1. Learning About Student Mathematical Discourse: Case Study of a Middle-School Lesson Study Group Betsy Elizabeth King eking@mills.edu Mills College Aki Murata amurata@mills.edu Mills College

2. Participants and Setting Washington Middle School (SF Area) 5 Mathematics Teachers Lesson Study during school year (2003-2004; meeting 2 - 3 times a month) Research lesson - 3 lesson study cycles.

3. Research Question How does the teachers' thinking (beliefs, goals, knowledge) about student mathematical conversations evolve over the course of their participation in lesson study?

4. Research Framework Students learn to communicate mathematically (NCTM, 1989, 1991 & 2000).

5. Research Framework Students learn to communicate mathematically (NCTM, 1989, 1991 & 2000). Classroom social processes for mathematics reasoning (Cobb, et al., 1997).

6. Research Framework Students learn to communicate mathematically (NCTM, 1989, 1991 & 2000). Classroom social processes for mathematics reasoning (Cobb, et al., 1997). Orchestrating classroom discourse, model for instructional practice (Sherin, 2002).

7. Research Framework Students learn to communicate mathematically (NCTM, 1989, 1991 & 2000). Classroom social processes for mathematics reasoning (Cobb, et al., 1997). Orchestrating classroom discourse, model for instructional practice (Sherin, 2002). Lesson Study process to support teacher learning

8. Teachers’ Lesson Study Goal: “To promote student mathematical conversations ”

9. Research Methods Data Collected –Lesson study meetings, lessons and debriefings were video/audio-taped –Lesson artifacts –Teacher interviews were audiotaped

10. Research Methods Data Analysis Transcribed meetings, lessons, interviews Coded dialogue identified themes clustered into components Traced learning threads, shifts in teachers’ thinking

11. Components of Teacher Learning Facilitation of Student Mathematics Conversation Use of Representations and Tools to Guide Student Mathematics Conversation

12. Components of Teacher Learning Facilitation of Student Mathematics Conversation –Classroom Norms and Culture

13. Components of Teacher Learning Facilitation of Student Mathematics Conversation –Classroom Norms and Culture –Making On-the-Spot Decisions

14. Components of Teacher Learning Facilitation of Student Mathematics Conversation –Classroom Norms and Culture –Making On-the-Spot Decisions –Time and Pacing of the Lesson

15. Components of Teacher Learning Facilitation of Student Mathematics Use of Representations and Tools to Guide Student Mathematics Conversation –Representation Use –Scaffolding

16. Teacher Questions: How to make on-the-spot decisions? How do you know which students’ questions to follow up? What if a student tells the answer early in lesson? How do you know when to move on? How do you know which students to call on and in what order?

17. Research Lesson 1: Gauss’ Houses 1.Find apt # when given bldg# and floor. 2.Find floor # for student’s apt #. 3.Create algorithm for floor # 4.Class generates formula or rule

18. Anticipate Student Strategies Before the lesson: –Anticipate student strategies –Decide sequence of strategies During the lesson: –Observe students’ strategies –Students explain strategies in sequence

19. Lesson 1 Teachers anticipated student strategies for question 1: –Counting with drawing –Multiply building # x 6, subtract 1 Strategies for other questions –Divide by 6, interpret remainder –Multiples No discussion about sequencing

20. “How do you decide which students to call on, and in what order?”

21. Lesson 2: Planning Examined student strategies - lesson 1, anticipated strategies - lesson 2 Planned sequence of students’ strategies Start with most accessible, more common, and build to more sophisticated

22. Lesson 2: More Focus on Strategies Teacher facilitated students discussing their strategies. Strategies are named after student who suggests them.

23. Lesson 3 Common vocabulary and knowledge of student strategies and learning trajectory: –observe, easily record and discuss student strategies –more visible student thinking and learning

24. Lesson 3: Shift in Teachers’ Questions “… students engaging in mathematical conversations?” “… successful students…helping … struggling students?” “… evidence and proof … in their arguments - 8th grade teacher

25. Shifts in Teacher’s Thinking “ …instead of intervening and saying, ‘oh no, no, let me help you,’ I just tried to think about, ‘What’s going on in this person’s mind?’ …So that’s very different.” -- 6th grade teacher

26. Shifts in Teachers’ Thinking “…it takes this sort of very careful orchestration of the conversation. And just saying, ‘ok, talk about this, solve this problem and talk about it while you’re solving it,’ isn’t enough.” - 8th grade teacher

27. Connecting Student Strategies Teacher makes connections between solutions –pictorial to table representation –table representation to numbers “Students share solutions” “Kids main talkers in whole-class discussions” “Move from concrete operations towards formal operations” -- 6th grade teacher

28. Summary Student mathematical conversations need careful orchestration

29. Summary Student mathematical conversations need careful orchestration Knowledge of student strategies –facilitating discussions –connections between strategies –teachers’ observations/ student thinking

30. Summary Student mathematical conversations need careful orchestration Knowledge of student strategies –facilitating discussions –connections between strategies –teachers’ observations/ student thinking Shift in teachers’ questions –from what teacher does –to what students do

31. Learning in Lesson Study Research lessons and debriefings visible Lesson study planning meetings largely invisible