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AERA2005 Betsy King and Aki Murata 1 WHAT DO LESSONS TELL US ABOUT TEACHERS’ LEARNING FROM LESSON STUDY? AERA Interactive Symposium, April 12, 2005 Presenters: Betsy King, Catherine Lewis, Aki Murata, Rebecca Perry, Mills College Lesson Study Group Discussant: Brian Lord, EDC Learning About Student Mathematical Discourse: Case Study of a Middle-School Lesson Study Group Elizabeth (Betsy) King, Aki Murata, Mills College, Oakland CA www.lessonresearch.net Case 2 comes from a one year self-initiated lesson study by five middle school math teachers in a Northern California middle school. The teachers conducted three research lessons which progressed through three lesson study cycles. It was taught by the 8th grade teacher to his algebra students, a 7th grade teacher of pre-algebra, and by a 6th grade teacher. The goal of the lesson study group was to promote student mathematical conversations. Lesson study is the major form of professional development used by teachers in Japan and is credited with the steady improvement of Japanese elementary mathematics and science instruction over the last 50 years (Lewis, 2002a, b; Lewis & Tsuchida, 1997, 1998; Stigler & Hiebert, 1999; Yoshida, 1999). Lesson study consists of an iterative design cycle of collaborative teacher activities: (a) formulating goals for student development, (b) studying existing instructional materials, (c) planning a lesson designed to make the goals visible in the classroom, (d) having one team member teach the lesson while others observe and collect data, and (e) using the data to redesign instruction in response to student learning (Lewis, 2002b). The lesson study process is relatively new in the United States, yet it has emerged in at least 32 states since its 1999 introduction (Lesson Study Research Group1, 2004). Even with this proliferation, the published research on lesson study has primarily been limited to only two sites in the United States (Chokshi & Fernandez, 2004; Fernandez & Yoshida, 2004; Lewis, 2002a). This study seeks to add to that body of research, specifically concerning teacher learning in the lesson study process. Lesson study is believed to be a key-supporting factor for Japanese teachers to change their teaching style from traditional and directive to student centered and inquiry based. In the United States, teachers are also facing the challenge to shift their ways of thinking about classroom teaching and learning, and introduce more student-centered practices. Practice-based professional development, AERA2005 Betsy King and Aki Murata 2 which is situated in practice (the classroom) and engages teachers in collaboration, inquiry and reflection (Ball & Cohen, 1999; Darling-Hammond, 1998; Loucks-Horsley, 1998; Putnam & Borko, 2000; Smith, 2001; etc.) such as lesson study, has the potential for supporting such a shift in teachers. The middle school mathematics teachers in this study wanted to use the lesson study process to help them understand and learn more about mathematics teaching practices. Of particular interest to some of the teachers was developing more student discourse in their classrooms. They believed student mathematical conversations would help their students become more successful learners of mathematics. Consequently, the group of teachers articulated “promoting student mathematical conversations” as their guiding principle for their work in the lesson study process. This study investigates the teachers’ thinking (e.g., knowledge, beliefs, goals) about student discourse and how it changed over time. In order to access the teachers’ thinking about student mathematical discourse and how the process of lesson study may facilitate shifts in that thinking we now review the literature on student discourse and lesson study. Perspectives on Student Discourse The National Council for Teachers of Mathematics’ (NCTM) landmark document, Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), described the vision of mathematics learning for all students and presented five goals, one being that all students learn to communicate mathematically. It was followed up by two more highly regarded documents that elaborated the goal of communication by describing the meaning of classroom discourse, what it looks like and how to promote it (NCTM, 1991), and then elevating mathematics communication as one of the standards in the Principles and Standards for Mathematics Teaching (NCTM, 2000). Many others in the education research community have also investigated the use of student discourse, and orchestrating classroom discourse is now seen as a central model for instructional practice (Sherin, 2002). Researchers have described children actively constructing their mathematical understandings as they participate in classroom social processes (Cobb, Boufi, McClain, and Whitenack, 1997), where not only are students experiencing mathematical ideas and concepts but also making sense of and sharing these concepts with others. When students attempt to articulate their thoughts and reason with others about mathematics, they are pressed to clarify their own thinking (Chapin, O’Connor, & Anderson, 2003; Lampert & Cobb, 2003; NCTM, 1991; Ball, 1991). During classroom discourse, AERA2005 Betsy King and Aki Murata 3 students develop ideas and improve their reasoning when they conjecture, explain, and justify their solutions to others (Ball, 1991; Carpenter, Blanton, Cobb, Franke, Kaput and McClain, 2004; Lampert & Cobb, 2002; Manouchehri, & Enderson, 1999; Gravemeijer, 1994; Hufferd-Ackles, Fuson, & Sherin, 2004; Kilpatrick, Swafford and Findell, 2001; NCTM, 1991; Rittenhouse, 1998; Sherin, Louis, and Mendez, 2000; Rittenhouse, 1998). Teachers play a crucial role in supporting communication and guiding classroom discussions by posing appropriate questions for students to carry meaningful discourse. (NCTM, 2000) Studies have identified new demands on teachers who orchestrate productive mathematical discourse in the classroom. Teachers need to gauge their interventions during the discussion and cultivate an adaptive style of teaching, as their decision-making is dependent upon a number of issues, such as shifting back and forth between the role of participant in the discussion and the role as commentator about the discussion. Teachers must find a balance between being open to students’ ideas and ensuring that students are learning specific mathematical content. Since many more students’ ideas surface than are possible to pursue, teachers must select and direct the students' explorations accordingly. Teachers must make delicate decisions about when to let students struggle in making sense of an idea or a problem, when to ask leading questions, and when to insert mathematical ideas and tell students directly. Teachers promote discourse by establishing a classroom culture and environment that is inquiry-oriented in which everyone's thinking is respected and valued by both the teacher and students. By modeling problem solving, exploring relevant contexts, and allowing students time to explore, create, discuss, argue, hypothesize, and investigate, teachers stimulate deeper student insight and understanding (Chazan & Ball, 1999; Frykholm & Pittmann, 2001; Manouchehri & Enderson, 1999; NCTM, 1991, 2000; Sherin, 2000, 2002; Rittenhouse, 1998). Establishing a discourse community requires teachers to adopt new teaching strategies, and requires students to make a shift as learners. Members of the discourse community are encouraged to take ownership of the discussions at hand, therefore students have to be more self-reliant and take responsibility for each other’s learning (Frykholm & Pittmann, 2001; Gravemeijer, 1994; Manouchehri & Enderson, 1999; Sherin, Louis, & Mendez, 2000). Classroom culture is established in the discourse community through negotiation, in which reasoning, explaining, and arguing about mathematical meanings is the norm (Ball, 1991; Chapin, O’Connor & Anderson, 2003; Manouchehri & Enderson, 1999; NCTM, 1991, 2000; Yackel and Cobb, 1996). The Principles and Standards for School Mathematics describes the students’ role as follows: AERA2005 Betsy King and Aki Murata 4 Each student should be expected not only to present and explain the strategy he or she used to solve a problem but also to analyze, compare, and contrast the meaningfulness, efficiency, and elegance of a variety of strategies. Explanations should include mathematical arguments and rationales, not just procedural descriptions or summaries. (NCTM, 2000) Making the transition to bring the above into practice and orchestrate rich mathematical discussions has been difficult for teachers. Researchers and educators that describe these teaching practices also acknowledge the difficulties and dilemmas they produce in the teachers (Edwards, 2000; Fennema, et al., 1996; Hufferd-Ackles, Fuson, & Sherin, 2004; NCTM, 2000; Sherin, 2002; Smith, M., 2001). The teachers in this study saw lesson study as a means to assist this transition. The Lesson Study Process Teachers have reported being drawn to lesson study because it engages them in the selection, study, and solution of issues that are important to them (Lewis, Perry, and Murata, 2003). It is the teachers who choose their goals and determine how they will use research lessons to explore these goals. Outside experts in mathematics education, often called “knowledgeable others” or “facilitating educators,” and administrators may observe lessons and provide feedback, but they do not control the lesson study process (Chokshi, 2002). What are lesson study practitioners learning? Does lesson study help teachers understand student thinking and target instruction