Facilitator as Math Expert Facilitator as Student Work Facilitator Facilitator as Group/Materials Manager Facilitator as

1. Facilitator as Math Expert • Facilitator as Student Work Facilitator • Facilitator as Group/Materials Manager • Facilitator as Advocate for Change

2. Facilitator as Math Expert • Mathematics may be over my head (Rotations) • I’ll need to answer questions about the mathematics • I will need to articulate the math big ideas • I’ll need to make sure the G-HOMs are understood • How to transfer to teaching geometry • Algebra-only teachers will need to see relevance

4. Facilitator as Math Expert • Facilitator as Student Work Facilitator • Facilitator as Group/Materials Manager • Facilitator as Advocate for Change

5. Facilitator as Student Work Facilitator • Foggy on the goals of student work discussions

6. Facilitator as Math Expert • Facilitator as Student Work Facilitator • Facilitator as Group/Materials Manager • Facilitator as Advocate for Change

7. Facilitator as Group/Materials Manager • I’ll need to answer questions • Teachers are at different levels • Perceived level of difficulty of the problems • Timing of “sessions expected to be taught” • Time constraints for sessions

9. Facilitator as Group/Materials Manager, Cont’d • Burden of videotaping • Pacifying teachers who want to see solutions • Getting participants in ‘explore’ mode • Order of materials this week? • Correcting math mistakes

11. Facilitator as Math Expert • Facilitator as Student Work Facilitator • Facilitator as Group/Materials Manager • Facilitator as Advocate for Change

12. Facilitator as Advocate for Change • Making materials meaningful for long-term change • Perceived level of difficulty of problems • Correcting math mistakes • Algebra-only teachers will need to see relevance

13. Major Areas of Facilitator Attention • Productive mathematical explorations and discussions • Productive analyses and discussions based on evidence of student thinking • Managing Group Processes and Group Dynamics

14. Productive mathematical explorations and discussions At the end of the discussion, all participants have the sense of making progress, if not achieving a full understanding of a problem solution. Often, a hallmark of a productive exploration and discussion is the desire by participants to continue thinking about the problem after the session is over. Participants describe their thinking to each other in small groups as well as in the full group discussion.

15. Productive mathematical explorations and discussions • Multiple ways of solving a particular problem are valued, particularly in the full-group discussion, and links are made among them. • During explorations and discussions it is safe to make mistakes and errors are valued because they open doors to examining thinking. • Collaboration on solving problems is valued and pursued. • In the context of FGT group discussions, opportunities to explore the geometric habits of mind are seized frequently.

16. Productive analyses and discussions based on evidence of student thinking • Participants feel safe. Discussions are guided by ground-rules that keep people focused on the evidence, not on the teachers involved or particular students. • The primary focus—in both analysis and discussion—is on student thinking and learning, not on evaluating student achievement, nor on teaching. • Alternative interpretations of student evidence are valued and encouraged. • Discussions present challenges to each person’s beliefs, assumptions, and mindsets.

17. Productive analyses and discussions based on evidence of student thinking • Discussions tease out important mathematical ideas underlying the evidence of student thinking. • Participants base their interpretations of student thinking on the evidence provided. • Based on their interpretations of evidence, participants make explicit connections to classroom practice.

18. Features of productive student-work discussions • Emphasis on thinking • Emphasis on mathematics content

19. Student work discussions: Emphasis on thinking • Expressing curiosity about student and teacher thinking • Looking for strengths and potential, not just weaknesses, in student's work • Developing plausible and consistent storylines about the student thinking represented in the work • Acknowledging when the line has been crossed between description and interpretation of the student work • Grounding interpretations of thinking in the evidence • Comparing/contrasting the thinking to the thinking in other students' work

20. Student work discussions: Emphasis on mathematics • Considering the mathematical ideas represented in the student work • Using a guiding framework (e.g., G-HOMs) to analyze the mathematics in the student work • Describing the perceived student potential and storyline in the work in terms of what is important mathematically • Considering how convincing are the mathematical explanations in the student work • Comparing/contrasting the mathematical representations across students' work

21. Managing Group Processes and Group Dynamics • A sense of group direction or focus for work, developed and sustained by a capacity to be reflective about practice • A threefold change strategy, paying attention to the development of knowledge, beliefs, and attitudes • Group norms that encourage questioning, challenging, and supportive interactions • A genuine collaboration among all members, including the group leader