Explaining Contrasting Solution Methods Supports Problem-Solving

Flexibility and Transfer

Bethany Rittle-Johnson

Vanderbilt University

Jon Star

Michigan State University

Explanation is Important, But…

• Students often generate shallow explanations (e.g. Renkl, 1997)

• Generating explanations does not always improve learning (e.g. Mwangi & Sweller, 1998)

• How can we support effective explanation?

Explaining Contrasting Solution Methods

• Share-and-compare solution methods core component of reform efforts in mathematics (e.g. Silver et al, 2005)

• But does it lead to greater learning?

Comparison as Central Learning Mechanism

• Cognitive science literature suggests it is:– 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

• Does contrasting solution methods support effective explanation in k-12 classrooms?

• Is it effective for mathematics learning?

• Does it support high-quality explanations?

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 & Siefert, in press

Predicted Outcomes

• Students in compare condition will – Generate more effective explanations– Make greater knowledge gains:

• Greater problem solving success (including transfer)

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

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

Compare Condition

Sequential Condition

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Overview of Results: Gains in Problem Solving

0

5

10

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45

Learn Transfer

Problem Solving

Post - Pre Gain Score

CompareSequential

F(1, 31) = 2.12, p < .05

Gains in Flexibility

• Greater use of non-standard solution methods– Used on 17% vs. 10% of problems *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) = 2.78, p < .05

Sample ConversationHigh Learning Pair

Sample ConversationModest Learning Pair

Sample Dialogue for5(y+1) = 3(y+1) + 8

2(y+1) = 8 (see preceding slides)

HIGH LEARNING PAIR (Compare Condition)

Krista: “What’d they <Erica> do?”

Ben: “Subtracted 3(y + 1) and they had that as one whole term, so

they … and then over here was (y + 1). Subtracted 3(y + 1)

from 5(y + 1) to get 2(y + 1). And this wasn’t over here, so 2(y +

1) = 8.”

Krista: “Oh, I getcha.”

Ben: “That’s correct. Subtracted them on both. So then y + 1 = 4,

they divided this by two and divided this by two…. These are

both correct.”

Krista: “I believe, because when they divided it by two, what

happened to, they just divided it by two and that kinda makes

the two go bye-bye? Or”

Ben: “Because if you have two of this and you divide by two, you

only have one y + 1, correct? And over here you divide 8 by two

and have four.

Krista: “Right. Or you could also multiply by the reciprocal and

basically get the same thing.”

Krista: <reading prompt> “Mandy and Erica solved the problem

differently, but they both got the same answer. Why?”

<begin with shallow answer and push each other>

Mandy just kinda did a few extra steps, I believe. She did like”

MODEST LEARNING PAIR (Sequential Condition)

Allison: <begins to read question> “Check Erica's solution…so

let's pretend…10x, 30 equals 6x, 18…she didn't get the right

answer…”

Matt: “Yeah, so, no.”

Allison: “No, she didn’t distribute.”

Matt: “She didn't distribute at all,”

Allison: “which gave her the wrong answer.”

(3:10-3:39; side B)

General Characteristics of Written Explanations

Explanation

Characteristic

Sample Explanations Compare Sequential

Reference multiple

methods

“It is okay to do it either way.” 92%

25% **

Focus

on method

shortcut

on answer

“He divided each side by 2.”

“Mary combined like terms.”

“The answer is right.”

90%

11%

29%

77%**

4%*

27%

Judge

Efficiency

Accuracy

“Jame’s way was just faster.”

“Sammy’s solution is also correct because

she distributed correctly.”

47%

32%

37% *

26%

Justify Mathematically “Used the right properties at the right times.” 30% 46% *

Difference between conditions were signi ficant with df (1, 31) as marked: * p < .05. ** p < .01.

Explicit Comparisons

Explanation Characteristic Sample Explanations Compare Sequential

Compare methods “Jessica distributed and Mary

combined like terms” or “You

could have combined first”

11 12

Compare answers “They end up with the same

answer after all the steps”

16 0**

Compare efficiency of steps “Jill used more steps” 19 2**

Any comparison At least one of the above done 41 12**

Note. Difference between conditions were significant with df (1, 31), **p < .01.

Summary

• Comparing alternative solution methods rather than studying them sequentially– Improves problem solving accuracy and flexibility– Focuses students’ explanations on the viability of

multiple of solutions and their comparative efficiency.

How Contrasting Solutions Supports Explanation

• Guide attention to important problem features– Reflection on:

• Joint consideration of multiple methods leading to the same answer

• Variability in efficiency of methods

– Acceptance of multiple, non-standard solution methods

Educational Implications

• Teachers need to go beyond simple sharing of alternative strategies– Support comparative explanations

It pays to compare!