thereby deepening the intervention? Teachers have said they benefit from the process and are different teachers after participating in lesson study. "It has totally changed my practice. I do not look at a lesson the same way. I always think, 'what is the student response going to be? What do I want students to show so I will know they have learned this?'” (Becky LaChapelle as reported by Richardson, 2004). "Once you've had that kind of thinking put before you, it's hard to turn around and teach any other way," (Jill Precel, as reported by Viadero, 2004). Lesson study helps shift teachers’ focus from what is being taught to how students learn. Teachers are learning to be researchers right in their own school buildings. They are moving from reviewing student work simply to provide a grade to developing an understanding of what students are or are not learning in class through careful observation of lessons (Kikuchi & Nagai, 2002). The lesson study cycle affords multiple opportunities for teachers to collaboratively consider different facets of their teaching and their students’ learning. It provides the context and process for teachers to think about, discuss, and make connections among different kinds of knowledge. During lesson study teachers: AERA2005 Betsy King and Aki Murata 5 • Formulate goals for student development, imagining and articulating what they would like to see their students accomplish • Collaborate with other teachers – time to think, share, listen, consider, and examine their own and others’ knowledge in the context of actual classroom lessons. • Plan lessons, anticipate student responses, construct thought-revealing tasks • Teach in new ways, getting feedback from teachers and knowledgeable others • Observe students during actual lessons, not videotapes, and experience the entire lesson, not only seeing what the videographer chose to record. • Discuss those lessons and students conversations, sharing their own perceptions and hearing what others’ perceived, including perceptions and perspectives of outside experts and educators. • Analyze student work, gaining more insight and understanding of student thinking and a deeper conceptual understanding of the mathematics involved. • Revise and reteach according to what was observed and learned about previous lesson This study traced the work of lesson study practitioners through three iterations of the lesson study cycle that occurred over one school year. The purpose of the study is to provide a coherent picture of how teacher thinking and learning may develop and change in a lesson study case where teachers work together and think deeply about promoting student mathematical conversations. The focus of the study is the shifts in the teachers’ thinking and how that influenced the planning and teaching of the research lessons, and conversely, how the teachers’ observations of the research lessons influenced additional shifts in their thinking. Specifically, the study examined the discussions of the teachers, the development of their written and actual research lessons, and different aspects of lesson study to determine how the phases of the lesson study process and teacher learning influenced each other. By analyzing changes in teachers’ discussions, changes in actual lessons, and the interaction between the two, the study aims to help us better understand what influences teachers’ thinking and learning during lesson study and be able to devise ways to measure this learning more accurately. Method AERA2005 Betsy King and Aki Murata 6 Participants of the Study Washington Middle School is one of three middle schools in a mid-sized city in the California. (Washington Middle School is a pseudonym, as are all the names of the teachers and the students in this paper.) The middle school serves a diverse student population in racial and ethnic identity, socioeconomic group, and academic skills and performance. Of the 578 students, 43% identify themselves as African-American, 23% white, 11% Hispanic or Latino, 7% Asian, and the remaining 16% of the students chose other categories, including “multi-ethnic.” Ten percent of the students have a parent who receives subsidies and supportive services from the state of California and 68% of the students qualify for free or reduced price meals. Student test scores are divergent, with students’ academic skills and performance ranging from exceptionally low to exceptionally high. 2 For student demographic data and Standardized Testing and Reporting - California Standards Tests (STAR-CST) test scores for math and algebra 1 for the school year 2003-04, see table 1. The district school board decided the year before the study to not track students anymore and also mainstream students with learning difficulties who had previously been pulled out for reading and math. The sixth and seventh grade classes were all heterogeneous for the first time the year of the study. The study focused on a group of math teachers and their lesson study work during the 20032004 school year. The teachers in this study included three white males, Mr. Doyle, Mr. Ray, and Mr. Rodgers, one Asian American male, Mr. Linn, and one African-American female, Ms Hayes. A summary of their background and their experience is shown in Table 2. Also joining the lesson study group sporadically was an outside math curriculum developer who served as a “knowledgeable other.”3 Mr. Doyle took the responsibility for organizing the group and sending out email reminders of the meetings, which he facilitated and held in his classroom. He brought in the first draft of the research lesson and taught the first research lesson at the end of October. He went on paternity leave the first week of November and was not present at any of the planning meetings for lesson 2 which was taught in December. He resumed teaching after the winter break and was present at all of the planning meetings for lesson 3. Background of Lesson Study at Washington Middle School The staff at Washington Middle School voted in the spring of 2002 to do lesson study as their professional development, and one of the mathematics teachers, Mr. Doyle, contacted an educational researcher in the area who had expertise in lesson study for guidance. She gave an introductory AERA2005 Betsy King and Aki Murata 7 workshop on lesson study to the entire staff and offered her support if desired during the year. The first author documented their lesson study activities, offered general support, and facilitated communication between the researchers and teachers. In the first weeks of their lesson study process the faculty of the entire school met together, with Mr. Doyle facilitating, to discuss and formulate school goals for student development. The first author of this paper videotaped those meetings and answered general questions about lesson study. Later teachers broke into their lesson study groups, and she followed and videotaped the lesson study group of math teachers. She was invited to also participate in the discussions, and she took on the role of participant/observer. The teachers planned and taught two research lessons the first year. That summer, June 2003, both authors traveled to Japan with three of the Washington mathematics teachers and other American teachers and researchers and participated in a Japanese lesson study immersion course. The Year of the Study The following school year, 2003-2004, the first author of this paper continued meeting with the math lesson study group in the role of participant/observer. The teachers typically met in one of the teacher’s classrooms usually for one and one half hours, two to three times a month for a total of 20 meetings not including the three research lessons conducted in the classrooms. The first four meetings were mainly for goal setting, nine meetings were for planning and revising the research lessons, three meetings were for the post-lesson discussion of the observed research lessons, and there were three additional meetings after the third post-lesson discussion. The first author took field notes and periodically communicated with the teachers through email. She spent at least three days observing each teacher’s math class over the course of the year and observed one teacher extensively in the fall and spring. She offered support to the teachers by suggesting and making available materials to augment their lesson study work (e.g., articles about student discourse, summaries of their goal-setting and post-lesson discussions, summarized student strategies, sample mathematics problems, etc.). The knowledgeable other was present at most meetings for planning lesson 1. Her attendance was sporadic the rest of the year due to other commitments. Data Collection AERA2005 Betsy King and Aki Murata 8 Data for this study were collected during the 2003-2004 school year, beginning with the first math lesson study group meeting September 4, 2003 and ending with final interviews of the teachers in June of 2004. The first author videotaped all meetings and research lessons and most were also audiotaped. Teachers were interviewed and audio-taped, three times, in the fall, winter, and spring. The teachers’ mathematics classes were observed at the beginning and end of the year. Other data sources included summaries of the lesson study meetings, field notes and journal entries throughout the year. This study is a part of a larger NSF-funded project following the development of lesson study sites in the U.S. Data Analysis Before initiating the data analysis, a start list of codes (Miles & Huberman, 1984) was compiled: Teachers’ knowledge or beliefs concerning student mathematical conversations (SMC) - How SMC are defined - Effective