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

• 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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• 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

• 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

• 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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• 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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x
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
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x
x
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
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• 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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• 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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• 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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• Comparison condition
– compare and contrast alternative solution methods
• Sequential condition
– study same solution methods sequentially
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Comparison
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Sequential
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• Procedural knowledge
• Conceptual knowledge
• Flexibility
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• 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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• 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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• 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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(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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(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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• 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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• Procedural knowledge
• Flexibility
– Independent measure
– Strategy use
• Conceptual knowledge
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0.3
0.2
0.1
0
Students in the comparison condition made greater gains in procedural knowledge.
Sequential
0.5
Compare
0.4
Familiar
Algebra
Transfer Familiar Transfer
Estimation
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(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
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(algebra)
Strategies used on procedural knowledge items:
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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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• 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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• What kinds of comparison are most beneficial?
– Comparing problem types
– Comparing solution methods
– Comparing isomorphs
• Improving measures of conceptual knowledge
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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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