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

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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

- 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

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- 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

- 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)

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- 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

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- 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

- Linear equation solving
- Multiple strategies for solving equations
- Some are better than others
- Students tend to memorize only one method

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Strategy #1:

3(x + 1) = 15

3x + 3 = 15

3x = 12

x = 4

Strategy #2:

3(x + 1) = 15

x + 1 = 5

x = 4

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- Comparison condition
- compare and contrast alternative solution methods

- Sequential condition
- study same solution methods sequentially

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- 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.

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- 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

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- 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

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- 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

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- 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

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- 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

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- 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

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- No condition difference in equation solving accuracy, on familiar or transfer problems

Gain scores post - pre

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- 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)

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- 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?

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- Recall that in Study 2:
- Assignment to condition by class

- Whole class discussion

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- 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

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- 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

Thanks!

You can download this presentation and other related papers and talks at www.msu.edu/~jonstar

Jon Star

Bethany Rittle-Johnson