The Nature and Development of Experts’ Strategy Flexibility for Solving Equations

1. The Nature and Development of Experts’ Strategy Flexibility for Solving Equations Jon R. StarHarvard University Kristie J. NewtonTemple University PME-NA

2. Thanks to… • Research assistants at Temple University and Harvard University for help with all aspects of this work • Seed grant from Temple University to Newton PME-NA

3. Plan for this talk • Background • Strategy flexibility • Development of flexibility • Nature of flexibility • Current study which explored the nature and development of experts’ flexibility • Implications of this work PME-NA

4. BACKGROUND PME-NA

5. Strategy flexibility • Flexibility not a consistently defined construct • Same as adaptability? • Ease in switching solution methods? • Selection of the most appropriate method? • In this study, flexibility defined as • knowledge of multiple solutions • ability to selectively choose the most appropriate ones for a given problem (Star, 2005; Star & Rittle-Johnson, 2008; Star & Seifert, 2006) PME-NA

6. Studies on students’ flexibility • Second grade students in a program that emphasized conceptual understanding and skill showed more flexibility in their preference and use of procedures than those in traditional programs (Blote, et. al. 2001) • Students asked to solve previously completed algebra problems in a new way showed greater use of multiple strategies than that of the control group, although accuracy was similar (Star and Seifert 2006) PME-NA

7. But what about experts? • How and when do experts become flexible? • We define experts as individuals with substantial practice extending over 10 years • Existing research on experts’ flexibility is sparse and also somewhat inconsistent PME-NA

8. Research on experts’ flexibility • Experts’ approach to solving problems is not only efficient, but also determined by the characteristics of the problem (Cortéz, 2003) • However, the primary difference between experts and less able solvers was not the knowledge and use of multiple strategies, but that they make fewer errors (Carry et. al. 1979) • Existing research on experts does not explore the development of flexibility PME-NA

9. Our questions • What are the conditions by which experts’ flexibility emerged? • Do experts attribute their flexibility to prior instruction? • How did experts become flexible? • Do experts believe instruction has a role? • Do experts consistently use the strategies they prefer or do they sometimes use standard algorithms? PME-NA

10. CURRENT STUDY PME-NA

11. Rationale • Attempts to address current weaknesses in the literature on experts’ flexibility • Designed tasks to provide opportunities for experts to demonstrate flexibility • Probed experts about their approaches • Asked experts to reflect on the emergence of this capacity in their own learning • Included experts from different fields PME-NA

12. Research questions • What is the nature of experts’ flexibility for solving algebra problems? • Use of multiple strategies • Knowledge of multiple strategies • Preferences for certain strategies • How do experts become flexible, and what are their views about the role of instruction in developing flexibility? PME-NA

13. Participants • Eight experts in school algebra • 2 mathematicians, 2 mathematics educators, 2 secondary mathematics teachers, and 2 engineers • Experts’ education range from bachelor’s degrees to doctorate degree • 5 of the experts were male and 3 were female • Experts chosen to provide a range of perspectives on flexibility and potentially different approaches to solving problems PME-NA

14. Measures • 55-item algebra test • Originally designed as a final exam for a three-week summer course for high school students, adapted to assess flexibility • Covered solving and graphing both linear and quadratic equations as well as simplifying expressions with exponents and square roots • Interview about a sub-set of the items PME-NA

15. Sample test questions PME-NA

16. Interviews with experts • How did they solve a subset of the problems? • Why did they chose the strategies they used? • What do they mean when they used terms such as “easy” and “better”? • Do they know of other ways to solve the problems? • How did they become flexible? • Should or could flexibility be taught in schools? PME-NA

17. Procedure • One-on-one setting with the experts • Focus on the methods they used to solve the problems, asked them to show work • Test was not timed, but most completed within 20 minutes • Interviews were conducted immediately following the completion of the test PME-NA

18. Results • Nature of experts’ flexibility • Choice of strategies • Failure to choose optimal strategy • Development of experts’ flexibility PME-NA

19. Nature of experts’ flexibility • Experts in the study rarely made errors • Exhibited knowledge of and use of multiple strategies for solving a range of problems • Generally used and/or expressed a preference for the most efficient strategies for a given problem PME-NA

20. Choice of strategies • Overwhelmingly based on ease • “Faster, quicker, less steps” • Minimizing effort • Avoiding fractions • Considered structure and features of problem • Complete squares • Divisibility • Etc. PME-NA

