A MAtheMAtics stAff DevelopMent course - Heinemann

Newton, MA

Portsmouth, NHheinemann

Lesson study in Prac tice

A M At h e M At i c s s t A f f D e v e l o p M e n t c o u r s e

Jane GormanJune MarkJohannah Nikula

Education Development Center, Inc.

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“Dedicated to Teachers” is a trademark of Greenwood Publishing Group, Inc.

This work is supported by the National Science Foundation under Grant Numbers ESI-0554527 and ESI-0138814. Any opinions, findings, conclusions, or recommendations expressed here are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Library of Congress Cataloging-in-Publication DataGorman, Jane. Lesson study in practice : a mathematics staff development course / Jane Gorman, June Mark, Johannah Nikula. p. cm. Includes bibliographical references. ISBN-13: 978-0-325-02800-2 (3-ring binder) ISBN-10: 0-325-02800-1 (3-ring binder) 1. Mathematics—Study and teaching. 2. Mathematics—Outlines, syllabi, etc. I. Mark, June. II. Nikula, Johannah. III. Title. QA11.G6748 2010 510.71—dc22 2009046256

Editor: Victoria MereckiProduction: Lynne CostaCover and interior designs: Shawn Girsberger Video producer: David Smith ProductionsDVD developers: Sherry Day and Anthony MoreschiTypesetter: Shawn GirsbergerManufacturing: Valerie Cooper

Printed in the United States of America on acid-free paper14 13 12 11 10 ML 1 2 3 4 5

iii

contents

Acknowledgments v

Part I IntroductIon

Introduction 1Orientation to Lesson Study in Practice: A Mathematics Staff Development

Course 1What Is Lesson Study? 1What Is the Course? 2

Goals 2Course Structure and Design Features 3Outline of the Course Sessions 5

Facilitating the Course 6Overview of the Facilitator Role 6

What Does the Facilitator Do? 6Resources to Support the Facilitator and Participants 8

Session-by-Session Facilitator Notes 8Recording Tools 9Video, Readings, and Other Resources 10

Implementing the Course 12Getting Started 12

Recruiting the People 12Planning the Logistics 14

Start-up Materials 14References 15

Part II SeSSIon FacIlItator noteS and PartIcIPant MaterIalS

SeSSIon 1 Introduction to lesson Study in Mathematics

SeSSIon 2 topic Selection and research

SeSSIon 3 research and task analysis

SeSSIon 4 the lesson design emerges: Problem Selection and envisioning the Flow of the lesson

SeSSIon 5 What Will Students do and think?

iv

SeSSIon 6 Preparing to observe Students

SeSSIon 7 the First lesson observation and discussion

SeSSIon 8 revising and reflecting

SeSSIon 9 the Second lesson observation and discussion

SeSSIon 10 analyzing, Sharing, and Planning

dVd contentS

Participant Handouts and Team Meeting Logs (Sessions 1–10)Case Study Team Materials

Video Clips from Case Study Team (for use in Session 1 and Session 6)Sample Student Work from the Case Study Team Research LessonCase Study Team Research Lesson ReportCase Study Team Mathematical Context Chart

Recording ToolsDraft Research Lesson Report TemplateResearch Lesson Report Template with Callouts: This version includes

notes for the facilitator and/or participants about what goes in each section of the template.

Mathematical Context Chart Blank Team Meeting Log Template (Session-specific team meeting logs

are included with the handouts, but this template can be used for recording notes at future team lesson study meetings.)

Course Readings “A Lesson Is Like a Swiftly Flowing River”Selection from TIMSS Frameworks

Start-up MaterialsLetter to AdministrationLetter to Prospective ParticipantsSuggested Agenda for Informational Recruitment MeetingSession CalendarAttendance/Contact Sheet

PowerPoint PresentationPowerPoint Slides as PDFs (Sessions 1–10)

a c k n o w L e d g m e n t s v

acknowledgmentsThese materials are the result of years working together with many teachers and students to improve mathematics teaching and learning using lesson study. We are particularly grate-ful for the teams of teachers who participated in our Lesson Study Communities in Secondary Mathematics project, which provided the basis for many of the ideas and activities included in Lesson Study in Practice: A Mathematics Staff Development Course. The voices of teachers from these teams are heard throughout the Course in quotes that launch the Course sessions. That project included teachers from the following Massachusetts schools:

Ahern Middle School, Foxborough

Andover High School

Bartlett Middle School, Lowell

Brookline High School

Carver High School

Danvers High School

Dartmouth High School

Everett High School

Lexington High School

MATCH School, Boston

Newton South High School

Norfolk County Agricultural High School, Walpole

Old Rochester Regional, Mattapoisett

Scituate High School and Gates Intermediate School, Scituate

Uxbridge High School and Whitin Middle School, Uxbridge

Wareham Middle and High Schools

Watertown Middle and High Schools

The teachers’ experiences learning about lesson study in their efforts to make their teaching of mathematics more effective for their students were invaluable to our work, and their thought-fulness and willingness to share their learning with others have been a tremendous contribu-tion. These teachers demonstrated through their work a powerful commitment to their col-leagues, to their students, and to their own learning. We are very thankful for the support of school and district administrators in these schools for making teachers’ participation possible. We also want to thank the teams and facilitators from the following districts, who pilot-tested this Course and helped us identify activities and approaches that were working well and those that needed improvement: Baltimore City Schools, MD; King’s Highway School, Westport, CT; Murdoch High School, Winchendon, MA; New York City Public Schools, NY; Watertown Public Schools, MA; West Virginia STEM Center; and Windsor High School, CT.

The actual preparation of the Course materials was helped immensely by the efforts of Euthecia Hancewicz, Hillary Dockser Burns, and Terry Leverich. An advisory group of coaches and teach-ers provided us with invaluable guidance on the development of the Course, piloting materials and providing careful feedback. This group included: Debra Casey, Sue Chlebus, Melanie Kellum, Andrea Plate, Joanne Smith, Liz Van Cleef, and Dan Wulf. Ken Levasseur at the University of Massachusetts, Lowell, provided a critical review of the mathematical content of the Course and we are appreciative of his feedback.

Throughout our work in lesson study, we have benefited from the support of our advisors Thomas Judson, Catherine Lewis, Deborah Schifter, Akihiko Takahashi, Phyllis Tam, and Tad Watanabe. We also wish to recognize the powerful ideas contributed by Brian Lord, who led our

a c k n o w L e d g m e n t svi

project research. These advisors have taught us about lesson study, teaching, mathematics, and colleagueship. The knowledge they so generously shared with us is embedded in every page of these materials. We also wish to thank our evaluator, Joan Karp at the Program and Evaluation Group at Lesley University, for providing valuable feedback and advice. We gratefully acknowl-edge the support of the National Science Foundation for funding the projects (ESI 0138814 and 0554527) that made this work possible.

i n t r o d u c t i o n 1

and Share LearningRe�ect, Consolidate,

Teach, Observe, and Discuss the Research Lesson

Develop the

Research Lesson

Research the TopicFocus, Set Goals, and

introduction

orientation to Lesson Study in Practice: A Mathematics Staff Development Course

What Is Lesson Study?Lesson study is a model for teacher research and learning built around collaborative lesson plan-ning and observation. Its primary goals are to improve students’ opportunities to learn and to build teachers’ understanding of content and pedagogy. Introduced to the United States from Japan in the late 1990s and widely publicized by Stigler and Hiebert (1999) in The Teaching Gap, lesson study attracted immediate attention among mathematics educators for its potential to develop and sustain changes in teaching practice and culture. Using lesson study, teachers care-fully consider how their teaching can best support student learning, deepen their understand-ing of mathematics and how students learn it, use evidence of student thinking to inform the design of instructional lessons, and build strong collegial connections. Lesson study, through a cycle of inquiry, provides a structure for establishing a school-based professional learning com-munity centered on teacher-driven research and learning.

In one lesson study cycle, a team of teachers meets regularly at their school to plan a detailed “research lesson” on a research theme or goal the team selects. The first phase of the lesson study cycle begins with goal setting and identifying a research focus. Broad goals such as encouraging students to develop multiple strategies for solving mathematics problems are chosen to address stu-dents’ greatest needs. In a second phase of the cycle, the focus shifts to developing and planning the research lesson, which embodies the team’s ideas for improving student understanding of mathematics. The third phase centers on teaching, observation, and discussion of the lesson. One team member teaches the team-developed research lesson while the rest of the team observes. The team and guests share their observation data at a post-lesson meeting that is focused on dis-cussions of how students were thinking and learning during the lesson. A revised lesson is then prepared and taught again for observation and discussion. In some cases, the team chooses to invite other teachers from their school or teachers from other schools to attend the research lesson observation and post-lesson discussion—cre-ating a rich learning opportunity for all involved. The lesson study cycle culminates in a fourth phase that includes time for reflection and analysis of the team’s learning, and completion of a written report containing the lesson plan, observation notes, and a summary of that learning (see Figure I–1).

