1. NCTM Presession 2005 1 Research Issues in Developing Strategic Flexibility: What and How Presenters: Christine Carrino Gorowara  University of Delaware Dawn Berk  University of Delaware Christina Poetzl  University of Delaware Jon R. Star  Michigan State University Susan B. Taber  Rowan University Discussant: John K. Lannin  University of Missouri-Columbia

2. NCTM Presession 2005 2 Symposium Overview Audience Task Introduction Description of Projects  MSU Project  UD Project Challenges of Researching Strategic Flexibility Discussant’s Comments Audience Feedback

3. NCTM Presession 2005 3 Audience Task Problem #1: Find x if 4(x + 5) = 80. Problem #2: Joan drove 200 miles in 3.5 hours. How far can she drive in 14 hours? For each problem—  Solve, using any strategy you like.  Solve again, using a different strategy.  Determine which strategy is better, and why.  Try to change the problem so that the other strategy is now better.

4. NCTM Presession 2005 4 Problem #1: Find x if 4(x + 5) = 80 4(x + 5) = 80 x + 5 = 20 x = 15 4(x + 5) = 79 x + 5 = 79/4 x = 59/4 Strategy 1: Strategy 2: Strategy 1 with changed problem: Strategy 2 with changed problem: 4(x + 5) = 80 4x + 20 = 80 4x = 60 x = 15 4(x + 5) = 79 4x + 20 = 79 4x = 59 x = 59/4 [Find x if 4(x + 5) = 79]

5. NCTM Presession 2005 5 What counts as different? Different number of steps  3 lines? 4 lines? Different sequence of steps  Distribute first? Divide first? Other characteristics?

6. NCTM Presession 2005 6 Problem #2: Joan drove 200 miles in 3.5 hours. How far can she drive in 14 hours? ? x 3.5 hours = 14 hours 4 x 3.5 hours = 14 hours 4 x 200 miles = 800 miles Joan drives 200/3.5 mph. In 14 hours, she drives 14 x 200/3.5 = 800 miles. ? x 5 = 14 hours 14/5 x 5 = 14 hours 14/5 x 200 miles = 560 miles Strategy 1: Strategy 2: Strategy 1, with changed problem: Strategy 2, with changed problem: Joan drives 200/5 mph, or 40 mph. In 14 hours, she drives 14 x 40 miles, or 560 miles. [Joan drove 200 miles in 5 hours…]

7. NCTM Presession 2005 7 What counts as different? Different number of steps Different multiplicative relationships used  Scale factor between two sets of hours vs. scale factor between hours and miles Other characteristics?

8. NCTM Presession 2005 8 Introduction Many problems can be solved with a variety of strategies Important goal for students is to develop flexibility in the use of strategies, which means that they:  Know multiple strategies for solving a class of problems  Select from among those strategies the most appropriate for solving a particular problem Audience task demonstrated your flexibility

9. NCTM Presession 2005 9 Defining Flexibility Proficiency in executing a range of strategies AND Ability and disposition to choose wisely among those strategies with respect to a particular goal

10. NCTM Presession 2005 10 Related Work Our notion of “flexibility” is related to but distinct from other terms  Adaptive expertise (Baroody & Dowker)  Procedural fluency, strategic competence (NRC, 2001)  Conceptual knowledge, procedural knowledge (Hiebert, 1986) Informed by research on:  Strategy development in developmental psychology (e.g., Siegler )  Problem solving (e.g., Schoenfeld, Silver)  Strategy choice in arithmetic (e.g., Baroody, Fuson, Carpenter)

11. NCTM Presession 2005 11 Importance of Flexibility When flexible, students are more successful on transfer problems (e.g., Resnick, 1980; Schwartz & Martin, 2004; Carpenter et al, 1998) When not flexible, teachers are less likely to promote flexibility in their students (Hines and McMahon, 2005)

12. NCTM Presession 2005 12 Terms in this Talk Variations in terms across two projects  MSU: Flexibility Appropriate or Best  UD: Strategic Flexibility Wise/Unwise

13. NCTM Presession 2005 13 Challenges (Preview) #1: What strategies are different? #2: What strategies are “best”? #3: How can we tell when a student is flexible? #4: How do we develop strategic flexibility?

