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The two-and a half-day workshop The Study of Teaching Practice as a Medium for Professional Development focused on the use of teaching practice as a way to study what elementary mathematics teachers need to know to teach well. The opening sessions were designed to share background information about the education of mathematics teachers in the two countries. The first whole day of the workshop addressed practice by studying the preparation for and enactment of an actual lesson, through an investigation and analysis of Japanese Lesson Study. On the second day, participants considered records of teaching that included a video of a classroom lesson and an analysis of a case describing teachers and their work. After the workshop, two of the participants wrote papers describing the events from their perspective as mathematics educators. The following two papers reflect their overviews of the workshop, their sense of what they learned, and how the workshop experience related to their own background and work as mathematics teacher educators. Observations from the Study of Teaching Practice as a Medium for Professional Development Henry S. Keener, Jr., University of Wisconsin-Milwaukee Building an International Community: Sharing Knowledge and Experiences in Professional Development for Mathematics Education Carol E. Malloy, University of North Carolina-Chape! Hill

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red ~} . ~cq ream Bi - - - ~ . 11 , ml Henry S. Kepner, Jr., University of Wisconsin-Milwaukee The workshop considered the study of teaching practices as a medium for professional development. Three main approaches were identified and discussed at length during the conference: the lesson study a frequent practice in Japan, and two evolving practices using records of study in the United States- classroom video and written case study. The overall workshop activities repre- sented a clear difference in approach to professional development priorities and background. First, the Japanese partici- pants reflected an almost unanimous awareness and acceptance of content and its placement in the national curriculum. When a mathematical topic was identified, they responded by knowing the grade level of its presentation and what students should already know at that time. This clear and consistent assumption of stu- dent mathematical content background was not evident among the U.S. partici- pants. As Deborah Schifter and others noted, in the United States the local control of schools combined with diverse teaching strategies and curricula often supports professional (levelopment that focuses on isolated topics that may not be closely related to the mathematics in the curriculum such as group work, use of ~_. manipulatives, or problem solving. Second, throughout the workshop, Japanese participants focused more consistently on the mathematical structure and clear student performance expecta- tions in a lesson. Each move made by the teacher was first addressed with regard to the mathematical goal of a lesson. Again, this single purpose was not as evident among the U.S. participants. Often, other issues related to the observations, such as a teacher's skill in eliciting student ideas, arose first before the attention to the mathematics and its structure. This stu(ly of teacher (levelopment as advocated in two countries brought together (lifferent perspectives on what teachers coul(1 learn from two (lifferent, yet similar, sets of evidence with a focus on teacher planning, instructional materials, classroom activity, and student work both through videos or observations and written work. The positions, presente(1 by several members of Japanese anti U.S. mathematics educators, teachers, anti mathematicians, showe(1 both perspectives. The lesson stu(ly format, with slight variations, is con(lucte(1 by a group of teachers, often with a university educator, who develop a lesson in great detail for self-stu(ly anti sharing among colleagues

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who come to observe the lesson taught by a member of the group. The lesson study has an accepted place as professional development in the elementary schools and to a lesser extent in the middle schools. It is not accepted as needed by most secondary teachers of mathematics. In both plenary and small group discus- sions, the Japanese university participants repeatedly saw the lesson study as a way to help elementary teachers improve their reportedly minimal mathematical back- ground. The lesson study team makeup and purpose vary (lepen(ling on the primary reason for a lesson study. The activity may be for the professional growth of the teachers at a school or district or for a focus group of teachers where the activity is primarily a demonstration lesson for the engagement and growth of those partici- pating the team and observers on that day. It was reported that a university person was often invited to be a team member or a reactor for a local lesson study. University participants at the workshop noted that a university person who does not contribute significantly to the team or review process would not be invited back to future lesson study efforts. Lesson study is constructed to do research on the feasibility or effectiveness of a lesson. The lesson might be necessary for a new mathematical topic inserted in the curriculum, a different lesson approach or structure for a standard topic, or a new approach for a topic that is perceived as (difficult to teach. Such lessons are often done at demonstration schools associated with universities anti less often at other schools. In the lesson study process, teachers from the host school and frequently from surrounding schools and universities observe the class and participate in professional discussion following the lesson. Although