1. Does Comparison Support Transfer of Knowledge? Investigating Student Learning of Algebra Jon R. Star Michigan State University (Harvard University, as of July 2007) Bethany Rittle-Johnson Vanderbilt University

2. April 2007AERA Presentation, Chicago2 Acknowledgements Funded by a grant from the Institute for Education Sciences, US Department of Education, to Michigan State University Thanks also to research assistants at Michigan State: ◦ Kosze Lee, Kuo-Liang Chang, Howard Glasser, Andrea Francis, and Tharanga Wijetunge And at Vanderbilt: ◦ Holly Harris, Jen Samson, Anna Krueger, Heena Ali, Sallie Baxter, Amy Goodman, Adam Porter, and John Murphy

3. April 2007AERA Presentation, Chicago3 Comparison... Is a fundamental learning mechanism Lots of evidence from cognitive science ◦ Identifying similarities and differences in multiple examples appears to be a critical pathway to flexible, transferable knowledge Mostly laboratory studies Not done with school-age children or in mathematics (Gentner, Loewenstein, & Thompson, 2003; Kurtz, Miao, & Gentner, 2001; Loewenstein & Gentner, 2001; Namy & Gentner, 2002; Oakes & Ribar, 2005; Schwartz & Bransford, 1998)

4. April 2007AERA Presentation, Chicago4 Central tenet of math reforms Students benefit from sharing and comparing of solution methods “nearly axiomatic”, “with broad general endorsement” (Silver et al., 2005) Noted feature of ‘expert’ math instruction Present in high performing countries such as Japan and Hong Kong (Ball, 1993; Fraivillig, Murphy, & Fuson, 1999; Huffred-Ackles, Fuson, & Sherin Gamoran, 2004; Lampert, 1990; Silver et al., 2005; NCTM, 1989, 2000; Stigler & Hiebert, 1999)

5. April 2007AERA Presentation, Chicago5 Comparison support transfer? Experimental studies of learning and transfer in academic domains and settings largely absent Goal of present work ◦ Investigate whether comparison can support transfer with student learning of algebra ◦ Experimental studies in real-life classrooms

6. April 2007AERA Presentation, Chicago6 Why algebra? Students’ first exposure to abstraction and symbolism of mathematics Area of weakness for US students (Blume & Heckman, 1997; Schmidt et al., 1999) Critical gatekeeper course Particular focus: ◦ Linear equation solving 3(x + 1) = 15 Multiple strategies for solving equations ◦ Some are better than others ◦ Students tend to memorize only one method

7. April 2007AERA Presentation, Chicago7 Solving 3(x + 1) = 15 Strategy #1: 3(x + 1) = 15 3x + 3 = 15 3x = 12 x = 4 Strategy #2: 3(x + 1) = 15 x + 1 = 5 x = 4

8. April 2007AERA Presentation, Chicago8 Current studies Comparison condition ◦ compare and contrast alternative solution methods Sequential condition ◦ study same solution methods sequentially

9. April 2007AERA Presentation, Chicago9 Comparison condition

10. April 2007AERA Presentation, Chicago10 Sequential condition next page

11. April 2007AERA Presentation, Chicago11 Predicted outcome Students in the comparison condition will make greater procedural knowledge gains, familiar and transfer problems By the way, there were other outcomes of interest in these studies, but the focus of this talk is on procedural knowledge, especially transfer.

12. April 2007AERA Presentation, Chicago12 Procedural knowledge measures Intervention equations 1/3(x + 1) = 15 5(y + 1) = 3(y + 1) + 8 Familiar equations -1/4(x - 3) = 10 5(y - 12) = 3(y - 12) + 20 Transfer equation 0.25(t + 3) = 0.5 -3(x + 5 + 3x) - 5(x + 5 + 3x) = 24

13. April 2007AERA Presentation, Chicago13 A tale of two studies... Study 1 ◦ Rittle-Johnson, B. & Star, J.R. (in press). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology. Study 2 ◦ not yet written up

14. April 2007AERA Presentation, Chicago14 Study 1: Method Participants: 70 7th grade students Design ◦ Pretest - Intervention - Posttest ◦ Intervention during 3 math classes ◦ Random assignment of student pairs to condition ◦ Studied worked examples with partner ◦ Solved practice problems on own ◦ No whole class discussion

15. April 2007AERA Presentation, Chicago15 Study 1: Results ComparisonSequential Procedural knowledge 34 * 24 Familiar 3527 Transfer 32 ~ 20 Gain scores post - pre; * p <.05 ~ p =.08 Comparison students were more accurate equation solvers for all problems ◦ almost significant when looking at transfer problems by themselves

16. April 2007AERA Presentation, Chicago16 Study 1 Strategy use Solution MethodComparisonSequential Conventional.61 ~.66 Demonstrated non-standard.17 *.10 Solution Method at Posttest (Proportion of problems) ~ p =.06; * p <.05 Comparison students more likely to use non-standard methods and somewhat less likely to use the conventional method

17. April 2007AERA Presentation, Chicago17 Study 2: Method Participants: 76 students in 4 classes Design: ◦ Same as Study 1, except ◦ Random assignment at class level ◦ Minor adjustments to packets and assessments ◦ Whole class discussions of partner work each day

18. April 2007AERA Presentation, Chicago18 Study 2 Results ComparisonSequential Procedural knowledge 3435 Familiar 4542 Transfer 2228 Gain scores post - pre No condition difference in equation solving accuracy, on familiar or transfer problems

19. April 2007AERA Presentation, Chicago19 Study 2 Strategy use Solution MethodComparisonSequential Conventional.17 *.38 Demonstrated non-standard.73 +.48 Solution Method at Posttest (Proportion of problems) * p <.05 ; + After controlling for pretest variables, the estimated marginal mean gains were.67 and.55, respectively, and there was no little of condition (p =.12) Comparison students less likely to use conventional methods No difference in use of non-standard methods

20. April 2007AERA Presentation, Chicago20 In Study 2 Advantage for comparison group on problem solving accuracy disappears ◦ Condition effect on transfer problems disappears Use of non-standard methods equivalent across conditions ◦ Sequential students much more likely to use non- standard approaches in Study 2 than in Study 1 Why?

21. April 2007AERA Presentation, Chicago21 Our hypothesis Recall that in Study 2: ◦ Assignment to condition by class Whole class discussion

22. April 2007AERA Presentation, Chicago22 Discussion  comparison Multiple methods came up during whole class discussion Sequential students benefited from comparison of methods ◦ Even though teacher never explicitly compared these methods in sequential classes Legitimized use of non-standard solution methods ◦ As evidence by their greater use in Study 2 in both conditions, but especially sequential

23. April 2007AERA Presentation, Chicago23 Closing thoughts Studies provide empirical support for benefits of comparison in classrooms for learning equation solving Whole class discussion, which inadvertently or implicitly promoted comparison, led to greater use of non-standard methods and also eliminated condition effects for procedural knowledge gain

24. Thanks! Jon Star jonstar@msu.edu Bethany Rittle-Johnson Bethany.Rittle-Johnson@vanderbilt.edu You can download this presentation and other related papers and talks at www.msu.edu/~jonstar