Contrasting Cases in Mathematics Lessons Support

Procedural Flexibility and Conceptual Knowledge

Jon R. StarHarvard University

Bethany Rittle-JohnsonVanderbilt University

EARLI Invited Symposium: Construction of (elementary) mathematical knowledge: New conceptual and methodological developments, Budapest, August 29, 2007

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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

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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)

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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)

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“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

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Solving 3(x + 1) = 15

Strategy #1:3(x + 1) = 153x + 3 = 15

3x = 12x = 4

Strategy #2:3(x + 1) = 15

x + 1 = 5x = 4

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

Strategy #1:3(x + 1) + 2(x + 1) = 10

3x + 3 + 2x + 2 = 105x + 5 = 10

5x = 5x = 1

Strategy #2:3(x + 1) + 2(x + 1) = 10

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

x = 1

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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

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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

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Comparison condition

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Sequential condition

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Outcomes of interest• Procedural knowledge• Conceptual knowledge• Flexibility

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Procedural knowledge• Familiar: Ability to solve problems similar to those

seen in intervention

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

Algebra Estimation

-1/4(x - 3) = 10 Estimate: 12 * 24

5(y - 12) = 3(y - 12) + 20 Estimate: 37 * 17

Algebra Estimation

0.25(t + 3) = 0.5 Estimate: 1.92 * 5.08

-3(x + 5 + 3x) = 5(x + 5 + 3x) = 24 Estimate: 148 ÷ 11\

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Conceptual knowledge• Knowledge of concepts

Algebra Estimation

If m is a positive number, which of these is equivalent to (the same as) m + m + m + m? (Responses are: 4m; m4; 4(m + 1); m + 4)

What does “estimate” mean?

For the two equations:213x + 476 = 984

213x + 476 + 4 = 984 + 4 Without solving either equation, what can

you say about the answers to these equations? Explain your answer.

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

• “Independent” measure– Multiple choice and short answer assessment

• Direct measure– Strategies on procedural knowledge items

(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)

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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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Procedural knowledge

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Familiar Transfer Familiar Transfer

Procedural Gain Score (Post - Pre)

Sequential

Compare

Algebra Estimation

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

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Flexibility (independent measure)

Students in the comparison condition made greater gains in flexibility.

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Algebra Estimation

Flexibility Gain Score (Post - Pre)

Sequential

Compare

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Flexibility in strategy use (algebra)

Strategies used on procedural knowledge items:

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Conceptual knowledgeComparison and sequential students achieved similar and modest gains in conceptual knowledge.

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Algebra Estimation

Conceptual Gain Score (Post - Pre)

SequentialCompare

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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 StarJon_Star@Harvard.edu

Bethany Rittle-JohnsonBethany.Rittle-Johnson@vanderbilt.edu