1. Discussant Remarks Jon R. Star Michigan State University

2. Tues April 11, 2006AERA 2006 San Francisco2 My role... provide constructive criticism push your work forward I’ve chosen to assume role of journal reviewer allows for a conversation between reviewer and reviewee (I know that these are ‘just’ AERA papers!)

3. Tues April 11, 2006AERA 2006 San Francisco3 For starters... papers were great! quality was appropriate for discussant-as- reviewer comments 3 of 4 full and complete papers submitted on website prior to deadline! apologies to Moreno and colleagues - no comments on your paper in my remarks

4. Tues April 11, 2006AERA 2006 San Francisco4 Amy Ellis paper anticipated outlet: Journal for Research in Mathematics Education decision: reject, but strongly encourage revision and resubmission, taking into account my comments

5. Tues April 11, 2006AERA 2006 San Francisco5 1. Clarify phenomena distinction between types of generalization is not sufficiently clear –presence of real-life context? –or nature of reasoning? particularly important because curriculum A looks a lot like many that we think are good! –yet it did not necessarily support kind of generalization that you think is productive

6. Tues April 11, 2006AERA 2006 San Francisco6 1. Clarify phenomena what is quantitative reasoning? –“creating rules that had little quantitative meaning for them” what features of curriculum A do or do not support quantitative reasoning?

7. Tues April 11, 2006AERA 2006 San Francisco7 2. Clarify importance why does it matter that students reason quantitatively in this study? what does it enable students to do that other forms of generalization do not? evidence presented is somewhat weak in absence of good evidence, distinction is interesting but not obviously educationally relevant

8. Tues April 11, 2006AERA 2006 San Francisco8 2. Clarify importance is the goal to have students reason quantitatively, or by reasoning quantitatively does this allow students to do other things? –The study is not clear, either theoretically or empirically, on this issue

9. Tues April 11, 2006AERA 2006 San Francisco9 3. Clarify claims a comparison study? yes or no? method section says no but results do make comments about comparison

10. Tues April 11, 2006AERA 2006 San Francisco10 3. Clarify claims “This study’s results suggest that reasoning with quantitative relationships can support more sophisticated mathematical activity, which is a claim suggested by researchers but as yet rarely backed by empirical work examining students’ generalizations.” does this study’s empirical results support this suggestion, since it wasn’t comparative?

11. Tues April 11, 2006AERA 2006 San Francisco11 3. Clarify claims ‘Ellis (2006) found that middle school students who were pushed to produce generalizations focused on quantitative reasoning (as opposed to number patterns and procedures) ultimately were more likely to develop understanding of linearity.’ does your evidence support the claims made with this sentence and citation?

12. Tues April 11, 2006AERA 2006 San Francisco12 Overall... interesting, promising, and important paper you have a clear and well-articulated research agenda and this paper would be another important contribution

13. Tues April 11, 2006AERA 2006 San Francisco13 Rozy Brar paper anticipated outlet: Mathematical Thinking and Learning decision: Revise and resubmit good (or bad) news: reviewer thinks a lot about conceptual and procedural knowledge! comments and suggested revisions revolve around this issue

14. Tues April 11, 2006AERA 2006 San Francisco14 1. Know vs. understand? What is the difference (for you) between knowledge and understanding? –you use conceptual and procedural understanding, rather than knowledge –not an esoteric point for this paper –future work to do clinical interviews to assess conceptual and procedural knowledge - how you assess these competencies is critical to this and future studies, so you need to be clearer

15. Tues April 11, 2006AERA 2006 San Francisco15 1. Know vs. understand? What does conceptual knowledge (or understanding?) look like in this domain? What conceptually is missing from the students that you interviewed? Not enough to say that students don’t understand what they are doing What exactly don’t they understand? –additional clarity on this would strength paper

16. Tues April 11, 2006AERA 2006 San Francisco16 2. Issue to think more about possible to ‘proceduralize’ almost anything what is the potential value of things such as area models, manipulatives, algebra tiles, real-life contexts, various software packages? conceptual representational crutch - an alternative representation that aims to give students insight into the conceptual underpinnings (the ‘why’) of a procedure

17. Tues April 11, 2006AERA 2006 San Francisco17 2. Issue to think more about is a conceptual representational crutch sufficient? your study says NO some teachers don’t take full advantage of the connections to underlying concepts that the conceptual representational crutch affords

18. Tues April 11, 2006AERA 2006 San Francisco18 2. Issue to think more about is a conceptual representational crunch necessary? can a teacher provide accessible conceptual explanations for procedural actions to students without such a crutch? yes - we see it a lot in TIMSS videos

19. Tues April 11, 2006AERA 2006 San Francisco19 2. Issue to think more about what is the potential value of the conceptual representational crutch? we want students to understand why they do what they do - are some representations better at achieving this goal than others? the conceptual representational crutch often becomes just another way to do the problem

20. Tues April 11, 2006AERA 2006 San Francisco20 3. Situate work in context excellent job of connecting work to broader rational number literature seems related to investigation of children’s fraction schemes - Les Steffe, Amy Hackenberg, Eric Tillema

21. Tues April 11, 2006AERA 2006 San Francisco21 4. Policy issues of work problem: students fail to develop conceptual knowledge worse problem: students fail to see the value of conceptual knowledge worst problem: we are not sufficiently clear in our research to document the benefits of having conceptual knowledge

22. Tues April 11, 2006AERA 2006 San Francisco22 4. Policy issues of work what can students do (or say) when they understand that they cannot when they don’t understand? Do you have empirical evidence that this is the case in your study?

23. Tues April 11, 2006AERA 2006 San Francisco23 Overall... interesting particularly (to me) for the way you push on (my words) the proceduralization of a conceptual representational crutch

24. Tues April 11, 2006AERA 2006 San Francisco24 Van Dooren paper anticipated outlet: Learning and Instruction decision: accept, pending revisions very tight study - nice contribution to a productive line of research on this topic by authors

25. Tues April 11, 2006AERA 2006 San Francisco25 1. Explanation for posttest? analogical reasoning literature (Gentner) –difficulties that learners face in determining the similarities between a new problem and a previously solved one transfer literature (Gick & Holyoak) how might meaningful, performance-based tasks be used AND impact posttest performance on traditional tasks?

26. Tues April 11, 2006AERA 2006 San Francisco26 2. Time on task potential confound between conditions? P spent most time on task just reading P problem takes longer? time differences small enough - need to explain away this possibility

27. Tues April 11, 2006AERA 2006 San Francisco27 3. Clarify analysis what is a “contrast analysis” more detail and clarity on the statistical tests that you used

28. Tues April 11, 2006AERA 2006 San Francisco28 Overall... good work! impressed generally with the work that has come out of this Center at University of Leuven

29. Tues April 11, 2006AERA 2006 San Francisco29 Closing... view my remarks as suggestions –feel free to disagree or challenge me if you do pursue publication, I am happy to read additional drafts of your work, if you would find this helpful

30. Thanks! (Questions?) Jon Star Michigan State University jonstar@msu.edu www.msu.edu/~jonstar