and ineffective SMC - The value and benefits of SMC - How to promote SMC - How to facilitate SMC - The students’ role in SMC - The teacher’s role in SMC - The classroom environment, culture - Challenges of facilitating SMC Teachers’ goals for SMC The audio-tapes of teacher interviews and video-tapes of the lesson study meetings, post-lesson discussions and parts of the research lessons were transcribed and examined. During the first reading of the transcripts the first author highlighted all statements relating to student mathematical conversations (SMC). The highlighted data was examined and divided into chunks. Successive readings of the data necessitated modification and further development of the coding categories (Bogdan & Biklen, 2003). New themes emerged, for example “Rushing students through the lesson and through the year,” some themes diminished or were collapsed, for example, “Effective and AERA2005 Betsy King and Aki Murata 9 ineffective SMC” became part of “How SMC are defined.” The theme “Challenges of facilitating student mathematical conversations” grew in significance and subsets of the challenges theme were developed. The subsets of the themes to be analyzed for the study were then put into the form of the questions raised by the teachers during lesson study: Making on-the-spot decisions while facilitating SMC - How many and what kinds questions to ask? - Whom to call on and in what order? Which students’ strategies to use? - Whether or not to follow up a student’s question that digresses? - How to know when to “move on” to the next math problem or the next part of the lesson? Teachers are worried they won’t have time to finish the lesson - How does the teacher know who understands? And what do you do about the students who don’t? - Is it when most (some?) hands are raised? How many hands need to be raised? - What if one student tells the answer (“the punch line”) early in the lesson, does that “give it away” or “spill the beans” and ruin the lesson? Rushing students through the lesson and through the year - Trying to finish all of the lesson or wanting to fit a lot of content into 45 minutes - Trying to “cover the curriculum (or standards)” Setting up the classroom culture - What changes does the teacher have to make in the classroom norms? - How to get more students to talk? …to participate? - How does the teacher handle classroom management? The data were organized into time periods to separate, compare, and analyze the data establishing when or how the teachers’ thinking about student mathematical conversations evolved and/or shifted. The basic elements of lesson study were grouped into five basic phases: 1. Goal-setting: formulating goals for student development 2. Research and planning: studying instructional materials and planning a research lesson designed to make the goals visible in the classroom AERA2005 Betsy King and Aki Murata 10 3. Teaching and Observing: having one team member teach the research lesson while others observe and collect data 4. Post-lesson Discussion: team members and other observers share data collected 5. Revising: using collected data to redesign instruction in response to student learning. Because the teachers planned and taught the research lesson three times, phases 3 through 5 were repeated after the initial five phases and phases 3 and 4 were repeated one more time after lesson 2 making 10 time periods. The three additional meetings after the post-lesson discussion of lesson 3 and the end-of-year interviews made up the final time period. The data was examined and put into a table by time periods and the changes in the teachers’ discussions, the changes in the research lessons, and the interaction between the two were traced and analyzed to understand better the connections and influences between the teachers’ thinking and learning and the lesson study phases. In the following Results and Discussion section, we first summarize the lesson as it was written for lesson 1. Next, we describe the challenges of facilitating student mathematical conversations during the lessons and guiding student communication and thinking. Then we follow the teachers’ discussions and activities through the lesson study process, describing any shifts in the teachers’ thinking and when it occurred, and suggesting how the phases in the lesson study process supported the changes in teachers’ thinking about student mathematics discourse. Results and Discussion The Research Lesson, Gauss’ Houses To give context to the results and discussion section, a brief summary of the basic research lesson follows. Algebra was the focus of the research lesson with the lesson objective: Students will use patterns and knowledge of sequences to discover and discuss relationships between building height, building number, apartment number and apartment floor level. They will derive formulas for determining the set of apartment numbers that share a given floor level. The research lesson plan included a drawing of a row of numbered apartment buildings to be given to the students and shown on the overhead projector (see figure 1). The first question in the lesson is, “Mr. Doyle moved into the 5th building, in the apartment just below the penthouse. What is AERA2005 Betsy King and Aki Murata 11 the apartment number and how you know?” Students discuss strategies in whole class discussion for determining that apartment number, and other numbers, when given the location of building and floor level. Together the class generalizes the rule, such as 6n - 1 when n = building number. Students are then given their own apartment number and asked to determine the building and floor level of their apartment in small groups. The class would then share their strategies for finding their apartment location and discuss formulas for finding the floor and/or building number when given any number. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Fig. 1 Drawing of apartment buildings for Gauss’ Houses lesson At this initial stage, the lesson and its goal were mainly developed by the teacher who taught the first lesson, Mr. Doyle. Discussion questions about the goal, how teachers felt about it, and what they wanted students to get out of the lesson continued during the year. This stimulated in-depth discussions about multiples and division, number patterns and sequences, mathematical rules and algorithms, recursive or closed formulas, and what knowledge about these concepts do students bring into the classroom. As the lesson was revised the goal evolved. Facilitation of Student Mathematical Conversations During the Lesson At the beginning of the lesson study cycle the teachers chose “promoting student mathematical conversations” as their lesson study goal and envisioned it as a guiding principle for their work throughout the lesson study process. They wanted their activities with lesson study to further their AERA2005 Betsy King and Aki Murata 12 knowledge and understanding of SMC enabling them to guide and support students in talking about mathematics in their classrooms. In order to effectively facilitate student conversations teachers need to consider, among other things: classroom culture and norms (e.g., level of mutual trust and respect, teacher’s and students’ roles); purpose and format of discourse; questioning types and strategies, techniques and participation structures; and keeping the focus on the mathematics (NCTM, etc.). The teachers in the study questioned, discussed, shared ideas about, and sought solutions to a number of factors and aspects of facilitating mathematics conversations. These teachers specifically considered their classroom culture and norms, the purpose and format of the mathematical conversations, time and pacing of the lesson; and decisions about which students to call on, whose questions and comments to follow up, and when to move on to the next question or part of the lesson. The teacher’s knowledge and beliefs about these aspects of student discourse influenced how they guided student mathematical conversations in their classrooms. Classroom Norms and Culture The teachers thinking about facilitating and guiding SMC in lessons changed over time. The teachers discussed how they were beginning to make changes in their classroom norms and culture, incorporating new student-centered teaching practices to promote SMC. New practices included: using open-ended questioning strategies and encouraging students to be questioners; shifting to the students more responsibility for their learning and evaluating the reasonableness of math ideas; allowing students more time to explore and wrestle with mathematical concepts, not jumping in to tell students how to solve the problems; and anticipating student responses, planning in advance how and in what order students would share their strategies and solutions, and encouraging students to explain another student’s strategy or solution to the class. Goal setting. At the beginning of the year the teachers discussed formulating goals for student development. They talked about what they wanted their students to be able to do by the end of the year and what should be their focus during lesson study. As suggestions surfaced, the teachers talked about their classroom norms and culture and what they might do differently in their classrooms. The teachers differed in their teaching styles, some were more traditional (i.e. teacher-directed, using lecture and demonstration for transmitting mathematical concepts) than others, and there was a range in what teachers felt comfortable changing in their practice. Mr. Ray, from the beginning of the year, AERA2005 Betsy King and Aki Murata 13 experimented with more student-centered practices and felt more comfortable relinquishing some of his authority. At a lesson study meeting in September he told the other teachers about what happened in his classroom that day. The first author happened to be observing his class that morning. Mr. Ray: Today I did not know what to do.