Star, J.R., & Rittle-Johnson, B. - Harvard Graduate School of Education

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Contrasting Cases in

Mathematics Lessons Support

Procedural Flexibility and

Conceptual Knowledge

Jon R. Star

Harvard University

Bethany Rittle-Johnson

Vanderbilt University

EARLI Invited Symposium: Construction of (elementary) mathematical knowledge:

New conceptual and methodological developments, Budapest, August 29, 2007

Acknowledgements

• Funded by a grant from the United States

Department of Education

• Thanks to research assistants at Michigan State

University and Vanderbilt University:

– Kosze Lee, Kuo-Liang Chang, Howard Glasser,

Andrea Francis, Tharanga Wijetunge, Holly Harris, Jen

Samson, Anna Krueger, Heena Ali, Sallie Baxter, Amy

Goodman, Adam Porter, and John Murphy

2

Comparison

• Is a fundamental learning mechanism

• Lots of evidence from cognitive science

– Identifying similarities and differences in multiple examples appears to be a critical pathway to flexible, transferable knowledge

• Mostly laboratory studies

• Not done with school-age children or in mathematics

(Gentner, Loewenstein, & Thompson, 2003; Kurtz, Miao, & Gentner, 2001;

Loewenstein & Gentner, 2001; Namy & Gentner, 2002; Oakes & Ribar,

2005; Schwartz & Bransford, 1998) 3

Central tenet of math reforms

• Students benefit from sharing and comparing of solution methods

• “nearly axiomatic,” “with broad general endorsement”

(Silver et al., 2005)

• Noted feature of ‘expert’ math instruction

• Present in high performing countries such as

Japan and Hong Kong

(Ball, 1993; Fraivillig, Murphy, & Fuson, 1999; Huffred-Ackles, Fuson, & Sherin

Gamoran, 2004; Lampert, 1990; Silver et al., 2005; NCTM, 1989, 2000; Stigler

& Hiebert, 1999) 4

“Contrasting Cases” Project

• Experimental studies on comparison in academic domains and settings largely absent

• Goal of present work

– Investigate whether comparison can support learning and transfer, flexibility, and conceptual knowledge

– Experimental studies in real-life classrooms

– Computational estimation (10-12 year olds)

– Algebra equation solving (13-14 year olds)

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Why algebra?

• Area of weakness for US students; critical gatekeeper course

• Particular focus: Linear equation solving

• Multiple strategies for solving equations

– Some are better than others

– Students tend to memorize only one method

• Goal: Know multiple strategies and choose the most appropriate ones for a given problem or circumstance

6

Solving 3(

x

+ 1) = 15

Strategy #1:

3( x + 1) = 15

3 x + 3 = 15

3 x = 12 x = 4

Strategy #2:

3( x + 1) = 15 x + 1 = 5 x = 4

7

Similarly, 3(

x

+ 1) + 2(

x

+ 1) = 10

Strategy #1:

3( x + 1) + 2( x + 1) = 10

3 x + 3 + 2 x + 2 = 10

5 x + 5 = 10

5 x = 5 x = 1

Strategy #2:

3( x + 1) + 2( x + 1) = 10

5( x + 1) = 10 x + 1 = 2 x = 1

8

Why estimation?

• Widely studied in 1980’s and 1990’s; less so now

• Viewed as a critical part of mathematical proficiency

• Many ways to estimate

• Good estimators know multiple strategies and can choose the most appropriate ones for a given problem or circumstance

9

Multi-digit multiplication

• Estimate 13 x 44

– “Round both” to the nearest 10: 10 * 40

– “Round one” to the nearest 10: 10 * 44

– “Truncate”: 1█ * 4█ and add 2 zeroes

• Choosing an optimal strategy requires balancing

– Simplicity - ease of computing

– Proximity - close “enough” to exact answer

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Flexibility is key in both domains

• Students need to know a variety of strategies and to be able to choose the most appropriate ones for a given problem or circumstance

• In other words, students need to be flexible problem solvers

• Does comparison help students to become more flexible?

