to the teaching practice. In our study we surveyed teachers, interviewed teachers, and collected class observation cards from the participants of the class. Using these different methods, we tried to analyze the improvement in these three different levels of teaching practice of the teacher. Mr. A is a junior high school teacher, who admitted that he had very little interest in mathematics education (Box 3~. He was more inter- ested in guiding his students in the extracurricular activities of rugby games. After 15 years of experience, Mr. A wanted to improve his skills and his performance as a school teacher. How slid his teaching practice actually change? The improvement of Mr. A's teaching practice and his professional development was initially assessed by observing his class before he took this graduate course. Then we went back to his class, after he took this two-year course at the graduate school. The contents of the two classes that were observed were almost identical. We tried to compare what we noticed had changed over the two-year period (Box 31. We analyzed these observations from three perspectives. First, we analyzed ~ _ 1. To characterize a school teacher's change through participation in mathematics education program at the graduate level. 2. To assess the change of the teacher from the perspective of teaching practice in their mathematics class. Mathematics teachers' teaching practices are construed as the ability that enables teachers to conduct activities such as: Prelesson: To organize and transform content knowledge along with the purpose. To have the "eyes" to evaluate the result of such a process. Midlesson: To execute the plan. To create activities that lead to the goal, taking notice of a student's situation all the time, sometimes by applying various routines and other times by inventing them promptly. Postlesson: To reflect on the lesson using as a basis information about the attainment of a teacher's goal. To work out instructions concretely for the next lesson. BACKGROUND CONTEXT FOR TEACHER PREPARATION

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The teaching practice of a teacher was assessed twice, before and after the participa- tion in the graduate school program offered by Nara University of Education. Mr. A;'s Profile A junior high school teacher working at a public secondary school. In addition to teaching mathematics, he arranges student club activities every Jay aher school and also on the weekend. He does not participate in any mathematical study groups. "I have been teaching for 15 years. To keep up the times, I want to study how to make use computers toward mathematics education in new ages. I also need to develop my teaching practice in class through this program. When I return to my school I need to show my improved teaching practice to my colleagues./' Lesson Observation Dates ant! Participants Before: February 17, 1998, at a junior high school first-graders' class (seventh-grade) and a seconJ-graJers' (eighth-grade) class. Observers: 5 persons (2 university instructors, 1 high school mathematics teacher, 2 graduate students). Target students: 6 first graders and 6 second graders. Alder: February 15, 2000, at a junior high school first-graders' class. Observers: 5 persons (2 university instructors, 1 high school mathematics teacher, 2 graduate students). Target students: 5 first graders. Data Collection Pre/esson To teacher: interview questions on mathematics and ~earning. To target students: interview on their belief and interest of mathematics. Lesson Observation Recording of the lesson by videotape recorder and microphones. Observation by a script form from two aspects: the flow of lesson and students' activities. Post/esson To teacher: interview on se~f-assessment of the lesson. To target students: interview on their interest and motivation toward the content they learned in the lesson. A STUDY OF TEACHER CHANGE

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an ad 1.0- ~5 O 0.8- Q In an o o an Tar 0.6 - 0.4- 0.2 - o- from the observer's viewpoint. Observers were fellow graduate students, the teach- ers who had already finished the graduate school program, and ourselves, the researchers. We made three findings. We did not see any dramatic change; however, Mr. A's attitude with respect to the students had clearly changed. There are some specific data to indicate this change in attitude, which is shown in Figure 2. This figure shows the rate of correspondence among different observers before, during, and after this graduate course. Before Mr. A took the graduate course, most observers agreed that Mr. A was just teaching with a one-way method, teacher to students. This is shown by the overall high rate of correspondence among the observers. After the two-year course, this approach changed dramatically to become more interactive (Box 41. Notice the decrease in the rate of correspondence in most items. Mr. A's teacher-(lirecte(1 way of teaching ease(1 a little. That was a unanimous observation. The second perspective is the observa- tion made by the students themselves (Table I). Here again, (lepen(lence on textbooks has change(l. (In a (lifferent research study, we found that the Japanese teachers who do not like mathematics would inevitably depend on the textbooks in teaching the mathematics classes.~. But as you can see, after this two-year course, Mr. A (lepen(le(1 less on textbooks, which meant that he was trying to cater to the actual needs of the students in his class, and students noticed the difference. An(1 the thir(1 perspective is the self- assessment by Mr. A (Table 2~. Through- out the graduate course, Mr. A learned that what is most important is for the students to learn mathematics, not for the teacher to learn mathematics. What is most important in organizing the math- FIGURE 2 In the eyes of the observers. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 11 1 1 1 4 7 10 13 16 19 22 25 28 Item Before After NOTES: Mr. A's teacher-directed ways of teaching eased a little. Communication with students was brought into the class. The time for students' activities was put in the lesson. Rate of correspondence among the observers on each item, or what was the percent of observers who agreed that the item was oberved in the lesson. (1) items No. 1 to 14 concern of the teacher, (2) items No. 16 and 17 concern of the classroom atmosphere and (3) items No. 19 to 28 concern students in the class. BACKGROUND CONTEXT FOR TEACHER PREPARATION

