1. Explaining Contrasting Solution Methods Supports Problem-Solving Flexibility and Transfer Bethany Rittle-Johnson Vanderbilt University Jon Star Michigan State University

2. Explanation is Important, But… Students often generate shallow explanations (e.g. Renkl, 1997) Generating explanations does not always improve learning (e.g. Mwangi & Sweller, 1998) How can we support effective explanation?

3. Explaining Contrasting Solution Methods Share-and-compare solution methods core component of reform efforts in mathematics (e.g. Silver et al, 2005) But does it lead to greater learning?

4. Comparison as Central Learning Mechanism Cognitive science literature suggests it is: –Perceptual Learning in adults (Gibson & Gibson, 1955) –Analogical Transfer in adults (Gentner, Loewenstein & Thompson, 2003) –Cognitive Principles in adults (Schwartz & Bransford, 1998) –Category Learning and Language in preschoolers (Namy & Gentner, 2002) –Spatial Mapping in preschoolers (Loewenstein & Gentner, 2001)

5. Extending to the Classroom Does contrasting solution methods support effective explanation in k-12 classrooms? Is it effective for mathematics learning? Does it support high-quality explanations?

6. Current Study Compare condition: Compare and contrast alternative solution methods vs. Sequential condition: Study same solution methods sequentially

7. Target Domain: Early Algebra Star & Siefert, in press

8. Predicted Outcomes Students in compare condition will –Generate more effective explanations –Make greater knowledge gains: Greater problem solving success (including transfer) Greater flexibility of problem-solving knowledge (e.g. solve a problem in 2 ways; evaluate when to use a strategy)

9. Method Participants: 70 7th-grade students and their math teacher Design: –Pretest - Intervention - Posttest –Replaced 2 lessons in textbook –Intervention occurred in partner work during 2 1/2 math classes Randomly assigned to Compare or Sequential condition Studied worked examples with partner Solved practice problems on own

10. Compare Condition

11. Sequential Condition --Next Page --

12. Overview of Results: Gains in Problem Solving F(1, 31) = 2.12, p <.05

13. Gains in Flexibility Greater use of non-standard solution methods –Used on 17% vs. 10% of problems *p<.05

14. Gains on Independent Flexibility Measure F(1,31) = 2.78, p <.05

15. Sample Conversation High Learning Pair

16. Sample Conversation Modest Learning Pair

17. Sample Dialogue for 5(y+1) = 3(y+1) + 8 2(y+1) = 8 (see preceding slides)

18. General Characteristics of Written Explanations

19. Explicit Comparisons

20. Summary Comparing alternative solution methods rather than studying them sequentially –Improves problem solving accuracy and flexibility –Focuses students’ explanations on the viability of multiple of solutions and their comparative efficiency.

21. How Contrasting Solutions Supports Explanation Guide attention to important problem features –Reflection on: Joint consideration of multiple methods leading to the same answer Variability in efficiency of methods –Acceptance of multiple, non-standard solution methods

22. Educational Implications Teachers need to go beyond simple sharing of alternative strategies –Support comparative explanations

23. It pays to compare!