Each lesson study cycle is one link in a larger chain of research lessons. Over time, this builds knowl-edge for the individual teachers, and contributes to the profession more broadly—as gradual improvement of instructional practice, better curriculum, and profes-sional research. Figure i–1

i n t r o d u c t i o n2

What Is the Course? The Lesson Study in Practice: A Mathematics Staff Development Course materials (referred to as the Course) grew out of our work with hundreds of teachers as they learned about using lesson study to improve the teaching and learning of mathematics.1 In that work, we learned that most new, and even experienced, teams and coaches face the same dilemma: Lesson study can incorporate an astounding array of activities. It is easy to get bogged down, lose track of what matters most, or fail to ask the hard questions. Although some teams are successful work-ing independently, most would benefit from the support of a guidebook, concrete images of lesson study practice, and a coach or facilitator. Underlying this support, there needs to be a vision of good lesson study practice, detailed descriptions of this practice, and structured tools that help teachers maintain momentum and mathematical focus. We, at Education Development Center, Inc. (EDC), have developed this Course and the accompanying resource, A Mathematics Leader’s Guide to Lesson Study in Practice (henceforth called the Leader’s Guide) to provide teams with just this kind of support. We envision that the Course and Leader’s Guide will be of use to beginners, and also provide teams with long-term support for building high-quality lesson study and for supporting teachers as they grow into coaching roles.

The Course is intended to help participants learn about what lesson study is and how they can use it to improve mathematics teaching and learning. The focus on mathematics and student thinking in the Course encourages teams to establish lesson study practices that build math-ematical knowledge for teaching. The Course takes the form of a guided first cycle of lesson study, and so in addition to learning about lesson study, each team of participating teachers will develop and teach one research lesson and complete a study of its mathematical content.

Drawing primarily on our work with middle and high school teams, the Course provides a much-needed resource for lesson study in secondary mathematics. Field-testing has demonstrated, however, that it is also fully accessible and appropriate for elementary school teams. Sessions are designed for use by single or multiple teams within a school district setting, but coaches, preser-vice teachers, university faculty, and professional developers will find the materials amenable for use in other settings where a structured introduction to lesson study is desired. We expect that teachers will participate in the Course in teams of three to six teachers, with the assistance of a facilitator (e.g., a teacher leader, coach, mathematics specialist, teacher, or university faculty).

GoalsThe Course goes well beyond a basic how-to guide. It highlights the underlying principles of les-son study so that participants understand the reasons why studying and testing one lesson can impact other lessons, and teaching practice more generally. The Course goals support teachers, individually and as a team, in:

� Building an understanding of lesson study and an interest in continuing it. In this Course, teach-ers learn about lesson study, including its basic structure, processes, and larger purposes. Teachers experience the phases of a lesson study cycle, and develop an understanding of how that cycle is part of an ongoing process of learning that leads to improvement in instruction and in their opportunities to develop as professionals.

1 Teachers participated in the Lesson Study Communities Project in Secondary Mathematics project, NSF Project ESI 0138814, and the Resources for Supporting Lesson Study in Mathematics project, NSF Project ESI 0554527. Both are projects at Education Development Center, Inc. (EDC), Newton, MA. Project information is available at www.edc.org/lessonstudy. Teachers and coaches from these project are quoted with permission throughout the Leader’s Guide.


� Building a professional learning community. By pooling their expertise about student under-standing and the difficulties students have learning mathematical concepts, teachers form a professional community focused on the improvement of instruction and learning. This Course also fosters a culture of collaboration and the building of a mathematical commu-nity needed to sustain ongoing learning and reflection through lesson study.

� Building an understanding of mathematical knowledge for teaching. Researchers and educators discuss “mathematical knowledge for teaching . . . teachers’ ability to understand and use subject matter knowledge to carry out the tasks of teaching” (Hill, Rowan, and Ball 2005, 372). In this Course, teachers study content with colleagues and experts; examine curricu-lum standards, instructional materials, and exemplary lessons; and use post-lesson discus-sions to dissect the intended and enacted mathematics in lessons.

� Building an understanding of how students think about and learn mathematics. Teachers need a strong mathematics content background themselves, but they also need to understand how students come to develop mathematical ideas and concepts. From setting goals for stu-dent learning to careful observation of research lessons, teachers in lesson study open up a strong focus on their students’ mathematical understanding and use data about student learning to inform revisions to lessons and thinking about teaching.

� Building a practice of reflection, analysis, recording, and sharing of ideas. In each lesson study cycle, teachers test new ideas for promoting student learning of mathematics through the design of a research lesson. In the Course, teachers share ideas about improving student learning, collect data to inform their hypotheses about learning processes, and reflect on instructional decisions through discussion with colleagues.

Course Structure and Design FeaturesThe Course is designed for use by one or more teams of three to six teachers working with a facilitator, and includes participant materials and facilitator notes for ten sessions. Course activ-ities are sequenced to follow the natural progression of inquiry in the lesson study cycle. The ten sessions total approximately twenty-five hours, including a full-day introduction and nine, two-hour sessions. Participants in the Course conduct research or do preparation between some sessions bringing the total commitment to about thirty hours. This structure is supportable in a variety of settings that are accessible to teachers:

� A one- or two-week summer institute, depending on the length of the day

� A summer workshop or professional release days—to launch the Course in one or two full days and complete remaining sessions in weekly or biweekly meetings

� A one-semester professional learning activity within the school—meeting weekly or biweekly after school, or at in-school meetings during release time, common planning time, department meetings, or team meetings

� A professional development course for university graduate or continuing education credits

The design of the Course activities and materials reflects what we, as coaches, have learned from our teams. We have included the kinds of experiences that our teams appreciated most as they first encountered this new form of collaborative research. We have also tried to give guidance for


those aspects of lesson study that we and our teachers found most perplexing or challenging in our first cycles. The following design features are the result.

Learning by doing. We believe that the best way to really learn about lesson study is to actually experience lesson study. Therefore, this Course is designed as a guided first cycle; participat-ing in the cycle of lesson study is embedded in the Course experience. Each Course session includes activities focused on learning about lesson study as well as Team Meetings, in which teams engage in doing lesson study: developing a research lesson, analyzing the cognitive demand of tasks, observing the lesson.

Learning from the experience of others. The experiences of other team members appear through-out the Course facilitator notes and activities. Course materials also include a Case Study: an extensive set of work samples from one high school algebra team (the Case Study team). Course participants will follow the Case Study team’s development of a research lesson about linear and exponential growth. At each new step in the lesson study cycle, Course participants will see how the Case Study team approached their work, and begin to build concrete images of lesson study processes, products, and challenges through study of team artifacts, including video.

Focusing on mathematical thinking and learning. The Course is designed to foster the investiga-tion of students’ mathematical thinking and learning as integral to the lesson study process. One of the best ways to focus on student thinking is for teachers to engage in and reflect on their own mathematical thinking, so teachers doing and learning mathematics together is a central activity in the Course. We intend for the Course to help build a collegial mathematical learning community among the team of teachers participating in the Course.

Focusing on the sharing of ideas about teaching. Even though lesson study does include a sequence of steps for teams to follow, we found that teachers appreciate the opportunity to discuss and debate ideas about teaching. Consequently, every Course session includes Guiding Questions that point to the central content of the session and encourage teachers to engage in these important discussions. For example, after the team has selected a topic for their lesson, the questions include, “What do we want our students to understand about this concept?” and “What does it mean to understand this concept?” After choosing the main problems for the lesson, teachers will discuss, “How do we as teachers approach this problem mathematically?” and then “What are the likely student methods and misconceptions?”

Using lesson study as lifelong learning. We envision the Course as the beginning of a team’s lesson study experience. The Course aims to convey the rich potential of sustaining lesson study as part of one’s professional practice and to prepare new teams to continue in this work independently. Facilitator notes and activities are designed to address challenges typically faced by new teams and also to provide transition support for future cycles. Course Session 10 materials include reflection opportunities and tools to guide the team in planning its next cycle. Furthermore, the Leader’s Guide that accompanies this Course (A Mathematics Leader’s Guide to Lesson Study in Practice) is expressly designed as support for deepening team learn-ing and building a sustainable lesson study practice.


Outline of the Course SessionsP h a s e 1 F o c u s , s e t G o a l s , a n d R e s e a R c h t h e to P i c

Full-day IntroSession 1(6 hours plus lunch)

An introduction to the Course; learn about lesson study; begin working with the team to develop a mathematical learning community; do mathematics together; discuss team’s goals; begin selection of mathematical topic for research lesson

Session 2(2 hours)

Select mathematics topic for research lesson; learn about research in lesson study; share expertise about topic; discuss “What do we want students to understand about this topic?”

Session 3(2 hours)

Research topic; create learning path for chosen topic based on that research; consider how tasks require mathematical thinking

P h a s e 2 d e v e lo P t h e R e s e a R c h l e s s o n

Session 4 (2 hours)

Explore possible mathematics problems for chosen topic; select problem(s) for research lesson

Session 5(2 hours)

Develop research lesson flow; anticipate student responses

Session 6 (2 hours)

Learn about observation and post-lesson discussions; prepare for first teaching of lesson, observation, and post-lesson discussion

P h a s e 3 t e ac h , o b s e R v e , a n d d i s c u s s R e s e a R c h l e s s o n

Session 7 (1.5 hours plus 1 hour observing the lesson)

First teaching of research lesson; post-lesson discussion focused on sharing observation data, learning about students’ thinking about mathematics, and team goals

Session 8 (1.5 hours)

Discuss revisions to research lesson and rationale for revisions; prepare for second teaching; begin consolidating learning about research hypotheses

Session 9(1.5 hours plus 1 hour observing the lesson)

Second teaching of research lesson; post-lesson discussion focused on sharing observation data, learning about students’ thinking about mathematics, and team goals

P h a s e 4 R e F l e c t, co n s o l i d at e , a n d s h a R e l e a R n i n G

Session 10(2 hours)

Analyze and record learning from first cycle of lesson study; plan for future lesson study work; celebrate!


