it pays to compare: effectively using comparison to support student learning of algebra bethany...
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It Pays to Compare: Effectively It Pays to Compare: Effectively Using Comparison to SupportUsing Comparison to Support Student Learning of AlgebraStudent Learning of Algebra
Bethany Rittle-Johnson
Jon Star
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Our approach to improving students’ mathematics learning
Identify instructional practices used in exemplary and typical classrooms
Use cognitive science literature to focus on practices most likely to help student learning
Experimentally evaluate impact of the instructional practice on student learning and develop instructional guidelines
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Potential of comparison
Mathematics Education: Central tenet of reform efforts; used by teachers
Cognitive Science: A fundamental learning mechanism
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Central tenet of math reformsStudents benefit from sharing and
comparing solution methods “nearly axiomatic”, “with broad general
endorsement” (Silver et al., 2005)
Noted feature of ‘expert’ math instructionPresent in classrooms 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; Richland et al 2007; Stigler & Hiebert, 1999)
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Used in some Algebra textbooks
Sobel, M.A., Maletsky, E. M., Lerner, N., & Cohen, L.S. (1985) Algebra One, Harper and Row Inc.
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But does comparison improve student learning?
No evidence that comparison improves student learning in mathematics
Cognitive science research suggests that it should…
Comparison in cognitive science
“The simple, ubiquitous act of comparing two things is often highly informative to human learners…. Comparison is a general learning process that can promote deep relational learning and the development of theory-level explanations” (Gentner, 2005, pp. 247, 251)
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Fundamental learning mechanism
Lots of evidence from cognitive science◦ Identifying similarities and differences in multiple
examples is an important pathway to flexible, transferable knowledge
Mostly laboratory studiesRarely done with school-age children or in
mathematics
(Catrambone & Holyoak, 1989; Gentner, Loewenstein, & Thompson, 2003; Gick & Holyoak, 1983; Kurtz, Miao, & Gentner, 2001; Loewenstein & Gentner, 2001; Namy & Gentner, 2002; Oakes & Ribar, 2005; Schwartz & Bransford, 1998)
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Does comparison support math learning?
Goal of our IES grant◦ Investigate whether comparison can support
conceptual and procedural knowledge of equation solving (and estimation)
◦Explore what types of comparison are most effective
◦Experimental studies in intact classrooms
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Why equation solving?Often students’ first exposure to abstraction and
symbolism of mathematicsArea of weakness for US students (Blume & Heckman,
1997; Schmidt et al., 1999)
According to NCTM and National Math Panel Report, linear equation solving should be a focal point of math instruction in middle school
Although real-world contexts and informal solution methods are powerful for simple problems, equations and equation solving are more effective for complex problems (Koedinger, Alibali & Nathan, 2008)
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Multiple methods for solving equations
Method #1:3(x + 1) = 153x + 3 = 15
3x = 12x = 4
Method #2:3(x + 1) = 15
x + 1 = 5x = 4
◦ Some are better than others◦ Students tend to memorize only one method◦ Example: Solving 3(x + 1) = 15
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Study 1Research question: Does comparing
solution methods improve equation solving knowledge?
Research design: Randomly assigned to:◦Comparison condition
Compare and contrast alternative solution methods
◦Sequential condition Study same solution methods sequentially
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.
