1. Teachers’ views about multiple strategies in middle and high school mathematics Jon Star and Kathleen Lynch Harvard Graduate School of Education compare@gse.harvard.edu

2. Teaching with multiple strategies Significant body of research indicates that the practice of generating, comparing and discussing multiple strategies for solving mathematics problems has clear benefits for students’ learning and performance (e.g., Gentner & Namy, 1999; Silver, Ghousseini, Gosen, Charalambous, & Strawhun, 2005; Star & Rittle-Johnson, 2008) October 29, 2010PME-NA2

3. 3 Central precept of reform pedagogy “It is nearly axiomatic among those interested in mathematical problem solving as a key aspect of school mathematics that students should have experiences in which they solve problems in more than one way” (Silver et al., 2005, p. 288) Formalized as recommended practice in the NCTM's Curriculum Focal Points (2006) and the NRC’s report Adding it Up (2001) October 29, 2010

4. Multiple strategies in the elementary grades PME-NA 4 Prior research almost exclusively in the elementary grades Projects such as Cognitively Guided Instruction (CGI) Developmental appropriateness to accepting multiple solution strategies from young children Progression from intuitive approaches to strategies that are increasingly complex and abstract for the same sorts of problems October 29, 2010

5. 5 Multiple strategies in the elementary grades In addition to describing models of effective instruction with multiple strategies, also identified ways that a focus on multiple strategies can go awry “Serial sharing” Without the crucial opportunity to investigate, compare, and justify multiple solutions, strategies, and algorithms, the pedagogical value of presenting multiple strategies is unclear October 29, 2010PME-NA

6. 6 Multiple strategies in the middle and secondary grades Significantly less research at middle and secondary levels What would a focus on multiple strategies in the secondary grades mean? Should algebra students learn more than one way to solve linear equations and proportion problems? Should students be asked to invent, share, and compare multiple strategies for solving quadratic equations? October 29, 2010

7. Current study Research Questions 1.Why do teachers think it is important to teach multiple strategies in middle and secondary school mathematics? 2.What attitudes and concerns did teachers have about teaching with multiple strategies? 3.How did teachers report using multiple strategies in the classroom? PME-NA7October 29, 2010

8. Method PME-NA8October 29, 2010

9. Setting One-week professional development workshop held in July 2009 The goal of the workshop was to prepare participating teachers to implement a set of researcher-developed curriculum materials in their Algebra I courses Curriculum materials were developed as part of an NSF-funded research project whose goal was to 'infuse' multiple strategies into Algebra I courses PME-NA9October 29, 2010

10. Participants 13 experienced Algebra I teachers from nine middle and high schools in the Boston metropolitan area, representing a mix of urban and suburban schools 6 high school teachers; 7 middle school teachers 11 of the teachers taught in public or charter schools, and the other 2 teachers were from private schools Mean number of years of teaching experience for the thirteen teachers was 10 (range 3 to 25) PME-NA10October 29, 2010

11. Data sources Semi-structured introductory interviews that were conducted with the teachers as the first activity of the first day of the workshop In particular, the teachers were asked about whether they felt it was useful to expose students to multiple strategies while teaching math, what they believed the advantages and disadvantages were to this approach, and whether and how they used multiple strategies in their current teaching Teachers were also asked what strategies they would teach students to use for several specific mathematics problems PME-NA11October 29, 2010

12. Results PME-NA12October 29, 2010

13. Teachers’ views on the importance of teaching with multiple strategies PME-NA13October 29, 2010

14. “Everybody’s brain is different, so if you only teach it in one way, you’re going to miss some kids. So the more ways you give them to access it, the more likely the majority of them are to get it... So the more the merrier, basically.” (Bernadette) PME-NA14 Success-oriented responses: Focus on giving multiple entry points into a problem for students who learn differently October 29, 2010

15. Understanding-oriented responses PME-NA15 “I think if you can look at two sides, and make a decision of, hey, you know this doesn’t work because of this, then you know what you’re doing.” (Robert) October 29, 2010

16. PME-NA16 Teachers’ concerns about teaching with multiple strategies October 29, 2010

17. How did teachers report using multiple strategies in the classroom? PME-NA17 All teachers reported that they used multiple strategies in their teaching But only for certain topics and as time permitted Characteristics of instruction with multiple strategies: 1. Sequential presentation 2. Brief discussions 3. Focus on student preferences October 29, 2010