Facilitating the Course

Overview of the Facilitator Role This Course structure assumes that the facilitator will prepare for and lead session activities, and keep an eye on the overall progress of the team(s) throughout the lesson study cycle. Using the facilitator notes and other support material provided in the Course, the facilitator assumes the role of a coach, guiding the team through their first lesson study cycle. The facilitator may come from a variety of backgrounds: mathematics coach, mathematics specialist, classroom teacher, university professor, and so on. We share our vision of the facilitator’s role in order to help all facilitators prepare to lead the Course, and also to help facilitators think about ways that this role is similar to or different from their other primary roles. We expect that skill in facilitation will develop over time, with experience and increased understanding of the lesson study process.

What Does the Facilitator Do?Facilitates the group. In addition to leading and preparing for each session, the facilitator has practical responsibilities such as maintaining the schedule and calendar, sending reminders, making arrangements for release time, organizing the teachings and post-lesson discussions, and cultivating support from administrators. Good facilitation also involves summarizing the ideas of others, inserting a focusing question, helping the team make decisions, modeling col-laborative norms, and encouraging the team to keep good written records.

Encourages the team and supports broad leadership development. The first cycle of lesson study is new for most people, and may present a few bumps and hurdles. Facilitators may want to emphasize the long-term view to their teams—that the lesson study process grows stronger over many cycles—and be patient as the team tries it for the first time. An important goal is for the team to end the first cycle wanting to continue with another cycle. The facilitator should be supportive of team members as they rotate the Team Leader and Recorder roles at Team Meetings, and encourage team members to talk about their lesson study work with other people in the school who might provide long-term support for their work.

Helps the group learn about lesson study. The facilitator’s role is to be aware of the goals and purposes of the session activities and keep those in mind as the team progresses through the Course. It may be necessary to help the team understand how lesson study is different from reg-ular lesson planning or other professional development. Teachers use research lessons to learn things that will apply to many lessons, to understand student learning, and to develop their own mathematical inquiry and understanding. Facilitators will want to look for ways to increase their own knowledge of lesson study, as well—through reading, attending lesson study events, or forming a coaches’ group. One source for learning about lesson study is the Leader’s Guide that accompanies this Course.

Serves as a resource to the group. Facilitators identify appropriate resources such as curriculum materials, mathematics problems, or research articles or offering a different perspective in a


discussion. This might also mean locating someone who has special expertise on the team’s topic or goals, and inviting them to consult with the team as a knowledgeable other.2

Keeps an eye on the big picture. Most new lesson study teams and coaches find it difficult to stay focused on the big ideas and to deeply explore important mathematical questions when they are just starting out and trying to keep track of the activities and steps in the lesson study process. The Course materials engage teams in activities designed around a set of important themes, described next, and the facilitator plays a major role in making sure that these big ideas3 are kept in focus.

Big Ideas in Lesson Study

� Lesson study as research. Lesson study is a research process. Ideas about teaching and learn-ing are crafted into a research lesson—they become hypotheses about how best to teach the lesson mathematics. These ideas are then tested and refined through the teaching and discussion of the research lesson. The Course materials and facilitator support the team in such things as adopting this research stance by encouraging reflection, keeping goals in mind, and making decisions based on observation data.

� Focus on mathematics. A key role of the facilitator is to keep mathematics present in team discussions. Mathematical learning is embedded in structured Course activities as teach-ers work on problems being considered for the lesson, share different solutions with col-leagues, anticipate student responses to problems, and study textbooks. The facilitator can also encourage discussion of how students learn particular mathematical ideas and of the topic’s broader mathematical context. The lesson observation brings insight into students’ mathematical thinking. The facilitator can help the team consider mathematical reasons for their pedagogical choices and the mathematical impact of these choices.

� Focus on students. In lesson study, we work to improve instruction by deepening our knowl-edge of how students think about and learn mathematics. The Course supports this student focus in its activities: framing team goals in terms of student understanding, unpack-ing what student understanding of the particular topic looks like, anticipating student responses, developing a lesson that encourages students to think, and using data on stu-dents to inform the lesson design. Facilitators help teams use student thinking and under-standing as a guiding force in their work.

� Professional community. The facilitator’s role is to support the team in creating a culture of collaboration and inquiry. Lesson study offers teachers a huge benefit: the opportunity to spend time with colleagues working collaboratively on their teaching. Facilitators try to cre-ate a culture that is safe and encouraging but also has room for critical debate and analysis of teaching and learning. Facilitators encourage development of a community of co-learn-ers who see each other as resources and work together to pursue a research agenda.

2 Knowledgeable other is a lesson study term for an individual from outside the team who shares their special expertise with the team in some way. He or she might attend a meeting, run a mini-workshop on the mathematics, attend the lesson and make special commentary, or simply be available to talk with team members. See also Chapter 16, Incorporating Expertise from Outside the Team in the Leader’s Guide.3 See Chapter 1 of A Mathematics Leader’s Guide to Lesson Study in Practice for additional explanation of these big ideas that drive lesson study.


These big ideas are embedded in the Guiding Questions that appear at the beginning of each Course session. Facilitators should keep these Guiding Questions on hand and raise them at apt moments during team discussions, or perhaps to use them to help remember what the main point of that session is.

Resources to Support the Facilitator and ParticipantsThe Course includes the following resources for facilitators and teams: detailed session-by-session facilitator notes; a collection of handouts, slides, and other resources available on the Course DVD; and a set of recording tools. In addition, facilitators can refer to the essays in the Leader’s Guide and may explore websites and publications in the Video, Readings, and Other Resources sections.

To prepare to lead the Course, the facilitator should begin with some background reading (i.e., this introductory chapter, Chapter 1 of the Leader’s Guide, the article “A Lesson Is Like a Swiftly Flowing River” from the Course DVD) and investigation of lesson study websites and publications. A list of lesson study readings and links is located in the Leader’s Guide Resource Appendix. Talking with colleagues who have done lesson study, or visiting a team or lesson study leader in another school, would also be an excellent way to learn what this process is about and what teachers and leaders have to say about getting started. Next, read the Getting Started section that follows, which gives some practical direction about administrative support, team formation, recruitment, and scheduling. This foundation should clarify what the Course is about, help facilitators think through their course goals, anticipate what the team(s) will be working toward, and set in place the logistics of who, what, when, where, and why. Session-by-session notes will provide the guidance that is needed from there.

Session-by-Session Facilitator NotesFacilitators should prepare for each session by reading all the activity instructions, thinking about the guiding questions, and gathering materials and handouts. At the beginning of each session’s notes, key information about the session is summarized. This summary material will quickly tell explain what will happen in the session and what is most important. It includes:

� A quote that captures a key idea of the session.4

� An overview of the activities, goals, and Guiding Questions frame the session. The over-view contains Guiding Questions that highlight the focus of the session and are a useful resource for facilitators. The overview also notes related readings from A Mathematics Leader’s Guide to Lesson Study in Practice. These readings go into more depth than is possible within the facilitator notes format, and provide examples of team practice.

� An agenda for the session lists the session activities and provides a recommended timeline. A thumbnail description of each of the session activities is included.

� Notes on preparation include information about any handouts, slides, and other materials that need to be gathered or prepared.

4 All unattributed quotes in the Course are from the teachers and coaches who participated on lesson study teams from the EDC NSF projects: Lesson Study Communities in Secondary Mathematics and Resources to Support Lesson Study in Mathematics.


Facilitation notes follow the session-by-session summary; these detail each activity with copies of all handouts participants will use. These activity notes include approximate timing, a list of materials needed, and the following support for leaders:

� Purpose of the activity: The main focus of the team’s work is explained and any expected products or outcomes are pointed out.

� Activity instructions and handouts: Expected timing and step-by-step instructions for all activities are provided. Of course, facilitators may need to adapt activities to meet the spe-cific needs of their team(s). We recommend that facilitators initially follow the substance of the session as given, reflect on the effectiveness of the activity, and note any changes needed for the next time the Course is taught.

• Bulleted notes with shading within the instructions give suggestions for facilitation. These notes include practical tips, suggestions for managing the group’s work together, strategies for encouraging discussion and participation, examples of challenges that teams typically face during their first cycle of lesson study, or things to keep in mind as a facilitator during the activity.

• In multiteam implementations, all teams come together for the facilitator-led activities, and encourage sharing of ideas in the full group.

� Team Meeting instructions and handouts: In every session there is at least one Team Meeting in which the teacher team works on their own research lesson. These meetings are led by the teachers themselves, usually as a rotating job.

• A Team Meeting handout provides the teacher-leader with an outline of the goals and activities of the Team Meeting.

• The facilitator also receives a set of notes about the Team Meeting, with suggestions for playing a supportive role while the teachers provide the primary leadership.

