1. 1 Cathy Carroll & Judy Mumme WestEd Developing Mathematical Knowledge for Teaching

2. 2 Purposes for Doing Mathematics in PD Some commonly identified purposes… To have a common experience for a group of people To prepare for using with students An opportunity to increase teacher content knowledge of math Begin to address particular math issue(s) Opportunity to reconstruct one’s notion of what it means to do mathematics, nature of mathematical activity Identify certain kinds of pedagogy that support learning in particular situations How do these resonate with your experience?

3. 3 Session Overview Consider a purpose of developing teachers’ mathematical knowledge for teaching (with a focus on specialized content knowledge) through  Work on a mathematics task  Watch a video  Consider next steps from the video to work on MKT with teachers

4. 4 Doing Mathematics in PD Pairs (1 minute each) What supports you in doing math in PD? What gets in the way? Small group (5 minutes) Share what things are supportive and what are not. (Note: these may be different for each person) Pairs/Small Group With this in mind, it is important to respect individual differences and provide everyone equal access in group work

5. 5 Assume the pattern continues to grow in the same manner. Find a rule or formula to determine the number of tiles in a figure of any size. Logos What are the different ways that the geometric model can be decomposed and how can those ways be connected to symbolic expressions? How do those different expressions represent the growth of this model?

6. 6 Considering the Task What are some expressions you came up with to represent the growth (un-simplified versions)? Whole Group

7. 7 Considering the Task Take each of the expressions and see if you can figure out how that group was thinking about how the model was growing  What is the relationship between these expressions and the logos model? Small Group

8. 8 Considering the Task How did you map the expressions to the logos model? Whole Group

9. 9 Considering Purpose What potential mathematical ideas could teachers work on through the use of this task?  Why would those be good goals for teachers? Would you do that with students?  How might one’s purposes be similar/different there? Whole Group

10. 10 Mathematical Knowledge for Teaching (MKT) Frame: knowledge “entailed by the work of teaching”  Knowledge used or needed in practice What do we mean by “knowledge”?  Mathematical knowledge, skills, habits of mind What do we mean by the “work of teaching”?  The activities in which teachers engage, and the responsibilities they have, to teach mathematics, both inside and outside of the classroom

11. 11 Pedagogical Content Knowledge Common Content Knowledge (CCK) Specialized Content Knowledge (SCK) Knowledge of Content and Students (KCS) Knowledge of Content and Teaching (KCT) Subject Matter Knowledge Knowledge at the mathematical horizon Knowledge of curriculum Mathematical Knowledge for Teaching Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal for Teacher Education, 59(5), 389-407.

12. 12 Common Content Knowledge (CCK) Mentally find the answer to 92-56: 92-56 = 36 Knowing how to estimate that it is greater than 30 and less than 40

13. 13 What meanings of subtraction might each represent? Do these methods always work? If so, why do they work? 56 = 50 + 6 92 – 50 = 42 42 - 6 = 36 56 = 52 + 4 92 – 52 = 40 40 – 4 = 36 92 = 80 + 12 80 – 50 = 30 12 – 6 = 6 30 + 6 = 36 92 = 32 + 60 (32 + 60) – 56 32 + (60 – 56) 32 + 4 = 36 +4 +30 +2 56 60 90 92 Specialized Content Knowledge (SCK) 92 - 56

14. 14 92 - 56 Pedagogical Content Knowledge Knowledge of content and students (KCS)  Knowing which approaches students are likely to use  Knowing which ways of decomposing the numbers are likely to lead to confusion (e.g. rounding the two numbers in different directions - 92 to 90 and 56 to 60 Knowledge of content and teaching (KCT)  Choosing numbers which invite particular approaches or stumbles  Choosing contexts or models to illustrate different approaches  In a whole-class, choosing which approaches or methods you want to pursue and in what order

15. 15 Tasks of Teaching Mathematics That Require SCK Unpacking and decomposing mathematical ideas Explaining and guiding explanation Using mathematical language and notation Generating examples Making mathematical practices explicit Choosing and using representations Comparing the affordances of different representations or methods Analyzing and interpreting alternative solutions Analyzing errors Interpreting and evaluating alternative solutions and thinking Analyzing mathematical treatments in textbooks

16. 16 SCK in Logos What SCK might it be possible to develop using the Logos task?

17. 17 SCK in Logos How did our framing of the task focus your attention on SCK? In what ways did our discussion of the task enable consideration of SCK? Our framing of the task: What are the different ways that the geometric model can be decomposed and how can those ways be connected to symbolic expressions? How do those different expressions represent the growth of this model?