14. NCTM Presession 2005 14 MSU Project Description

15. NCTM Presession 2005 15 MSU Project Team PI: Jon Star Graduate research assistants at MSU:  Howard Glasser  Mustafa Demir  Kosze Lee  Beste Gucler  Kuo-Liang Chang Collaborator :  Bethany Rittle-Johnson, Vanderbilt University

16. NCTM Presession 2005 16 My Research Paradigm 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 Conduct problem solving interviews to explore rationales behind strategy choices Implement and evaluate instructional interventions

17. NCTM Presession 2005 17 Instructional Interventions Alternative ordering task  Students asked to re-solve previously completed problems using a different ordering of steps (Star, 2001; 2002) Explicit strategy instruction  Strategy instruction is provided after students have achieved basic fluency and differentiated domain knowledge (Schwartz & Bransford,1998)

18. NCTM Presession 2005 18 Method 134 6th graders (83 girls, 51 boys) 5 1-hour classes in one week (Mon - Fri) Class size 8 to 15 students; worked individually Pre-test (Mon), post-test (Fri) Domain was linear equation solving 3(x + 1) = 12 2(x + 3) + 4(x + 3) = 24 9(x + 2) + 3(x + 2) + 3 = 18x + 9

19. NCTM Presession 2005 19 Instruction 30 minute benchmark lesson  Combine like terms, add to both sides, multiply to both sides, distribute How to use each step individually  Not shown how to chain together steps No strategic or goal-oriented instruction  Not told when to use a step No worked examples of solved equations

20. NCTM Presession 2005 20 Alternative Ordering Treatment Random assignment by class  AO treatment vs. AO control Solve this problem again, but using a different ordering of steps AO control group solved new but isomorphic problem 3(x + 1) = 12 4(x + 2) = 24

21. NCTM Presession 2005 21 Explicit Strategy Instruction Random assignment by class  Strategy instruction vs. no strategy instruction At start of 2nd problem solving class, 3 worked examples presented to strategy instruction classes  “This is the way I solve this equation.”  Problems solved with atypical, ‘better’ strategy  No notes taken by students Total time was 8 minutes of supplemental instruction

22. NCTM Presession 2005 22 Assessing Flexibility When I work on equations, I always use the same steps, in the same order (true/false) Figure out ALL possible NEXT steps that can be done on 2(x + 3) + 6 + 4x + 8 = 4(x + 2) + 6x + 2x Use the “combine like terms” step on 2(x + 1) + 5(x + 1) = 14 Given this partially solved equation, what step did the student use to go from the first line to the second line?

23. NCTM Presession 2005 23 Results AO Treatment students significantly more flexible  54% treatment  41% control Strategy instruction led to significantly more flexibility  53% strategy instruction  45% no strategy instruction No significant interaction effect

24. NCTM Presession 2005 24 UD Project Description

25. NCTM Presession 2005 25 UD Project Associates Research Collaborators:  Jim Hiebert  Yuichi Handa Instructional Collaborators:  Eric Sisofo  James Beyers  Laurie Goggins

26. NCTM Presession 2005 26 Context of the Study Domain: Missing-value proportion problems Participants: Pre-service K-8 teachers (n = 148)  Familiar with missing-value proportion problems  Many identified cross-multiplication as THE strategy for solving missing-value proportion problems Setting: Mathematics content course  Semester-long focus on rational number concepts  4 proportional reasoning lessons taught at end of semester

27. NCTM Presession 2005 27 Example of Students’ Thinking It takes 9 minutes to read 10 pages. How many minutes will it take to read 500 pages? The criteria the group used to decide "best" strategy was instructive. They seemed to choose strategies that had a more formal appearance or ones that were similar to what someone remembered having learned before. In solving one problem, a student suggested early in the conversation that:  because 10 pages would take 9 minutes to read, and  because 500 pages is 50 times that number of pages,  500 pages should take 50 times longer to read, and  50 x 9 = 450, so it would take 450 minutes to read 500 pages.