• In multiteam implementations, the facilitator should circulate among teams during the Team Meeting portion of each session, and make time at the end of the meeting for cross-team sharing of ideas.

� Slides: Slides are provided to help guide all sessions.

Recording ToolsThroughout the Course, participants will be collecting information, generating ideas, developing a lesson plan, making decisions, sharing observation data, and having important discussions about what all of this means. Our experience with many teams tells us that good record keep-ing helps the team’s learning, momentum, and general satisfaction with the process. Nothing is more frustrating than revisiting the same decision two or three times, or losing a great idea because it didn’t get written down. The following tools are included to help teams keep good records during the Course, with a minimum of effort. These are all tools that teams can use in their ongoing lesson study practice after the Course ends.

Meeting Log. Use this template to keep basic records of team meetings. At each meeting, a Recorder should enter important decisions, ideas, and learning of the team. If the team takes five minutes at the end of the meeting to think about the next meeting’s agenda, the


Recorder can jot it down on the log. Keeping a log may sound too formal, or the contents may seem trivial. But most teams meet weekly or biweekly and without a log, they might start each meeting having forgotten where they left off. Also, the log provides the team with a record of the development of their thinking, which will be helpful as they end the cycle and reflect on what they have learned.

Draft Research Lesson Report. This is a lesson plan template for recording the team’s research lesson plan as it is developed. It is based on a multicolumn lesson plan format used widely in lesson study in the United States that was based on similar plans used in Japan. It is called a report rather than just a lesson plan because it includes important additional information about the team’s research and about the mathematics of the lesson. By the end of the Course (or the end of any given cycle) the Draft Report contains everything needed for a Final Report except the last section on findings and a bit of polishing. The team’s Final Research Lesson Report is an important contribution of teacher research that can be shared professionally, and can be the basis for future research. Details about when and what to enter into the Draft Report are noted throughout the Course. Participants will also have access on the Course DVD to the complete Case Study Team Research Lesson Report to study as an example.

Mathematical Context Chart. Teams can use this graphic organizer to support the team when thinking about the mathematical content and learning trajectories for the team’s topic. First appearing in Session 2, teachers use the chart to record their initial research about the topic, then revisit it periodically throughout the Course. As they gradually refine their vision of the mathematical context and how students learn about it across the grades, the chart may change, or the team may replace it with a different representation of this mathematical and pedagogical knowledge. The chart can also be shared with colleagues to spark discussion of vertical articulation.

The formats for these recording tools are certainly not unique, and facilitators may want to adapt or revise them based on their own experience and their teams’ needs. The Mathematical Context Chart, for example, could be replaced by a concept map or “knowledge package” (Ma 1999) representation. We do believe that these versions are particularly well crafted to suit teams’ needs in the Course, and would encourage teams to wait until their second cycle before considering revisions or replacements.

Video, Readings, and Other ResourcesThe following additional resources are included with the Course materials. They are intended to play a role in certain Course sessions and to provide general support for leaders and teams in devel-oping broad understanding of lesson study and in providing practical ideas for implementing les-son study in your own setting. Unless noted otherwise, these resources are on the Course DVD.

Case Study Team Materials

Throughout the Course, participants will use the Case Study team and their research lesson to clarify what lesson study products and practices look like.

� Video. By watching the video, which shows the Case Study team at work, teams can visual-ize the nature of lesson study in secondary mathematics, and learn about particular new practices, like the post-lesson discussion.


� The Case Study Team Research Lesson Report. This is the Case Study team’s final research les-son plan with background material and post-lesson findings. Participants may refer to this report at any point in the Course to learn more about the Case Study team’s work or to view as an example of a completed report.

� Case Study Team Mathematical Context Chart. This chart is available as a handout in the Session 2 materials and provides a sample of topic research.

Readings for the Team and Facilitator

The following readings are provided as resources to accompany the Course materials.

� A Mathematics Leader’s Guide to Lesson Study in Practice. This companion book to the Course contains essays on how to develop strong lesson study in mathematics practice as well as stories of real lesson study teams’ work. The chapters follow the phases of the lesson study cycle and each one elaborates on the challenges of deepening a particular aspect of lesson study practice. Reading chapters that relate to particular Course sessions provide facilita-tors with additional insights, examples of team practice, and ideas for coaching the team. The Leader’s Guide will be particularly useful for teams and leaders as they move into their second and continuing cycles.

� “A Lesson Is Like a Swiftly Flowing River” (Lewis and Tsuchida 1998). This article is one of the early publications that introduced lesson study to educators in the United States. Course participants will read and discuss this article in Session 1. It offers a good introduc-tion to lesson study process and philosophy, as well as images of its Japanese origins.

� Selection from the TIMSS 2007 Assessment Frameworks (Mullis, Martin, Ruddock, O’Sullivan, Arora, and Erberber 2005). This reading provides background for the facilitator on mathematics cognitive domains. Course participants will use these domains in Session 3 in reviewing mathematics tasks for the potential to spark student thinking and learning in the team’s research lesson.

Additional Web Resources

There are a number of key websites that facilitators and teams will find particularly rich sources of material and knowledge about lesson study, including lesson video, lesson plans, and read-ings. Here are a few we recommend:

� www.edc.org/lessonstudy This is the website for the EDC Lesson Study Center and contains general resources for lesson study, as well as information and resources from the Lesson Study Communities Project in Secondary Mathematics.

� www.lessonresearch.net This website, from the Mills College Lesson Study Group, con-tains a number of excellent lesson study videos. Two of these (Can Your Lift 100 Kilograms and How Many Seats) are introductions to lesson study, and can be used as alternate videos in Course Session 1. The website also has links, workshop resources, and an extensive set of excellent articles on lesson study. We highly recommend the following:

• “A Deeper Look at Lesson Study” (Lewis et al. 2004). This article discusses the pathways that connect lesson study and improvement in instruction.

• “Lesson Study Comes of Age in North America” (Lewis et al. 2006). This article describes what lesson study looks like in the United States and challenge for the future.


• Lesson Study: A Handbook of Teacher-Led Instructional Change (Lewis 2002). This book gives an overview of lesson study, its origins and philosophy, as well as practical guid-ance for teams and leaders.

� http://hrd.apecwiki.org/index.php/Classroom_Innovations_through_Lesson_Study This website is from the APEC (Asia Pacific Economic Community Human Resource Development Working Group) lesson study initiative. It contains a glossary of lesson study terms, videos of research lessons from many countries (with subtitles), and many research articles about lesson study and mathematics teaching. It also provides a connection to the international lesson study community.

� www.lessonstudygroup.net This website is from the Chicago Lesson Study Group and contains information about a national annual lesson study conference hosted by this group, as well as many research lessons and resources from the group.

� www.globaledresources.net This website has information, textbooks, and teacher guides from Japanese curricula (translated into English) as well as resources on lesson study.

Please see the Resources Appendix of the Leadership Guide for a more extensive bibliography of readings, resources, and links, including resources on mathematics and mathematics teaching.

implementing the CourseTo plan for implementation, one has to think locally and strategize for the long term. Who are the people in my school who might want to participate in the Course and learn about lesson study? Who are the people who support improvement in mathematics teaching and learning and might want to support the Course as part of this long-term goal? Who will lead the Course? Who might provide internal and external support for the teams in their next cycle? What logistical plans, permissions, or publicity need to be in place? The purpose of this final section of the Introduction is to address questions like these, so that when the Course begins, organi-zational challenges will be taken care of, and the focus can be on implementing it to achieve your goals.

Getting StartedRecruiting the PeopleFor the Course to happen, you will need one team, one facilitator, and two classes of students to participate in the research lesson. Most teams also need at least one school administrator to provide a measure of support. Take time to talk to people in the school about lesson study, perhaps hosting an informational meeting or speaking at a department meeting. Beginning the Course already connected to teachers’ goals and school initiatives can make its impact greater. A sample informational meeting agenda and letters for this purpose are included on the Course DVD and described later in the Start-up Materials section.

The Team A team size of four to six mathematics teachers is optimal. Smaller teams are also viable, but may bring fewer perspectives to the discussions. Larger teams face differ-ent challenges, including building agreement about approach and direction, or simply hav-ing time for everyone to speak. The Course can also be used in settings where multiple teams


participate—usually with one main facilitator. Facilitators using the Course for the first time may want to start by working with one team of teachers. Teams can be comprised of teachers from one grade or course (e.g., all sixth-grade teachers or all geometry teachers), or be made up of teachers from several grades (e.g., a sixth to eighth middle school team or a high school math department team). Although the Course was designed with middle and high school math-ematics teachers in mind and draws heavily on examples from these levels, it has also been suc-cessfully used in elementary school settings. Whether starting with one team of volunteers or introducing a program to a whole district, the Course represents a major time commitment for teachers. Spending time explaining the nature and benefits of the work, its long-term poten-tial impact, and its direct connection to classrooms helps teachers assess what the value of the Course will be.

The Facilitator The facilitator can be a mathematics teacher, teacher leader or math special-ist, coordinator or department chair, instructional coach, administrator, or professional devel-opment provider. The Course materials include such detailed guidance so that the facilitator does not necessarily need to have previous experience with lesson study or with facilitation. But he or she should be someone who is curious about lesson study, interested in mathemat-ics, and eager to find ways to improve teaching and learning through collaboration with col-leagues. The facilitator must commit to reviewing the facilitator notes and materials before each session.