21. Example of non-optimal strategy: • Two experts distributed the first step instead of dividing both sides by 7 PME-NA

22. Not the most efficient strategy… • “Not thinking” – just “blew through it” • Used well-practiced, automated approaches • Experts agreed that in general, a less efficient strategy was the result of not looking carefully at particular structures/features of a problem PME-NA

23. Avoiding fractions and other pitfalls • Experts had a preference for efficient or elegant strategies even if they did not use these strategies to solve the problem • Experts preferred to combine like terms, even when they multiplied by 3 first to avoid calculating with fractions PME-NA

24. Experts’ reflections on flexibility • Considered themselves to be flexible • Did not believe flexibility was an overt instructional goal for their K-12 instructors • Believed their own flexibility emerged naturally from seeing problems over and over again • Their search for easiest strategies was based on their own initiative PME-NA

25. Flexibility as a result of teaching • Experts attributed the development of their flexibility to their own teaching practice • Explaining problems in multiple ways to struggling students • Exposure to the idiosyncratic, original, or even erroneous strategies that students produce PME-NA

26. Is flexibility important to experts? • Most believe that flexibility is an integral part of doing mathematics • However, the importance of teaching students flexibility is mixed • Flexibility can be taught to help students understand mathematics more deeply • Teaching for flexibility may confuse students and students should learn from trial and error PME-NA

27. Summary • Experts exhibited flexibility, overwhelmingly chose strategies based on ease • Considered specific features of problems and chose elegant strategies based on these • Agreed on which strategies were optimal • Yet did not always select optimal strategies, despite knowledge of and preference for them PME-NA

28. Remaining Questions • How does an expert’s facility with arithmetic interact with concerns about accuracy and ease? • What does “easier” really mean? • ‘‘faster, quicker, less steps’’ • ‘‘It’s not about extra steps. I don’t mind putting in extra steps if extra steps makes it easier’’ PME-NA

29. IMPLICATIONS PME-NA

30. Implications for Education • Experts were in agreement that flexibility was not an explicit focus of their K-12 and university mathematics experience • Should flexibility be an instructional focus in K-12? PME-NA

31. Personal development of flexibility • One interpretation of the experts’ views would be that flexibility should not be an instructional target K-12 • Best developed implicitly and individually • Requires a significant amount of personal initiative, including a propensity to seek easier and quicker strategies • Available only to those with significant mathematical talent and drive PME-NA

32. Flexibility through instruction • Another interpretation is that flexibility is attainable and critical for students of all ability levels • K-12 mathematics instruction has likely changed a lot since the experts have attended K-12 • May be more possible now to consider flexibility as an instructional goal for all students • Are experts critiquing the lack of mathematical instruction they have experienced in the past? PME-NA

33. Helping teachers teach flexibility • Teachers’ own flexibility need to be addressed first (Yakes and Star, in press) • “so focused on one solution method” • “never taken the time to express why or even let the students suggest why” • Gains in teachers’ conceptual knowledge does not yield flexibility (Newton, 2008) • Flexibility needs to be a specific goal of teacher education PME-NA

34. Implications for research • Prior work with students shown that knowledge of innovative strategies often precedes the ability to implement these strategies (Star & Rittle-Johnson, 2008) • This study and others suggest that experts do not always use the most efficient strategy for solving a given problem, even when it is clear that they know the most efficient strategy PME-NA

35. Implications for research • Tasks that assess students’ flexibility • Interviews conducted to accompany problem solving • Different kinds of problems incorporated to better assess participants’ flexibility • Asked participants to solve the same problem in more than one way and to identify which strategy was optimal • Other innovative tasks and methodologies are needed to investigate strategy choices and the development of flexibility PME-NA

36. Conclusion • Strategy flexibility is an important mathematical instructional goal at all levels • Emergence of research base on students’ flexibility • Experts’ flexibility is yet mostly unexplored • Experts agree that flexibility was not emphasized in their own learning, but the experts demonstrated and valued efficient and elegant strategies to solve algebra problems PME-NA

37. Thank you! Kristie Newton Temple Universitykjnewton@temple.edu Jon R. Star Harvard UniversityJon_Star@harvard.edu Star, J.R., & Newton, K.J. (in press). The nature and development of experts' strategy flexibility for solving equations. ZDM - The International Journal on Mathematics Education. PME-NA