1. This study was funded in part by MSU Lilly Teaching Fellowship awarded to the first author

2. 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. 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. 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. 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. 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. 5 Collaborative Learning Logs, delivered as Google Forms Goals Development Topic Study Observation Guide Lesson Reflection Post-Lesson Debrief

8. 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. 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. Cycle 1 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan Lead by MT Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper Cycle 2 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan * Lead by PST- Ms Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper

11. Cycle 1 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan Lead by MT Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper Cycle 2 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan * Lead by PST- Ms Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper Cycle 1 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan Lead by MT Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper Cycle 2 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan * Lead by PST- Ms Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper

12. Cycle 1 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan Lead by MT Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper Cycle 2 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan * Lead by PST- Ms Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper

13. Cycle 1 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan Lead by MT Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper Cycle 2 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan * Lead by PST- Ms Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper

14. Cycle 1 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan Lead by MT Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper Cycle 2 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan * Lead by PST- Ms Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper

15. Cycle 1 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan Lead by MT Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper Cycle 2 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan * Lead by PST- Ms Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper

16. Cycle 1 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan Lead by MT Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper Cycle 2 1: Setting Goals *2: Planning3: Teaching4: Post-Lesson Discussion Goals Development Log Topic Study Log Observation Guide Log Mathematical Context Chart Lesson Plan * Lead by PST- Ms Post-Lesson Reflection Log Post-Lesson Debrief Log Revised lesson plan End-of-Cycle Reflection Paper

17. 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. “Students were applying their previous knowledge of patterns to their predictions, assuming the total would double.”

19. 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. 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. 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 Attending to Student Thinking Analyzing Teaching Moves Analyzing Mathematics

22. Attending to Student Thinking Analyzing Teaching Moves Analyzing mathematics related to:  students’ thinking  teaching moves Observations of:  students’ errors  student understanding  mathematical procedures  students’ prior knowledge  student engagement  student discourse  students’ ability to apply new knowledge Questioning/wondering about student thinking Analyzing teaching moves related to:  students’ errors  students’ prior knowledge.  students’ thinking  students’ mathematical procedures  students’ discourse  students’ engagement “The teacher graphed both the function and inverse function on the same set of axes, but this is mathematically incorrect”.

23. Analyzing Mathematics Attending to Student Thinking Analyzing Teaching Moves Analyzing mathematics related to:  students’ thinking  teaching moves Observations of:  students’ errors  student understanding  mathematical procedures  students’ prior knowledge  student engagement  student discourse  students’ ability to apply new knowledge Questioning/wondering about student thinking Analyzing teaching moves related to:  students’ errors  students’ prior knowledge.  students’ thinking  students’ mathematical procedures  students’ discourse  students’ engagement “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. Analyzing Mathematics Attending to Student Thinking Analyzing Teaching Moves Analyzing mathematics related to:  students’ thinking  teaching moves Observations of:  students’ errors  student understanding  mathematical procedures  students’ prior knowledge  student engagement  student discourse  students’ ability to apply new knowledge Questioning/wondering about student thinking Analyzing teaching moves related to:  students’ errors  students’ prior knowledge.  students’ thinking  students’ mathematical procedures  students’ discourse  students’ engagement “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. Analyzing Mathematics Attending to Student Thinking Analyzing Teaching Moves Analyzing mathematics related to:  students’ thinking  teaching moves Observations of:  students’ errors  student understanding  mathematical procedures  students’ prior knowledge  student engagement  student discourse  students’ ability to apply new knowledge Questioning/wondering about student thinking Analyzing teaching moves related to:  students’ errors  students’ prior knowledge.  students’ thinking  students’ mathematical procedures  students’ discourse  students’ engagement “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.” “Since students were having problems plotting points, then the lesson before should have covered how to construct a graph.”

34. Cycle 2 Cycle 1

35. 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. 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. MT With Some Experience 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.” 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 I was 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.”

38. MT With Most Experience 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.” 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.”

39. 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. 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. Kristen Bieda (kbieda@msu.edu)kbieda@msu.edu Jillian Cavanna (cavannaj@msu.edu)cavannaj@msu.edu Xueying Ji (jixueyin@msu.edu)jixueyin@msu.edu Lilly Teaching Fellowship program