It Pays to Compare! The Benefits of Contrasting Cases on Students’ Learning of MathematicsJon R. Star1, Bethany Rittle-Johnson2, Kosze Lee3, Jennifer Samson2, and Kuo-Liang Chang31Harvard University, 2Vanderbilt University, 3Michigan State UniversityIntroductionFor at least the past 20 years, a central tenet of reform pedagogy in mathematics has been that students benefit from comparing, reflecting on, and discussing multiple solution methods (Silver et al., 2005). Case studies of expert mathematics teachers emphasize the importance of students actively comparing solution methods (e.g., Ball, 1993; Fraivillig, Murphy, & Fuson, 1999). Furthermore, teachers in high-performing countries such as Japan and Hong Kong often have students produce and discuss multiple solution methods (Stigler & Hiebert, 1999). While these and other studies provide evidence that sharing and comparing solution methods is an important feature of expert mathematics teaching, existing studies do not directly link this teaching practice to measured student outcomes . We could find nostudies that assessed the causal influence of comparing contrasting methods on student learning gains in mathematics. There isa robust literature in cognitive science that provides empirical support for the benefits of comparing contrasting examples for learning in other domains, mostly in laboratory settings (e.g., Gentner, Loewenstein, & Thompson, 2003; Schwartz & Bransford, 1998). For example, college students who were prompted to compare two business cases by reflecting on their similarities were much more likely to transfer the solution strategy to a new case than were students who read and reflected on the cases independently (Gentner et al., 2003). Thus, identifying similarities and differences in multiple examples may be a critical and fundamental pathway to flexible, transferable knowledge. However, this research has not been done in mathematics, with K-12 students, or in classroom settings.Current Study. We evaluated whether using contrasting cases of solution methods promoted greater learning in two mathematical domains (computational estimation and algebra linear equation solving) than studying these methods in isolation. The research focused on three core learning outcomes: (1) problem-solving skill on both familiar and novel problems, (2) conceptual knowledge of the target domain, and (3) procedural flexibility, which includes the ability to generate more than one way to solve a problem and evaluate the relative benefits of different procedures.Algebra equation solving. The transition from arithmetic to algebra is a notoriously difficult one, and improvements in algebra instruction are greatly needed (Kilpatrick et al., 2001). Algebra historically has represented students’ first sustained exposure to the abstraction and symbolism that makes mathematics powerful (Kieran, 1992). Regrettably, students’
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Elementary algebra, Gentner, solution methods, E. A. Silver