Exploring Adaptive and Representational Expertise

1. Exploring Adaptive and Representational Expertise Short Oral Discussant Remarks Jon R. Star Michigan State University

2. Common questions • What strategies or representations should students know how to use when solving proportion problems? • How do we interpret students’ failure to use strategies or representations intelligently or adaptively? • How do we assess students’ knowledge of and use of particular strategies and representations? PME-NA 2005

3. 1. What strategies or representations should students know how to use when solving proportion problems?

4. 1. Known strategies. • Very detailed framework of strategies in Alatorre & Figueras (A & F) • Framework for representations and concepts in Ledesman & Alvarez (L & A) • Difficult for me to get a handle on from brief short oral papers • (A bit more clear after hearing short oral presentations) • I would push authors about the following: PME-NA 2005

5. 1. Known strategies.. (a) To what extent does categorization of strategies in A & F help in exploring a particular issue? • ‘Landscape’ of A & F strategy domain can be divided up into 2, 5, 10, 21 different strategies. How to map this landscape is not an exercise in itself but rather is a tool for exploring a particular issue. • In the little that I read/heard from A & F, connection between how the landscape of strategies is mapped and the problem that is being explored is not particularly clear. PME-NA 2005

6. 1. Known strategies... • For example in A & F: simple (centrations or relations) vs. compound. Centrations can be on totals, on antecents, on consequents. Relations can be order, subtractive, or proportionality. Proportionality relations can be semi-formal or formal. Compound can take four forms. • How is difference between centration and relation (or semi-formal and formal relations) critical to issue under exploration? PME-NA 2005

7. 1. Known strategies..... • In L & A, similar issue in classification scheme using registers and concepts • Three registers (table, graph, numbers) • verbal? hybrid representations? • Concepts - classification scheme not well-articulated • what is a proportion concept? • “concepts and meanings” - what is the difference? • Do these classification schemes help you to explore particular issues? Are there better classification schemes? PME-NA 2005

8. 1. Known strategies.... (b) Reliability of framework • To what extent is the framework for classifying strategies or representations one that someone other than you, if trained, could use reliably? • Is this framework useful for the field, or primarily for you? PME-NA 2005

9. Summary - known strategies • What strategies or representations should students know how to use when solving proportion problems? (a) To what extent does categorization of strategies or representations help in exploring a particular issue? (b) Reliability of framework PME-NA 2005

10. 2. How do we interpret students’ failure to use strategies or representations intelligently or adaptively?

11. 2. Interpret failure to use. • What do we think it means when someone doesn’t use strategies or representations in the way that an expert does? • Consider Vicente in A & F • PhD in Chemistry • “Behaved as a routine expert” - used same strategy on almost all problems PME-NA 2005

12. 2. Interpret failure to use.. • Was Vicente’s routine expertise a concern? • Likely not, because • It apparently didn’t prevent Vicente from doing the kinds of math problems he does in his work • It apparently didn’t prevent Vicente from doing the A & F problems correctly either • No evidence that Vicente’s purported routine expertise affected his ability to problem solve in this study PME-NA 2005

13. 2. Interpret failure to use... • Is ability to vary strategies according to context and structure (a key feature of adaptive expertise, or procedural flexibility, in my work) a benefit in itself? • Or is it beneficial because it enables one to solve problems that a less flexible or more routine expert is not able to? • If the latter, where is the evidence that this is the case in Vicente? PME-NA 2005

14. 2. Interpret failure to use.... • Case of Nuria in L & A • She demonstrated an ability to make connections across the registers, which enabled her to construct knowledge of the concepts • Ability to use multiple registers, rather than rely upon a single register - why is this useful? • If using one representation only allowed Nuria to solve “photograph” problem, is this a concern? PME-NA 2005

15. 2. Summary - failure to use • Perhaps not sufficient to merely say that adaptive expertise (using multiple strategies) is better than routine, or using multiple representations is better than reliance on a single one • Need to demonstrate the limits of routine expertise and reliance on single representations • What kinds of tasks cannot be done when one has only these less-desirable outcomes? PME-NA 2005

16. 3. How do we assess students’ knowledge of and use of particular strategies and representations?

17. 3. Assessing knowledge. • Need to demonstrate benefits to problem solving of use of multiple strategies and representations • Give tasks where routine experts have trouble, but adaptive experts do not • Give problems where reliance on a single representation leads to difficulty, but use of multiple representations leads to solutions • Coming up with such tasks is difficult but critical PME-NA 2005

18. 3. Assessing knowledge.. • Examples of tasks that “directly” assess benefits of adaptive expertise: • Problem itself includes multiple representations, and participant must use information from different representations to complete problem • We can be creative about the problems that we ask students to do PME-NA 2005

19. 3. Assessing knowledge... • Another level of creativity involves creating tasks that “indirectly” assess benefits of knowledge of multiple strategies • Solve a problem in more than one way • Show a problem solved with an unfamiliar strategy and then ask participant to interpret, analyze, or use this new strategy • Show two problem strategies and solutions and ask which one is better and why • Show a very hard worked example and see which students learn from it PME-NA 2005

20. 3. Assessing knowledge.... • Keep the case of Vicente in mind • Weakens claim that adaptive expertise is goal if you say that (a) PhD in chemistry has routine expertise, and (b) his problem-solving performance (accuracy) is not different than someone with adaptive expertise • If other kinds of tasks used, either Vicente does more poorly because of routine expertise, or... PME-NA 2005

21. 3. Assessing knowledge..... • Vicente might have adaptive expertise, but assessment did not push him to demonstrate it • Did current assessment tap performance (what Vicente chose to do on these problems) or competence (what he was capable of doing)? • Did he have knowledge of multiple strategies but choose not to use them, because it was not necessary to do so? PME-NA 2005

22. 3. Summary - assessment • Create tasks that show limits of routine expertise • Choose tasks that indirectly assess benefits of knowledge of multiple strategies and representations • In addition to assessing performance, choose tasks that specifically address competence PME-NA 2005

23. This presentation and other related papers can be downloaded at: www.msu.edu/~jonstar Jon R. Star Michigan State University jonstar@msu.edu