Learning About Student Mathematical Discourse:

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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,
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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,
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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:
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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:
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•
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
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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
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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
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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
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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
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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
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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
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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,
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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.
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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.
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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
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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’
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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.
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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
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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
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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
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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.
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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
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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
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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.
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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
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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.
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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
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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
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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
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