1. Using a Model Teaching Activity to Help Teachers Learn to Use Comparison in Algebra Kristie J. Newton, Temple University Jon R. Star, Nataliia Perova Harvard University This work is supported by the National Science Foundation

3. Practice-Based Professional Development Aims to situate mathematics teacher learning within the profession of teaching (Silver, 2009; Smith, 2001) Integrates content, pedagogy, and student thinking Utilizes artifacts of teaching to create professional learning tasks (PLTs)

4. Professional Learning Tasks Artifacts/materials include "mathematical tasks, episodes of teaching, and illuminations of student thinking" (Smith, 2001, p. 10). PLTs engage teachers in aspects of their work and draw on their experiences as teachers.

5. Examples of PBPD Generally involve one or more aspects of the teaching cycle - planning, teaching, and reflecting (Smith, 2001) Examples: ▫Solve a mathematical task, read a narrative case about a teacher using a similar task, collaboratively plan lessons, debrief after lessons are taught (Silver et al., 2007) ▫Lesson Study (e.g., Perry, & Lewis, 2009)

7. Model Teaching Participants collaboratively plan a lesson, teach the lesson to a "class" of fellow teachers, discuss the demonstration lesson as a group Involves “active” viewing of a lesson – observers are also playing the role of students

8. Potential Significance Videos/narrative cases from teachers' own classrooms are especially effective (Fernandez, 2005; Lin, 2002; van Es & Sherin, 2010) but may not be available in the initial stages of a professional development program. Simulated classroom allows teachers to obtain feedback prior to trying ideas with their own students.

9. Research Questions How do teachers implement the model teaching activity? In what ways do teachers' implementation of the model teaching activity foster the integration of mathematics content, student thinking, and pedagogy?

11. Participants 13 teachers: 8 high school, 5 middle school Participated in a one-week professional development institute for algebra teachers Taught in public urban, public suburban, private suburban schools Number of years teaching ranged from 2 to 25, with an average of almost 10 years

12. Professional Development Week Goals for the week: ▫introduce teachers to comparison as a tool for fostering understanding in algebra ▫train teachers to use curriculum materials designed to help them include comparison in their teaching ▫assist teachers in finding ways the materials could be easily implemented in their own classrooms Activities ranged from solving mathematics problems to watching videos to planning and enacting lessons.

13. Worked Example Pairs (WEPs) Side-by-side presentation Discussion phases: ▫understand (what are the two methods?) ▫compare (what are the similarities and differences between the methods?) ▫make connections (why/when is one method more efficient?)

14. Our Model Teaching Activity Demonstration lesson by PD leader Collaborative lesson planning “Teach” a lesson, debrief, repeat (5 groups)

15. Data and Data Analysis Lessons were videotaped, field notes were taken Analysis ▫Phase one: writing detailed descriptions of the lessons and identifying features of the implementation. ▫Phase two: coding the lessons for mathematics content, student thinking, and pedagogy and identifying trends in the way that these were addressed within the model teaching activity.

17. Implementation of the Model Teaching Activity Lesson sequence Use of WEPs Enactment of the lesson

18. Lesson Sequence A general trend was found: ▫warm-up ▫introduction and discussion of WEPs ▫opportunities to apply the methods presented in the WEPs and draw conclusions about them ▫an “exit ticket” problem to assess understanding

19. Use of WEPs All groups included the understand, compare, and make connections phases. Some variation occurred in make connections phase, in part depending on: ▫Comparison type ▫Teachers’ school/classroom context

20. Enactment of the Lesson None of the presenting groups stayed “in character” throughout the lesson. In contrast, teachers did act like "students" when they were in that role. Initial discomfort: ▫“Are you alright with that [explanation]? It’s a good thing because you will be presenting after me!” ▫One teacher said she wouldn’t answer a question because it would have previously been addressed.

21. Integration of Mathematics, Student Thinking, and Pedagogy The presence of "students" appeared to play a particularly critical role. Also facilitated by parenthetical “asides” used to provide underlying rationale for pedagogical decisions

22. Presence of “Students” Questions: “Can you substitute that into either equation?” Comments: “Well, I noticed that when Morgan solved for y, she got two negatives and then when Alex solved for x, he only got one. And then Morgan had to plug in a negative, and like, that's just a lot of negatives.”

23. Parenthetical Asides "And I would actually, in the lesson - sorry, this is just an aside - at this point, I wouldn't finish that one. Because my whole point, with the next couple of these, is just getting the kids to think about what am I changing and why am I changing it.“ "When kids do problems like this, this is one of the main places they fall down. They see the y, and they see the other y, and they say ok, 'I have what I need. They cancel each other out.'"

25. Means of Assessing Teachers' Learning Lesson sequences indicated that teachers were able to integrate the WEPs into a typical classroom routine, making it more likely they will do so within their own classrooms. Use of WEPs suggest teachers generally understood how to implement our comparison- based lessons using the three-phase discussion model.

26. Opportunity to Practice, Learn Pedagogy from Others "It was good for me to get up and do it myself, and it was beneficial to me to see how other people do it. It was also beneficial to work in the team that I did, not simply to do it by myself." Watching others present "was incredibly helpful, and it was the springboard for a lot of ideas that are sort of percolating right now."

27. Immediate Focus on Student Thinking Prior research suggests that getting teachers to focus on student thinking can be challenging (Perry & Lewis, 2009). "I think it is good that we are doing this because it is really making me think more about what it looks like to the kids."

28. Future Studies Include larger and more diverse groups of teachers, including elementary school teachers and novice teachers Examine other contexts for the activity, including other topics and other purposes Examine actual practice to understand the ways that knowledge learned from the activity did or did not transfer to the classroom

29. kjnewton@temple.edu