there was substantive professional discussion along with chal- lenge or disagreement among the teacher, lesson study team, and observers, for the most part there was a politeness and cautious consideration for the teacher and team-(levelope(1 lesson. When pressed, several of the Japanese participants indicated that the actual conclusion of a lesson study process was "a party" at which there might arise substantive criticism of, or disagreement about, the lesson and/or its delivery. When participants viewed or rea descriptions of lessons and student responses, there was a marke(1 (1ifference in U.S. and Japanese reactions about the students' content background. While U.S. participants often had varying perspec- tives on student mathematical background, the Japanese participants reflected apparent uniformity about where that content would be presented in the school curriculum anti the backgroun(1 students woul(1 have experienced. This (lifference, attribute(1 to a national curriculum by Japanese participants, ma(le an e(luca- tional (liscussion much more focused by the Japanese. Although the lesson stu(ly activities experienced and discussed at the workshop encouraged each observer to bring their own professional perspec- tive, the presentations anti (discussions in(licate(1 that the Japanese teachers hall a common set of expectations anti points of reference. For the Japanese, there was an over- ri(ling concern for the mathematics, its structure, anti attempts to motivate stu- dents to learn it. This was critiqued within the lesson plan and the way the lesson was implemente(l. Most questions starte(1 with issues of mathematical purpose along with effectiveness of student motivation. In small group anti follow-up (liscus- sions, the function of a lesson stu(ly as a REFLECTIONS ON THE WORKSHOP

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device for professional development was probed in depth. Conference participants reported the development of both a set of observation criteria and a language of discussion useful in the professional conversation. This was a means of communicating the criteria or expectations of the observers and the lesson study team, including the classroom instructor. Examples of pedagogical jargon, in the English translation provided by Toshiakira Fujii and other Japanese participants, led to implied observation criteria. The criteria, also associated with the develop- ment of the lesson plan, included the opening problem setting with its motivational focus; the teacher's questioning, both the . . . sequence of questions and attention to hatsumon, "thought-provoking ques- tions" important to mathematical development and connections; kikan-shido, "between desk walking" or "purposeful scanning" referred to the teacher's purposeful observations and interactions while "walking among the desks" observing student work; the teacher's skill in anticipating student thinking; hansho, or "blackboard writing," stressed the organization of student work and key mathematical statements or results recorded by the teacher on the blackboard; neriage, "raising the level of the whole class discussion," through the orches- tration and probing of student solutions, usually in the whole class format; and matome, the teacher's mathematical summary of the lesson for the whole class, with attention to students' ideas and contributions. The extensive discussion of such criteria was seen by all participants as important to help both preservice and experienced teachers to be productive observers of classroom practice. Participants from both countries indicated that training in(livi(luals to be effective observers of the content development was a difficult task that occurs over time. In particular, Deborah Ball noted the difficulty experi- enced in helping teachers observe what the demonstration teacher is really doing to develop the mathematics. From the Japanese participants, there was a non-deviating focus on the correct- ness of the mathematics during the lesson and teacher summaries. It was frequently noted that an underlying purpose of lesson studies was to assist preservice anti professional teachers to increase their mathematical knowledge which univer- sity participants reporte(1 as weak. This purpose for lesson study presented a challenge with respect to lesson study team composition. While the team typically included a group of teachers in the host school, in some cases there was a reference to the involvement of a univer- sity mathematics educator or mathemati- cian on the team. It would that if seem this member is not an integral member of the team, the lesson might lack the mathematical precision and development expected. Such team makeup is often an area of concern in U.S. professional evelopment when it is school based. The Japanese e(lucational process demonstrated in the lesson study examples stressed the challenge to the teacher of motivating the students about the math- ematical topic without forcing the teacher's view on the students in the developmental portion of the lesson. Although this position ha(1 similarities to the lessons illustrate(1 at the workshop, it was noted that this was in contrast to the most common mo(le of U.S. instruction, variations on a (lirect instruction format. OBSERVATIONS FROM THE STUDY OF TEACHING PRACTICE

