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 Algebra Method 1 Method 2 3(

x

+ 1) = 15 3

x

+ 3 = 15 3

x

= 12 3(

x

+ 1) = 15

x

+ 1 = 5

x

= 4

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

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Sequential Condition

15 10 5 0 30 25 20 35 40 45 Overview of Results: Gains in Problem Solving Compare Sequential Learn Transfer

Problem Solving

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

45 40 35 30 25 20 15 10 5 0 Gains on Independent Flexibility Measure Compare Sequential Flexiblity

F

(1,31) = 2.78,

p

< .05

Sample Conversation High Learning Pair

Sample Conversation Modest Learning Pair

Sample Dialogue for 5(y+1) = 3(y+1) + 8 2(y+1) = 8 (see preceding slides) HIGH LEARNING PAIR (Compare Condition) Krista: “What’d they do?” MODEST LEARNING PAIR (Sequential Condition) Allison: “Check Erica's soluti on…so Ben: “Subtracted 3(y + 1) and they had that as one whole term, so the y … and then over here was (y + 1). Subtracted 3(y + 1) let's pre tend answ er…” …10

x

from 5(y + 1) to get 2(y + 1). And this wasn’t over here, so 2(y + Matt: “ Yeah, so, no.” , 30 equals 6

x

, 18…she didn't get the right 1) = 8.” Krista: “Oh, I getcha.” Ben: “That’s correct. Subtracted them on both. So then y + 1 = 4, the y divided this by two and divided this by tw o…. These are both correct.” Krista: “I believe, because when they divided it by two, what Allison: “No, s he didn’t distribute.” Matt: “S he didn't distribute at all,” Allison: “ which gave her the wrong answer.” (3:10-3:39; side B) happened to, they just di vided it by two and that kinda makes the two go b ye-bye? O r” 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 fo ur. Krista: “Ri ght. Or you could also multiply by the reciprocal and basically get the same thing .” Krista: “Mandy and Erica solved the proble m differently, but they both got the same answer. Why ?” Mandy jus t kinda did a few extra steps, I believe. She did like”

General Characteristics of Written Explanations Explanation Characteristic Reference multiple methods Focus on metho d shortcut on answer Judge Efficiency Accuracy Sample Explanations “It is okay t o do it eithe r way.” “He divided each side by 2.” “Mary com bined like terms.” “The answer is righ t.” “Jame’s way was just faster.” “Sammy’s solution is also correc t because she distributed correctly.” “Used the right prop erties at the righ t times.” Compare Sequential Justify Mathematically 92% 90% 11% 29% 47% 32% 30% 25% ** 77%** 4%* 27% 37% * 26% 46% * ficant with

df

(1, 31) as marked : *

p

< .05. **

p

< .01.

Explicit Comparisons Explanation Characteristic Compare methods Compare answers Compare effic iency o f steps Any comparison Sample Explanations “Jessica distributed and Mary combined like terms ” or “You could have combined f irst” “They end up w ith the same answer after all th e steps” “Jill used more steps” At least one o f the above done Note. Diff erence between conditions were signifi cant with

df

(1, 31), **

p

< .01. Compare 11 Sequential 12 16 19 41 0** 2** 12**

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!