Contrasting Examples in Mathematics Lessons Support Flexible and Transferable Knowledge

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Contrasting Examples in Mathematics Lessons
Support Flexible and Transferable Knowledge

• Bethany Rittle-Johnson
• Vanderbilt University
• Jon Star
• Michigan State University

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Comparison helps us notice distinctive features
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Benefits of Contrasting Cases

• 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

• 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

• Compare condition Compare and contrast
  alternative solution methods vs.
• Sequential condition Study same solution
  methods sequentially

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Target Domain Early Algebra
Star, in press
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Predicted Outcomes

• 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

• 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

• 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 Content of Explanations

• Compare It is OK to do either step if you know
  how to do it. Marys way is faster, but only
  easier if you know how to properly combine the
  terms. Jessicas solution takes longer, but is
  also ok to do.
• Sequential Yes its a good way. He
  distributed the right number and subtracted and
  multiplied the right number on both sides.

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Intervention Flexible Strategy Use

• Practice Problems Greater adoption of
  non-standard approach
• Used on 47 vs. 25 of practice problem, F(1, 30)
  20.75, p lt .001

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Gains in Problem Solving
F(1, 31) 4.88, p lt .05
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Gains in Flexibility

• Greater use of non-standard solution methods
• Used on 23 vs. 13 of problems, t(5) 3.14,p
  lt .05.

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Gains on Independent Flexibility Measure
F(1,31) 7.51, p lt .05
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Summary

• Comparing alternative solution methods is more
  effective than sequential sharing of multiple
  methods
• In mathematics, in classrooms

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Potential Mechanism

• 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

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

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It pays to compare!
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Assessment

• 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

• Conceptual Knowledge

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Explanations During Intervention
Difference between groups p lt .01 p lt .05