Pathways to Flexibility: Leveraging Comparison and Prior Knowledge Bethany Rittle-Johnson Jon Star Kelley Durkin 1

Goal of This Research Program Focus on one basic learning process to: ◦ Better understand how and when it aids learning ◦ Contribute to improvements in mathematics education Focus on Comparison 2

Comparison is Ubiquitous In Our Lives “It’s not fair! She got more than me.” 3

Comparison is a Key Learning Process “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). 4

Comparison Aids Learning Across Variety of Tasks and Ages Comparing multiple examples supports abstraction of common structure ◦ 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 is Used in Mathematics Education 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) Occasionally presented in textbooks 6

Comparison Aids Math Learning Topics: ◦ 7 th graders learning about equation solving ◦ 5 th graders learning about computational estimation Comparing solution methods vs. studying methods one at a time ◦ During partner work in their classrooms For both topics, comparison led to ◦ Greater procedural knowledge gain ◦ Greater flexibility – recognize and use non- standard and more efficient methods (Rittle-Johnson & Star, 2007; Star & Rittle-Johnson, 2008)

What More is There to Know? What should be compared? When in the learning process are different types of comparison most effective? 8

In Our Original Research: Compare Solution Methods Equation Solving

Is Comparing Solution Methods Optimal? What are benefits and drawbacks to different types of comparisons? ◦ In mathematics education, focus on comparing solution methods. ◦ Analogy literature suggests that comparing two problems solved with the same solution method should also benefit learning.  e.g., Comparing two isomorphs to the Dunker radiation problem greatly facilitated spontaneous transfer of the solution to the radiation problem (Gick & Holyoak, 1983)

IES Co nfe ren ce 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)

12 Results: Comparing Solution Methods was Optimal Comparing Solution Methods supported the largest gains in conceptual knowledge and procedural flexibility for equation solving ◦ Supported attention to multiple methods and their relative efficiency, which both predicted learning. Rittle-Johnson, B. & Star, J. (2009). Compared to what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology.

When to Compare? Role of Prior Knowledge Hypothesis: Novices will not benefit from comparing solution methods even though more knowledgeable learners do. ◦ Characteristics of the learner, such as prior knowledge, influence which instructional approaches are most effective (Kalyuga, 2007; Snow, 1992) ◦ For people with low prior knowledge, comparison of worked examples requires interpreting each example as well as comparing the similarities and differences between the examples. Too much! ◦ Learning from comparing unfamiliar examples is often difficult (Gentner et al., 2007; Kotovsky & Gentner, 1996) ◦ Participants in our past studies had moderate prior knowledge – used an algebraic method at pretest

Method Participants: th & 8 th -grade students in classes with limited algebra instruction. Design ◦ Pretest - Intervention - Posttest ◦ Intervention occurred in partner work during 3 math classes Randomly assigned to: ◦ Compare solution methods ◦ Compare problem types ◦ Study sequentially  (no comparison)

15 Types of comparison Solution Methods (one problem solved in 2 ways) Problem Types (2 different problems, solved in same way) Sequential (same examples shown on separate pages)

16 Procedural knowledge assessment Equation Solving ◦ Intervention: 1/3 (x + 1) = 15 ◦ Posttest Familiar: ½ (x + 3) = 10 ◦ Posttest Novel:

17 Procedural flexibility Use of more efficient solution methods on procedural knowledge assessment Knowledge 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

18 Conceptual knowledge assessment

Prior Knowledge Measure Identified whether students used algebra at pretest ◦ 40% did not attempt algebra ◦ 60% attempted algebra  Only 20% of students accurately used algebra 19 Rittle-Johnson, B., Star, J. & Durkin, K. (in press). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology.

20 Procedural Knowledge Comparing methods harmed learning for students who did not use algebra at pretest, but aided learning for students who did use algebra.

Flexible Use 21 Same finding

Flexible Knowledge 22 Pattern of results similar, but less strong

Conceptual Knowledge Comparing methods harmed learning for students who did not use algebra at pretest; otherwise condition had minimal influence

Why? Clues from Performance During the Intervention For students who did not use algebra at pretest, those in the compare methods condition: ◦ Got through less intervention material:  Solved 2 fewer practice problems (out of 12)  Answered fewer explanation prompts ◦ Were less likely to implement an efficient solution method correctly when prompted to try one. ◦ Focused less on comparing problem features, although frequency of this type of comparison predicted success at posttest. 24

Summary Prior knowledge matters! For students who did not use algebra at pretest: ◦ Sequential study of examples or comparing problem types was best across measures ◦ Students in these conditions showed fewer signs of confusion For students who attempted algebra ◦ Comparing solution methods tended to be most effective

An Expertise Reversal Effect: Integrating Across Studies Expertise-reversal effect (Kalyuga, 2007): Instructional approach that is most effective for more experienced learners is not most effective for novices If know one solution methods: ◦ Comparing multiple solution methods is better than sequential study or other types of comparison (Rittle- Johnson & Star, 2007, 2009; Star & Rittle-Johnson, 2009) If attempting to master one solution method ◦ Comparing solution methods helps a little If do not know one solution method ◦ Comparing solution methods harms learning ◦ Comparing problem types neither helps nor harms 26

Implications for How Comparison Aids Learning Type of analogy differs based on whether one example is familiar ◦ Classic view: Learn new example through analogy from familiar to unfamiliar example (when have prior knowledge of one method)  Make inferences by identifying similarities and differences to known example and making projections about new example (e.g., Gentner, 1983; Hummel & Holyoak, 1997) ◦ Learn from mutual alignment of two unfamiliar examples (when do not know one method)  Identify their similarities and differences and make sense of them (Gentner, Lowenstein & Thompson, 2003; Schwartz & Bransford, 1998) Mutual alignment might be more difficult and less effective. Develop familiarity with one example first? ◦ Our recent study suggests that delayed introduction to multiple solution methods is not a good idea – harmed learning. 27

Conclusion Comparison is an effect way to support learning, and particularly procedural flexibility It matters what is being compare – comparing solution methods can be most effective for mathematics learning. It matters when we promote comparison – students with little prior knowledge may not benefit from comparing solution methods.

What’s Next? Support teachers’ use of comparison throughout Algebra I. Identifying types of comparisons and the strengths of each 29

30 Acknowledgements Funded by a grant from the Institute for Education Sciences, US Department of Education Thanks to research assistants at Vanderbilt: ◦ Kelley Durkin, Holly Harris, Shanelle Chambers, Jennifer Samson, Anna Krueger, Heena Ali, 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