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

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

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

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  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


  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


  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


  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 Jon R. Star Harvard 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