Transcript What do YOU NOTICE? Using Contrasting Examples to Support

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
• 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 Algebra
Method 1
Method 2
3(x + 1) = 15
3(x + 1) = 15
3x + 3 = 15
x+1=5
3x = 12
x=4
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
45
Compare
Sequential
Post - Pre Gain Score
40
35
30
25
20
15
10
5
0
Learn
Transfer
Problem Solving
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
45
Compare
Sequential
Post - Pre Gain Score
40
35
30
25
20
15
10
5
0
Flexiblity
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
Reference multiple solutions
Compare
Sequential
92%
26% **
on method alone
71%
73%
on answer alone
10%
23% **
on method & answer
19%
4% **
Focus
Judge
Efficiency
47%
37% *
Accuracy
32%
26%
30%
46% *
Justify Mathematically
Difference between groups
** p < .01; * p < .05