Does Comparison Support Transfer of Knowledge? Investigating Student Learning of Algebra

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. 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 AERA Presentation, Chicago

3. 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) AERA Presentation, Chicago

4. 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) AERA Presentation, Chicago

5. 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 AERA Presentation, Chicago

6. 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 AERA Presentation, Chicago

7. 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 AERA Presentation, Chicago

8. Current studies • Comparison condition • compare and contrast alternative solution methods • Sequential condition • study same solution methods sequentially AERA Presentation, Chicago

9. Comparison condition AERA Presentation, Chicago

10. next page next page next page Sequential condition AERA Presentation, Chicago

11. 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. AERA Presentation, Chicago

12. 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 AERA Presentation, Chicago

13. 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 AERA Presentation, Chicago

14. 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 AERA Presentation, Chicago

15. Study 1: Results • Comparison students were more accurate equation solvers for all problems • almost significant when looking at transfer problems by themselves Gain scores post - pre;*p < .05 ~ p = .08 AERA Presentation, Chicago

16. Study 1 Strategy use • Comparison students more likely to use non-standard methods and somewhat less likely to use the conventional method Solution Method at Posttest (Proportion of problems) ~p = .06; * p < .05 AERA Presentation, Chicago

17. 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 AERA Presentation, Chicago

18. Study 2 Results • No condition difference in equation solving accuracy, on familiar or transfer problems Gain scores post - pre AERA Presentation, Chicago

19. Study 2 Strategy use • Comparison students less likely to use conventional methods • No difference in use of non-standard methods 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) AERA Presentation, Chicago

20. 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? AERA Presentation, Chicago

21. Our hypothesis • Recall that in Study 2: • Assignment to condition by class • Whole class discussion AERA Presentation, Chicago

22. 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 AERA Presentation, Chicago

23. 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 AERA Presentation, Chicago

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