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The teacher's intent to bring multiple student presentations together controlled the conclusion of a Japanese lesson. However, in contrast to constructivist lessons where student summaries often are encouraged, the Japanese participants adamantly stated that the final summary of the mathematical conclusion must be made by the teacher to ensure that students heard correct mathematics correctly stated. A perspective of both approaches centered on observing and reflecting upon student work in the learning of mathematics. In the lesson study process, the teacher observers were intent on observing what students did during the observed class. Frequently each observer would take responsibility for looking at the seat work of a particular student and even asking questions quietly during the period, gathering information about students' reactions to the lesson. Observers are careful not to become teacher's aids. In the case study approach, reflections of videotapes and samples of student work were used for a similar goal. The professional development culture of lesson study provides a real-time obser- vation and discussion for the observers, a portfolio notebook kept at the host school, and perhaps a short article on the lesson in a local Japanese education periodical. The best of these lessons becomes part of the teaching resources for others to use in teacher preparation or professional development. Although there was an educational history of lesson study as an ongoing activity, participants noted that the use and involvement were uneven. Discussion indicated that schools attached to univer- sities do many lesson studies while public schools (lo far less. Several participants from public schools indicate that lesson studies were often omitted due to other priorities of the school staff or administra- tion. Workshop participants indicated that many teachers serve as the lesson study teacher only once or a few times in a career. The impact of lesson study is primarily on the teacher team that created the lesson. They spend considerable time outside of their daily practice preparing the lesson and postlesson reflection and documentation. For observers, this is an opportunity to establish and continue professional communication and take away ideas for implementation or refine- ment. For preservice teachers, this can have a powerful impact on learning from the demonstration of more experienced teacher teams. For experience(1 teachers, a lesson may present a new curriculum piece or a new pedagogical strategy. During the workshop, (liscussions in small groups anti in(livi(lual conversations considered the merits of using the differ- ent forms of studying practice. The live observation of a teacher anti class in action around a focused lesson plan brings a strong cultural component to lesson stu(ly for the Japanese participants. Their continual reference to "seeing with the eyes" is the essence of professional growth. The use of videos of lessons along with samples of student work anti an opportunity to have reflections from the classroom teacher seemed to make a classroom video a possibility for reflection on a lesson. However, the written case approach was not seen as valuable by the Japanese because of the lack of an authentic teacher anti students. Also, the participants were concerned that the perspective of the writer limite(1 the opportunity of each participant to bring their own perspective to the (liscussion. Lesson stu(ly requires the actual enactment of the lesson stu(ly process for preservice or professional (levelopment for a set of in(livi(luals. In contrast, the REFLECTIONS ON THE WORKSHOP

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video and written case study records of practice, organized and packaged for on- demand use, could be used at the inshructor's choice in settings across time and geography. That is, a collection of records of practice could be used semes- ter after semester in a teacher preparation program for preservice teachers. Or a record of practice could be used for professional development in school dishricts across the counbry. One comparative shrength of the records of practice approach is the opportunity to develop these records over time with the same children or different children. How does their mathematical ~ commun~cahon grow over a year, or even over years? Records of practice (videos, written student work, hranscripts) could be collected and organized for such study. When used for research or professional development, the records could be revisited repeatedly for purposes of argument or clarification of a classroom act by student or teacher. The real-time observation in a lesson study would not have this opportunity to "see it again." Lesson study has its roots in the lesson on the day of observation, although lesson study is about much more than a single lesson. Japanese educators are concerned about students' growth in mathematics and communication, how lessons are sequenced and how ideas are built across lessons and grades despite the existence of a national curriculum. The workshop clearly presente(1 two approaches to professional (levelopment through the study of teaching practice. Two models of records of practice, video and written case studies, were identified as providing tools that multiple audiences could use at convenient times and loca- tions. The need to develop a package that coul(1 be use(1 by those (1istant from the teaching being recorded and with varying leaders was an important consideration in effective use of these tools for profes- sional development. Lesson study was con(lucte(1 with primary attention to professional development by those present at the lesson. The resulting lesson plan and indications of student responses might be written for a local or national professional journal to give teachers a plan (letermine(1 effective by the observers and those who conducted the lesson study. OBSERVATIONS FROM THE STUDY OF TEACHING PRACTICE