The Students During the Course, each team will teach its research lesson twice and, therefore, will need students to participate. This is not usually an issue for teachers who are taking the Course at their school during the school year. Students come with the territory and the deci-sion of which students will participate can happen midway through the Course. However, some coordination will be necessary to arrange for all the teachers on the team to be free to observe the research lesson during Session 7 and again around Session 9. Beginning this coordination early in the Course is recommended. If the Course is being offered outside a school or outside the school year, locating students can be difficult. Consequently, we recommend that the facili-tator arrange for student participation well in advance of the Course starting date. We highly recommend that the Course include live observation, and so suggest a high priority on advance planning to make that happen. If multiple teams are involved in the Course, it may not be prac-tical for all teams to observe all lessons. Preferably, all teachers on a given team will be able to observe their own lesson.

The Administrator As you begin to organize a group of teachers to work with the Introduction to Lesson Study in Mathematics Course (and later, as you work with those teachers) it is important to stay in communication with administrators in your district and to gain and maintain their support. The school principal can be a key resource for obtaining release time, meeting space, and also for providing academic leadership, and for generating and maintain-ing enthusiasm for the work of improving mathematics learning in the school. Work with the principal (or department chair) to create time for the Course sessions, and to arrange coverage for the research lessons and post-lesson discussions (e.g., through release time or arrangement of substitutes for teachers to observe and discuss together). This time for sessions and coverage for the observation may come with a cost. In the long run, working with the union and school administration to locate possible funding sources, or to negotiate duties, professional develop-ment release time, or scheduling changes is what it takes to support lesson study as an ongoing professional activity in the school.


Planning the LogisticsBefore beginning the Course, planning the where and when are necessary. (See Start-up Materials on the DVD for a sample planning calendar.)

Meeting Times Whether the Course is for a small group of colleagues meeting together informally after school, or a formal course for credit, set a fixed calendar for Course meetings in advance, in consultation with the team, if possible. Setting meetings one at a time never works. Session 1 is probably best accomplished as a summer workshop day or a professional development release day, and the two-hour sessions as weekly or biweekly meetings. Spreading meetings out to occur once a month is less successful for maintaining focus and momentum. A careful reading of the session goals and activities allows facilitators to identify a few places where two sessions could be combined, held back-to-back on one day to take advantage of an early release or professional development day. If you choose this strategy, look carefully at the sessions ahead of time to check if there is any work that needs to be done between meetings or that will require some intervening time (e.g., between early sessions team members will need to gather resources to study their topic and mathematics problems to explore). We highly recom-mend establishing a schedule that takes into account the reality that around holidays and test-ing periods it is difficult for everyone to meet.

Meeting Place Most teams meet in a classroom, library, or dedicated professional learning space. The meeting place should be comfortable but the most important factor should be conve-nience for the teachers. With multiple teams, make sure that you have space for each team to sit together for small-group work. Arranging the tables and desks so that participants can face each other during both large-group and small-group activity can facilitate discussion. It is helpful for the room to contain an overhead or LCD projector so that Course slides can be displayed, to have Internet access, and to have mathematics resources nearby.

Managing Materials Participants will work with and receive many handouts during the Course. Many facilitators prefer to produce the full set of Course handouts and distribute them to participants in a binder at the beginning of the Course. Some teams keep a single team binder with all Meeting Logs and documentation, leaving individuals free to keep personal notes as it suits them. Individuals taking the Course will complete many brief reflections, work on math-ematics problems, and make notes during the teaching of the lesson. Encourage individuals to keep these notes in some way—from keeping a simple folder, to making entries in a journal for each session.

Start-up MaterialsThe following materials can be adapted for your local setting to help you get started with the Course. They are located in the Start-up Materials section of your Course DVD.

Outreach:

� Letter to Administration—A sample letter to administrators that can be used to introduce administrators to the Course.

� Letter to Prospective Participants—A sample recruitment letter that can be used to build the participant group for the Course.


� Suggested Agenda for Informational Recruiting Meeting—A sample agenda for an infor-mational meeting to recruit teachers to participate in the Course.

� Readings and Slides—The article (“A Lesson Is Like a Swiftly Flowing River”) and the slides from Session 1 support a brief introduction to lesson study at the informational meeting.

Organizational

� Introduction to Lesson Study in Mathematics Session Calendar—A sample schedule that can be used to list the dates and times of Course sessions.

� Contact Sheet—A contact sheet for collecting email and phone information from the group.

referencesHill, H. C., B. Rowan, and D. L. Ball. 2005. “Effects of Teachers’ Mathematical Knowledge for

Teaching on Student Achievement.” American Educational Research Journal 42(2): 371–406.

Lewis, C. 2002. Lesson Study: A Handbook of Teacher-Led Instructional Change. Philadelphia: Research for Better Schools.

Lewis, C., R. Perry, and J. Hurd. 2004. “A Deeper Look at Lesson Study.” Educational Leadership (Feb): 18–22.

Lewis, C., R. Perry, J. Hurd, and P. O’Connell. 2006. “Lesson Study Comes of Age in North America.” Phi Delta Kappan (December): 273–281.

Lewis, C., and I. Tsuchida. 1998. “ ‘A Lesson Is Like a Swiftly Flowing River’: Research Lessons and the Improvement of Japanese Education.” American Educator (Winter): 14–17, 50–52.

Ma, L. 1999. Knowing and Teaching Elementary Mathematics. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

Mullis, I. V. S., M. O. Martin, G. J. Ruddock, C. Y. O’Sullivan, A. Arora, and E. Erberber. 2005. TIMSS 2007 Assessment Frameworks. Boston, MA: International Association for the Evaluation of Educational Achievement (IEA): TIMSS and PIRLS International Study Center.

Stigler, J. W., and J. Hiebert. 1999. The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom. New York: The Free Press.

Facilitation Notes 3

Getting Started (5 minutes) 5

Introduction to Topic Research in Lesson Study (25 minutes) 6

Team Meeting: Choose Topic, Share Expertise, and Form Understanding Goals (75 minutes) 9

Prepare for Next Session (15 minutes) 11

Team Meeting Log 13

Topic Research Overview 14

Researching Topics in Lesson Study 15

Case Study Team Mathematical Context Chart— First Draft 16

Team Meeting: Choose Topic, Share Expertise, and Form Understanding Goals (75 minutes) 17

Mathematical Context Chart 18

PowerPoint Slides 19

s e s s i o n 2

topic selection and Research

3s e s s i o n 2 • Facilitation Notes

Facilitation notesWhen a new mathematical topic is approaching, our collaborative group gathers to “fess up” to our strengths and weaknesses in this particular area of mathematics. If there is a masterful mathematician among us to help bring us all onboard, we tap into the strength and experi-ence of that particular instructor. If there is not . . . we go out and find one so that we all can become more masterful in our teaching. This was not the case before lesson study.1

overviewDuring Session 2, teams make their final selection of a mathematics topic for the research les-son and begin researching and investigating that topic. Before investigating their own topic, teams will review a sample of the Case Study team’s topic research and discuss how this research was conducted, and how it served as a foundation for the research lesson. With this background on the process, teams finalize their topic choice, share their expertise and experience about the teaching and learning of the topic, and frame a goal for student understanding. Every lesson study cycle begins with this process of researching and sharing expertise about the research les-son topic. Ideas and questions about the trajectory of learning for the topic form the basis for the team’s research throughout the lesson study cycle.

Guiding Questions � What is the mathematical focus of our study during this lesson study cycle?

• Within this focus area, what topic will be the subject of the research lesson?

• Why is this mathematical focus and specific topic important?

� What is our current understanding of this mathematics? What more can we learn about it?

� What do we want our students to understand as a result of the research lesson?

� What trajectories do students generally follow as their understanding of this topic develops over time?

Recommended Readings from the Leader’s Guide � Chapter 4, Topic Selection

� Chapter 5, Topic Study

1 All unattributed quotes in the Course are from the teachers and coaches who participated on lesson study teams from the EDC NSF projects: Lesson Study Communities in Secondary Mathematics and Resources to Support Lesson Study in Mathematics.

s e s s i o n 2 • Facilitation Notes4

agendat i M e ac t i v i t y d e s c R i P t i o n

5 min Getting Started Review session agenda. Prepare for lesson study work.

25 min Introduction to Topic Research in Lesson Study

Learn how lesson study teams conduct research about the mathematics topic of their research lessons.

75 min Session 2 Team Meeting: Choose Topic, Share Expertise, and Form Understanding Goals

Select a mathematical topic for the research lesson. Share and record team members’ expertise and experience about the teaching and learning of the chosen mathematical topic through a combination of written and verbal discussions. Use this sharing of knowledge to begin capturing what the team knows about the teaching and learning of the topic in the Mathematical Context Chart, and to form understanding goals for students.

15 min Prepare for Next Session Prepare for Session 3 and reflect on Session 2.