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~ or.. s~.~.~.~.I~.1881.'.r.~el~_ Before 1998 ~ The teacher does not suggest students try various approaches to solve a problem. The teacher does not relate mathematics with other subjects or daily life. The teacher Joes not instruct to develop individually. The teacher teaches the content exactly as it is in the textbook. The teacher gives a teacher-centered telling of the lesson. Many students do not volunteer to state their opinions. Many students do not think in various ways. None of the students in the class chat among themselves. All of the students in the class put their textbooks and notebooks on their desks. After 2000 The teacher doesn't relate mathematics with other subjects or daily life. Communication with students is seen. Communication among students is not seen. Some students in the class have private charter. Many students tackle to compute, J raw figures and graphs, or use compasses and rulers. TABLE 1 In the Eyes of Students Items Recognized By Students Before (/0) After (/0) Many of his lessons go with textbooks Materials are ohen used in his lessons I ask him questions frequently Pace of his lessons seems fast Frequently students attempt to first solve the problem by themselves We ohen have a discussion Two teachers have taught me a lesson I write the lesson summary with my own words 100 o 67 83 100 17 o 25 80 20 ~0 20 100 60 o 60 NOTES: Fewer students feel that the pace of his lessons is fast. More students think they have discussions in the class. A STUDY OF TEACHER CHANGE

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TABLE 2 In the Eyes of the Teacher Before 1998 After 2000 First Year Second Year First Year of Junior of Junior of Junior Items for the Interview High School High School High School Aims of the lesson To understand To understand To promote how to construct the Miclpoint stuclents' figures. Connection exploratory To learn how to theorem and activities. use the compass. to use it. Special attention To make all To make Tovaluestuclents' students learn students voluntary statement. the proper use understand Not to force teachers' of a compass. the theorem. views. To give comments on contents that require direct explanations. Diclyou value stuclents' cliscussion? No No Yes Dicl you arrange time for stuclents No No Yes to solve problems by themselves? Diclyou giveteacher-centerecl lesson? Yes Yes No Dicl you communicate with the Sometimes Sometimes Yes stuclents? Dicl you consider connecting the No No No subject with claily life examples? Dicl you con lecture possible Yes Yes Yes questions the stuclents may have prior to the lesson? Dicl you make use of wrong answers No No Yes proviclecl by stuclents? NOTES: He gave a better evaluation to his "aher/' lesson than "before/' lesson. The aims of the lesson focused on stuclents' thinking and attitucle. Special attention to teaching instructions can be seen in the "aher/' lesson, while they were restricted solely to the teaching contents in the "before/' lesson. BACKGROUND CONTEXT FOR TEACHER PREPARATION

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emetics class is to make sure that the students learn, that the class is more student focused, and student oriented. And through his self-assessment and through interviews, we can see that he has actually learned that difference. The biggest change within Mr. A was that he learned that he needs to communicate with the students (Box 51. He learned that what is often termed the mathemati- cal activities of the students need to be supported. And to do this, you have to provide enough time for the students to engage in mathematical activities. Mr. A learned that as well. As we saw at the beginning of this discussion, these changes did not take place at once. Rather, the changes were supported by Mr. A's long-term investigation into his own framework of mathematics classroom practice characterized by the four phases of growth. While this is a case study involving just one teacher, we are continuing our research involving elementary school teachers and senior high school teachers. Our preliminary findings indicate that we can see similar improvements in teaching practices and professional development based on the fact that the teachers need to have an open min(l, identify and recognize their shortcomings and areas for improve- ment and be walling to try to cope with the changes. The responsibility of the gradu- ate school is to help teachers tackle those issues and challenges. Conclusions Change in the teacher's teaching practice in mathematics class was mainly observed in two aspects. The teacher came to communicate with his students with respect to mathematics. The teacher came to have more time for the students to work on his or her own. The emergence of this "pedagogical reasoning" was supported by his long-term investigation into his own framework of mathematics classroom practice. The process of investigation was characterized by four phases. The graduate program influenced his change by providing opportunities in observing, practicing, and discussing mathematics classes in junior high school and in planning and conducting a research study for the purpose of writing a master's thesis. Tasks for the Future By accumulating the information from more cases, the process of teacher change needs to be clarified further. Construction and modification of the classes and the program that enhance the smooth shift along the four phases need to be investigated from different angles. A STUDY OF TEACHER CHANGE