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Carol E. Malloy, University of North Carolina-Chapel Hill Mathematics educators in the United States and in Japan are working to provide preservice and practicing teachers with exemplary professional development in mathematics teaching. However, situations confronting the United States and Japan in mathematics education are quite different. The United States is faced with a teacher shortage in mathematics education (Riley, 2000), and numerous mathematics classes at the elementary and middle grade levels are taught by teachers without substantial mathematics training (Dossey and Usiskin, 2000~. A(l(litionally, over 30 percent of U.S. students at grade 4, 6, and 12 perform below the basic level of achievement on the National Assessment of E(lucational Progress (Dossey, 2000~. In Japan, students have historically achieved at high levels in mathematics; however, the results of the Third Inter- national Mathematics and Science Study show that 47 percent of Japanese students do not like mathematics (Hashimoto, 1999~. In response to the national con- cern that schooling should respond to the needs of children, "in 199S, the Ministry of Education, Culture, Sports, Science and Technology reformed the Course of Study for K-12 to emphasize the well-being of human development of the child in a changing international society" (National Science Foundation, 2000~. This plan reduces the number of (lays students are in school and the mathematics content hours that they experience. The plan also mandated changes in university teacher preparation programs that reduce the preservice content course credits while increasing the pedagogy course credits. These changes in philosophy anti pro- grams come to Japan at a time when there is a surplus of mathematics teachers and a decline in the population of school-age children. GOALS OF THE WORKSHOP Both the Unite(1 States an(l Japan are face(1 with challenges in the mathematics education of their students. One strategy is to address the challenges through a dialogue of mathematics educators about the delivery of professional development to teachers. To promote this dialogue, the National Research Council (NRC) held a U.S. - Japan Workshop on Professional Development in Makuhari, Japan, cen- tered on the questions: What knowledge of both content and pedagogy do teachers need to teach well and how can teachers

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come to acquire this knowledge in ways that are usable in practice? The major goal of the workshop was to use the expertise of the participants to investigate these questions to develop ideas and insights that could address the issues surrounding the preparation of mathematics teachers. This workshop used an accessible medium, teaching practice, as a too} to address professional development of mathematics teachers in both countries. Before the conference, participants received papers that explained and demonstrated three forms of professional development using teaching practice: (a) lesson stu(ly, or jugyokenkyu as it is practiced in Japan, and (b) two types of records of practice cases and fairly complete records of instruction including classroom video called record study, from the United States. In this paper, ~ chronicle the events of the conference through my experiences as a mathematics teacher and teacher educator from the United States. ~ begin by explaining my position anti the approach to professional development and teacher preparation coming into the workshop. Next ~ share my journey through the workshop, the insight ~ gained, the confusion ~ felt, and how this experience reshaped my view of teacher preparation. MY POSITION AS A TEACHER EDUCATOR In teacher education, as in many disciplines, teaching and learning are 2. intertwined. We learn our disciplines- pe(lagogy and content and we learn more about teaching as we practice our discipline. To foster this type of teaching and learning within the methods and mathematics classes ~ teach and the professional development ~ offer to B U I LD I N G AN I ATE R NATI O CAL COMMU N ITY practicing teachers, ~ use a "community of learners" approach where the participants and ~ interchange the roles of the teacher and the learner. The tools that ~ use are case discussion with narratives of teach- ing and learning, lesson records through the medium of video and student work, reflection, and study groups. My goals are to help preservice and practicing teachers learn strategies to understand how their students' learn, to develop strategies to enhance student learning, and to have a strong foundation in and conceptual un(lerstan(ling of the math- ematics they are teaching. As a commu- nity, students in classes and teachers in professional (levelopment sessions explore educational topics and issues through inquiry and critique, striving to find answers to questions and formulate positions. We use inquiry to explore the mathematics content and related issues. We develop and model what we think are appropriate pedagogy strategies for teaching mathematics and content for middle and high school students. The most important phase is our reflection. Each of us personally reflects on our teaching to assess how we are (loin" our work to improve our practice and achieve the goals we have individually set for ourselves. Over the past 30 years ~ have isolate(1 three essential components that support this metho(1 of professional evelopment: love of anti passion for teaching anti learning; knowle(lge of content with the neces- sary pe(lagogical skills anti ability to assemble, synthesize, anti convey course content to students; anti continuous reflection that gui(les learning anti teaching. These three components were central

