Mathematical flexibility: What is it, why it is important, and how can it be studied?

1. Mathematical flexibility: What is it, why it is important, and how can it be studied? Jon R. Star Michigan State University

2. Plan for this talk • Mostly a theoretical talk with some empirical results sprinkled in • Flexibility represents a new direction or emphasis in research on students’ learning of mathematics • Since not bound by constraints of a 12-minute conference talk, I will devote more attention to defining and motivating this construct • Time for questions at the end, and hopefully answers! UDel SoE Colloquium

3. Definitions of terms • Procedure • a step-by-step plan of action for doing something • Strategy • a plan of action • I use these two terms interchangeably to mean the same thing (a la Siegler) UDel SoE Colloquium

4. A strategy is either a • Heuristic • a helpful procedure for arriving at a solution; a rule of thumb • Algorithm • a procedure that is deterministic; when one follows the steps in a predetermined order, one is guaranteed to reach the solution UDel SoE Colloquium

5. An example from TIMSS • TIMSS 2003; 8th graders • If 4(x + 5) = 80, then x = ? • Hong Kong: 90% • Korea, Singapore: 82% • Chinese Taipei, Japan: 80% • 10 countries between 60% and 79% • US: 57% • Note that this problem is • “routine” - an algorithm exists • other strategies also can be used successfully • more on this in a moment UDel SoE Colloquium

6. Why did US do so poorly? • A lot of ways to answer this question • sociological, political, cultural, psychological, anthropological lenses • focus on teachers, schools, policy, Standards, curriculum, learners • Focus here on explanations relating to student cognition • Why didn’t the average student remember the algorithm or a strategy for solving this problem? • assume the average student did cover this material but still got the problem wrong UDel SoE Colloquium

7. Explanation #1 • Student learned algorithm by rote but didn’t really understand it, and thus forgot it • If instruction had emphasized the underlying concepts that are related to the algorithm, procedural knowledge would have been more tightly connected to the conceptual knowledge and thus the algorithm would have been more likely to be remembered UDel SoE Colloquium

8. This may be true, but... • I find the ‘underlying concepts’ explanation to be vague and only weakly supported by empirical research (and almost exclusively at the elementary school level) • At least as important as more connected knowledge of ‘underlying concepts’ is developing more and deeper knowledge of the procedures UDel SoE Colloquium

9. Explanation #2 • If the problem is students’ rote knowledge of procedure, a solution is for students to develop deeper, more strategic knowledge of the procedure • Students should • know multiple ways that problems can be approached • know which of these ways are most productive or appropriate for particular problem variations (and why) UDel SoE Colloquium

10. Focus: Procedural knowledge • With deeper, more strategic, more flexible knowledge of procedures, students will be more likely to remember and use the strategies that they learned UDel SoE Colloquium

11. Strategies for 4(x + 5) = 80 • Symbolic #1 (‘standard’ algorithm) 4(x + 5) = 80 4x + 20 = 80 4x = 60 x = 15 • Symbolic #2 4(x + 5) = 80 x + 5 = 20 x = 15 UDel SoE Colloquium

12. Strategies for 4(x + 5) = 80. • Tabular UDel SoE Colloquium

13. Strategies for 4(x + 5) = 80.. • Graphical UDel SoE Colloquium

14. Strategies for 4(x + 5) = 80... • Informal (unwinding) 80 divided by 4 is 20 20 minus 5 is 15 so x is 15 UDel SoE Colloquium

15. Which strategy is best? • Does ‘best’ mean fewer steps? quickest to implement? easiest? • What if the problem were changed to: • 4(x + 5) = 79 • 4.124(3.51x + 5.795) = 80.102 • (4/17)(x + 5) = (80/17) • 4(sin x + 0.5) = 0 • ‘Best’ depends on the individual (preferences, what is automatic, goals) as well as on the problem to be solved UDel SoE Colloquium

16. Flexibility • A solver who knows multiple approaches and can choose among these approaches depending on which ones she thinks are best is going to be more likely to remember at least one way to solve this problem when it is seen on TIMSS (e.g., Resnick, 1980; Schwartz & Martin, 2004; Carpenter et al, 1998) • Such a solver has deeper knowledge of the procedures or strategies for solving this class of problems or has mathematical flexibility UDel SoE Colloquium

17. More formally, flexibility is • Knowledge of how to use and apply multiple strategies for completing mathematical problems • multiple tools in the toolbox • Ability to adaptively select the most appropriate strategies for completing a specific problem • knowledge of how to select most appropriate tool for a given task UDel SoE Colloquium

18. Comparing to related terms • Procedural and conceptual knowledge • Popularized as a result of this colloquium series 20 years ago and a subsequent book (Hiebert, 1986) • Longer conversation about why I think these terms are problematic (Star, JRME, forthcoming) • Flexibility seems to fall between the cracks UDel SoE Colloquium