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Intervention

• Comparison condition

– compare and contrast alternative solution methods

• Sequential condition

– study same solution methods sequentially

12

Comparison

condition

13

Sequential

condition

14

Outcomes of interest

• Procedural knowledge

• Conceptual knowledge

• Flexibility

15

Procedural knowledge

• Familiar: Ability to solve problems similar to those seen in intervention

Algebra

-1/4( x - 3) = 10

5( y - 12) = 3( y - 12) + 20

Estimation

Estimate: 12 * 24

Estimate: 37 * 17

• Transfer: Ability to solve problems that are somewhat different than those in intervention

Algebra

0.25( t + 3) = 0.5

-3( x + 5 + 3 x ) = 5( x + 5 + 3 x ) = 24

Estimation

Estimate: 1.92 * 5.08

Estimate: 148 ÷ 11\

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Conceptual knowledge

• Knowledge of concepts

Algebra

If m is a positive number, which of these is equivalent to (the same as) m + m + m + m ? (Responses are: 4 m ; m 4 ; 4( m + 1); m + 4)

For the two equations:

213 x + 476 = 984

213 x + 476 + 4 = 984 + 4

Without solving either equation, what can you say about the answers to these equations? Explain your answer.

Estimation

What does “estimate” mean?

Mark and Lakema were asked to estimate 9 * 24. Mark estimated by multiplying 10 * 20 = 200. Lakema estimated by multiplying 10 * 25 =

250. Did Mark use an OK way to estimate the answer? Did Lakema use an OK way to estimate the answer?

(from Sowder & Wheeler, 1989)

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Flexibility

• Ability to generate, recognize, and evaluate multiple solution methods for the same problem

(e.g., Beishuizen, van Putten, & van Mulken, 1997; Blöte, Klein, &

Beishuizen, 2000; Blöte, Van der Burg, & Klein, 2001; Star & Seifert, 2006;

Rittle-Johnson & Star, 2007)

• “Independent” measure

– Multiple choice and short answer assessment

• Direct measure

– Strategies on procedural knowledge items

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Flexibility items

(independent measure)

Algebra

Solve 4( x + 2) = 12 in two different ways.

For the equation 2( x + 1) + 4 = 12, identify all possible steps (among 4 given choices) that could be done next.

A student’s first step for solving the equation 3( x + 2) = 12 was x + 2 = 4.

What step did the student use to get from the first step to the second step? Do you think that this way of starting this problem is (a) a very good way; (b) OK to do, but not a very good way; (c) Not OK to do?

Explain your reasoning.

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Flexibility items

(independent measure)

Estimation

Estimate 12 * 36 in three different ways.

Leo and Steven are estimating 31 * 73. Leo rounds both numbers and multiplies 30 * 70. Steven multiplies the tens digits, 3 █ * 7█ and adds two zeros. Without finding the exact answer, which estimate is closer to the exact value?

Luther and Riley are estimating 172 * 234. Luther rounds both numbers and multiplies 170 * 230. Riley multiplies the hundreds digits 1 █ █ * 2█

█ and adds four zeros. Which way to estimate is easier?

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Method

• Algebra: 70 7th grade students (age 13-14)*

• Estimation: 158 5th-6th grade students (age 10-12)

• Pretest - Intervention (3 class periods) - Posttest

– Replaced lessons in textbook

• Intervention occurred in partner work during math classes

– Random assignment of pairs to condition

• Students studied worked examples with partner and also solved practice problems on own

*Rittle-Johnson, B, & Star, J.R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology , 99(3), 561-574.

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Results

• Procedural knowledge

• Flexibility

– Independent measure

– Strategy use

• Conceptual knowledge

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0.3

0.2

0.1

0

Procedural knowledge

Students in the comparison condition made greater gains in procedural knowledge.

Sequential

0.5

Compare

0.4

Familiar

Algebra

Transfer Familiar Transfer

Estimation

23

Flexibility

(independent measure)

Students in the comparison condition made greater gains in flexibility.

0.5

Sequential

Compare

0.4

0.3

0.2

0.1

0

Algebra Estimation

24

Flexibility in strategy use

(algebra)

Strategies used on procedural knowledge items:

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Conceptual knowledge

Comparison and sequential students achieved similar and modest gains in conceptual knowledge.

0.5

Sequential

Compare

0.4

0.3

0.2

0.1

0

Algebra Estimation

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Overall

• Comparing alternative solution methods rather than studying them sequentially

– Helped students move beyond rigid adherence to a single strategy to more adaptive and flexible use of multiple methods

– Improved ability to solve problems correctly

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Next steps

• What kinds of comparison are most beneficial?

– Comparing problem types

– Comparing solution methods

– Comparing isomorphs

• Improving measures of conceptual knowledge

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Thanks!

You can download this presentation and other related papers and talks at http://gseacademic.harvard.edu/~starjo

Jon Star

Jon_Star@Harvard.edu

Bethany Rittle-Johnson

Bethany.Rittle-Johnson@vanderbilt.edu

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