Compared to What?How Different Types of

Comparison Affect Transfer in Mathematics

Bethany Rittle-Johnson

Jon Star

What is Transfer?• Transfer

– “Ability to extend what has been learned in one context to new contexts” (Bransford, Brown & Cocking, 2000)

– In mathematics, transfer facilitated by flexible procedural knowledge and conceptual knowledge

• Two types of knowledge needed in mathematics– Procedural knowledge: actions for solving problems

• Knowledge of multiple procedures and when to apply them (Flexibility)

• Extend procedures to a variety of problem types (Procedural transfer)

– Conceptual knowledge: principles and concepts of a domain

How to Support Transfer:Comparison

• Cognitive Science: A fundamental learning mechanism

• Mathematics Education: A key component of expert teaching

Comparison in Cognitive Science

• Identifying similarities and differences in multiple examples is a critical pathway to flexible, transferable knowledge– Analogy stories in adults (Gick & Holyoak, 1983; Catrambone & Holyoak,

1989)

– Perceptual Learning in adults (Gibson & Gibson, 1955)

– Negotiation Principles in adults (Gentner, Loewenstein & Thompson, 2003)

– Cognitive Principles in adults (Schwartz & Bransford, 1998)

– Category Learning and Language in preschoolers (Namy & Gentner, 2002)

– Spatial Mapping in preschoolers (Loewenstein & Gentner, 2001)

– Spatial Categories in infants (Oakes & Ribar, 2005)

Comparison in Mathematics Education

– “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)

Comparison 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 transfer in mathematics?

• Experimental studies of learning and transfer in academic domains and settings largely absent

• Goal of present work– Investigate whether comparison can support

transfer with student learning to solve equations– Explore what types of comparison are most

effective– Experimental studies in real-life classrooms

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

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

Study 1

• Compare condition: Compare and contrast alternative solution methods vs.

• Sequential condition: Study same solution methods sequentially

Rittle-Johnson, B. & Star, J.R. (in press). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology.

Compare Condition

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 non-

standard procedures; evaluate when to use a procedure)

– Conceptual knowledge (e.g. equivalence, like terms)

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

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

Gains in Procedural Knowledge: Equation Solving

0

5

10

15

20

25

30

35

40

45

Familiar Novel

Equation Solving

Post - Pre Gain Score

CompareSequential

F(1, 31) =4.88, p < .05

Gains in Procedural Flexibility

• Greater use of non-standard solution methods to solve equations– Used on 23% vs. 13% of problems,

t(5) = 3.14,p < .05.

Gains on Independent Flexibility Measure

0

5

10

15

20

25

30

35

40

45

Flexiblity

Post - Pre Gain Score

CompareSequential

F(1,31) = 7.51, p < .05

Gains in Conceptual Knowledge

0

10

20

30

Conceptual

Post - Pre Gain Score

CompareSequential

No Difference

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 Study 1

• Comparing alternative solution methods is more effective than sequential sharing of multiple methods– In mathematics, in classrooms

Study 2: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 problems with the same solution– 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)

Study 2 Method• Participants: 161 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– Assessment adapted from Study 1

Gains in Procedural Knowledge

Gains depended on prior conceptual knowledge

Gains in Conceptual Knowledge

Compare Solution Methods condition made greatest gains in conceptual knowledge

0

5

10

15

20

25

Surface Problems MethodsCompare Condition

Post - Pre Gain Score

Gains in Procedural Flexibility: Use of Non-Standard Methods

Greater use of non-standard solution methods in Compare Methods and Problem Type conditions

0

10

20

30

40

50

60

Surface Problems Methods

Compare Condition

Frequency of Use at Posttest

Gains on Independent Flexibility Measure

0

5

10

15

20

25

30

Surface Problems Methods

Condition

Post - Pre Gain Score

No effect of condition

Summary

• Comparing Solution Methods often supported the largest gains in conceptual and procedural knowledge

• However, students with low prior knowledge may benefit 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 matters

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