… It was a blown lesson, was the thing… Half way through the lesson I looked around and I realized that not a single person in the class knows what I’m talking about…and I said, “Well, what do we do here guys?” So a few hands went up and I said, “OK, why don’t you kids take over, because I don’t have a clue.” In addition, I just totally abdicated. King: It looked planned, Mr. Ray, like you planned it that way. Mr. Ray: What? Looked planned? It wasn’t planned at all. I was going to do a step-by step logical process.… King: … the kids then took ownership of figuring out how to explain this to the girl who wasn’t understanding it. And there were others that didn’t get it. And then she was starting to get it, and then it was starting to make sense to other girl. And I thought it was very exciting. Mr. Ray : … So how do we create that kind of class as a norm?4 Other teachers were more reticent about trying new practices and changing their teaching style. Also in September at a lesson study meeting the teachers were deciding on “promoting student mathematical conversations” for their guiding principle, and Mr. Rodgers expressed his concern: I have had a certain amount of success with what I’m doing, but sort of setting up the classroom, like we’re talking about now, it’s pretty difficult, for me. It’s not that it’s threatening, but I’m just thinking through … what shifts and changes am I going to have to make? How I’m going to have to reorient myself in my classroom… changing the climate of the classroom, my role, power structures, which is separate from the math content, pedagogy, curriculum.5 Planning lesson 1. As the teachers were planning and discussing the research lesson and its iterations, they were cognizant of their guiding principle. The teachers wanted the structure of their research lesson to be more inquiry-based and have open-ended questions to encourage discussion of multiple strategies. Mr. Doyle, who was to teach the research lesson first, wrote the first draft of the lesson with questions intended to stimulate student conversations. The teachers discussed the parts of the lessons and the questions, and agreed on having non-directed student exploration followed by mathematical conversations about their strategies and solutions, in both small groups and the whole class. Mr. Doyle said he had been striving to develop a more student-centered class culture in his own classroom where students participate more, share their math ideas and take more responsibility for math learning. The intent was for the research lesson to further these capabilities in his students. AERA2005 Betsy King and Aki Murata 14 Lesson 1. During the research lesson teachers observed that some student groups were having mathematical conversations while in other groups student work individually. During whole class discussions there were times when some students had difficulty explaining their thinking and the other students’ attention drifted as they could not follow the explanations. One boy pointed to a multiplication chart high on the wall and struggled to demonstrate his use of multiples before losing the class’s attention. Post-lesson discussion of lesson 1. One of the outside observing teachers commented on some of the students’ ability to carry on the discussion. He said Mr. Doyle’s questioning procedures and process made students comfortable with thinking and responding. Even so, because some students had difficulty communicating their thinking, an outside observer who was experienced in lesson study made reference to this during the post-lesson discussion. She asked the teachers, “Are there representations that would enable teachers to see students’ work and students see one another’s work and catch onto explanations better?” She suggested they think of representations that “would enhance the capacity of some kids to explain to other kids.”6 Mr. Ray was concerned that not all students were having mathematical conversations and said he came away wondering how students learn to talk about math. Revising lesson 1 and planning lesson 2. Mr. Rodgers was next to teach the research lesson. He told the others that he is more teacher-directed than Mr. Doyle, and he was unsure about teaching the lesson. He was concerned because his students were not used to whole class discussions and could be disoriented with open-ended questions. The other teachers were supportive and shared ideas about guiding discussions. Changing teacher/student response patterns was mentioned and Mr. Rodgers agreed they have to get away from teacher response, student response and then the teacher evaluation of that response. Mr. Ray proposed having four students put their strategies on the board and then ask other students to explain those strategies to the class. Although Mr. Rodgers did not try this idea in lesson 2, two other teachers in the group did use it in their classrooms. The teachers also discussed the idea of providing a representation to students to use when explaining their strategies. They wanted to see how it would influence mathematical conversations. Teachers decided a large grid would be posted on the board during the lesson. AERA2005 Betsy King and Aki Murata 15 Lesson 2. When students were to determine their floor number and find someone in the class who would be their neighbor in the apartment building, two students got into a heated disagreement about one of their floor numbers which attracted the attention of five other students. Mr. Rodgers approached the students to “break it up” but found the argument was about math and let it continue. He offered the grid representation later at the end of the lesson and there was only time for one student to use. Post-lesson discussion of lesson 2. Mr. Rodgers expressed satisfaction with the lesson and said at the beginning of the post-lesson discussion: So over all level of engagement, and their overall behavior, sort of “on-task-ness” was good, I think. I thought in terms of promoting mathematical conversation, it was pretty good, especially earlier on. As they were all trying to find their, at first, you know, find # 59 and then their own numbers. I think that was pretty good too. … So, I guess I was somewhat pleased.7 During the lesson Mr. Ray observed that students conversed freely in their small groups. But when the class was together he reflected that the discussion was carried by a small number of students. Teachers wondered if the students who were arguing had the grid available, would they have been able to more effectively communicate their differing perspectives on the problem. Revising lesson 2 and planning lesson 3. After the winter break, the teachers spent the beginning of one lesson study meeting to share how they have been promoting SMCs in their classrooms. Each teacher felt they had been able to bring into their classroom some new teaching practices. For Mr. Rodgers and Ms Hayes it was adding more student-centered questions to their lessons, asking students to give rationalizations or justifications, explain why an answer makes sense, or if they agreed or disagreed with another student’s answer. Mr. Ray talked about the importance of establishing norms for SMC and gave examples from his class on how he talks with his students about what it looks like to have a class discussion. How you sit, how you look, “Look around the room, Mickey, do you see anybody…ready to have a class discussion?” “He’s ready, he’s ready,” “Well, what does it look like?” “Well, he’s listening, he’s paying attention,” and then modeling that and practicing that…it does take a lot of time.8 Mr. Doyle said he now realized that it’s not enough to just put out an interesting problem to get students to have mathematical conversations, “…it takes this sort of very careful orchestration of the AERA2005 Betsy King and Aki Murata 16 conversation. And just saying, ‘ok, talk about this, solve this problem and talk about it while you’re solving it,’ isn’t enough.”9 Mr. Linn said that instead of just having students put up and show the class their solutions, this year he is asking students to explain someone else’s strategy, similar to the idea Mr. Ray suggested for lesson 2. Mr. Ray said he was now more interested in the students who were doing a math problem incorrectly than those who were getting it right. He said, …instead of intervening and saying, “oh no, no, let me help you,” I just tried to think about, “what is this person thinking?” What’s going on in this person’s mind?” “Where is this coming from?”