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COG EM ~ ~ ~ _a_r ~ C.l,{.~. my Toru Handa, Waseda University, Honjo Senior High School Waseda University has an annual summer seminar for inservice teachers at what we call recurrent education centers. In 1996, Waseda University Education Research and Development Center launched this program. Since then, four or five days in August are dedicated to this seminar, with about ten courses. The basic course is for beginning users of personal computers. There is a class management and an attainment manage- ment course; a course of English with personal computers; a course of Japanese language with personal computers; courses in mathematics, social studies, anti sciences; a course to make computers; and a course on networks in schools. This seminar is for inservice teachers. We now have information technology everywhere, but teachers need to use the technology and information media to make their classes more interesting and easy to understand by students. Therefore, for each subject, teachers must first improve their computer literacy; however, Japanese teachers are busy in daily routines. From early in the morning to late in the evening, they have to manage extracurricular courses. Before classes anti after classes, there are teachers' meetings. So they (lo not have enough time for studying infor- mation technology. So how can teachers change their classroom teaching to use technology anti the information me(lia? First, how can they get information that will be useful in their teaching? How can they introduce the technologies and information me(lia into the classrooms? How can they create self-ma(le successful classes? To achieve these things, they need the basic knowI- edge and expertise for the new technolo- gies and information processing media. Specifically, some of my remarks focus on how to improve and change the mathematics classes. The two mathematics courses I selected are an introduction to MATHEMATICA, a software package that processes numerical formulas and does mathematical integration, and how to make a web page. A web page is a future textbook, and that means the teacher must be able to use HTML. On the first day of the seminar, basic MATH EMATICA and the basics of HTML are presented exclusively for the real beginners. So these are really the basics. And on the second and third days they use MATH EMATICA to process formulas and to enjoy the program. At the end of

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the third day they learn about software other than MATHEMATICA. On the fourth day they create a web page using MATH EMATICA and then develop the teaching materials for their own classes. Finally they make presentations and have a discussion about their work. Word processing software in Japan for information technology literacy is Excel and Ajitalo. Our participants vary in skill level. We had one person who had never touched the keyboard of a computer, 72 percent of the participants didn't know about MATH EMATICA at all, and about 83 percent had no web page experiences. Despite this, the participants finish the seminar with some achievements. The computer beginners in last year's seminar created a spiral structure of shells and used formulas to make simulations. By changing the value of a constant, one participant created different types of shells. Some showed a tangent of the function as an example of simulation. Last year we had a three-day seminar, and after completing the three days, participants could create such simulations. Finally, ~ would like to address some of the problems or the challenges that we have now. The budget for the education center is limited. We do not receive many grants, so the participants have to pay approximately $100 for four (lays. Maybe this is too expensive. A second problem is that the number of trainee teachers does not increase in the way we would like to see. There could be several reasons. First we do not have a preparation initia- tive tied to promotion. In Japan this kind of recurrent training is not so popular, and it is not generally accepted. Also in Japan, once teachers get a teaching certificate, they can stay there without studying or learning anything new until retirement age. In addition, even during the summer holidays, Japanese teachers are very busy because they have to take care of extra- curricular activities. Teachers also have to ask for permission of the principal or head teachers to participate in this kind of a seminar, which introduces all sorts of procedures. Another possible reason is that there are some people who take this course every year, many of whom are very experienced teachers. We probably need to prepare several different levels of the courses anti a follow-up service. We have to develop the mailing list of the participants anti make a (1atabase system so that we can continue to provide infor- mation about teaching and education to the teachers. For these reasons, it is (difficult to motivate the teachers to participate in this kind of seminar. The important thing is considering how we can create new Pleas anti how we can create a good image about participation in the seminar. Demonstration lessons may be an effective method in this respect. RECURRENT EDUCATION IN JAPAN: WASEDA UNIVERSITY

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Mamora' Takezawa, Kanagawa Prefectural Education Center The Kanagawa Prefecture, the local government, carries out programs for the inservice programs for teachers. The education center is located in the middle of the prefecture so teachers in Kanagawa Prefecture can come to it easily. This is where teachers receive training, both during summer vacation and during the school terms. They also come for training as a long-term researcher or student for one or two years. We provide training for between 40,000 to 50,000 teachers from 1,644 schools. The center has a staff of 50. belong to the information technology and education department. The education center is organized into four divisions: Japanese art, social science or social studies, and foreign languages for English and Spanish (Kanagawa has a relationship with Peru and Brazil in South America, and we have many children from Peru and Brazil); mathematics; informa- tion technology and education depart- ment; and principal and vice principal school management. The basic programs in information technology anti education start with general seminars, followe(1 by specialized seminars like networks, multime(lia,robot programming. For example, Logo or Mindstorms can be used. We also have a leader seminar to create information technology coordinators, anti we nurture lea(lers throughout the year. There is a license seminar for the new evaluation of the national curriculum. We also act as a help desk for supporting other (lepartments such as science, social studies, an(l Japanese language in how they can utilize information technology. What is the license seminar? In 2003, Japan will have a new national curriculum. The subject of information or informatics will be a(l(le(1 to the high schools' pro- grams in the new curriculum. However, we have no teachers to teach that subject. So for the next three years, we have to train or develop teachers in informatics. The program is targeted at math and natural science teachers and is a three- week summer seminar to give them the license to get really for the new curricu- lum in 2003. One of the issues is that many of the better mathematics teachers are using computers. They will receive a license for information technology, anti when they are ready to teach it, the number of goo(1 mathematics teachers will be (leplete(l.