18. 18 Pause Point What issues does this session have you thinking about now?  How are those related to your practice?

19. 19 We’ll start back promptly at 10:30 BREAK

20. 20 We are looking at this We are here

21. 21 Context Group of 27 high school and middle school teachers from 6 districts Session 7 of of an ongoing series--8 Saturday sessions over the school year Small groups have worked on the logos task and posted charts with their solutions 3 groups have shared their work, and Shamshir is asked to talk about his poster Mike is the PD leader We drop in here

22. 22 Mike and the teachers are offering us a gift of allowing us to carefully examine a real instance of practice. We are examining their practice, not critiquing them. Caveat

23. Mike

24. 24 Viewing the Video We will watch the clip one time, then use it as a jumping off spot for connecting to practice, with a focus on helping teachers develop SCK

25. 25 Frame for Viewing What approaches are teachers sharing? Suggestion: Use transcript to think about issues after viewing the video

26. What approaches are teachers sharing?

27. 27 Small Group Discussion What approaches were teachers sharing?

28. 28 Whole Group Discussion How did the approaches that were shared relate to the approaches we came up with?

29. 29 Connecting to Practice Assume a goal of developing teachers’ understandings of how various ways of decomposing or transforming the model relate to different symbolic expressions--possibly including how they show aspects of quadratic growth (i.e., square, linear, and constant components)  How would you frame the next steps in sharing?  What solutions would you select to pursue to help develop teachers’ SCK? Why? How do they relate to your goal?  How would you sequence and connect these to achieve your goal with teachers?  How would you highlight aspects of SCK? What ideas did you consider in your process? What issues arose for you in this task? Small Group

30. 30 Whole Group Connecting to Practice What ideas did you consider in planning? What issues arose for you?

31. 31 Reflecting on the Experience What are you thinking about now with regard to developing SCK with teachers in PD? Small Group/Whole Group

32. 32 A High Leverage Purpose We’re placing our bets that a focus on developing mathematical knowledge for teaching (specifically specialized content knowledge) will better enable teachers to be effective in the classroom

33. 33 Assume the pattern continues to grow in the same manner. Find a rule or formula to determine the number of tiles in a figure of any size. Logos What are the different ways that the geometric model can be decomposed and how can those ways be connected to symbolic expressions? How do those different expressions represent the growth of this model?

34. 34 The Mathematical Task Framework (MTF)* Tasks as set up by teachers Tasks as they appear in curricular materials Tasks as enacted by teacher and students Student learning * Smith & Stein (1998) Stein, Smith, Henningsen & Silver (2000)

35. 35 The Mathematical Task Framework Adapted to PD* Tasks as set up by PD leader Tasks as they appear originally Tasks as enacted by PD leader & teachers Teacher learning * Kazemi 2009

36. 36 Unpacks, makes explicit, and develops a flexible understanding of mathematical ideas that are central to the school curriculum Opens opportunities to build connections among mathematical ideas Provokes a stumble due to a superficial “understanding” of an idea Lends itself to alternative/multiple representations and solution methods Provides opportunities to engage in mathematical practices central to teaching (explaining, representing, using mathematical language, analyzing equivalences, proving, proof analysis, posing questions, writing on the board) * Suzuka, K., Sleep, L., Ball, D. L., Bass, H., Lewis, J., & Thames, M. (in press). Designing and using tasks to teach mathematical knowledge for teaching. Association of Mathematics Teacher Educators (AMTE) Monograph. Important Features of SCK Task Design*

37. 37 Challenges of Teaching SCK Staying focused on the mathematics, and not on how to teach the math Keeping the problems focused on SCK and not just sliding to Knowledge of Content & Students or Curriculum Unpacking the mathematics sufficiently and convincingly helping teachers see what there is to learn and do Making visible the connections to the kinds of mathematical thinking, judgment, reasoning one has to do in teaching 37

38. 38 Enactment What are key questions and moves that leaders can use to keep a task focused on developing SCK?  Asking teachers to Explain their solutions to others Make correspondences between solutions and/or representations Explain someone else’s thinking Figure out what might be confusing/difficult for someone else about the problem  Having teachers Explain what is/was confusing to them Ask questions to become clearer about their colleagues’ solutions  Providing opportunities to “talk mathematics” and write on the board  Narrating how something a teacher does/says relates to or is a skill used in teaching

39. 39 To Learn More About… Learning to Lead Mathematics Professional Development Leadership Development Seminars  Contact Information Email:  ccarroll@wested.org  jmumme@wested.org Web:  WestEd.org/LLMPD  WestEd.org/PLMPD