…So that’s very different.10 Lesson 3. The third research lesson was taught over two days giving students more time for discussion, and the teachers observed the second day only. A larger number of students participated in the whole class discussions, some students quite energetically. At one point the discussion became intense as two students defended their ideas. The students easily worked together in their groups, helping each other, and there were conversations in every group. The grid representation was posted on the board both days and students used it and the desk copies that were available. Post-lesson discussion of lesson 3. Each teacher commented on observing energy and excitement in the class during the heated SMCs. They said they would like more discussions like that. Mr. Ray related, I kept thinking, “how can we get other people to join in that conversation…rather than stifle it?” ….We can’t control it so we kind of put it off to the side,…And, because of time constraints and because it wasn’t going in the direction that we had anticipated for the lesson, you had to kind of stifle it. That got me thinking about how we can cultivate and encourage that sort of exchange.11 Mr. Doyle suggested having a “fish bowl” with those students at the front of the class. The teachers remarked on the students use of the representations and the amount of learning shown. End of year. At the end of the year each teacher was interviewed and asked how, if at all, they had been able to change their teaching practice because of their participation in lesson study. Three of the teachers said they had made progress in incorporating more student-centered practices. Mr. Doyle talked about allowing students to “wrestle with the things that aren't clear and make sense of them themselves.” Mr. Ray felt he was a better listener and more patient with his students’ AERA2005 Betsy King and Aki Murata 17 misunderstandings and was more open to letting students construct their own kinds of meanings. Mr. Linn described how he changed his practice with the support of the other teachers: … So I shifted somewhere in the year where kids would place their solution on the board, but they themselves would not do the explanation. I would ask someone else to explain. I think that asks kids to be a little bit more engaged, because they had to listen to someone else, and I tried to make it a point to call on people who are not always talking…And, again, that whole pedagogy came about through talking with my seventh and eighth grade colleagues. Mr. Rodgers and Ms Hayes were more limited in being able to change their teaching practice. They both are more traditional teachers who are more comfortable with a teacher-directed style. Ms Hayes said her students had behavioral issues and with a higher number of special education students in her classes she was reticent to try new practices this year. Mr. Rodgers also had difficult classes and said he was able to be less teacher-directed with students working in small groups than during whole class discussion. “I keep in mind that I want them to be talking with their partner about [the math problem], rather than me coming over and enlightening them.”12 Even in light of these issues, Ms Hayes and Mr. Rodgers said they felt supported by the other teachers in their lesson study group. They thought promoting student conversations had been a worthwhile guiding principle and would support continuing with it the following year. Summary. Changing the culture and norms in their classrooms proved to be difficult for two of the five teachers. Nevertheless, all of the teachers shared their ideas and experiences and gave support to each other at each meeting. The three teachers who taught the research lessons had varying teaching styles. Their lessons were observed by the other teachers, which gave the group common experiences and contexts on which to base their questions, ideas, and discussions. The lesson study planning meetings were the setting for many rich discussions not only about the research lessons but also their own teaching practice. If a teacher reported having success with a particular method, at least one other teacher would try it too. When teachers described challenging classroom situations, suggestions were generated. Overall, teachers became more confident incorporating student-centered teaching practices. Mr. Ray, Mr. Linn, and Mr. Doyle periodically gave accounts of classroom episodes where they relinquished some authority to their students, not insisting on being the only source of knowledge. They related the times when their students took on more responsibility for their learning, found solutions and explained their reasoning, and worked together helping each other. AERA2005 Betsy King and Aki Murata 18 Making on-the-Spot Decisions: What if the “Punch Line” Comes Out Too Early? Trying new teaching practices such as guiding SMCs can be disquieting because of their unpredictable nature, and the teachers were concerned about making effective on-the-spot decisions. During their lesson study work, they gained new understanding and confidence in thinking about what questions to ask and handling situations such as a student giving the answer, “the punch line,” early in the lesson or deciding which students to call on and in what order. Planning lesson 1. As the teachers started planning the first research lesson they considered what questions the teacher should ask. Mr. Doyle wondered if he could think of the right questions on his feet. He said, “...making those snap decisions is the tough part of facilitating discussions.” For instance, he was concerned about what to do if one student says the formula at the beginning of the lesson. “…what if the punch line comes out too early, what do we do from there?”13 The teachers talked about these and other difficulties in making on-the-spot decisions though no one had any suggestions. Lesson 1. Within the first five minutes one student did verbalize the formula. Teachers wondered if that was going to spoil the lesson. Post-lesson discussion of lesson 1. Mr. Linn shared his reaction with the other teachers. “When Mahmoud said he did 6 times 4 minus one, I thought, ‘oh well, we can all go home now, they understand a big chunk of it.’” Then Mr. Linn noticed that the next student said she just counted. So then he wondered “in what direction the flow of the conversation was going to go?”14 The teachers observed that although one student could articulate the formula there were others in the class who were not ready to understand that level of abstraction, and the lesson was not spoiled. Post-lesson discussion of lesson 3. In lesson 3 Mr. Linn twice faced that same situation, and he reflected on his decision at that moment during the post-lesson discussion: …do you stop and explain something to the rest of the class or do you sort of ignore, or validate for the one kid, and then move on? …Jonathan, within the first two minutes says, “Oh, no, it’s 6 times the building minus one gives you the apartment below the penthouse.” …five maybe AERA2005 Betsy King and Aki Murata 19 eight kids understood what he said, and 20 something kids who probably did not understand. I decided to not address that, … Jonathan gets it…his group of friends understands what he is talking about…I thought it would have been fruitless for those kids who don’t understand it to try to explain that formula.15 The other teachers noticed how Mr. Linn handled that and the other decisions he made in the lesson. Mr. Rodgers expressed that they thought he made all the “right calls.” Summary. When planning lesson 1 the teachers worried about the answer, the punch line, coming out too early in the lesson. It was Mr. Linn who was thinking “we can all go home now” at the beginning of lesson 1 when one student called out the equation. If teachers believe that the point of a lesson (or a problem) is to get to the formula or some kind of “the punch line, ” then if one student articulates that punch line, they feel the lesson is spoiled. In this study, after lesson 1 teachers did not talk about a punch line or worry about higher-achieving students spoiling the lesson. The post-lesson discussions after both lesson 1 and lesson 3 showed a shift in the teachers’ thinking. In both lessons at least one student articulated the equation to answer the problem, and the teachers observed the lesson continued with meaning for all students. The objective of the lesson was not so much as find the formula to solve a problem, as it was for each student to understand the mathematical situation, find a strategy that makes sense to answer the question, and then discuss the different strategies moving towards generalizing an efficient rule or formula. Instead of wondering about what to do if a student gives the answer (“the punch line”) early in the lesson, one teacher noted, “when one student gets it, and they get what you want them to get, [and you] try to push that onto the rest of the class, it sort of kills the conversation, because everybody else’s conversation is at a different level.”16 So the teachers observed how their lesson allowed all of the students to access the problems in ways that were meaningful to them and remain engaged in the lesson. Making on-the-spot decisions: Which students do you call on and in what order? Another issue about making decisions while facilitating discussions was deciding which students to call on and in what order. The lesson was intended to be open-ended, and teachers wanted to highlight a variety of student strategies. They were unsure about how to bring that about. The trick is how do you find out who is doing what method, … and in what order do you unveil it. I think that would be the tricky part, how would you find out who’s doing what during that AERA2005 Betsy King and Aki Murata 20 part, and how are you going to get that information from the students? And once you get that information, in what order are you going to introduce it? That’s the tricky part.17 The teachers had seen Japanese lessons live or on videotape, where educators used a studentcentered approach, walking around the classroom taking notes while students worked on mathematics problems. The educators wrote on a seating chart which strategy each student was using and called on them in a predetermined order. To use this method teachers first anticipate the students’ strategies and answers, categorize them, and become familiar enough with the strategies to quickly recognize them while scanning student work during the walk-around. There are challenges with this method, but after