28. NCTM Presession 2005 28 Example of Students’ Thinking (cont’d) After the student explained this several times so the others understood, the group wondered whether this was right. They had convinced themselves that the answer was right but wondered whether this is what they should be doing. "I'm not sure this is right because I don't think it's even a method." "Yeah, it seems too easy." "I think we should do it another way.” The group ended up writing a proportion and using cross- multiplication, congratulating the student who thought of this because they all agreed this looked much better. They were surprised to find they got the same answer both ways (even though they seemed convinced that the first strategy had given them the right answer).

29. NCTM Presession 2005 29 Instructional Goals Develop a recognition of, appreciation for, and ability to use multiple strategies  Cross-multiplication  Unit rate  Scale factor  Scaling up/down Develop an ability to analyze a given problem and choose “wisely” from among a range of strategies  Both computational and conceptual benefits

30. NCTM Presession 2005 30 Examples It takes 9 minutes to read 10 pages. How many minutes will it take to read 500 pages? x 50 500 pages: x minutes Scale Factor Strategy: 10 pages: 9 minutes 500 pages: 450 minutes = 450 minutes 500 pages x 9/10 minutes per page 9/10 minutes per (1) page Unit Rate Strategy:

31. NCTM Presession 2005 31 Instructional Design Multiple strategies were identified and named. Students were asked to compare and contrast strategies for solving a given problem. Students were encouraged to choose strategies that capitalized on particular number relationships in the problem.

32. NCTM Presession 2005 32 Data Collection Pre/Post Tests (n = 148), Delayed Post Test (n = 53)  6 items: Solve using 1 strategy  2 items: Solve using 2 strategies Pre/Post Interviews (n = 22)  4 items: Solve using 1 strategy  1 item: Solve, given first step of strategy  1 item: Choose “best” among 3 worked solutions

33. NCTM Presession 2005 33 Coding Scheme Correctness Three measures of flexibility  Number of strategies used across problems  “Wise” choice of strategy on a given problem  Ability to solve same problem using multiple strategies

34. NCTM Presession 2005 34 Results Significant increase in: Number of problems solved correctly Number of strategies used across all problems Number of problems on which a “wise” strategy was used Use of multiple strategies on a given problem

35. NCTM Presession 2005 35 Comparing our Projects MSUUD Instructional Goals No discussion of strategy choice “Wise” use an explicit focus Populations6th-graders Prospective teachers Students’ Prior Knowledge No prior instruction on symbolic approaches Competence in domain Students’ Prior Values No knowledge of established values Value for “formal” strategies

36. NCTM Presession 2005 36 Challenges of Researching Strategic Flexibility

37. NCTM Presession 2005 37 Challenges #1: What strategies are different? #2: What strategies are “best”? #3: How can we tell when a student is flexible? #4: How do we develop strategic flexibility?

38. NCTM Presession 2005 38 Challenge #1: What strategies are different? Different ways of representing the solution (e.g., graphically vs. symbolically) Number of lines/steps Different sequence of steps Steps ‘chunked’ vs. not ‘chunked’ Different structural elements (e.g., number relationships) used

39. NCTM Presession 2005 39 Addressing Challenge #1: What strategies are different? MSU:  Different sequence of steps UD:  Different number relationships used

40. NCTM Presession 2005 40 Joan drove 200 miles in 3.5 hours. How far can she drive in 14 hours? 4  Joan will drive 4 times as many hours. This means she can drive 4 times as many miles, or 4 x 200 = 800 miles Joan drove 200/3.5 miles in one hour This means she can drive 14 x 200/3.5 miles in 14 hours. 14 (200) = 3.5 ( x ) Scale Factor Cross-Multiplication Unit Rate  200/3.5  200/3.5

