When it pays to compare: Benefits of comparison in mathematics classrooms

Bethany Rittle-Johnson

Jon 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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Familiar Novel

Equation Solving

Post - Pre Gain Score

CompareSequential

F(1, 31) =4.49, p < .05

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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 Methods

Condition

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 Methods

Compare Condition

Estimated Marginal Mean Use

Procedural Knowledge

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Surface Problems Methods

Compare 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: b.rittle-johnson@vanderbilt.edu

• 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