observing lesson 1, teachers wanted to use their data on the student strategies to enable them to try it with the research lesson. Planning lesson 1. The teachers discussed possible strategies the students might use for solving the first problem, finding the apartment number. They anticipated some students would count and others would divide by six. However, there was minimal talk about what strategies students might use for solving the second and more difficult problem, finding the floor level when given the apartment number. There was no discussion about how to bring out the strategies and how to decide what order to call on the students during the lesson. Revising lesson 1 and planning lesson 2. After observing lesson 1 and collecting data about student work, the teachers had a better idea about the ways students approached the problems. The first author supplied summarizing data on the students’ strategies as taken from student notes and journal writing. As the teachers perused the students’ work and the summarizing tables they anticipated the strategies grade 7 students might use and planned the discussions. At first Ms Hayes was doubtful about the process: … all the kids are different, you don’t know how they’re going to respond. Because I’m thinking, the grid would be great to have after you go through the counting process, and you give them a couple examples so they feel confident, …and then I would give them the grid, so they would have something visually in hand to see, concretely work with, and see what it stimulates, or what kind of discussion as you said, …and maybe one of these methods will come out, but it’s just very difficult to anticipate because the kids are so different.18 AERA2005 Betsy King and Aki Murata 21 Yet later in the discussion she is interested in analyzing the methods students used and joined in planning the order of strategies: Mr. Rodgers: So what order might they, what makes the most sense to present them?… Counting looks like the most accessible, right? Don’t we think, is that developmentally sort of Mr. Ray: And everyone could follow that in a discussion. King: Okay, so you think counting first. (PA announcement; teachers talked off-task for a minute) …But anyway, we’re going to start off with counting first. So counting would be first, what would be the next one? Developmentally or in terms of --? Mr. Ray: Common, in terms of how many you’re going to see maybe Ms Hayes: I’d say multiply by six. King: Yeah, if you look at the number of strategies here, the multiply by six, the M-S, multiply then subtract, has ten total kids that did that. Ms Hayes: um-hm Mr. Ray: So we’re going to anticipate seeing Mr. Rodgers: So what else would we present next? Ms Hayes: King: (teachers perusing the student work and student strategy table) Mr. Rodgers: …But see, it says multiply six by the building number, they’ve already figured out the building number somehow. King: But they’ve shown on the paper Mr. Ray: There’s fifteen, aren’t there? King: See, like this one, they show on the paper the multiplication. And then they subtract it. Mr. Rodgers: But the question is, where did they get this, where did they get the eight from? King: Well, I don’t know, maybe, (looking at the papers) well here, they drew the buildings, they drew more. Mr. Ray: …Are we talking M-S and M-A? Mr. Rodgers: We were talking about M-S but now we’re looking at D-R. Ms Hayes: D-R Mr. Ray: Oh, D-R. Mr. Rodgers: … The fact that so many kids did it wrong, should we leave that one till the end, so it doesn’t - is there going to be, obviously Mr. Ray: It’s the most sophisticated one, right? Mr. Rodgers: Lets say that…that my kids do mirror this, um, the fact that so many kids did it incorrectly - first did it and then did it incorrectly - would that imply, suggest that we should present that one last? Mr. Ray: I think it should be last. Mr. Rodgers: so it doesn’t confuse things? Because there’s obviously a lot of confusion around that one. Ms Hayes: um-hm. Mr. Ray: It’s also the one that points to the formula, right? I mean, it’s the one you’re looking for. AERA2005 Betsy King and Aki Murata 22 Mr. Rodgers: This is the one that you don’t need to know the building number or the nearest multiple, it takes the broad data, divides it by six, do this with the answer. (one minute later) Mr. Linn: … the thing I like about it is that if so many people did it, had the idea to do it like that, then the concept is there, it’s the procedural stuff and the detail stuff that gets them confused. During that discussion teachers developed taken-as-shared19 understanding and language about the different strategies and could recognize, name, and talk about them. This familiarity with the strategies brought more focus to lesson 2. Over the course of that lesson teachers began to identify the strategies they observed students doing and started to address what that might mean for that student’s understanding. (see figure 2) Fig. 2. Student Strategies for Gauss’ Houses lessons Student Strategies – “Gauss’ Houses” (10.30.03) abbreviation CT# CT-bf M-S M6 M-A N-M D-R D6 Strategy type Counting to find apartment # Counting to find own building & floor Multiply 6 times blg#, then subtract Multiply by 6 only; S’s apt# is a multiple of 6 Multiply 6 times blg#, then add Find nearest multiple Divide apt# by 6, then +1=blg#; remainder is floor# Divide apt# by 6; S’s apt# is a multiple of 6 N=30 1 Incorrectly calculated or inverse 0 3 1 1 2 0 6 [didn’t +1] 0 4 -2 -- -3 -1 Correctly calculated 12 4 9 2 3 4 6 Drew representations ExH ExH-I ChG ChG-I Extended houses; wrote in numbers Extended houses; wrote in numbers inversed Chart or grid; wrote in numbers Chart or grid; wrote in numbers inversed Post-lesson discussion of lesson 2. After lesson 2, Mr. Rodgers talked with Mr. Doyle who observed the lesson but could not attend the post-lesson discussion. Mr. Rodgers shared Mr. Doyle’s observations of the lesson later at the Post-lesson discussion: I saw [Mr. Doyle] afterwards, and he seemed to think that there was a whole different focus, that it was much more focused on discussing strategies for finding your own number, where it is in the building, then the previous iteration of this, that this was more on the general form that we never really got to.20 AERA2005 Betsy King and Aki Murata 23 Planning lesson 3. The teachers were now using a common vocabulary when discussing the strategies. As he reviewed the plan for lesson 3, Mr. Linn said, “I was thinking of circulating around and watching for kids’ strategies…I’ll try to keep my ear out to what strategies the kids are using, and then decide on the fly who’s going present their strategy first, from the simplest to the more complex explanation.”21 The lesson plan articulated that the teacher “will search for students who have answered correctly and ask that they share their strategies…from simpler to more complex. [The teacher] will record strategies in word form on a white board." Teachers also planned to collect data using recording sheets for student strategies. Lesson 3. The third research lesson was taught over two days and some students explained their strategies and solutions both days. The second day Mr. Linn started the lesson by calling on two students to demonstrate the strategies they had used the previous day. This introduction identified the strategies for the teachers gave a review for the students. Post-lesson discussion lesson 3. Because of their familiarity with the strategies, teachers could easily record which strategy each student was using. Mr. Rodgers reported, “As I was walking around I was…looking at the representations that were being used, and I haven't looked at the chart to see if this verifies what I was seeing, but I got a lot of 3’s everywhere, and said “3” was the one where students would make a representation, basically the chart. I hardly saw anyone making pictures of the apartment.”22 Teachers were also able to observe when students progressed from one strategy to another, thereby making their students’ learning visible. Mr. Linn was able to describe how one of his students moved to a more sophisticated strategy on the first day of lesson 3. Planning a lesson that allows for different strategies and knowledge about those strategies allows teachers to report their success with differentiated instruction: What we talked about yesterday in our staff meeting was this whole differentiated discussion kind of a deal about kids entering the problem at their own level. And Mrs. Mann [the special education teacher who assists in Mr. Linn’s class] pointed out that with this problem, there were many kids, you know, working on it, as was discussed, but doing it in their own way. And so some kids were using the counting strategy and some kids were doing a counting on strategy, and some were doing multiples, and some were doing formulas and multiplication.23 End of Year. Some of the teachers had become comfortable using the student strategies to guide the lesson and were working on helping students see the connections between the strategies. During the AERA2005 Betsy King and Aki Murata 24 end-of-the-year interview, Mr. Linn shared how he had been trying to make the connections between the students’ strategies: … I've been cognizant of trying to draw the threads between the solutions…the connections from the pictorial representation to the table representation. Again, it's a higher order of thinking, it's a higher level of abstraction, to go from the pictures of caterpillars and leaves to just the number of caterpillars and leaves….And then going from the table to the numbers, again, is another level of abstraction. And I think a goal of middle-school