Developing Mathematical Knowledge for Teaching through Mentor-Guided Lesson Study

1. Developing Mathematical Knowledge for Teaching through Mentor-Guided Lesson Study Kristen Bieda, Jillian Cavanna, XueyingJi Michigan State University PMENA 2013 Chicago This study was funded in part by MSU Lilly Teaching Fellowship awarded to the first author

2. The role of early field experiences in learning to teach • Early field experiences for novice teachers may involve observations, clinical interviews with students, tutoring, or leading short homework or warm-up sections of lessons (Anderson Barksdale & Hite, 2005).

3. The role of early field experiences in learning to teach • Early field experiences for novice teachers may involve observations, clinical interviews with students, tutoring, or leading short homework or warm-up sections of lessons (Anderson Barksdale & Hite, 2005). • Provide opportunity for prospective teachers to enact practices learned in teacher preparation coursework • Understand the complexity of teaching • Understand the complexity of schooling • Examine assumptions about factors related to school and student success

4. Challenges of mentoring in early field experiences • Mentors and novices rarely get time to debrief and reflect on those experiences; mentors lack involvement [the laboratory class syndrome] (Zeichner, 2010, citing Valencia et al., 2009) • Need to get mentors and novices talking about teaching, and need for mentors’ tacit knowledge about teaching to be made explicit to novices. )

5. Challenges of mentoring in early field experiences • Mentors and novices rarely get time to debrief and reflect on those experiences; mentors lack involvement [the laboratory class syndrome] (Zeichner, 2010, citing Valencia et al., 2009) • Need to get mentors and novices talking about teaching, and need for mentors’ tacit knowledge about teaching to be made explicit to novices. ) “Carefully constructed field experiences that are coordinated with campus courses are more influential and effective in supporting student teacher learning…” (Zeicher, 2010, citing Darling-Hammond, 2006; Tatto, 1996)

6. Mentor-Guided Lesson Study • Prospective secondary teachers (PSTs) attend placement classroom as pairs for four hours per week • In mentor-guided lesson study, the mentor and 2 PSTs collaboratively: • Establish goals for a lesson (content and process goals based on Common Core Standards for Mathematical Practice (CCSSI, 2010)) • Plan a lesson • Teach and Observe the enactment • Reflect, Debrief, and Revise the lesson.

7. Supports for Collaborative Inquiry 5 Collaborative Learning Logs, delivered as Google Forms Goals Development Topic Study Observation Guide Lesson Reflection Post-Lesson Debrief

8. Lesson Reflection Log List one or two observations you would like to share with the team. Be as specific as possible about the evidence of student thinking that you observed. What questions about students' mathematical thinking about this topic were raised for you through this observation? In what ways did the lesson seem effective (or ineffective) in helping students understand the main mathematical ideas in the lesson? What did you notice about how the lesson helped or hindered the team's work toward its broad lesson study goal?

9. Post-Lesson Debrief Log The major revisions your team is making are... Discuss how your team came to make decisions about the revisions. You will want to discuss what seemed to be the most important pieces of evidence collected about what students learned. Discuss your impressions of how well your team collaborated during the debrief and revision process. Also mention any aspects of how you participated in the debrief process that you would like to improve upon in the next cycle.

10. Opportunities for Developing MKT

11. Opportunities for Developing MKT

12. Opportunities for Developing MKT

13. Opportunities for Developing MKT

14. Opportunities for Developing MKT

15. Opportunities for Developing MKT

16. Opportunities for Developing MKT

17. Developing an analytic framework for measuring MKT Began with initial open-coding of written logs to identify instances where PST-Ms attended to and discussed features of instruction pertinent to those knowledge domains

18. Developing an analytic framework for measuring MKT Began with initial open-coding of written logs to identify instances where PST-Ms attended to and discussed features of instruction pertinent to those knowledge domains “Students were applying their previous knowledge of patterns to their predictions, assuming the total would double.”

19. Developing an analytic framework for measuring MKT Began with initial open-coding of written logs to identify instances where PST-Ms attended to and discussed features of instruction pertinent to those knowledge domains “Students were applying their previous knowledge of patterns to their predictions, assuming the total would double.” “I think the lesson was effective in helping the students understand approximating area under a function because they were able to both think on their own and collaborate with others.”

