Transcript Slide 1

The Role of Comparison in the
Development of Flexible Knowledge
of Computational Estimation
Jon R. Star (Harvard University)
Bethany Rittle-Johnson (Vanderbilt University)
AERA, New York City. Tuesday, March 25, 2008. Session 33.074 Developing Mathematical
Understanding paper session, Crowne Plaza Times Square, Room 509/510, 4:05 – 5:35 pm
Thanks to...
• Research supported by Institute of Education
Sciences (IES) Grant # R305H050179
• All participating teachers and schools in
Nashville, Tennessee and Hale, Michigan
• Graduate and undergraduate research
assistants at Vanderbilt, Michigan State, and
Harvard
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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
– Algebra equation solving
– Computational estimation
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Rittle-Johnson & Star, 2007
• Experimental study in algebra classrooms with
70 7th grade students on equation solving
• Intervention (Comparison condition)
– Comparing and contrasting alternative solution
methods
• Control (Sequential condition)
– Reflecting on same solution methods one at a time
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 (from 2007 study)
• At posttest, students in comparison condition
made significantly greater gains in procedural
knowledge and flexibility and comparable gains
in conceptual knowledge
• The intervention worked! But need to replicate
and extend these findings...
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Computational estimation
• Widely studied in 80’s and early 90’s
– Less so in recent years
• Process of mentally generating an approximate
answer for a given arithmetic problem (Rubenstein, 1985)
– (Distinct from “mental computation,” which means
finding the exact answer)
• Estimates of 2-digit multiplication problems
13 x 27
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Strategies for estimating 13 x 27
• Round both (to the nearest 10)
10 x 30 or 300
• Round one (to the nearest 10)
10 x 27 or 270
(Alternatively, 13 x 30, or 390)
• Trunc (truncate) (Sowder & Wheeler, 1989)
1 x 2, or 2
Then append 2 zeros, or 200
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Why estimation for replication?
• Estimation is different from algebra equation
solving in several ways that play a potentially
important role whether comparison will help
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Equation solving vs. estimation
• Equation solving
– Problems have a single correct answer
• Estimation
– Correctness or “goodness” of estimate depends on
two sometimes competing goals
– Simplicity: how easy it is to compute
– Proximity: how close the estimate is to the exact
answer
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Complexity of “goodness”
• Sometimes easy-to-compute estimate is not very
proximal to the exact value, and vice versa
• Some strategies present false illusion of
consistent proximity
• Is round one always more proximal than round
two?
– Intuitively, yes? The less you round, the closer you
get
– Actually no! It depends on the problem
– Try it for 39 x 41 versus 39 x 37
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Solution efficiency
• Algebra equation solving
– Easy and visually apparent to judge relative efficiency
of two compared solutions
– For example, just count the number of steps!
• Estimation
– Not at all clear how one would judge efficiency of two
compared solutions
– Efficiency is more of an individual or subjective
judgment
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Summary of rationale
• Evidence for effectiveness of comparison in
algebra, but replication needed
• Estimation is a good domain in which to replicate
– Certain features of estimation raise legitimate
questions about whether comparison of strategies
can have the same positive impact
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Method
• Estimation: 158 5th-6th grade students
• 69 5th graders in urban private school
– 4 classes, taught by the same teacher
• 44 5th and 45 6th graders in rural public school
– Two 5th classes, taught by same teacher
– Two 6th classes, taught by same teacher
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Design
• Pretest - Intervention – Posttest – Retention test
– 3-day intervention replaced lessons in textbook
• Intervention occurred in partner work during
math classes
– Random assignment of pairs to condition
– Both conditions present in all classrooms
• Students studied worked examples with partner
and also solved practice problems on own
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Comparison materials
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Sequential materials
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Results
• Procedural knowledge
– Ability to compute accurate estimates
• Flexibility
– Knowledge of multiple strategies and ability to select
most appropriate strategies for a given problem and
problem-solving goal
– Direct measure (from procedural knowledge items)
– Independent measure
• Conceptual knowledge
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Procedural knowledge
• Mental
– Estimate 32 x 17 mentally and quickly
• Familiar
– Estimate 12 x 24 and 113 x 27
• Transfer
– Estimate 1.19 x 2.39 and 102 ÷ 27
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Procedural knowledge results
• No difference between conditions on:
• Accuracy of estimates
– Assessed accuracy a number of ways, none of which
showed a difference for intervention students
• Both conditions improved the accuracy of
students’ estimates
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Accuracy of estimates
1
0.9
Proportion correct
0.8
0.7
0.6
Compare
Sequential
0.5
0.4
0.3
0.2
0.1
0
Pretest
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Posttest
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Retention
22
Conceptual knowledge sample
• What does “estimate” mean?
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Conceptual knowledge results
• Comparison improved conceptual knowledge for
students who began with some initial procedural
knowledge, but not for those who began with
very little procedural knowledge
• In other words, comparison was particularly
beneficial for students who began the study with
some initial ability to compute estimates
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Conceptual knowledge
Adjusted mean proportion correct
(posttest + retention)
0.52
0.5
0.48
0.46
Compare
Sequential
0.44
0.42
0.4
0.38
Low
Modest
Procedural knowledge pretest score
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Flexibility sample items
• Independent measure
• Estimate 12 x 36 in three different ways
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Flexibility results
• Comparison led to greater flexibility
• Independent measure: Comparison students...
– Were more likely to be able to produce estimates for
the same problem in multiple ways
– Were better at making judgments about which
strategy would led to an easier or a closer estimate
for a given problem
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Flexibility independent measure
1
0.9
Proportion correct
0.8
0.7
0.6
Compare
Sequential
0.5
0.4
0.3
0.2
0.1
0
Pretest
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Posttest
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Retention
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Flexible strategy use
• Direct measure
– Strategies on procedural knowledge assessment
• Comparison students were:
– More likely to use trunc, optimizing their strategy use
for ease more than sequential student
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Flexibility
• Recall that students in both conditions saw the
same strategies demonstrated on the same
problems
• Yet comparison students were
– Better at generating multiple ways to find estimates
– Were more likely to use the easiest strategy for a
given problem
– Were better at predicting which strategy would led to
a close or easy estimate for a given problem
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In sum...
• It pays to compare!
• Comparison led to:
– Greater flexibility (2007 and present study)
– Improved procedural knowledge (2007 study)
– Improved conceptual knowledge (for students with
modest procedural knowledge at pre-test; present
study)
– In two very different mathematical domains, algebra
and computational estimation
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National Math Panel (p. 27)
• “Teachers should broaden instruction in
computational estimation beyond rounding. They
should insure that students understand that the
purpose of estimation is to approximate the
exact value and that rounding is only one
estimation strategy.”
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National Math Panel (p. 27)
• “Textbooks need to explicitly explain that the
purpose of estimation is to produce an
appropriate approximations. Illustrating multiple
useful estimation procedures for a single
problem, and explaining how each procedure
achieves the goal of accurate estimation, is a
useful means for achieving this goal. Contrasting
these procedures with others that produce less
appropriate estimates is also likely to be helpful.”
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Thanks!
This presentation and other related papers and
presentations can be found at:
http://gseacademic.harvard.edu/~starjo/
or
by contacting Jon Star ([email protected])