When it pays to compare: Benefits of comparison in mathematics classrooms Bethany Rittle-Johnson Jon R. Star
Feb 22, 2016
When it pays to compare: Benefits of comparison in mathematics classrooms
Bethany Rittle-JohnsonJon R. Star
Common Ground:Comparison
• Cognitive Science: A fundamental learning mechanism– This symposium!
• Mathematics Education: A key component of expert teaching
Comparison in Mathematics Education
– Compare solution methods– “You can learn more from solving one
problem in many different ways than you can from solving many different problems, each in only one way”
– (Silver, Ghousseini, Gosen, Charalambous, & Strawhun, p. 288)
Compare Solution Methods
• Expert teachers do it (e.g. Lampert, 1990)
• Reform curriculum advocate for it (e.g. NCTM, 2000; Fraivillig, Murphy & Fuson, 1999)
• Teachers in higher performing countries help students do it (Richland, Zur & Holyoak, 2007)
Does comparison support mathematics learning?
• Experimental studies on comparison in K-12 academic domains and settings largely absent
• Goals of initial work– Investigate whether comparing solution methods
facilitates learning in middle-school classrooms• 7th graders learning to solve equations• 5th graders learning about computational estimation
Studies 1 & 2
• Compare condition: Compare and contrast alternative solution methods vs.
• 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.
Compare ConditionEquation Solving
Sequential Condition
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Predicted Outcomes
• Students in compare condition will make greater gains in:– Procedural knowledge, including
• Success on novel problems• Flexibility of procedures (e.g. select efficient
procedures; evaluate when to use a procedure)– Conceptual knowledge (e.g. equivalence)
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
Randomly assigned to Compare or Sequential condition
Studied worked examples with partner
Solved practice problems on own
Knowledge Gains
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5
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Familiar NovelEquation Solving
Post - Pre Gain Score
CompareSequential
F(1, 31) =4.49, p < .050
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Flexiblity
Post - Pre Gain Score
CompareSequential
F(1,31) = 7.73, p < .01
Compare condition made greater gains in procedural knowledge and flexibility; Comparable gains in conceptual knowledge
Study 2:Helps in Estimation Too!
• Same findings for 5th graders learning computational estimation (e.g. About how much is 34 x 18?)– Greater procedural knowledge gain– Greater flexibility– Similar conceptual knowledge gain
Summary of Studies 1 & 2
• Comparing alternative solution methods is more effective than sequential sharing of multiple methods– In mathematics, in classrooms
My Own Comparison of the Literatures
• Comparing the cognitive science and mathematics education literatures highlighted a potentially important dimension:– What is being compared?
Study 3:Compared to What?
Solution Methods
Problem Types
Surface Features
Compared to What?
• Mathematics Education - Compare solution methods for the same problem
• Cognitive Science - Compare surface features of different examples with the same solution or category structure– e.g., Dunker’s radiation problem: Providing a solution in 2
stories with different surface features, and prompting for comparison, greatly increased spontaneous transfer of the solution (Gick & Holyoak, 1980; 1983; Catrambone & Holyoak, 1989)
– e.g., Providing two exemplars of a novel spatial relation greatly increased extension of the label to a new exemplar (Gentner, Christie & Namy)
Similarity May Matter
• Comparing moderately similar examples is better (Gick & Paterson, 1992; VanderStoep & Seiffert, 1993) – But: Comparing highly similar examples is
sometimes better (Reed, 1989; Ross & Kilbane, 1997)
– Comparing highly similar examples can facilitate success with less similar examples (Kotovsky & Gentner, 1996; Gentner, Christie & Namy)
Study 3:Compared to What?
Solution Methods• (M = 3.8 on scale from 1 to 9)
Problem Types• (M = 6.6)
Surface Features• (M = 8.3)
Predicted Outcomes
• Moderate similarity/dissimilarity is best, so Compare Solution Methods and Compare Problem Types groups will outperform compare surface features group.– But, students with low prior knowledge may
benefit from high similarity, and thus learn more in compare surface features condition.
Study 3 Method
• Participants: 163 7th & 8th grade students from 3 schools
• Design:– Pretest - Intervention - Posttest - Retention– Replaced 3 lessons in textbook– Randomly assigned to
• Compare Solution Methods• Compare Problem Types• Compare Surface Features
– Intervention occurred in partner work
Conceptual Knowledge
Compare Solution Methods condition made greatest gains in conceptual knowledge F (2, 154) = 6.10, p = .003)
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Surface Problems MethodsCompare Condition
Estimated Marginal Mean
Flexibility: Flexible Knowledge of Procedures
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Surface Problems MethodsCondition
Estimated Marginal Mean
Solution Methods > Problem Type > Surface FeatureF (2, 154) = 4.95, p = .008)
Flexibility: Use of Efficient Procedures
Greater use of more efficient solution methods in Compare Methods and Problem Types conditionsF (2, 135) = 3.35, p = .038)
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Surface Problems MethodsCompare Condition
Estimated Marginal Mean Use
Procedural Knowledge
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Surface Problems MethodsCompare Condition
Estimated Marginal Mean Use
No effect of condition on familiar or transfer equationsBut…
Procedural Knowledge and Prior Knowledge
Posttest performance depended on prior conceptual knowledge
Explanation Characteristics• Explanations offered during the intervention:• Very similar for Compare Solution Methods and
Problem Types:– Mostly focus on solution methods, and often on multiple
methods– Most common comparison is of solution steps– Evaluations usually focus on efficiency of methods
• Compare Surface Features – more likely to focus on and to compare problem
features– Evaluations are rare
Summary
• Comparing Solution Methods often supported the largest gains in conceptual knowledge and flexibility.– Comparing Problem Types sometimes as
effective for flexibility.• However, students with low prior
knowledge may learn equation solving procedures better from Comparing Surface Features
Conclusion
• Comparison is an important learning activity in mathematics
• Careful attention should be paid to:– What is being compared– Who is doing the comparing - students’
prior knowledge may matter
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, Jennifer Samson, Anna Krueger, Heena Ali, 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
Two Equation Solving Procedures
Method 1 Metho d 2
3(x + 1) = 15
3x + 3 = 15
3x = 12
x = 4
3(x + 1) = 15
x + 1 = 5
x = 4
Why Equation Solving?
• Students’ first exposure to abstraction and symbolism of mathematics
• Area of weakness for US students – (Blume & Heckman, 1997; Schmidt et al., 1999)
• Multiple procedures are viable– Some are better than others– Students tend to learn only one method
Procedural Knowledge Assessments
• Equation Solving– Intervention: 1/3(x + 1) = 15– Posttest Familiar: -1/4 (x – 3) = 10– Posttest Novel: 0.25(t + 3) = 0.5
• Flexibility– Solve each equation in two different ways– 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
Conceptual Knowledge Assessment