Preparation

Gather Supplies

� Copies of the session handouts for everyone

� Current version of the Draft Research Lesson Report (electronic version is the easiest to update, but a paper copy with decisions to date will work; referred to as Draft Report in remainder of materials)

� PowerPoint slides for the session and equipment or a means for sharing those PowerPoint slides (e.g., a projector, or copies of the slides to hand out)

� Blank Team Meeting Log for this session (one copy)

� Completed Team Meeting Logs and notes from Session 1

� Session agenda to display throughout session (e.g., on chart paper or blackboard)

� Name tags and markers, if necessary

Additional Materials and Preparation Tasks for This Session

� Textbooks from your own curriculum to support discussions about research lesson topics; state or national standards for use in Sessions 2 and 3

� Have available copies of the Team Meeting Roles handout and the Choosing a Research Lesson Topic handout from Session 1

5s e s s i o n 2 • Getting Started

getting started (5 MINUTES)

Preparation Slides 1–4

Instructions and Notes1. Launch Session 2 by asking the team to think back to Session 1 (show Slide 2 to remind

them about the lesson study process) and invite the team to say something about their thinking as they enter today’s session. Ask them “What are you hoping for during this lesson study cycle (or this Course), based on what you experienced in Session 1?” (This prompt is on the bottom of Slide 2.)

2. Review the agenda for Session 2 (Slide 3).

• You may want to post the agenda so participants can refer to it during the session.

• Explain that Session 2 will continue the Phase 1 Lesson Study Cycle work (Slide 4) in particular focusing on content goals and background research.

6 s e s s i o n 2 • Introduction to Topic Research in Lesson Study

introduction to topic research in Lesson study (25 MINUTES)

PurposeThe purpose of this activity is to give the team an overview of the “topic research” process in les-son study. To teach any lesson well, a teacher needs a solid understanding of the mathematics in the lesson. Given that mathematics understanding of any topic builds on prior understanding, it is also important to understand the wider mathematical context of a topic. The mathematical context for a topic includes how students progress in their understanding of this mathematics over time: what mathematics ideas students have studied prior to the lesson and how they think about that mathematics, what students must understand and think about during the lesson, how the lesson leads to more advanced mathematics, and any other relevant mathematical con-nections. All of these connected mathematical ideas form a broad mathematical context for the topic.

As part of the lesson study process, teachers work together to build this broad understanding for their chosen lesson topic. To develop and represent the mathematics context for a given topic, teachers draw on their own experiences, and study national and state standards, instruc-tional materials, and published research on the teaching and learning of the topic. Throughout the lesson study process, teachers continue to develop and modify their vision of the mathemat-ical context.

PreparationBe sure to study the handouts carefully to gain an overview of the goals and process of topic research.

Handouts: Topic Research Overview; Researching Topics in Lesson Study; Case Study Team Mathematical Context Chart—First Draft

Slides 5–10

Instructions and NotesExplain the Nature of Topic Research in Lesson Study (15 MINUTES)

1. Discuss why teams conduct background research on their topic.

• Use the quote (Slide 5) to spark a brief team discussion about what knowledge it takes to plan an effective lesson and to understand why and how that lesson promotes understanding. Ask the team to share their thoughts about what the quote is saying, and what they might want to know in order to achieve the goal stated in the quote.

• Summarize the purpose of topic research (Slide 6). The statements on Slide 6 will probably mirror closely what the team has just discussed. The knowledge needed to create effective lessons and understand why they work is exactly what topic research builds.

s e s s i o n 2 • Introduction to Topic Research in Lesson Study 7

2. Explain the process of topic research in lesson study using the Topic Research Overview handout (Slide 7 if you wish to project it). The handout portrays graphically what the team does as they research their topic. The main steps in that research include the following:

a. Consulting resources to learn about the topic. These resources help participants build knowledge of the mathematical context.

b. Developing products of team’s topic research. The knowledge is organized to show the mathematical context and learning trajectory. At this stage, the team will also be articulating understanding goals for their students.

c. Informing the design of the research lesson. The team develops ideas about how to teach this topic effectively in a way that reflects this vision of the mathematical context.

• Some points to make in your explanation include:

♦❏ Purpose of topic research: All teams do this research because it not only pro-vides a foundation for their lesson, but it also helps them to learn more about the teaching and learning of the topic.

♦❏ Scale of effort: New teams often select just a few of the resources shown on the Topic Research Overview handout. As teams become more experienced, they tend to extend their reach to other sources, especially an in-depth study of textbooks, use of outside expertise, and research readings.

♦❏ Timeline of topic research: Slide 8 shows a timeline for the team’s topic research. The team begins today by sharing their own knowledge of the topic and then researching the scope and sequence. In future sessions, the team will continue to deepen their knowledge about the mathematics topic by exploring mathematics problems together and studying textbooks and lessons. Teams have the option to choose other resources from this list to extend their under-standing even further, as team interest and time allow.

♦❏ Important outcomes of topic research (Slide 9). One outcome of the research will be the articulation of a goal for student understanding. Another outcome will be a description of the mathematical context—including a trajectory of learning. This will be a representation of how students’ understanding of the topic develops, starting in earlier grades with related mathematics, building during the students’ direct study of the topic, and eventually leading to explo-ration of more advanced or related mathematics. The team may choose what-ever form makes most sense to them to represent the mathematical context and learning trajectories.

♦❏ Goal for this session: The expectation for the team during this session is that they learn about the research process and start gathering information relevant to the mathematical context for their own topic. At the end of this session, the team will decide what additional research they wish to pursue in Session 3 and plan to obtain the necessary resources.

• Optional: If there is time, share some additional insights about topic research from material on the Researching Topics in Lesson Study handout. Or, you can ask the team to read it.

s e s s i o n 2 • Introduction to Topic Research in Lesson Study 8

Review Case Study Sample (10 MINUTES)

3. To complete this overview of topic research, have all participants look at the Case Study Team Mathematical Context Chart—First Draft handout and discuss the questions on Slide 10.

• Remind teachers that the mathematics problem they explored in Session 1 was from this team’s research lesson. The Case Study Team Mathematical Context Chart—First Draft represents the beginning stage of their topic research. So far, they had talked together about their students’ prior knowledge, reviewed the placement of the topic in their textbook, and begun to frame the mathematical context.

9s e s s i o n 2 • Team Meeting: Choose Topic, Share Expertise, and Form Understanding Goals

team meeting: choose topic, share expertise, and Form understanding goals (75 MINUTES)

Purpose Choosing a Topic The team needs to select one mathematical topic that will be the primary content focus of the team’s research lesson. The mathematical topic should represent math-ematics that is important and difficult for students to learn, and that the team wishes to explore through the lesson study process.

Sharing Expertise and Articulating a Goal for Student Understanding of the Topic The members of the lesson study team together provide a rich resource and knowledge base about the teaching and learning of the chosen topic. Taking plenty of time to share what the team already knows, or has experienced in their teaching, will help the team to consolidate that knowledge and to clarify what questions about the topic they would like to pursue. The Mathematical Context Chart will then help the team organize their thinking. In sharing exper-tise and gathering information about the mathematical context, the team will begin to articu-late a learning trajectory and specific goals for student understanding.

PreparationHandouts: Team Meeting Roles (from Session 1); Session 2 Team Meeting: Choose Topic, Share Expertise, and Form Understanding Goals; Choosing a Research Lesson Topic (from Session 1); Mathematical Context Chart (at least one copy for Recorder)

Slides 11–13

Other: Session 2 Team Meeting Log (one copy for Recorder). (Session 1 Team Meeting Log lists the Session 2 Team Meeting Recorder and Team Meeting Leader as well as the short list of top-ics that team had chosen during Session 1.) Blank paper and pens; chart paper and markers; a chalkboard and chalk; sticky notes and pens for engaging in a written discussion on a wall that everyone can see. Recommended: Computer and electronic version of Mathematical Context Chart and of the Draft Report so that the Recorder can add information electronically.

Instructions and Notes1. Give a quick overview of the purpose, steps, and timing for the Team Meeting, and then

turn the meeting over to the team and to the Team Meeting Leader. The team will work from either the Session 2 Team Meeting Log handout or the Team Meeting Agenda slide (Slide 11) for the remainder of the meeting. A summary of the steps in the Team Meeting is provided here, followed by bulleted notes that highlight things you may want to consider during the Team Meeting in your role as Course facilitator. (If there is any confusion about the roles of Leader and Recorder, refer to the Team Meeting Roles hand-out and Team Meeting Log from Session 1.)

s e s s i o n 2 • Team Meeting: Choose Topic, Share Expertise, and Form Understanding Goals 10

a. The team will select a mathematical topic from the short list created during Session 1.

• If the team needs an additional resource, use Slides 12 and 13 and/or the Choosing a Research Lesson Topic handout from Session 1.

• You may want to remind the team to discuss how “big” the topic is (i.e., how many lessons it involves). In the end, they will plan a detailed research lesson for one les-son within the topic. If your team chooses a topic that is too big and has trouble narrowing it down, encourage them to proceed with sharing expertise—they will likely be able to narrow the topic down as they explore it.

• If your team is leaning toward a process standard (e.g., problem solving), encour-age them also to select a content topic within which to study the process standard.

• Your team’s instinct may be to start drafting the lesson as soon as you select your topic but one of the key elements of lesson study is taking time to develop the foundation for the research lesson. This foundation includes a formal exploration of your topic based on your own experiences as well as existing resources. This work of exploring the topic is the first step toward completing a research lesson plan (i.e., participants are already working on the research lesson, even if it does not look like typical lesson planning to them!).

b. The team will engage in a written conversation about the mathematics topic by silently posting responses to the following three questions, and building off notes posted by other team members:

❏ What are the important mathematical ideas related to this topic that you want students to learn?