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in the three forms of professional develop- ment discussed at the workshop. JOURNEY THROUGH THE WORKSHOP The workshop was planned to promote a collaborative atmosphere between the participants from the United States and Japan who were teachers, teacher educa- tors, and researchers from both countries. We were assigned to seats so that we could form small discussion groups that would include people from both countries and with varied backgrounds and careers. The format for most workshop sessions started with presenters sharing informa- tion about key topics followed with participants reacting to and discussing what was shared in a group as a whole, in small prearranged groups, or in clusters based on seating assignment. The first session was designed to give us an understanding of the educational systems, teacher preparation, and profes- sional development in the United States and Japan. There are clear similarities in the structures that support mathematics education in both countries. Neither country has a nationally organized system of professional development, although professional development does exist at the school, (listrict, university, regional, and national levels. At the elementary level teachers are generalists teaching all academic subjects, and at the high school level teachers are specialists teaching in only one curricular area. The textbook selection process is similar in both countries. In the United States individual states or school districts authorize textbooks that match the curriculum guidelines of states and districts. In Japan the Minister of Educa- tion, Science and Culture authorizes textbooks that he deems suitable to be used in schools, which match the Japanese national curriculum Japan Society of Mathematical Education, 2000~. With guidance from principals anti (1istrict (prefecture) administration, schools are then allowe(1 to select from the list of approve(1 texts for students. Differences in structured programs far exceed the similarities. The most obvious difference is that Japan has a national curriculum in mathematics, and the United States does not. In the United States individual states or districts control educational programs for students, resulting in varied educational curricula; whereas in Japan the Minister of E(luca- tion, Science and Culture controls educa- tional programs, ensuring uniformity throughout the country. Certification requirements are different. Mathematics teachers for the middle grades in the Unite(1 States are not always certifie(1 in a content area, while those teachers in Japan are certifie(1 in mathematics only. This enables Japanese preservice teachers for the middle grades to take more math- ematics content courses. Most certifica- tion programs in the Unite(1 States require that teachers participate in professional (levelopment programs to retain their certification. Atypicairequirementis six credit hours every five years in education courses not necessarily mathematics courses (Dossey and Usiskin, 20001. However, Japanese teachers obtain a lifetime certification and are not required to take a(l(litional courses (luring their teaching careers. Requirements for initial teacher preparation are (lifferent. Teacher preparation programs in the Unite(1 States require 12 to 15 weeks of student teaching under the supervision of a classroom teacher where teacher preparation students observe anti in(le- pen(lently teach the full sche(lule of their REFLECTIONS ON THE WORKSHOP

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supervising teacher for at least 6 to ~ weeks. It is expected that new teachers have gained the necessary skills to be independent in their first years of teach- ing, but many school districts have induction programs where new teachers have mentors who are available for support. In Japan the system is slightly reversed. Japan requires approximately three weeks of student teaching where they observe other prospective teachers, and then they teach alone and in teams with a master teacher. When most new teachers are hired, they are paired with experienced teachers for a year so that they can team teach with the master teacher as they develop their skills as teachers. They also are required to attend professional development sessions throughout the year. The forms and goals of professional development in the United States and Japan reflect the needs of teachers to improve their practice, the needs of their students, and the structure of each country's teacher preparation programs and certification requirements. Profes- sional development in the United States is structured to help teachers grow in a variety of ways. Some may address change in classroom practice to be more inclusive of all students, or focus on helping teachers understand the math- ematics they teach, while others may help teachers with pedagogical decisions and strategies for effective instruction. Gener- ally professional development is struc- tured to support the accumulation of continuing practice credits for certifica- tion. Fried et al. (1998) provide a list of professional development strategies used in the United States, from teacher-led learning within schools or classrooms such as action research, study groups, coaching, and mentoring to formal opportunities outside the classroom such B U I LD I N G AN I ATE R NATI O CAL COMMU N ITY as workshops, institutes, courses, and seminars. The most commonly used strategies include short sessions at meetings of professional organizations, school-based workshops on specific topics, or two- to three-week grant- supported workshops in the summers with follow-up sessions during the aca- demic year. In many cases teachers who participate in these forms of professional development are not offered indepth follow-up sessions to reflect on their practice. None of these strategies inclu(le avenues for teachers to use observation of colleagues' teaching as a tool for learning. Teachers in the United States are solitary practitioners, coming together to learn about teaching but working in isolation in their classrooms (Lewis, 20001. Similar to teachers in the United States, Japanese teachers use professional development to improve their practice, to learn more mathematics and pedagogical strategies, and to make better pedagogical (recisions in their instruction. However, teachers in Japan use observation anti collaboration as the core of their profes- sional development called konaikenshu through lesson stu(ly. Lesson stu(ly is an in-school teacher education strategy where teachers are engaged in action research about teaching (Yoshi(la, 19991. Lesson studies are held to educate preservice teachers, mentor anti instruct novice teachers, improve the skids of all teachers, maintain collaboration among teachers, anti share ideas anti new approaches. Shimizu (2000) explained that lesson study consists of precollaborative work among teachers, lesson observation, anti postcollaborative work. This cycle is repeated over time in an iterative process. Professional (levelopment also has other forms in Japan. Teachers participate in workshops at the university and in school- base(1 stu(ly groups where they stu(ly a