19. Comparing to similar terms... • “No sooner than we propose definitions for conceptual and procedural knowledge and attempt to clarify them, we must back up and acknowledge that the definitions we have given and the impressions they convey will be flawed in some way. As we have said, not all knowledge fits nicely into one class or the other. Some knowledge lies at the intersection. Heuristic strategies for solving problems, which themselves are objects of thought, are examples.” (Hiebert & Lefevre, 1986, p. 9) UDel SoE Colloquium

20. Problems to explore flexibility? • There are multiple solution strategies • Non-trivial differences exist in qualities of the multiple strategies • efficiency, elegance, cognitive ‘overhead’, generalizability, speed in which strategy can be performed, tendency of strategy to result in error, representation used... • Choice of strategy can be evaluated as to its appropriateness • Flexibility is relevant whenever it is possible to be strategic UDel SoE Colloquium

21. Examples • Gorowara, Berk, & Poetzl (Delaware): missing value proportion problems • Lannin (Missouri): pre-algebra pattern recognition problems • Addition and subtraction problems: Siegler, Baroody, Fuson; recent work by Blöte (2001), Torbeyns (2005) • huge and current literature on students’ strategies for solving problems such as 4+5 • Missing: symbolic problems from the algebra curriculum UDel SoE Colloquium

22. Symbolic algebra problems • Omnipresent in any high school or middle school algebra text • ‘Proficiency’ in algebra has been shown to be an important factor in students’ future success in college math classes • Most research on these problems focuses on students’ errors, not on the diversity of their strategies (e.g., Matz, Sleeman, Carry, Lewis) UDel SoE Colloquium

23. Flexibility and symbolic probs? • Many view these problems as algorithmic • Just because an algorithm exists doesn’t mean that • Students shouldn’t know other ways to solve the problems • The algorithm is always the best way • One doesn’t haven’t be strategic in deciding how to approach problems UDel SoE Colloquium

24. My method • Work with students with minimal knowledge of strategies in problem class • Provide brief instruction with no worked-out examples and no strategic instruction • Provide minimal feedback • Observe what strategies develop • Implement and evaluate instructional interventions • Conduct problem solving interviews to explore rationales behind strategy choices UDel SoE Colloquium

25. Challenges • How to define “adaptive” • Recent study on students’ conceptions of “best” strategies for solving equations • How to assess flexibility • Competence vs. performance and choice of evaluative tasks UDel SoE Colloquium

26. Adaptive strategy selection • “Ability to adaptively select the most appropriate strategies for completing a specific problem” • Adaptive selection boils down to the ability to identify some strategies as “better” than others • Contingent on what a student thinks it means for a strategy to be good or effective • Evaluating a student’s strategy choice is difficult unless one knows student’s conception of “best” UDel SoE Colloquium

27. Little prior work on “best” views • Franke and Carey (1997) • Investigated strategies for solving “3 + 4” • 1st graders: MY strategy is the best • No recognition of efficiency • Similar findings by McClain & Cobb (2001) • Overall, students’ conceptions of best are idiosyncratic and often implicit • Issue alluded to in other research but little data available outside of elementary school (Isaacs, 1999; Schoenfeld, 1985; Taplin, 1994) UDel SoE Colloquium

28. More complex than it appears • First glance: “best” is most efficient • But what is “efficient”? • requires fewest steps • quickest to execute • requires least mental effort to execute • Characteristics may not coincide • most practiced approach is quickest and requires least mental effort, but not shortest? UDel SoE Colloquium

29. Strategies for 4(x + 5) = 80 • Symbolic #1 (‘standard’ algorithm) 4(x + 5) = 80 4x + 20 = 80 4x = 60 x = 15 • Symbolic #2 4(x + 5) = 80 x + 5 = 20 x = 15 UDel SoE Colloquium

30. More complex than it appears. • “Best” also related to: • elegance, parsimony, symmetry, coherence, simplicity, and beauty • Even mathematicians have difficulty quantifying or categorizing aesthetics, yet such judgments are primary means of evaluating work (Wells, 1990; Penrose, 1974) • Cognizance of solution aesthetics is hallmark of math expertise (Silver and Metzger, 1989) UDel SoE Colloquium

31. Method • 23 6th graders (12 males, 11 females) • Videotaped while solving equations • Five hours over five days • Pretest, 3 hours of solving, Posttest • 20 minutes of instruction on linear equation solving transformations • adding to both sides, multiplying to both sides, distributing, and combining like terms • Not shown any worked out examples • largely left to discover own strategies for problem solving in this domain UDel SoE Colloquium

32. Method. • Interviewed during problem solving: • Suppose your friend told you that he/she had solved an equation in the best possible way. What do you think he/she means by the best possible way? • How do you know when you’ve solved an equation in the best possible way? • Each student asked these two questions at the end of Day 2, Day 3, and Day 4 UDel SoE Colloquium

