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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
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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
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& Hiebert, 1999)
“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) = 15
3x + 3 = 15
3x = 12
x=4
Strategy #2:
3(x + 1) = 15
x+1=5
x=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 = 10
5x + 5 = 10
5x = 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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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
Algebra
Estimation
-1/4(x - 3) = 10
Estimate: 12 * 24
5(y - 12) = 3(y - 12) + 20
Estimate: 37 * 17
• Transfer: Ability to solve problems that are
somewhat different than those in intervention
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
(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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Procedural knowledge
Students in the comparison condition made
greater gains in procedural knowledge.
Procedural Gain Score (Post - Pre)
Sequential
0.5
Compare
0.4
0.3
0.2
0.1
0
Familiar
Algebra
Transfer
Familiar
Transfer
Estimation
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Flexibility (independent measure)
Students in the comparison condition made
greater gains in flexibility.
Flexibility Gain Score (Post - Pre)
0.5
Sequential
Compare
0.4
0.3
0.2
0.1
0
Algebra
Estimation
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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.
Conceptual Gain Score (Post - Pre)
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
Bethany Rittle-Johnson
[email protected]
[email protected]
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