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variety of topics (Yoshida, 1999). Japan's large-scare meetings of mathematics educators may be focused around demon- stration lessons. ELEMENTS OF PROFESSIONAL DEVELOPMENT IN THE THREE RECORDS OF PRACTICE Lesson Stucly The first full day of the workshop was dedicated to lesson study and reflection. The (lay began with an optional viewing of a videotape of a s~xth-grade lesson for those who had not been able to attend the actual lesson. The day's schedule included presentations, small group discussions, and group viewing of a second videotaped lesson. The workshop then became a postconference for the lesson study where we reflected on the lesson with the teacher and lesson team. As the presenta- tions and conversations in small groups progressed, the U.S. participants learned that there are several forms of lesson study. Because of language differences and translations from Japanese to English, there was some confusion about the definition for lesson study. What follows is my understanding of the two forms of study. Lesson study is a school-based strategy primarily used at the elementary level directed by teachers within a school. Teachers (leci(le on the content of the lesson study and proceed through a process that generally has five components: (~) teachers plan the lesson collaboratively; (2) the lesson is taught by one teacher and observed by other teachers; (3) the team of teachers meet to reflect on the lesson and improve it; (4) the lesson is taught again, usually by another teacher, with refinement; and (5) the lesson is discussed again and made into a booklet that is available to other teachers. Lesson study is an accepted part of teaching in Japan at the elementary levels, although the workshop participants indicated that in some schools teachers had fewer than two lesson studies a year. Teachers interviewe(1 by Lewis (2000) sai(1 that if they did not do research lessons, that they woul(1 not be teachers. Teachers also participate in a public lesson study, called jugyokenky~, which is open to teachers anti educators from outside of the school and the prefecture (district). Most elementary and lower secondary schools conduct lesson study, but the universities anti national schools generally conduct public lesson stu(ly. Also, when schools receive grants to develop their educational programs, it is expecte(1 that they will conduct a lesson study to present their products and findings. Stu(lylessons thelesson itself at conferences become a metho of transferring or transmitting good teaching ideas from one teacher to another throughout the country. With a national curriculum anti textbooks autho- rize(1 by the Minister of Education, Science anti Culture, lesson stu(ly seems to offer an efficient form of professional (levelopment for teachers. Lewis (2000) indicated that Japanese teachers credit lesson study as the primary method they use(1 to learn to teach. Toshiakira Fujii, in his comments at the beginning of the (lay, sai(l, 'The lesson is the battlefiel(1 of teaching. An(1 the teacher is evaluate(1 by the quality of the lesson." Lesson studies are works in progress where teachers strive for perfec- tion through the iterative process of (lemonstration anti reflection. Thus the function of lesson stu(ly is twofol(l: a method of research and a place to present new approaches. The teacher's aim is to ask colleagues to identify flaws in teach- ing through their eyes anti to identify the REFLECTIONS ON THE WORKSHOP

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causes of the failure of the lesson (Hirabayashi, 2000~. For this process to be productive, the teacher has to prepare a carefully planned and detailed lesson that includes the purpose, topic, teaching process, student activities, and intended results. Hirabayashi (2000) describes lesson study as the "method of research in mathematics education that is the way to grasp the true state of affairs of the problem in its whole and bring the syn- thetic, totally recognized interpretation about it, being aware of many factors which are subtly interacting to each other as if it were in one organism" (p. I). He believes that because lessons are compli- cated processes and their results on student learning are too subtle to express in writing, lessons can only be evaluated through close observation. You have to see what actually occurred in the lessons- both teaching and learning. In this, my first observation of lesson study through a videotape of the lesson and a discussion of the participants in the process, ~ was amazed and pleased with the detail of the lesson and the reflection of the participants. The teacher's ability to pose the problem precisely, the func- tional thinking of the students, the ques- tioning that required each student to think of multiple ways to solve the problem, the varied representations of solutions and the students' interpretations of the differ- ent forms were impressive. The teaching of mathematics content