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Contrasting Examples in Mathematics Lessons Support Flexible and Transferable Knowledge. Bethany Rittle-Johnson Vanderbilt University Jon Star Michigan State University. Comparison helps us notice distinctive features. Benefits of Contrasting Cases.

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Contrasting examples in mathematics lessons support flexible and transferable knowledge l.jpg

Contrasting Examples in Mathematics Lessons Support Flexible and Transferable Knowledge

Bethany Rittle-Johnson

Vanderbilt University

Jon Star

Michigan State University



Benefits of contrasting cases l.jpg
Benefits of Contrasting Cases and Transferable Knowledge

  • Perceptual Learning in adults (Gibson & Gibson, 1955)

  • Analogical Transfer in adults (Gentner, Loewenstein & Thompson, 2003)

  • Cognitive Principles in adults (Schwartz & Bransford, 1998)

  • Category Learning and Language in preschoolers (Namy & Gentner, 2002)

  • Spatial Mapping in preschoolers (Loewenstein & Gentner, 2001)


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Extending to the Classroom and Transferable Knowledge

  • How to adapt for use in K-12 classrooms?

  • How to adapt for mathematics learning?

  • Better understanding of why it helps


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Current Study and Transferable Knowledge

  • Compare condition: Compare and contrast alternative solution methods vs.

  • Sequential condition: Study same solution methods sequentially


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Target Domain: Early Algebra and Transferable Knowledge

Star, in press


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Predicted Outcomes and Transferable Knowledge

  • Students in compare condition will make greater gains in:

    • Problem solving success (including transfer)

    • Flexibility of problem-solving knowledge (e.g. solve a problem in 2 ways; evaluate when to use a strategy)


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Translation to the Classroom and Transferable Knowledge

  • Students study and explain worked examples with a partner

  • Based on core findings in cognitive science -- the advantages of:

    • Worked examples (e.g. Sweller, 1988)

    • Generating explanations (e.g. Chi et al, 1989; Siegler, 2001)

    • Peer collaboration (e.g. Fuchs & Fuchs, 2000)


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Method and Transferable Knowledge

  • Participants: 70 7th-grade students and their math teacher

  • Design:

    • Pretest - Intervention - Posttest

    • Replaced 2 lessons in textbook

    • Intervention occurred in partner work during 2 1/2 math classes

      • Randomly assigned to Compare or Sequential condition

      • Studied worked examples with partner

      • Solved practice problems on own


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Intervention: and Transferable KnowledgeContent of Explanations

  • Compare: “It is OK to do either step if you know how to do it. Mary’s way is faster, but only easier if you know how to properly combine the terms. Jessica’s solution takes longer, but is also ok to do.”

  • Sequential: “Yes [it’s a good way]. He distributed the right number and subtracted and multiplied the right number on both sides.”


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Intervention: and Transferable KnowledgeFlexible Strategy Use

  • Practice Problems: Greater adoption of non-standard approach

    • Used on 47% vs. 25% of practice problem, F(1, 30) = 20.75, p < .001


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Gains in Problem Solving and Transferable Knowledge

F(1, 31) =4.88, p < .05


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Gains in Flexibility and Transferable Knowledge

  • Greater use of non-standard solution methods

    • Used on 23% vs. 13% of problems, t(5) = 3.14,p < .05.


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Gains on Independent Flexibility Measure and Transferable Knowledge

F(1,31) = 7.51, p < .05


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Summary and Transferable Knowledge

  • Comparing alternative solution methods is more effective than sequential sharing of multiple methods

    • In mathematics, in classrooms


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Potential Mechanism and Transferable Knowledge

  • Guide attention to important problem features

    • Reflection on:

      • Joint consideration of multiple methods leading to the same answer

      • Variability in efficiency of methods

    • Acceptance & use of multiple, non-standard solution methods

    • Better encoding of equation structures


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Educational Implications and Transferable Knowledge

  • Reform efforts need to go beyond simple sharing of alternative strategies


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It pays to compare! and Transferable Knowledge


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Assessment and Transferable Knowledge

  • Problem Solving Knowledge

    • Learning: -1/4 (x– 3) = 10

    • Transfer: 0.25(t + 3) = 0.5

  • Flexibility

    • Solve each equation in two different ways

    • Looking at the problem shown above, do you think that this way of starting to do this problem is a good idea? An ok step to make? Circle your answer below and explain your reasoning.


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Assessment and Transferable Knowledge

  • Conceptual Knowledge


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Explanations During Intervention and Transferable Knowledge

Difference between groups

** p < .01; * p < .05


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