Mathematical flexibility: What is it, why it is important, and how can it be studied? Jon R. Star Michigan State University 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! March 17, 2005 UDel SoE Colloquium 2 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) March 17, 2005 UDel SoE Colloquium 3 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 March 17, 2005

UDel SoE Colloquium 4 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 March 17, 2005 UDel SoE Colloquium 5 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 didnt 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 March 17, 2005 UDel SoE Colloquium 6 Explanation #1 Student learned algorithm by rote but didnt 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 March 17, 2005 UDel SoE Colloquium 7 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 March 17, 2005 UDel SoE Colloquium 8 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)

March 17, 2005 UDel SoE Colloquium 9 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 March 17, 2005 UDel SoE Colloquium 10 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 March 17, 2005 UDel SoE Colloquium 11 Strategies for 4(x + 5) = 80. Tabular x 1

5 10 15 March 17, 2005 4(x + 5) 24 40 60 80 UDel SoE Colloquium 12 Strategies for 4(x + 5) = 80.. Graphical 100 80 60 40 y=4(x+5)

y=80 20 0 -20-10 -6 -2 2 6 10 14 -40 March 17, 2005 UDel SoE Colloquium

13 Strategies for 4(x + 5) = 80... Informal (unwinding) 80 divided by 4 is 20 20 minus 5 is 15 so x is 15 March 17, 2005 UDel SoE Colloquium 14 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 March 17, 2005 UDel SoE Colloquium 15 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 March 17, 2005 UDel SoE Colloquium 16 More formally, flexibility is 1)

Knowledge of how to use and apply multiple strategies for completing mathematical problems 2) 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 March 17, 2005 UDel SoE Colloquium 17 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 March 17, 2005 UDel SoE Colloquium 18 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) March 17, 2005 UDel SoE Colloquium 19 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 March 17, 2005 UDel SoE Colloquium 20 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 Blte (2001), Torbeyns (2005) huge and current literature on students strategies for solving problems such as 4+5 Missing: symbolic problems from the algebra curriculum March 17, 2005 UDel SoE Colloquium 21 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) March 17, 2005 UDel SoE Colloquium 22 Flexibility and symbolic probs? Many view these problems as algorithmic Just because an algorithm exists doesnt mean that

Students shouldnt know other ways to solve the problems The algorithm is always the best way One doesnt havent be strategic in deciding how to approach problems March 17, 2005 UDel SoE Colloquium 23 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 March 17, 2005 UDel SoE Colloquium 24 Challenges How Recent study on students conceptions of best strategies for solving equations How to define adaptive to assess flexibility Competence vs. performance and choice of evaluative tasks

March 17, 2005 UDel SoE Colloquium 25 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 students strategy choice is difficult unless one knows students conception of best March 17, 2005 UDel SoE Colloquium

26 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)

March 17, 2005 UDel SoE Colloquium 27 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? March 17, 2005 UDel SoE Colloquium 28 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

March 17, 2005 UDel SoE Colloquium 29 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) March 17, 2005 UDel SoE Colloquium 30 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 March 17, 2005 UDel SoE Colloquium 31 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 youve 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 March 17, 2005 UDel SoE Colloquium 32 Students responses to best Category Shortest way (fewest steps) Quickest or fastest way

Easiest, least complicated, or least confusing way Most accurate way, fewest errors Way Im more comfortable with Way that is most neatly written Depends on solver, problem, goals March 17, 2005 UDel SoE Colloquium % 65% 61% 61% 57% 30% 9% 9% 33 Categorizing best responses Sophisticated

expressed more typical views emphasizing quickest, shortest, least complicated Naive conceptions of best 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%) March 17, 2005 UDel SoE Colloquium 34 Responses into categories Sophisticated Shortest way (fewest steps) Quickest or fastest way Easiest, least complicated way Depends on solver, problem, goals

March 17, 2005 Naive Most accurate way, fewest errors Way Im more comfortable with Way that is most neatly written UDel SoE Colloquium 35 Conceptions of best Sophisticated 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 conceptions 70% (16 of 23) students mentioned at least one of the naive features of best (accuracy, confidence, neatness) March 17, 2005 UDel SoE Colloquium 36 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) March 17, 2005 UDel SoE Colloquium 37 It depends as sophisticated

Probably the easiest way possible for her. ... So if she called me on the phone at home one night and shes 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 Ill be like, well, thats great, but I dont 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 doesnt really matter to her, she can just go through each problem the longest way possible, but at least she would know Ive only moved one thing so you dont have to do anything else. (Cathy) March 17, 2005 UDel SoE Colloquium 38 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%, its the right answer and youve done your best. March 17, 2005 UDel SoE Colloquium 39 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? March 17, 2005 UDel SoE Colloquium 40 Flexibility in action Even assuming sophisticated conceptions of best, assessing adaptive choice of

strategies is tricky Most idealistic case: In 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 practice, this does not occur frequently March 17, 2005 UDel SoE Colloquium 41 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 March 17, 2005 UDel SoE Colloquium 42

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) March 17, 2005 UDel SoE Colloquium 43 Problems and strategies Sample 3(x + 1) + 9(x + 1) = 6(x + 1) 2(x + 3) + 4(x + 3) = 24

Sample problems 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 March 17, 2005 UDel SoE Colloquium 44

Performance 44% of students who saw an explicit demonstration of CV used this strategy on at least one problem on the posttest March 17, 2005 UDel SoE Colloquium 45 Competence Other types of tasks were included on the post-test to capture students competence or ability to use the CV strategy March 17, 2005 UDel SoE Colloquium

46 Competence. 1) 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 March 17, 2005 UDel SoE Colloquium 47 Competence.. 2)

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 March 17, 2005 UDel SoE Colloquium 48 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 March 17, 2005 UDel SoE Colloquium 49 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 March 17, 2005 UDel SoE Colloquium 50 Wrapping up Flexibility is knowledge of how to use multiple strategies and the ability to adaptively select the most appropriate strategies for completing specific problems Flexibility offers an alternative solution to the problem of rote learning of procedures

Not only develop connections to underlying concepts but also develop deeper, more strategic, more flexible knowledge of the procedure Research on flexibility in symbolic algebra procedures is particularly needed March 17, 2005 UDel SoE Colloquium 51 Wrapping up. Challenges include

inherent in studying flexibility Students conceptions of best methods impact their ability to adaptively select appropriate strategies Competence vs. performance: A range of tasks, not just problem solving, need to be used to assess flexibility March 17, 2005 UDel SoE Colloquium 52 This presentation and other related papers can be downloaded at: www.msu.edu/~jonstar Jon R. Star Michigan State University [email protected]