41. NCTM Presession 2005 41 Reasons for our Choices MSU: Different sequence of steps  Relatively simple way to begin looking at multiple strategies  Bottom-up classification of “different” UD: Different number relationships  Different computations result  Different concepts about proportions are illuminated (e.g.,that the scale factors are equal, that the unit rates are equal, that the cross-products are equal)  Top-down classification of “different”

42. NCTM Presession 2005 42 Challenge #2: What strategies are “best”? For what purpose(s)?  Speed  Accuracy  Generalizability  Preference  Elegance  Conceptual illumination For whose purpose(s)?  Learner  Instructor/Researcher  Discipline

43. NCTM Presession 2005 43 Addressing Challenge #2: What strategies are “best”? MSU:  Strategies with fewer steps  Strategies with low cognitive load  Bottom-up classification of “best” UD:  Strategies taking advantage of simple (whole-number) multiplicative relationships between related values  Top-down classification of “best”

44. NCTM Presession 2005 44 Reasons for our Choices MSU:  Least mental effort  Efficiency/Elegance (Disciplinary values) UD:  Least mental effort  Accuracy  Conceptual illumination

45. NCTM Presession 2005 45 Challenge #3: How can we tell when a student is flexible? Competence vs. Performance: False Negative?  Lack of variety in strategies does not necessarily indicate an inability to use multiple strategies Compliance vs. Disposition: False Positive?  Greater variety of strategies following instruction does not necessarily indicate a disposition to be flexible

46. NCTM Presession 2005 46 Addressing Challenge #3: How can we tell when a student is flexible? MSU:  Competence vs. Performance: Creative assessments  Compliance vs. Disposition: Not so much of an issue UD:  Compliance vs. Disposition: Assessments over time  Competence vs. Performance: Not so much of an issue

47. NCTM Presession 2005 47 Challenge #4: How do we develop strategic flexibility? Awareness of other strategies and competence in executing other strategies are necessary, but not sufficient Prior knowledge may play a role

48. NCTM Presession 2005 48 Addressing Challenge #4: How do we develop strategic flexibility? Draw attention to strategy choice… Manipulate problem features  Problems must be complex enough to have multiple different solutions, yet simple enough so that the students can solve them  Structure of problem and number choice should highlight appropriateness of various strategies

49. NCTM Presession 2005 49 Addressing Challenge #4 (continued): How do we develop strategic flexibility? Draw attention to strategy choice… Name strategies or strategy elements  Legitimizing effect "I'm not sure this is right because I don't think it's even a method."  Reifying effect For example: performance of students in AO treatment

50. NCTM Presession 2005 50 Closing Thoughts Studying strategic flexibility involves making a set of judgments about what counts as different and what counts as best. Measuring strategic flexibility involves measuring students’ perceptions, motivations, and intentions in addition to strategy choice, and/or measuring strategy choice over time. Developing strategic flexibility involves shifting students’ focus from the solution product to the solution process.

51. NCTM Presession 2005 51 Thank You! Christine Carrino Gorowara  cargoro@udel.edu Dawn Berk  berk@udel.edu Christina Poetzl  cpoetzl@udel.edu Jon R. Star  jonstar@msu.edu Susan B. Taber  taber@rowan.edu

52. NCTM Presession 2005 52 What counts as different? [Examples with Original Problem] Different relationships between the given values being used: Strategy 1 uses the relationship between number of hours already driven and number of hours to be driven. 4  (# hours already driven) = # hours to be driven, so 4  (# miles already driven) = # miles to be driven Strategy 2 uses the relationship between number of hours already driven and number of miles already driven. 200/3.5  (# hours already driven) = # miles already driven, so 200/3.5  (# hours to be driven) = # miles to be driven

53. NCTM Presession 2005 53 What counts as different? [Examples with Changed Problem] Different relationships between the given values being used: Strategy 1 uses the relationship between number of hours already driven and number of hours to be driven. 14/5  (# hours already driven) = # hours to be driven, so 14/4  (# miles already driven) = # miles to be driven Strategy 2 uses the relationship between number of hours already driven and number of miles already driven. 40  (# hours already driven) = # miles already driven, so 40  (# hours to be driven) = # miles to be driven