33. Students’ responses to “best” UDel SoE Colloquium

34. Categorizing “best” responses • “Sophisticated” conceptions of best • expressed more typical views emphasizing quickest, shortest, least complicated • “Naive” conceptions of best • expressed less typical views emphasizing confidence, neatness, and accuracy • “naive” - showing a lack of subtlety, sophistication, or critical judgment • Many students’ views of “best” had both sophisticated and naive features • Exclusively naive conceptions of best (9%) • Exclusively sophisticated conceptions (30%) • “Mixed” conceptions of best (61%) UDel SoE Colloquium

35. Responses into categories UDel SoE Colloquium

36. Conceptions of “best” • Sophisticated conceptions • 87% (20 of 23) mentioned at least one of four features of this view (shortest, quickest, least complicated, it depends) • 35% mentioned shortest, quickest, and least complicated, showing recognition that these features may be correlated • Naive conceptions • 70% (16 of 23) students mentioned at least one of the naive features of best (accuracy, confidence, neatness) UDel SoE Colloquium

37. Sophisticated example • Suppose your friend told you that he/she had solved an equation in the best possible way. What do you think he/she means by the best possible way? • “Doing it the fastest, the easiest. (Tell me more?) Like getting the answers as quick as you can, like if you need it maybe on the math quiz or something, where they are quizzing you on algebra and like stuff. They did it the best way or something, that they did it the fastest, they got the right answer. (What does easiest mean?) Like the least amount of steps.” (Helen) UDel SoE Colloquium

38. “It depends” as sophisticated • Probably the easiest way possible for her. ... So if she called me on the phone at home one night and she’s like, I have found the most easiest way to do this problem. Even though it takes more time it is so easy, you could just make sure you do not miss one step. And I’ll be like, well, that’s great, but I don’t want to be up until 11, doing my homework. So her way might be easier than my way because her whole afternoon might be blank. It doesn’t really matter to her, she can just go through each problem the longest way possible, but at least she would know I’ve only moved one thing so you don’t have to do anything else. (Cathy) UDel SoE Colloquium

39. Naive examples • Brad: “Like he used the steps right, And like he added and subtracted right.” • Melanie: “That they used each step at the right time, and got down to the correct answer.” • Oscar: “[I know my answer is the best] when I am sure, like 100%, it’s the right answer and you’ve done your best.” UDel SoE Colloquium

40. Implications for flexibility study • While many students developed at least partially sophisticated views of “best”, substantial variation existed • Most students had “mixed” conceptions • Cannot assume students hold sophisticated or expert conceptions of what makes one strategy better than another • May be necessary to ask explicitly before evaluating strategy choices • “Which of these solution methods is better and why?” UDel SoE Colloquium

41. Flexibility in action • Even assuming sophisticated conceptions of “best”, assessing adaptive choice of strategies is tricky • Most idealistic case: • Give student a problem • She solves it in the best possible way • Change problem slightly • She solves it in a different and even better way • In practice, this does not occur frequently UDel SoE Colloquium

42. Competence vs. performance • Performance • What the student chooses to do in a test situation • Competence • What the student is capable of doing • Often unintentionally, we assess performance but we are interested in competence • Assessing competence requires creative kinds of tasks UDel SoE Colloquium

43. Method • Same procedure and tasks as “best” study • 134 6th graders (83 girls, 51 boys) • Class size 8 to 15 students • Students worked individually; no interviews • Could solve 10-11 equations per day • Pre-test, post-test, delayed post-test (6 months later) UDel SoE Colloquium

44. Problems and strategies • Sample problems • 3(x + 1) + 9(x + 1) = 6(x + 1) • 2(x + 3) + 4(x + 3) = 24 • Sample strategies • Standard algorithm - distribute first, then combine like terms • Change in variable (CV) - combine (x + a) terms first • A variety of instructional conditions, including explicit demonstration of the CV strategy UDel SoE Colloquium

45. Performance • 44% of students who saw an explicit demonstration of CV used this strategy on at least one problem on the posttest UDel SoE Colloquium

46. Competence • Other types of tasks were included on the post-test to capture students’ competence or ability to use the CV strategy UDel SoE Colloquium

47. Competence. • Combine the 2(x + 1) and the 5(x + 1) terms in this equation: 2(x + 1) + 5(x + 1) = 14 • Correct answer: 7(x + 1) = 14 • For students who saw explicit demonstration of CV, 58% right UDel SoE Colloquium

48. Competence.. • An equation is partially solved below. What step did the student use to get from the from line to the second line? 3(x + 2) + 4(x + 2) = 14 7(x + 2) = 14 • Correct answer: Combine like terms • For students who saw explicit demonstration of CV, 79% right UDel SoE Colloquium

49. Competence vs. performance. • Large competence vs. performance differences were found • When given a problem to solve, most students did not use CV • When given other tasks to determine whether students knew how to implement CV, most could do so • Similar results were found for a variety of other strategies that students could have used on selected problems UDel SoE Colloquium

50. Implications for flexibility study • Giving students a set of problems to solve and then looking at the strategies that they use in order to assess flexibility is, by itself, potentially problematic • Other types of questions and measures must be used to assess what students are capable of, not just what they do under test performance conditions UDel SoE Colloquium