18. 1. Sequential presentation PME-NA18 Strategies discussed sequentially, rather than simultaneously Textbook-prescribed order Some teachers expressed concern that with this sequencing, students failed to make connections among strategies October 29, 2010

19. PME-NA19 “Usually we would separate, like we would learn substitution, then we would learn elimination, then we would learn graphing, or in some order I can’t remember. Which I was never very happy with because they never, once we were done with substitution and had gotten through the others and got to graphing, let’s say, they had forgotten how to do substitution.” (Rachel) October 29, 2010

20. 2. Brief discussions PME-NA20 There was not always time for discussions, and if any discussion were held, it would be short Time pressures of the Algebra I curriculum October 29, 2010

21. PME-NA21 “I teach each one separately, and then we have at least a small, not a lot of time in our curriculum, because the book is paced such that you’re supposed to get through most of the material... So we don’t have a lot of time to review in each unit. But we do build in some time, I’d like it to be more, where they have the opportunity to use the methods that they’re learned.” (Maxine) October 29, 2010

22. 3. Focus on student preferences PME-NA22 Primary focus of discussions would focus on students’ strategy preferences Most teachers would not discuss whether one strategy was better than another “Polling” October 29, 2010

23. PME-NA23 “I’ll do one [method], then the other. And I’ll ask them, ‘Okay, who likes this method?’ And we’ll call this Andrew’s method. ‘Who likes this method?’ And we’ll call that Kayla’s method. And then for homework, I’ll ask the kids, ‘Which method do you like, Andrew’s or Kayla’s?’ And then I’ll go over it with the method they liked better.” (Kelley) October 29, 2010

24. Use of multiple strategies for specific Algebra I curriculum topics Universal agreement about topics for which Algebra I textbooks typically present multiple strategies (e.g. systems of equations) Much less agreement among teachers about whether they would expose students to multiple strategies for solving other types of problems PME-NA24October 29, 2010

25. For example, teachers were asked, What strategy or strategies would you teach students to use for this problem? PME-NA25October 29, 2010

26. PME-NA26 Maxine’s StrategyAlternative Strategy “I’d say let’s think about the distributive property, we sometimes refer to it as the water balloon at our school, so they’d go ahead and distribute...” And are there any other ways to solve this equation that you would consider teaching students? “No, that’s typically the way I would do that one.” October 29, 2010

27. PME-NA27 If you are trying to explain to students how to simplify an expression such as, what strategy or strategies would you teach students to use? October 29, 2010

28. Naomi’s Strategy “Well, the first thing I would teach students is that it means square root 75 over square root 3. And I would tell them the rule is, point-blank, no square roots in the denominator. And the very first thing they need to do is get rid of the square root in the denominator. So then they’d multiply both top and bottom by square root 3... So that’s how I’ve taught that in the past.” So would you show any additional methods? “I don’t think so.” Alternative Strategy PME-NA28

29. Discussion PME-NA29October 29, 2010

30. To what extent did the views of the middle and high school teachers in the current study align with the views of elementary teachers as presented in the literature? PME-NA30October 29, 2010

31. PME-NA31 Substantial differences Middle/high school teachers did not express views widely held at the elementary level Teachers should accept multiple strategies from students as they progress from intuitive to abstract methods in a developmentally appropriate progression? Key pedagogical value of presenting multiple strategies is that it allows students to investigate, compare and justify diverse mathematical solutions, methods and algorithms? October 29, 2010

32. PME-NA32 On the other hand: Speed and efficiency Key pedagogical value of presenting multiple strategies is that if you keep showing students more and more strategies, at least one will be found that appeals to each learner October 29, 2010

33. How can we interpret this disconnect? Secondary teachers are not as far along on the road toward adoption of reform practices as elementary teachers are? Teaching with multiple strategies is more important in elementary than in middle and high school? Teaching with multiple strategies is important for both elementary and secondary teachers, but for different reasons? PME-NA33October 29, 2010

34. In sum... Teachers in the current study generally held positive views on teaching with multiple strategies Very different rationale for why teaching with multiple strategies is important than that commonly expressed at the elementary level Teachers’ descriptions of how they used multiple strategies in their classrooms suggested a lack of routines and scaffolding to engage students in substantial thinking about multiple approaches Future studies should observe teachers’ practice in order to see how teachers implement a multiple strategies approach with students PME-NA34October 29, 2010

35. Thanks! Jon Star and Kathleen Lynch Harvard Graduate School of Education compare@gse.harvard.edu