dominated the lesson and the post-lesson reflection session continued the focus, with the teachers having a lengthy conversation of the different meanings of 4 x 48 and 48 x 4. Although ~ was surprised with the length of this conversation, it demon- strate(1 the importance given by the Japanese teachers to teacher and student content knowledge. After a few hours of learning about B U I LD I N G AN I ATE R NATI O CAL COMMU N ITY lesson study, we understood that the work required to prepare a lesson for observa- tion and review was extensive. Most of the U.S. participants wondered how teachers could take so much time with planning. When we inquired, we were told that lesson studies are not the norm for instruction. The Japanese teachers explained that most of their lesson plans were in the teacher's editions of the textbooks. Lesson studies were tools to improve instruction, not to develop daily lesson plans. The concept of lesson study is an effective tool to create a community of learners throughout Japan because of the capacity to share research and approaches to teaching within schools, prefectures, and the national education community. Knowing this, we were surprised to learn that even though many elementary and some lower secondary teachers participate in research lessons, it rarely occurs in the upper secondary schools. RecorcIs arc! Practice In Deborah Ball's opening comments on the first (lay of the workshop, she questioned how teachers in Japan and the Unite(1 States use practice to work on their teaching. She expanded the battle- field concept expressed by Toshiakira Fujii to the use of recor(1 stu(ly anti case stu(ly as tools to analyze anti learn the practice ofteaching. Deborah Ball believes that practice is not learne(1 by just (loin" it, nor is it learne(1 by just acquiring knowle(lge or watching expert performances. To learn a practice teachers have to progress through the steps of studying, trying, analyzing, improving, anti (1eveloping new knowle(lge, as is the case with lesson stu(ly anti recor(1 stu(ly. In the secon(1 (lay of the workshop, we learne(1 about professional (development through records of instruction. Records of

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instruction used in the United States include videotapes of classroom instruc- tion, written cases that describe actual classroom situations and issues that arise, student's written work, transcripts of lessons, teachers notes, and lesson plans. Record study is useful for many reasons. Records are used to provide a context for learning and place professional develop- ment in the context of practice. The ability to select records to be studied ensures that the knowledge generated is useful and usable in practice. Specifically, the use of record study allows teachers to select particular problems of mathematics teaching and learning to be studied and can provide exposure to practices that teachers have not seen or do not know. Most importantly for U.S. teachers, the use of records for analysis allows teachers to critique practice in a safe environment where they are not asked to criticize each other's teaching. Record study can be problematic because not all records are worthy of study. It is difficult to design a good task for learning that will help develop norms for the professional study of teaching. Additionally there are chal- lenges as teachers move from evaluation and judging to the analysis of teaching and in balancing the analytic work with practical outcomes (Ball and Bass, 20001. The challenges of the use of record study can be addressed through the appropriate (resign anti enactment of recor(1 stu(ly. For instance, video records and case records accompanied by focused questions about the records provide teachers with an opportunity to (levelop the ability to analyze and reflect on their practice. Record study can enable teachers to understand and improve their peda- gogical content knowle(lge. "Pe(lagogical knowledge is a special form of knowledge that bun(lles mathematical knowle(lge with knowle(lge of learners, learning, anti pedagogy" (Ball and Bass, 2000~. Records can expand the pedagogical content knowledge that teachers possess because records afford teachers the opportunity to view, un(lerstan(l, analyze, anti reflect on situations that they have not experienced in their classrooms. Margaret Smith explaine(l, in the session on case studies, that records "create generalities that teachers can use to think about their own teaching." They allow teachers to investi- gate instruction through records to develop generalities that might be applied to their practice. Moreover, recor(1 stu(ly can help teachers learn (a) how to pay attention to anti teach every student in the class, (b) how to know anti use math- ematical knowledge to help students learn, and (c) how to work with others on (leveloping knowle(lge for teaching (Ball, 20001. In my practice ~ use case anti video records, thus ~ was not surprise(1 with the information shared at the workshop. was please(1 that the readings anti the presentations on recor(1 stu(ly stressed the nee(1 for pe(lagogical content knowI- edge. ~ believe that questions asked of the participants both at the beginning of the workshop anti in our small groups coul(1 not be answere(1 without a foun(la- tion in pedagogy, content knowle(lge, anti pe(lagogical content knowle(lge. It was clear to me that the three forms of profes- sional (levelopment on teaching practice (lesson stu(ly, video records, anti cases) coul(1 be part of the answer, but we were just beginning a long journey. Reflection At the en(1 of the workshop we realize that we had only scratched the surface of (leveloping answers to the questions that focused the workshop: What knowle(lge of both content anti pedagogy (lo teachers nee(1 to teach well anti how can teachers REFLECTIONS ON THE WORKSHOP