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Translation to the classroomStudents study and explain worked
examples with a partnerBased on core findings in cognitive science
-- the advantages of:◦Worked examples (e.g. Sweller, 1988)
◦Generating explanations (e.g. Chi et al, 1989; Rittle-Johnson, 2006)
◦Peer collaboration (e.g. Fuchs & Fuchs, 2000)
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Comparison condition
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Sequential condition
next page
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Predicted outcomesStudents in comparison condition will make
greater gains in:◦Procedural knowledge, including success on
novel problems◦Procedural flexibility (e.g. use more efficient
methods; evaluate when to use a procedure)◦Conceptual knowledge (e.g. equivalence)
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Study 1 Method
Participants: 70 7th-grade students and their math teacher
Design:◦ Pretest - Intervention - Posttest◦ Replaced 2 lessons in textbook◦ Intervention occurred in partner work during 2 1/2 math
classes
Intervention:◦ Randomly assigned to Compare or Sequential condition
◦ Studied worked examples with partner◦ Solved practice problems on own
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Procedural knowledge assessment
Equation Solving◦ Intervention: 1/3 (x + 1) = 15◦Posttest Familiar: -1/4 (x – 3) = 10◦Posttest Novel: 0.25 (t + 3) = 0.5
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Procedural flexibilityUse of more efficient solution methods on
procedural knowledge assessmentKnowledge of multiple methods
◦ Solve each equation in two different ways◦ Evaluate methods: Looking at the problem shown above,
do you think that this way of starting to do this problem is a good idea? An ok step to make? Circle your answer below and explain your reasoning.
(a) Very good way (b) Ok to do, but not a very good way
(c) Not OK to do
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Conceptual knowledge assessment
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Gains in procedural knowledge
0
5
10
15
20
25
30
35
40
45
Familiar Novel
Equation Solving
Post - Pre Gain Score
CompareSequential
F(1, 31) =4.49, p < .05
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Flexible use of procedures
Solution Method Comparison Sequential
Conventional .61~ .66
Demonstrated efficient .17* .10
Solution Method at Posttest (Proportion of problems)
~ p = .06; * p < .05
Comparison students more likely to use more efficient method and somewhat less likely to use the conventional method
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Gains in flexible knowledge of procedures
0
5
10
15
20
25
30
35
40
45
Flexiblity
Post - Pre Gain Score
CompareSequential
F(1,31) = 7.73, p < .01
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Gains in conceptual knowledge
0
10
20
30
Conceptual
Post - Pre Gain Score
CompareSequential
No Difference
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Summary of Study 1Comparing alternative solution methods is
more effective than sequential sharing of multiple methods◦ Improves procedural transfer and flexibility◦ In mathematics, in classrooms
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Comparison can help: Now what?
Replicated findings for fifth graders learning computational estimation
Goal: Develop guidelines for using comparison to support mathematics learning
Starting Point: Standard classroom practices
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What are teachers doing? US teachers use comparison in 8th grade math
lessons (average 4 per lesson) Types of comparisons (used with equal
frequency):1. Compare two similar problems with same basic solution
2. Compare two moderately similar problems or solutions
3. Compare a problem to a mathematical rule or principle
4. Compare a problem to a non-mathematical situation
(Richland , Holyoak & Stigler, 2004 analysis of TIMSS videos)
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What are teachers doing?May not be using comparison well◦Teachers, rather than students, initiate
comparisons and make links between examples◦When they present multiple solutions, rarely
provide support for or discuss comparisons◦Don’t know which types of comparison support
learning e.g. Comparisons to contexts from different domains
rarely support learning in laboratory studies.
(Richland , Holyoak & Stigler, 2004; Richaland, Zur & Holyoak, 2007; Chazan & Ball, 1999)
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What about Algebra I textbooks?
Compare two similar problems with same basic solution method (Equivalent Equations)
Bellman,A.E., Bragg,S.C., Charles, R.I., Hall,B., Handlin, W.G., & Kennedy, D. (2007) Algebra 1, Pearson Education Inc, Pearson Prentice Hall
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Hollowell, K.A., Ellis, W., & Schultz, J.E. (1997). HRW Algebra. Holt, Rinehart, & Winston.
Algebra I textbooksCompare problems with different structures
(Different Problem Types)
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Algebra I textbooks
Sobel, M.A., Maletsky, E. M., Lerner, N., & Cohen, L.S. (1985) Algebra One, Harper and Row Inc.
