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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. Acknowledgements.

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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


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


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


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


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


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


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


Current studies

  • Comparison condition

    • compare and contrast alternative solution methods

  • Sequential condition

    • study same solution methods sequentially

AERA Presentation, Chicago


Comparison condition

AERA Presentation, Chicago


next page

next page

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Sequential condition

AERA Presentation, Chicago


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


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


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


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


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


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


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


Study 2 Results

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

Gain scores post - pre

AERA Presentation, Chicago


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


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


Our hypothesis

  • Recall that in Study 2:

    • Assignment to condition by class

  • Whole class discussion

AERA Presentation, Chicago


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


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


Thanks!

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

Jon Star

[email protected]

Bethany Rittle-Johnson

[email protected]


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