❏ What makes this topic difficult for students?

❏ What approaches to teaching this topic have you tried in the past?

• These three questions should be written on three pieces of chart paper on the wall, or in three spots on a blackboard/whiteboard.

• This written discussion will allow everyone to provide input on the topic while also creating a written record of the discussion. Although this format for discussion might feel unfamiliar, encourage teachers to give it a try as it allows team members to process and share information in a different way, which may be better suited to some people’s style, while also leaving a written record that can be referred back to.

• If possible, take a picture of your “written discussion” as a record for your team.

c. The team will take a few minutes to read through everything that has been written, then begin discussing the themes and questions that emerge, in order to work toward creating a trajectory of learning for the topic and understanding goals for students.

• The team will continue refining the information they have gathered about the mathematics context for the topic throughout the Course, but they should begin now to articulate what they know, wonder, and hypothesize about how students learn about this topic.

d. The team should finish by recording decisions on the Mathematics Context Chart and on the Session 2 Team Meeting Log.

11s e s s i o n 2 • Prepare for Next Session

Prepare for next session (15 MINUTES)

PreparationHandouts: Session 2 Team Meeting Log (same copy used during Team Meeting activity)

Slides 14–16

Instructions and Notes

End-of-Session Reflection (SLIDE 14) (5 MINUTES)

1. Spend a few minutes discussing one or both of the reflection questions on Slide 14: What is something about your team’s chosen mathematics topic that you are thinking differently or wondering about after sharing expertise with your colleagues? What is something about how students learn this topic that you hope to explore further through this lesson study cycle?

• If possible, give teachers a minute or two to collect their thoughts individually before sharing.

• This reflection will give teachers a chance to consolidate what they have been think-ing about during today’s session. In addition, hearing what the teachers have to say will help give you a sense of where they are in their learning and how the sessions are going.

• If you want additional feedback to help you plan future sessions, you could distribute index cards and have teachers jot down “something that worked today” and “some-thing you wish had been different today” on the cards before they leave the session.

Prepare for Session 3 (SLIDE 15) (10 MINUTES)

1. Confirm the date, time, and location for the next session.

2. Make sure that key decisions and notes have been recorded on the Team Meeting Log. Identify someone to make copies of Team Meeting Log and of the Mathematical Context Chart (in its current form) for everyone.

3. Identify team roles (Team Meeting Leader, Recorder, Snack Provider) for the next session.

4. Homework: There are two main activities in Session 3 that require preparation: continu-ing the research and examining mathematical tasks.

• To prepare for examining mathematical tasks, it is important that everyone brings one or two mathematics problems on the team’s topic to Session 3 (with copies for the team). The team should look for problems that might be appropriate for the team’s research lesson.

s e s s i o n 2 • Prepare for Next Session 12

• To prepare for the continuing research, determine “What additional resources we need?” to learn more about the progression of learning for our topic or about how students learn the topic? Use the Topic Research Overview handout to remember the variety of resources teams can access, but don’t feel limited to that list. Also decide who will gather these resources and bring them to Session 3. The team members who gather resources should also make photocopies of the resources, when possible, so that everyone on the team can have access to the materials during Session 3.

• If the team is not sure where to find resources, direct them to the NCTM resources such as the national standards (see www.nctm.org) and published journals, as well as textbooks from the teachers’ curriculum.

• You may wish to show the slide Catherine Lewis on Reviewing Resources (Slide 16) to motivate this brief discussion about resources, or as a way to close the session.

13

© 2010 by Education Development Center, Inc. from Lesson Study in Practice. Portsmouth, NH: Heinemann.

s e s s i o n 2 Handouts • Team Meeting Log

T e a m m e e T i n g L o g(Attach any additional notes supporting decisions recorded.)

Date: ________________________________ In attendance: ___________________________________

Recorder: _____________________________ Team Meeting Leader: _____________________________

❯ Team Meeting—Choose Topic, Share Expertise, and Form Goals for Student Understanding

The mathematical topic for our team’s research lesson is . . .

What we want our students to understand about the topic is . . .

Attach a picture of the “written discussion” about the topic mathematics from the Team Meeting (or note here where a record of that written discussion is located).

Attach the Mathematical Context Chart (or note here where the team’s current research notes are located).

❯ Other Team Decisions: ______________________________________________________________

❯ Parking Lot (items to remember to bring up at future meeting): _________________________

❯ Moving Forward

Session 3 Team Meeting Leader will be: ____________________________________________________

Session 3 Recorder will be: ______________________________________________________________

Preparation for Session 3:

Resources to gather (and who will gather them): _______________________________________

14


s e s s i o n 2 Handouts • Topic Research Overview

T o P i C r e s e a r C h o v e r v i e w

Resources the Team Consults to Learn About the Topic

Team knowledge Textbooks Standards and scope and sequence

Math problems Lessons NCTM resources Readings/research

Knowledgeable people from outside the team

Products of the Team’s Topic Research

knoWledGe oF tHe MatHeMatIcal context

The team articulates a trajectory of learning and other mathematical connections for the topic:

before this year this year and this lesson

after this year

GoalS For Student underStandInG

Informed by increased team knowledge of student thinking and student learning needs

Informing the Design of the Research LessonThe topic research informs the team’s ideas and theories about teaching the topic.

The team chooses problems, activities, and pedagogical strategies that reflect their new understanding of the mathematical context and student learning trajectories.

15


s e s s i o n 2 Handouts • Researching Topics in Lesson Study

r e s e a r C h i n g T o P i C s i n L e s s o n s T u d yThe goal is not only to produce an effective lesson but also to understand why and how the lesson works to promote understanding among students.1

This document provides detailed information about topic research in lesson study, and is intended to supplement the Topic Research Overview handout.

Resources: Throughout the lesson study cycle, you will be engaging in topic research activities that will help you learn about the mathematics of your topic. These activities early in the cycle provide a founda-tion for thinking about the lesson design and include:

� Sharing what you know about the teaching and learning of the mathematics topic that your team chooses. (Session 2)

� Studying additional resources (e.g., textbooks, state or national standards) that will help you learn more about how the mathematical ideas related to the topic are usually learned (e.g., in what order are the ideas presented in different curricula, what prerequisite knowledge is assumed, how do the ideas relate to more advanced mathematics or to mathematics taught in lower grades). (Session 3)

� Doing and discussing mathematics problems related to the topic. (Sessions 3 and 4)

� Anticipating students’ methods, approaches, and thinking for the topic. (Sessions 4 and 5)

Products: Discussing experiences related to teaching the team’s chosen topic, combined with studying resources such as those described and exploring mathematics problems, will help the team develop a shared understanding of the mathematical context for the topic. This investigation will be driven by ques-tions such as:

� How does this topic fit within a learning trajectory?

� What other mathematical connections are important when students study this topic?

� How do students currently think about or misunderstand the topic?

� What do we want students to understand at the end of the lesson (and unit)?

� What will this allow them to learn next?

The team will begin to articulate specific understanding goals for their research lesson, and to map out the trajectory of learning for the topic. The mathematical context, including the trajectory of learning, can be represented in many ways (e.g., a graphical representation, an annotated timeline, or a written narrative explaining the important ideas and sequences of learning for the topic). The team’s ideas about the tra-jectory of learning will probably look different at the beginning of the lesson study cycle when the team is first studying the topic than it will at the end of the cycle when the team is preparing to share what they have learned with other teachers. The team will gather ideas on the Mathematical Context Chart, but may ultimately represent their thinking in any way in their Draft Report.

Implications: The purpose behind studying the mathematical context for your topic is to allow creation of a research lesson that will foster learning by students (about the topic) and foster your own learning (about how students learn about the topic and about the mathematics of the topic). Therefore, the mathematical context should directly inform the theories, ideas, and questions behind the design of the team’s research lesson. For example, your team may learn about a different order to try presenting ideas, or a type of task that might help students explore an idea, or a theory about how students make connections between two ideas. The mathematical context is the backbone of a well-planned research lesson.

1 Stigler, J. W., and J. Hiebert. 1999. The Teaching Gap (p. 113). New York: The Free Press.

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s e s s i o n 2 Handouts • Case Study Team Mathematical Context Chart— First Draft

C a s e s T u d y T e a m m aT h e m aT i C a L C o n T e x T C h a r T — F i r s T d r a F TResearch Lesson Mathematical Topic: Comparing exponential growth to linear growth

Grade Level: Ninth grade

What do students learn related to the mathematical topic? What contributes to that learning? (ideas they explore/learn; information about how they think about or learn the topic; common misconceptions or challenges)

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Prior learning (in ninth grade and middle school): Linear functions, slope, intercept, graphing, tables of values. Our topic falls within the textbook unit on exponents so students will know how to compute expressions with exponents. Prior to the lesson, students will not have graphed exponential functions, and many will assume that all graphs are linear.

Textbook unit that contains our topic (8.1–8.4 will be covered before the research lesson):Section 8.1 Multiplication Properties of Exponents Section 8.2 Zero and Negative ExponentsSection 8.3 Division Properties of ExponentsSection 8.4 Scientific NotationSection 8.5 Exponential Growth Functions; Comparing Linear and Exponential Growth (the

lesson)Section 8.6 Exponential Decay Functions (after the lesson)

Question we want to research further: What kinds of patterns do students study in K–8?