Compare different solution methods to same problem (Solution Methods)
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Comparison in Algebra 1 textbooks
Type of Comparison Percent of worked examples
Equivalent equations (similar problems; same method) 33%
Different problem types (diff probs, solved same
way)
1%
Solution methods (one problem solved in two ways) 19%
None - single worked examples 47%
Informal analysis of 10 Algebra I textbooks - chapter on multi-step linear equations
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What should be compared?Variety of comparisons are being used in
math classroomsWhat are benefits and drawbacks to
different types of comparisons?◦Study 1 confirms that comparing solution
methods aids learning, as suggested by expert teaching practices
◦Cognitive science literature suggests that comparing two problems solved with the same solution method should benefit learning
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Study 2Research question: What are the relative
merits of comparing solution methods vs. comparing problems?
Research design: Randomly assigned to:◦Compare solution methods◦Compare problems that:
Are very similar (Equivalent) Have different problem features (Different problem types)
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Types of comparison
Solution Methods(one problem solved in 2
ways)
Problem Types(2 different problems,
solved in same way)
Equivalent(two similar problems,
solved in same way)
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Study 2 Method Participants: 161 7th & 8th grade students from 3
schools (more diverse sample) Design:
◦ Pretest - Intervention - Posttest - 2 week Retention◦ Replaced 3 lessons in textbook◦ Randomly assigned to
Compare Solution Methods Compare Problem Types Compare Equivalent
◦ Intervention occurred in partner work◦ Assessment adapted from Study 1
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Conceptual knowledge results
F (2, 153) = 5.76, p = .004, 2 = .07
0
10
20
30
40
50
60
Equivalent Problem Types Methods
Compare Condition
Percent Correct for Conceptual Knolwedge
*
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Procedural knowledge results
0
10
20
30
40
50
Equivalent Problem Types Methods
Compare Condition
Percent Correct for Procedural Knowledge
No differences, even on novel problem types
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Flexible use of procedures
F (2, 153) = 4.96, p = .008, 2 = .06
0
10
20
30
40
50
Equivalent Problem Types Methods
Compare Condition
Percent Use of Shortcut Strategies
*
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Flexible knowledge of procedures
F (2, 153) = 5.01, p = .008, 2 = .07
20
25
30
35
40
45
50
55
60
65
Equivalent Problem Types Methods
Compare Condition
Percent Correct for Flexibility Knowledge
*
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SummaryAcross studies, Comparing Solution
Methods often supported the largest gains in conceptual knowledge, procedural knowledge and procedural flexibility◦Supported attention to multiple methods and their
relative efficiency, which both predicted learning
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Guidelines for using comparisonProvide a written record of examples
◦ Leverage current use of worked examples in textbooksContrast important dimensions in the examples, such
as problem features or solution methods◦ Contrasting correct and incorrect solution methods can help too
(Kelley Durkin, IES pre-doc research)Have students compare a familiar method to an
unfamiliar method Invite comparisons by using common labels and
prompting for specific comparisons, including efficiency of the methods
Be sure students, not just teachers, are comparing and explaining
Incorporate some direct instruction
What’s next?Teacher Professional Development for
using comparison in Algebra I coursesType of comparison matched to prior
knowledge and sequencing different types of comparison
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Acknowledgements
For slides, papers or more information, contact: [email protected]
Funded by a grant from the Institute for Education Sciences, US Department of Education
Thanks to research assistants at Vanderbilt:◦ Holly Harris, Shanelle Chambers, Jennifer Samson, Anna Krueger,
Heena Ali, Kelley Durkin, Kelly Cashen, Calie Traver, Sallie Baxter, Amy Goodman, Adam Porter, John Murphy, Rose Vick, Alexander Kmicikewycz, Jacquelyn Beckley and Jacquelyn Jones
And at Michigan State:◦ Kosze Lee, Kuo-Liang Chang, Howard Glasser, Andrea Francis,
Tharanga Wijetunge, Beste Gucler, and Mustafa Demir
And at Harvard:◦ Martina Olzog, Jennifer Rabb, Christine Yang, Nira Gautam, Natasha
Perova, and Theodora Chang