Contrasting Examples in Mathematics Lessons Support

Flexible and Transferable Knowledge

Bethany Rittle-JohnsonVanderbilt University

Jon StarMichigan State University

Comparison helps us notice distinctive features

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)

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

Current Study

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

• Sequential condition: Study same solution methods sequentially

Target Domain: Early AlgebraMethod 1 Metho d 2

3(x + 1) = 15

3x + 3 = 15

3x = 12

x = 4

3(x + 1) = 15

x + 1 = 5

x = 4

Star, in press

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)

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)

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

Intervention: Content 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.”

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 < .001

Gains in Problem Solving

0

5

10

15

20

25

30

35

40

45

Learn TransferProblem Solving

Post - Pre Gain Score

CompareSequential

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

Gains in Flexibility

• Greater use of non-standard solution methods– Used on 23% vs. 13% of problems,

t(5) = 3.14,p < .05.

Gains on Independent Flexibility Measure

0

5

10

15

20

25

30

35

40

45

Flexiblity

Post - Pre Gain Score

CompareSequential

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

Summary

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

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

Educational Implications

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

It pays to compare!

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.

(a) Very good way

(b) Ok to do, but not a very good way

(c) Not OK to do

Assessment

• Conceptual Knowledge

Explanations During Intervention

Explanation Characteristic Compare Sequential

Reference multiple solutions 92%

26% **

Focus

on method alone

on answer alone

on method & answer

71%

10%

19%

73%

23% **

4% **

Judge

Efficiency

Accuracy

47%

32%

37% *

26%

Justify Mathematically 30% 46% *

Difference between groups** p < .01; * p < .05