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What students will learn or understand during the lesson: We are most interested in students understanding how exponential growth patterns differ from linear patterns. One is additive, the other multiplicative. But we also want our students to recognize differences between linear and exponential functions in all different representations.

Massachusetts Standards related to our topic:AI.P.4 Translate between different representations of functions and relations: graphs, equations,

point sets, and tabular.AI.P.11 Solve everyday problems that can be modeled using linear, reciprocal, quadratic, or

exponential functions. Apply appropriate tabular, graphical, or symbolic methods to the solution. Include compound interest, and direct and inverse variation problems. Use technology when appropriate.

What contributes to that learning? In our experience, direct comparisons (e.g., linear graph with exponential graph, table with table) are effective for noticing differences. Also, a real-life application should highlight the steep exponential pattern and engage students.

Question we want to research further: How do students go beyond surface differences to recognize underlying numerical growth patterns when they study tables and graphs?

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. . . After the lesson in Algebra 1:

What is exponential decay? (Text Section 8.6)Quadratic functions.In Algebra 2, students learn about exponential and logarithmic functions and expand their

knowledge of polynomial functions. Questions we want to research further: What else does this unit lead to in upper grades—Algebra 2,

Pre-calc, Calculus? What other “growth patterns” are important? How are they studied?

17


s e s s i o n 2 Handouts • Team Meeting: Choose Topic, Share Expertise, and Form Understanding Goals

T e a m m e e T i n g : C h o o s e T o P i C , s h a r e e x P e r T i s e , a n d F o r m u n d e r s T a n d i n g g o a L s (75 MINUTES)

1. Choose a topic for your team’s research lesson. (15 minutes)

Resources to Support Topic Selection:

• The short list of potential topics recorded on your Session 1 Team Meeting Log.

• The Choosing a Research Lesson Topic handout from Session 1.

Record your final topic selection on your Draft Report and on the Team Meeting Log.

2. Engage in a written (silent!) sharing of knowledge about the team’s topic. (15 minutes)

3. Discuss themes, questions, and ideas about the teaching and learning of your topic that arose dur-ing the written sharing of knowledge. Focus especially on articulating your understanding goals for students and on refining your ideas about the topic mathematical context and trajectory of learning. (45 minutes)

• Record ideas about goals for student understanding on the Draft Report and on the Team Meet-ing Log.

• Record ideas about the trajectory of learning and mathematical context on the Mathematical Con-text Chart.

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s e s s i o n 2 Handouts • Mathematical Context Chart

m aT h e m aT i C a L C o n T e x T C h a r TResearch Lesson Mathematical Topic:

Research Lesson Target Grade Level:

What do students learn related to the mathematical topic? What contributes to that learning? (ideas they explore/learn; information about how they think about or learn the topic; common misconceptions or challenges)

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. . What students will learn or understand during the lesson . . .

What contributes to that learning?

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s e s s i o n 2 • PowerPoint Slides 19


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s e s s i o n 2 • PowerPoint Slides20


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Topic Selection and Research

Session 2

2-1

“When a new mathematical topic is approaching, our collaborative group gathers to ‘fess up’ to our strengths and weaknesses in this particular area of mathematics. If there is a masterful mathematician among us to help bring us all on board, we tap into the strength and experience of that particular instructor. If there is not . . . we go out and find one so that we all can become more masterful in our teaching. This was not the case before lesson study.”


All unattributed quotes in the Course are from the teachers and coaches who participated on lesson study teams from the EDC NSF projects: Lesson Study Communities in Secondary Mathematics and Resources to Support Lesson Study in Mathematics.

2-2 © 2010 by Education Development Center, Inc. from Lesson Study in Practice. Portsmouth, NH: Heinemann.

Lesson Study Overview

What are you hoping for during this lesson study cycle (or this Course), based on what you experienced in Session 1?

Agenda

2-3

Getting Started (5 min)

Introduction to Topic Research in Lesson Study (25 minutes)

Team Meeting (75 minutes)

  Choose Topic

  Share Expertise

  Form Understanding Goals

Prepare for Next Session (15 minutes)


Lesson Study Overview Focus, Set Goals, Research the Topic (Main focus of Sessions 1–3)

  Deepen knowledge about lesson study.

  Set long-term goals for student learning.

  Select content-specific goals and select a topic.

  Make a plan for conducting the lesson study in your school.

  Do some background work about curriculum and student understanding of the topic.

2-4 © 2010 by Education Development Center, Inc. from Lesson Study in Practice. Portsmouth, NH: Heinemann.

2-5

“The goal is not only to produce an effective lesson but also to understand why and how the lesson works to promote understanding among students.”

J. W. Stigler, and J. Hiebert. 1999. The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom. New York: The Free Press.


The Purpose of Topic Research

2-6

  Build our knowledge of mathematics.

  Build our understanding of what our students know, and how they think about and learn this mathematics.

  Help the team articulate clear understanding goals and outcomes for students.

  Support the team in developing a lesson (materials, activities, pedagogy) that achieves these goals.

Based on a presentation by Yoshida, M. 2003. “Conducting Kyozaikenkyu (Instructional Materials Investigation) During Lesson Study: Toward Improving Teachers’ Content and Pedagogical Knowledge,” copyright GER.


2-7

Topic Research Overview


Timeline for Topic Research

2-8

Session 2 (today)

  Learn about topic research.

  Share expertise.

Session 3

  Study textbooks, standards, and lessons.

  Do, discuss, and study the mathematics.

Sessions 3 and 4

  Find and do problems that we might include in the research lesson.

Sessions 4 and 5

  Anticipate how students will approach the problems in the lesson.


Outcomes include . . .

2-9

A description of the mathematical context including:

  Learning trajectory: Prior understandings students will build on; what the lesson will add; what future learning it leads to

  Connections: How the topic connects to ideas within and outside mathematics.

Understanding goals for students

Ideas for possible pedagogical approaches


Case Study Research Discussion Questions

2-10

The Case Study lesson introduces exponential growth in an Algebra I course. (The chart represents the Case Study team’s initial review of resources and sharing expertise.)

  Based on this initial review, how do you think the Case Study team expects students to build understanding of exponential growth?

  What do you think the team might want to investigate further—about the topic mathematics and about student learning of the topic?


Team Meeting Agenda

2-11

Come to a consensus on a topic for your team’s research lesson. (15 minutes)

  Resources: Create a short list of potential research lesson topics from Session 1 Team Meeting Log and Choosing a Research Lesson Topic handout.

  Record your final topic selection on your Draft Report.

Engage in a written (silent!) sharing of knowledge about the team’s topic. (15 minutes)

Discuss themes, questions, and hypotheses about the teaching and learning of your topic that arose during the written sharing of knowledge. (45 minutes)

  Articulate your team’s goals for student understanding and record them on the Draft Report.

  Record ideas and questions related to the mathematical context and the learning trajectory for the topic on the Mathematical Context Chart and put it in the Team File.


Choosing a Topic Is Guided by Purpose

2-12

  Your goal is to find out how students think about (and learn) important mathematical concepts.

  At the same time, your goal is to increase your own understanding of mathematics.

These dual purposes will guide team decisions, including your choice of a mathematics topic for your team’s first research lesson!


Choosing a Topic—Guiding Questions

2-13

You do not need to address all of these questions, but choose one or two.

  Which topics do students have difficulty learning?

  Which topics do teachers find difficult to teach? In what ways?

  What mathematical importance does each topic have (e.g., in terms of connections to other topics in the curriculum)?

  As teachers, what mathematics do we want to better understand?

  For which topics do we want to learn more about how students think?

  Are there new additions to our curriculum that we want to explore?

  Which topics will students be studying around the date when the research lesson is expected to occur?


Reflecting on Session 2

2-14

What is something about your team’s chosen mathematics topic that you are thinking differently or wondering about after sharing expertise with your colleagues?

What is something about how students learn this topic that you hope to explore further through this lesson study cycle?


Prepare for Session 3

2-15

Review date, time, and location.

Assign roles (Recorder, Team Meeting Leader).

Homework: Bring problems and resources. Each person should gather one or more mathematics problems that address the team’s topic and that could possibly form the core of the team’s research lesson. Team members should also gather resources for researching your team’s topic. At a minimum, if the team has not already consulted their own textbooks, and relevant Standards or Scope and Sequence documents, then these items should be gathered before Session 3. In addition, the team should refer to the Topic Research Overview handout and decide if there are other particular resources you will want to consult, and make a plan to have access to those resources during Session 3.


Catherine Lewis on Reviewing Resources

2-16

“Lesson study is most productive when educators build on the best existing lessons or approaches, rather than reinventing the wheel. . . . Try to immerse yourself in others’ lessons through whatever means you can . . . textbooks, research lessons, books, video. . . . If your group searches out and studies the best existing lessons, it will result in a better research lesson and help create a system that learns rather than one in which every group of educators reinvents the wheel.”

Lewis, C. C. 2002. Lesson Study: A Handbook of Teacher-Led Instructional Change. Philadelphia, PA: Research for Better Schools, Inc.


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