math is that kids leave eighth grade being able to function in abstract representations of situations, mathematically speaking….And I think that sharing solutions and having kids talk with each other and be the main talkers of whole-class discussions – that's the way that you get kids to move from concrete operations towards formal operations.24 Summary. At the beginning of the year, most teachers were not certain about how to guide a discussion with students presenting their strategies and solutions to math problems. They were concerned about how to decide which students to call on and in what order. During the lesson study meetings the teachers discussed and encouraged each other in a strategy for orchestrating the discussion. Taking time to work on solving the problems, anticipate which strategies students might use, and identify them in students’ work familiarized the teachers with the strategies. This knowledge allowed teachers to quickly recognize which method students used and better gage the conceptual levels of the student and possible misunderstandings. As some of the teachers became comfortable with this teaching practice, they modified it and had other students explain the strategies. Time and pacing of the lesson and pressure to cover the curriculum. Teachers also recognized they are putting pressure on their students and rushing them through their lessons and through the curriculum, not allowing enough time for many students to construct meaning and develop mathematical understanding. Teachers also are feeling rushed. Beginning with the first meetings in the fall, teachers often commented about not having enough time for both getting through the lesson content in the 45-minute class period and the curriculum they needed to cover in the school year. When discussing ways to bring student discourse into their lessons, Ms Hayes said, “If I take out time to do mathematical conversations there will be some standards I won’t have time to teach the students this year.”25 She saw student mathematical conversations as adding on one more thing to teach, and not as a teaching practice that may facilitate student learning. AERA2005 Betsy King and Aki Murata 25 Planning lesson 1. As the teachers began planning lesson 1 the question emerged of giving students enough time to explore mathematical concepts in a lesson. The teachers wondered whether they would be able to complete a lesson in the 45-minute period if they were to give enough time for students to explore, share strategies in a thoughtful discussion, and write their thinking in a journal. Some of the teachers said they were expecting too much for one period. After perusing the first draft of lesson 1 two teachers told Mr. Doyle they didn’t think there was enough time in the lesson to allow for enough exploration and discussion and he would have to direct the discussion too much. Mr. Doyle took out one of the questions because he “wanted the students to wrestle with the problem, come back together, put some ideas up, and then take those ideas and discuss them until everybody has an understanding of them.”26 Lesson 1. Even so, lesson 1 went over the planned time, and instead of continuing the lesson the next day Mr. Doyle consolidated the ending to fit into one period. He felt the pressure to cover the curriculum. Post-lesson discussion lesson 1. Mr. Doyle said “this [lesson], itself, is taking a day away from what I need to do, which is be a unit ahead of where I am right now to get through the book by June.”27 The teachers noticed the lesson was not completed as planned and talked about their belief that lessons that include mathematical conversations, student exploration and develop meaning need more time. Lesson 2. Despite the teachers’ discussions about making sure Lesson 2 allowed enough time at the end for student discussion to develop meaning, it also ran out of time. Mr. Rodgers did follow up the lesson the next day, but said he only spent ten minutes for student discussion and did not have students write a final reflection. Planning lesson 3. Throughout the first semester teachers had been talking about the importance of giving students enough time to discover and investigate mathematical concepts and make meaning. Yet the teachers’ comments during the first meeting after the winter break illustrated their ambivalence. Two teachers used the term “play” when referring to students constructing their own meaning. Mr. Doyle said he wondered about finding an “equivalent alternative to letting students play with things and come to an understanding on their own.” Mr. Rodgers said that the teachers in AERA2005 Betsy King and Aki Murata 26 their school do not have the “luxury of a lot of time for [students] to play around.” Instead he said teachers have “to condense and keep pushing it more.” The teachers appeared to believe students could learn just as well if teachers could only find the right questions to ask or just push the students more. For lesson 3 the teachers talked about the possibility of the sixth grade teacher doing the lesson during his 90-minute period. They now wanted to see what would happen if the students had enough time to explore and discuss the problems in the lesson. Mr. Linn decided to extend the lesson over two days though the teachers could only observe the second day of the lesson. Lesson 3. The lesson was taught over a 2-day period. Students had multiple opportunities to explore and work together in small groups. At the beginning of the lesson on the second day, there was time for Mr. Linn to read to the class some of the students’ journal writings about what they had learned the day before and for students to share strategies used the first day. Both of these activities gave students a grounding on which to build their understanding the second day. Post-lesson discussion lesson 3. At the post-lesson discussion of lesson 3 the teachers indicated satisfaction with the additional time given for the lesson. They commented on being impressed with the amount of learning that they observed. End of the year. In his last interview Mr. Doyle stated he looks forward to the day when his students can “work on two problems thoroughly, or even one problem thoroughly, rather than having to do the ten problems that are assigned.”28 Teachers wanted to slow down the pace yet continued to experience the constant tension between trying move quickly to “cover the curriculum” or taking more time and reduce the number of topics to learn. Mr. Ray reflected on this dilemma in his last interview: [I think] in our effort to get through as much material as we possibly can, we just teach it really fast. And you know that your top 10% are going to get it, do well on the tests, and keep your test scores high enough that you won't get penalized. That's the strategy that I'm sure most teachers use, which is a really bad teaching strategy, but it's supported, condoned and encouraged by the educational structure that we've set up for ourselves.29 Mr. Ray was the only teacher that voiced that sentiment in an interview, though other teachers may have had similar feelings. In a casual conversation Ms Hayes told me she had to keep pushing ahead for the more capable students because of complaining parents. AERA2005 Betsy King and Aki Murata 27 Summary. Teachers recognized they have been rushing their students through lessons and through the curriculum, not allowing the time many students need to construct meaning and develop deep understanding of the mathematical concepts being addressed in the lesson. Even the teachers felt pressured and rushed. At the end of the year, most of the teachers still relented to pressures of “covering the standards” and tried to move through the lessons quickly, telling the students what to memorize for the high-stakes tests in the spring.. Those teachers made a concerted effort to take more time for student exploration and discussion by assigning fewer problems and not try to complete the expected material. Summary and Implications Over a six-month period the five middle school mathematics teachers of the lesson study group planned, taught, observed, and revised three research lessons, in which they anticipated, recorded, and discussed students’ responses and solution strategies. The teachers’ lesson study guiding principle, to promote student mathematical conversations, gave direction to their lesson planning and discussions. At the beginning of the lesson study process, the teachers’ thinking (knowledge, beliefs, goals) about student mathematical conversation was undeveloped. Many aspects of mathematical discourse had not been considered. From the beginning they were unclear about what type of discourse they meant when they used the word “conversation.” When planning the lesson they made little distinction between how the teacher would orchestrate the different types of conversations (e.g., between students as a whole class, in a small group, or in a dyad). Their comments implied a belief that if students were given a “rich” problem to solve there would be “rich” student mathematical conversations and student “engagement” which would yield “success.” If a student had an idea, s/he should be able to explain, demonstrate, or somehow communicate the idea to the class. Yet, there was some trepidation about how to orchestrate the SMC and they had questions about how to make all those on-the-spot decisions during the lesson. At that time the teachers’ focus was mainly on “how to do it ?” Their questions centered largely on themselves, thinking about what teachers should do during SMC. The lesson study planning and revising meetings functioned as a venue for raising concerns such as these. The teachers