it pays to compare: effectively using comparison to support student learning of algebra bethany...

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

IES Conference 2008 2

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


Potential of comparison

Mathematics Education: Central tenet of reform efforts; used by teachers

Cognitive Science: A fundamental learning mechanism


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)


Used in some Algebra textbooks

Sobel, M.A., Maletsky, E. M., Lerner, N., & Cohen, L.S. (1985) Algebra One, Harper and Row Inc.


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)


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)


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


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)


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


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.


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)


Comparison condition


Sequential condition

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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)


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


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


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


Conceptual knowledge assessment


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


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


Gains in flexible knowledge of procedures

0

5

10

15

20

25

30

35

40

45

Flexiblity


CompareSequential

F(1,31) = 7.73, p < .01


Gains in conceptual knowledge

0

10

20

30

Conceptual


CompareSequential

No Difference


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


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


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)


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)


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


Hollowell, K.A., Ellis, W., & Schultz, J.E. (1997). HRW Algebra. Holt, Rinehart, & Winston.

Algebra I textbooksCompare problems with different structures

(Different Problem Types)


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)


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


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


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)


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)


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


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

*


Procedural knowledge results

0

10

20

30

40

50


Compare Condition

Percent Correct for Procedural Knowledge

No differences, even on novel problem types


Flexible use of procedures

F (2, 153) = 4.96, p = .008, 2 = .06

0

10

20

30

40

50


Compare Condition

Percent Use of Shortcut Strategies

*


Flexible knowledge of procedures

F (2, 153) = 5.01, p = .008, 2 = .07

20

25

30

35

40

45

50

55

60

65


Compare Condition

Percent Correct for Flexibility Knowledge

*


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


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


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

it pays to compare: effectively using comparison to support student learning of algebra bethany...

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