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.

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does comparison support transfer of knowledge investigating student learning of algebra

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
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
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
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
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
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
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
Current studies
  • Comparison condition
    • compare and contrast alternative solution methods
  • Sequential condition
    • study same solution methods sequentially

AERA Presentation, Chicago

comparison condition
Comparison condition

AERA Presentation, Chicago

sequential condition

next page

next page

next page

Sequential condition

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predicted outcome
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
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
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
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
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
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
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
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
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
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?

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our hypothesis
Our hypothesis
  • Recall that in Study 2:
    • Assignment to condition by class
  • Whole class discussion

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discussion comparison
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
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

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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