in the study had developed a strong collaborative relationship and were open and willing to suggest, challenge, listen to, and reflect on individual teacher’s ideas and questions. They often had different perspectives but were respectful and willing to try ideas that were AERA2005 Betsy King and Aki Murata 28 different from their own. Before lesson 1 there were differing perspectives on scaffolding and the use of representations such as tables. Yet the teachers came to consensus and lesson 1 was taught without the support of a table or worksheet. Mr. Rodgers, a teacher with a teacher-directed style, was unsure about how successful the research lesson would be if he taught it. Yet, he was willing to teach lesson 2, even in front of observers and video cameras. The lesson study meetings were a place where Mr. Rodgers could reveal his uncertainties and receive support and suggestions. Because Mr. Rodgers and the others talked through the intricacies of the students’ strategies and the mathematics in the lesson, the lesson study meetings were educative for all of the teachers. The research lessons provided the teachers multiple opportunities to observe actual lessons and study students’ mathematical conversations up close. The laboratory-like situation allowed the teachers to gain insight into “research-based practice.” They took a big idea from research, student mathematics discourse supports mathematical reasoning, and studied it in the classroom. Their first questions and observations of lesson 1 focused on whether the students talked about mathematics, socialized, or if they worked alone. They reported that they were pleased so many students had “rich mathematical conversations,” without writing down what those conversations were or indicating what qualifies as “rich.” Two teachers noticed some students were not following others’ explanations and two students had lost the attention of the class. After they gathered their data about SMC and other issues of the lesson, the teachers participated in the post-lesson discussion. Here again a venue was provided for the teachers to share their thinking and suggest, challenge, listen to, and reflect on each other’s ideas and questions. The difference between this meeting and the regular planning meetings was its structure, the outside teachers, administrators, and researchers in attendance, the focus on what occurred during the lesson, and the data collected. The planning team has the added opportunity of hearing suggestions, insights, and issues raised by outside educators and knowledgeable others. For example, the idea of finding a representation that would facilitate student communication during the lesson was raised by an outside researcher at the post-lesson discussion after lesson 1. In the fall, the teachers were asking “how to do it?” questions, and by spring they asked “what’s happening here?” questions. The focus had shifted from thinking about what the teacher does to what the students do. Mr. Doyle said he had three questions in mind as he observed lesson 3: “how are students engaging in mathematical conversations?” “How are successful students or the students who are getting it helping and asking questions and pushing struggling students?” and “what evidence and proof are they utilizing in their arguments?” Teachers noticed the interaction between the students AERA2005 Betsy King and Aki Murata 29 and the quality and contents of the SMC. The teachers also discussed the events and class dynamics leading up to the decisions Mr. Linn made during the lesson. One teacher’s comments about Mr. Linn making the “right calls” came after their deliberation of the complex classroom events making the discussion more research-like than if he were just to say, “Mr. Linn taught a good lesson.” The experience of observing lessons and comparing those observations helped the teachers over time to develop a more analytical and reflective stance. Mr. Doyle said that while watching lesson 3 unfold he saw they needed to make a distinction between the whole class discussion, teacher-directed discussion, and mathematical conversations. He thought the difference might have to do with how controlled it is by the teacher. He wondered, “if there was something specific [the teachers] wanted to come out, would that be considered a mathematical conversation or not?” At the beginning of the year Mr. Doyle felt he could define and describe SMC, after lesson 3 he had many questions about SMC and he appreciated their complexities. The five phases of lesson study each contributed differently to alleviate concerns about facilitating student discussions. Each element played its own part in deepening and developing the teachers’ understanding of SMC and in nudging the shifts in the teachers’ thinking about student discourse. Teachers started the year with limited knowledge about student discourse. Observations of students having conversations during three lessons deepened their understanding and perspectives about student discourse and gave teachers a shared experience that they referred to in later discussions. The extensive discussions during the lesson study meetings were about what happens in real classrooms with real students. The meetings gave them a venue for sharing their observations and new ideas and wrestling with what students conversations are and could be. Teachers thinking about and discussing student discourse on a regular basis at lesson study meetings kept those ideas in their minds encouraging teachers to try them out in the classroom. The teachers learned how to construct a lesson and provide important tools so that each student could access the mathematical situation, find and use a strategy that makes sense, and then share and discuss them with the class. As Mr. Doyle learned, "it’s not enough to just put out an interesting problem to get students to have mathematical conversations…. And just saying, ‘ok, solve this problem and talk about it while you’re solving it,’ isn’t enough.”30 AERA2005 Betsy King and Aki Murata 30 Table 1 Washington Middle School Student Data School Enroll AfricanAmerican American Indian or not Hispanic Alaska Native Washingt on MS District 578 250 (43%) 3 (0.5%) 8,843 2,773 (31%) 23 (0.3%) Multiple or no response Spec ed English learners 84 (15%) 1,328 (15% ) 77 1,039 43 ( 7%) 1,204 (14%) STAR - CST Student Tested 2003-04 Grade 6 Grade 7 8th grade general math (6th & 7th standards) Algebra I STAR CST 2002-03 Grade 6 Grade 7 8th grade general math (6th & 7th standards) Algebra I Asian Filipino 42 (7% ) 647 (7 %) 1 (<1%) Fluent-Englishproficient students 64 ( 11%) 861 ( 10%) Hispanic Pacific or Latino Islander 61 (11%) 1,407 (16%) 53 (<1% ) Pupilteacher ratio 23.1 19.4 Avg. class size 30 29 % Advanced % Proficient % Basic 156 179 46 Mean Scale Score 332 318 272 13% 6% 0% 25% 17% 0% 131 69% 317 4% 21% % of Enrollment Mean Scale Score 22 (<1 %) Free & reduced price meals 392 (68%) 4,416 (50%) % of Enrollm ent 95% 95% 24% Students Tested 2 (<1%) White not Hispanic 135 (23% ) 2,590 (29 %) Cal WORKS (formerly AFDC) 46 (8%) 574 (7%) 25% 39% 17% % Below Basic 32%% 26% 50% % Far Below Basic 6% 12% 33% 28% 38% 8% % Advanced % Proficient % Basic % Below Basic % Far Below Basic 185 174 49 94% 90% 23% 329 322 264 7% 6% 0% 26% 22% 0% 30% 34% 8% 31% 22% 57% 6% 16% 35% 149 69% 308 5% 16% 28% 40% 12% gender Subjects taught Table 3 Teacher profiles name Ethnic group Previous lesson study experience (in years) Research Lesson taught Traveled to Japan Summer Lesson study workshop Mr. Doyle Mr. Rodgers white white male male Ms Hayes African-American female Mr. Linn Asian male Mr. Ray white male grade 8 algebra grade 7 & 8 pre-algebra & algebra 1a grade 7 prealgebra grade 6 math & science self-contained grade 6 class AERA2005 Betsy King and Aki Murata 31 Japan -Japan -- 2 2 Lesson 1 Lesson 2 1 -- -- -- 1 Lesson 3 -- workshop 1 -- Japan -- AERA2005 Betsy King and Aki Murata 32 References Ball, D. 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Unpublished doctoral dissertation, University of Chicago: IL. 1 see online database of lesson study groups at www.tc.columbia.edu/lessonstudy/lsgroups.html Source of Washington Middle School student demographic data is the California Department of Education website: http://www.cde.ca.gov/ds/ 3 knowledgeable other is a term used in lesson study for a person who is a resource for the math content of the lesson. 4 Lesson study meeting, 9.11.03, p. 6-7 5 Lesson study meeting, 9.15.03, p. 7 6 Lesson 1 post-lesson discussion, Oct 30, p 5-6 7 Post-lesson discussion, 12.10.03, p. 8 8 Lesson study meeting 1.22.04, p. 8 9 Lesson study meeting 2.5.04, p. 6-7 10 Lesson study meeting 2.5.04, p. 4-5 11 Post-lesson discussion, 2.5.04, p.7 12 Interview with Mr. Rodgers, 6.18.04 13 Lesson study meeting, Oct 20, 2003, p. ? 14 Post-lesson discussion, 2.26.04, 15 Post-lesson discussion, 2.26.04, p. 3 16 Post-lesson discussion, 2.26.04, p. 13 17 Lesson study meeting, 11.24.03, p. 10 18 Lesson study meeting, 11.24.04 p 12 19 The term “taken-as shared” mathematical meanings are constructed by a community through a social process and interaction. These cultural agreements and representations facilitate communication by that community. (Cobb & Bauersfeld, 1995) 20 Post-lesson discussion, 12.10.03, p. 8 21 Lesson study meeting, 2.23.04, p. 1 22 Post-lesson discussion, 2.26.04, p. 6-7 23 Post-lesson discussion, 2.26.04, p. 15 24 Interview with Mr. Linn, 6.17.04, p. 6-7 25 Lesson study meeting, 26 Lesson study meeting, Oct 23, 2003, p. ? 27 Post-lesson discussion, Oct 30, 2003, p. ? 28 Interview with Mr. Doyle, Jun 17, 2004, p. ? 29 Interview with Mr. Ray, Jun 17, 2004, p. ? 30 Lesson study meeting, 2.5.04, p. 6-7 2