Explaining Contrasting Solution Methods Supports Problem-Solving Flexibility and Transfer

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 ConversationHigh Learning Pair

16. Sample ConversationModest Learning Pair

17. Sample Dialogue for5(y+1) = 3(y+1) + 82(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!