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Compared to What? How Different Types of Comparison Affect Transfer in Mathematics. Bethany Rittle-Johnson Jon Star . What is Transfer?. Transfer “Ability to extend what has been learned in one context to new contexts” (Bransford, Brown & Cocking, 2000)

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Compared to what how different types of comparison affect transfer in mathematics l.jpg

Compared to What?How Different Types of Comparison Affect Transfer in Mathematics

Bethany Rittle-Johnson

Jon Star

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What is Transfer?

  • Transfer

    • “Ability to extend what has been learned in one context to new contexts” (Bransford, Brown & Cocking, 2000)

    • In mathematics, transfer facilitated by flexible procedural knowledge and conceptual knowledge

  • Two types of knowledge needed in mathematics

    • Procedural knowledge: actions for solving problems

      • Knowledge of multiple procedures and when to apply them (Flexibility)

      • Extend procedures to a variety of problem types (Procedural transfer)

    • Conceptual knowledge: principles and concepts of a domain

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How to Support Transfer:Comparison

  • Cognitive Science: A fundamental learning mechanism

  • Mathematics Education: A key component of expert teaching

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Comparison in Cognitive Science

  • Identifying similarities and differences in multiple examples is a critical pathway to flexible, transferable knowledge

    • Analogy stories in adults (Gick & Holyoak, 1983; Catrambone & Holyoak, 1989)

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

    • Negotiation Principles 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)

    • Spatial Categories in infants (Oakes & Ribar, 2005)

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Comparison in Mathematics Education

  • “You can learn more from solving one problem in many different ways than you can from solving many different problems, each in only one way”

  • (Silver, Ghousseini, Gosen, Charalambous, & Strawhun, p. 288)

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Comparison Solution Methods

  • Expert teachers do it (e.g. Lampert, 1990)

  • Reform curriculum advocate for it (e.g. NCTM, 2000; Fraivillig, Murphy & Fuson, 1999)

  • Teachers in higher performing countries help students do it (Richland, Zur & Holyoak, 2007)

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Does comparison support transfer in mathematics?

  • 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 to solve equations

    • Explore what types of comparison are most effective

    • Experimental studies in real-life classrooms

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Why Equation Solving?

  • Students’ first exposure to abstraction and symbolism of mathematics

  • Area of weakness for US students

    • (Blume & Heckman, 1997; Schmidt et al., 1999)

  • Multiple procedures are viable

    • Some are better than others

    • Students tend to learn only one method

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Study 1

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

  • Sequential condition: Study same solution methods sequentially

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.

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Compare Condition

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Sequential Condition

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Predicted Outcomes

  • Students in compare condition will make greater gains in:

    • Procedural knowledge, including

      • Success on novel problems

      • Flexibility of procedures (e.g. select non-standard procedures; evaluate when to use a procedure)

    • Conceptual knowledge (e.g. equivalence, like terms)

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Study 1 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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Procedural Knowledge Assessments

  • Equation Solving

    • Intervention: 1/3(x + 1) = 15

    • Posttest Familiar: -1/4 (x – 3) = 10

    • Posttest Novel: 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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Gains in Procedural Flexibility

  • Greater use of non-standard solution methods to solve equations

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

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Helps in Estimation Too!

  • Same findings for 5th graders learning computational estimation (e.g. About how much is 34 x 18?)

    • Greater procedural knowledge gain

    • Greater flexibility

    • Similar conceptual knowledge gain

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Summary of Study 1

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

    • In mathematics, in classrooms

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Study 2:Compared to What?

Solution Methods

Problem Types

Surface Features

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Compared to What?

  • Mathematics Education - Compare solution methods for the same problem

  • Cognitive Science - Compare surface features of different problems with the same solution

    • E.g. Dunker’s radiation problem: Providing a solution in 2 stories with different surface features, and prompting for comparison, greatly increased spontaneous transfer of the solution (Gick & Holyoak, 1980; 1983; Catrambone & Holyoak, 1989)

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Study 2 Method

  • Participants: 161 7th & 8th grade students from 3 schools

  • Design:

    • Pretest - Intervention - Posttest - (Retention)

    • Replaced 3 lessons in textbook

    • Randomly assigned to

      • Compare Solution Methods

      • Compare Problem Types

      • Compare Surface Features

    • Intervention occurred in partner work

    • Assessment adapted from Study 1

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Gains in Procedural Knowledge

Gains depended on prior conceptual knowledge

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Gains in Conceptual Knowledge

Compare Solution Methods condition made greatest gains in conceptual knowledge

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Gains in Procedural Flexibility: Use of Non-Standard Methods

Greater use of non-standard solution methods in Compare Methods and Problem Type conditions

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  • Comparing Solution Methods often supported the largest gains in conceptual and procedural knowledge

  • However, students with low prior knowledge may benefit from comparing surface features

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  • Comparison is an important learning activity in mathematics

  • Careful attention should be paid to:

    • What is being compared

    • Who is doing the comparing - students’ prior knowledge matters

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  • For slides, papers or more information, contact:

  • Funded by a grant from the Institute for Education Sciences, US Department of Education

  • Thanks to research assistants at Vanderbilt:

    • Holly Harris, Jennifer Samson, Anna Krueger, Heena Ali, Sallie Baxter, Amy Goodman, Adam Porter, John Murphy, Rose Vick, Alexander Kmicikewycz, Jacquelyn Beckley and Jacquelyn Jones

  • And at Michigan State:

    • Kosze Lee, Kuo-Liang Chang, Howard Glasser, Andrea Francis, Tharanga Wijetunge, Beste Gucler, and Mustafa Demir