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

3(x + 1) = 15 3x + 3 = 15 x+1=5 3x = 12 x=4 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 p

xt e n x ne p xt e n ag

e a tp ge ag e Predicted Outcomes Students in compare condition will make greater gains in:

Procedural knowledge, including Success on novel problems Flexibility of procedures (e.g. select nonstandard 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 45 Compare Sequential

40 35 30 25 20 15 10 Post - Pre Gain Score 5 0 Familiar

Novel Equation Solving 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 45 Compare Sequential 40 35 30 25 20 15 Post

10 - Pre Gain Score 5 0 Flexiblity F(1,31) = 7.51, p < .05 Gains in Conceptual Knowledge 30 Compare Sequential

20 10 Post - Pre Gain Score 0 Conceptual 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. Dunkers 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 25 20

15 10 Post - Pre Gain Score 5 0 Surface Problems Methods

Compare Condition Compare Solution Methods condition made greatest gains in conceptual knowledge Gains in Procedural Flexibility: Use of Non-Standard Methods 60 50 40 30 20 10

Frequency of Use at Posttest 0 Surface Problems Methods Compare Condition Greater use of non-standard solution methods in Compare Methods and Problem Type conditions

Gains on Independent Flexibility Measure 30 25 20 15 10 Post - Pre Gain Score 5 0 Surface Problems

Condition No effect of condition Methods 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: [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