20. Developing an analytic framework for measuring MKT Began with initial open-coding of written logs to identify instances where PST-Ms attended to and discussed features of instruction pertinent to those knowledge domains Saw a need to relate what PST-Ms described to their opportunities to develop MKT

21. Developing an analytic framework for measuring MKT Attending to Student Thinking Began with initial open-coding of written logs to identify instances where PST-Ms attended to and discussed features of instruction pertinent to those knowledge domains Saw a need to relate what PST-Ms described to their opportunities to develop MKT Analyzing Mathematics Analyzing Teaching Moves

22. Developing an analytic framework “The teacher graphed both the function and inverse function on the same set of axes, but this is mathematically incorrect”.

23. Developing an analytic framework “Students used a chart or graph to help them understand the properties of the equation.” “I wonder how the students had the inclination that the bounce height would double if the drop height doubles.”

24. Developing an analytic framework “My accommodations of group work and reading out loud had a positive effect with the student I included it for. For instance, he spoke up during class discussion and was talking openly during group work.”

25. Developing an analytic framework “Since students were having problems plotting points, then the lesson before should have covered how to construct a graph.” “My accommodations of group work and reading out loud had a positive effect with the student I included it for. For instance, he spoke up during class discussion and was talking openly during group work.”

26. Findings:To what extent does mentor-guided lesson study promote prospective teachers’ reflections on teaching related to their MKT?

27. Aspects of PSTs’ reflections across cycles

28. Changes in PSTs’ reflections across cycles

29. Changes in PSTs’ reflections across cycles

30. Findings: How does a mentor’s prior experience in doing lesson study contribute to the development of prospective teachers’ MKT during cycles of mentor-guided lesson study?

31. Influence of mentors’ experience on PSTs’ reflections

32. Influence of mentors’ experience on PSTs’ reflections

33. Influence of mentors’ experience on PSTs’ reflections

34. Cycle 1 Cycle 2

35. Nature of responses from Cycle 1 to Cycle 2 PROMPT: • List one or two observations you would like to share with the team. Be as specific as possible about the evidence of student thinking that you observed. • Attending to Student Thinking

36. Nature of responses from Cycle 1 to Cycle 2 MT With No Experience Cycle 1: “Students seemed to be lost on what the worksheet was asking them to do.” Cycle 2: “One big observation I made throughout the whole lesson was the difficulty students were having in making their own observations during the explore activity. They had a hard time figuring out what is was that they should have been noticing to be able to come up with the rules for the discriminant.”

37. Nature of responses from Cycle 1 to Cycle 2 MT With Some Experience Cycle 2: “The students seemed confused on how to make a graph off of their data since all the times were constant. Some of the students made a bar graph with a constant 30 seconds on the x-axis and how many time they jumped on the y axis. Some groups grasped what Iwas looking for and used the x axis as time and y axis as jumping jacks but used the three averages they found for 5,10 and 30 seconds as their points on the graph.” Cycle 1: “When students were shown a positive correlation graph relating their shoe size to their height, it was interesting that students weren’t able to think of a correlation. Many students didn’t realize the concept of an outlier.”

38. Nature of responses from Cycle 1 to Cycle 2 MT With Most Experience Cycle 2: “I noticed students struggling with calculating the theoretical probabilities on the worksheet. Students made some really good observations on the question "How does a student make it to Box A? B?...G? How many heads or tails would they have to get? Does order matter?” Two students only listed one possibility for each and said that there are more possibilities in the middle. Another student noticed that order did not matter as long as you had a certain number of heads or tails you would end at a certain spot.” Cycle 1: “I think students could see how using trapezoids was more accurate than using rectangles, so they could see that using different shapes to approximate the area under the function could make a large impact on the answer and create an under estimate or over estimate… I am not sure the students were aware that making the bases 0.5 instead of 1 to create twice the amount of rectangles or trapezoids would give them a better approximation of the area even though the mentor teacher briefly discussed the idea in class.”

39. Summary • MGLS does provide an opportunity to develop MKT • MGLS heightens PSTs’ attention to student thinking • Reflection on mathematics of study lesson is minimal • Mentors’ lesson study experience correlates to higher frequencies of MKT as evident in PSTs’ reflections • Increase in responses coded as Analyzing Student Thinking in Cycle 2 is not correlated to mentors’ lesson study experience

40. Implications • More work is needed to investigate evidence of PSTs’ shift in role from Cycle 1 to Cycle 2 • Mentors find MGLS beneficial: Additionally I find discussing practice with young teachers reminds me of why I do things and helps give me additional ideas.  It may take some more time but I find it beneficial for all of us.

41. Thank you! Kristen Bieda (kbieda@msu.edu) Jillian Cavanna(cavannaj@msu.edu) XueyingJi(jixueyin@msu.edu) Lilly Teaching Fellowship program