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

2. PME-NA 20052 Common questions 1. What strategies or representations should students know how to use when solving proportion problems? 2. How do we interpret students’ failure to use strategies or representations intelligently or adaptively? 3. How do we assess students’ knowledge of and use of particular strategies and representations?

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

4. PME-NA 20054 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:

5. PME-NA 20055 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.

6. PME-NA 20056 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?

7. PME-NA 20057 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?

8. PME-NA 20058 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?

9. PME-NA 20059 Summary - known strategies 1. 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

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

11. PME-NA 200511 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

12. PME-NA 200512 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

13. PME-NA 200513 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?

14. PME-NA 200514 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?

15. PME-NA 200515 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?

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

17. PME-NA 200517 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

18. PME-NA 200518 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

19. PME-NA 200519 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

20. PME-NA 200520 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...

21. PME-NA 200521 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?

22. PME-NA 200522 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

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