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come to acquire this knowledge in ways that are usable in practice? Questions and comments were shared at the end of each day as participants from each country reflected on what they had observed, heard, and learned. {chief Hirabayashi explained that curriculum development is composed of technology and humanity, and we had only addressed the tech- nology. He said that the humanity of the teacher was evident in the way students were attentive throughout the entire period. But he said, "Lesson study is not enough. We need good experienced teachers to guide us. ~ videotape myself every day as the mentor teachers instructed me." He challenged us by saying, "In the United States you have to determine how to incorporate lesson study into your school. What are you going to do and what problems do you anticipate when you return?" Deborah Schifter suggested some needs that teachers might have before they participate in lesson study, including understanding that mathematics is more than being able to apply a single algorithm and that understanding alternate forms of mathematical procedures is the basis of mathematical reasoning and valid think- ing. Haruo Ishigaki explained that teachers learn in the same ways that students learn. He commented that excellent teachers do not have perfect knowledge, '~hey are 80 percent confi- dence and 20 percent doubt. Their knowledge has to be updated and restruc- tured often." He stated that students see extraordinary things but may make mistakes. These mistakes, however, are valuable and should be treated as a resource by the teacher. As ~ listened and reflected on my experience, ~ was thinking, is it possible for us, in the United States, to use what we have learned to improve professional B U I LD I N G AN I ATE R NATI O CAL COMMU N ITY development for our preservice and practicing teachers? First ~ had to think about what ~ had learned. My overwhelm- ing realization was that mathematics educators from different parts of the world, with (lifferent spoken languages and cultures, spoke in unison as we discussed our goals and needs to improve mathematics education. Our strategies and tools of delivery were different, but we were seeking similar outcomes. The most exciting new knowledge was my personal un(lerstan(ling of the organiza- tion and execution of lesson study and the power of lesson study to improve content knowledge and pedagogy of teachers and the delivery of content to students. My beliefs about the importance of teacher collaboration and observation of each other's teaching to improve practice and the use of record study to help teachers construct new pedagogical, content, and pe(lagogical content knowledge were reinforced. Second, ~ had to personalize the ques- tions for my own practice. How could ~ use all that ~ ha(1 learne(1 in my practice as a teacher educator? How could my students benefit from my experience? PUTTING IT TOGETHER IN PRACTICE Clearly, the most important experience from the workshop to me was being part of a community of learners from the Unite(1 States an(l Japan coming together to begin to answer questions regarding professional (levelopment. Just as we ha exten(le(1 our community to include each other, ~ decided to try to broaden my preservice students' community of learners to inclu(le all of the mentor teachers and students using a modified lesson stu(ly approach. Normally each student interacts with anti learns from one

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mentor teacher in a full year practicum. In the fall semester the students observe their mentor teacher in all classes, tutor students, help in the classroom, and discuss lesson planning and teach a trial lesson. During the second semester they do their formal student teaching. ~ will only have three preservice students in mathematics this year, and ~ plan to place them all in the same school for their practicum. All mentor teachers agreed to partici- pate in a modified lesson study with the mentor as the teacher. The mentor teacher and student teacher will select a class that the three students and ~ will observe. Prior to the observation we will meet with the teacher to learn about the class and the lesson she plans to present to learn her rationale for the pedagogy and content and what she expects from the class. Next we observe the class, taking notes on what we see. After the observation we will meet with the teacher to discuss and critique the lesson. These discussions are beneficial for the teachers because they can see how to improve the lesson and they are benefi- cial for the students because they learn how the teacher implements what she had planned and begin to think as a teacher, reflecting on what worked, what did not, anti why. Another change in my practice this year will be the cooperative planning process for student demonstration lessons. Students will be require(1 to teach one lesson in their mentor teacher's class. In prior years each student would plan and teach a lesson. The mentor teacher and woul(1 observe anti critique the lesson. Now, instead of having the students plan their lesson independently, my three students and ~ will act as a team to plan three different lessons one for each student. Our team and the mentor teacher will observe and critique the lesson. Our goal this year is to build a community of learners that can depend upon one another for knowledge and support. We are trying to remove the myth, through our mo(leling, that teachers in the Unite(1 States have to be solitary professionals. ~ hope that through these two mollifications of lesson stu(ly, my students anti their mentors will learn not only more pedagogy, mathematics con- tent, anti pe(lagogical content knowledge but also the importance of peer observa- tion, collaboration, and group planning. REFLECTIONS ON THE WORKSHOP