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Algebra (Study 1)
Estimation (Study 2)
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 University
For 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 no studies that assessed the causal influence of comparing contrasting methods on student learning gains in mathematics.
There is a 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’ difficulties in algebra have been well documented in national and international assessments (Blume & Heckman, 1997; Schmidt et al., 1999).Current mathematics curricula typically focus on standard procedures for solving equations, rather than on flexible and meaningful solving of equations (Kieran, 1992). In contrast, prompting students to solve problems in multiple ways leads them to greater procedural flexibility (Star & Seifert, 2006).
Computational Estimation. A large majority of students have difficulty doing simple calculations in their heads or estimating the answers to problems (e.g., Case & Sowder, 1990; Reys, Bestgen, Rybolt, & Wyatt, 1980). This disuse or inability to use mental math or estimation is a significant barrier to using mathematics in everyday life. In addition to being a fundamental, real-world skill, the ability to quickly and accurately perform mental computations and estimations has two additional benefits: 1) It allows students to check the reasonableness of their answers found through other means, and 2) it can help students develop a better understanding of place value, mathematical operations, and general number sense (Kilpatrick et al, 2001).
Samples of Intervention Materials
Comparing and contrasting alternative solution methods led to greater gains in procedural knowledge and flexibility, and comparable gains in conceptual knowledge, compared to studying multiple methods sequentially. These findings provide direct empirical support for one common component of reform mathematics teaching. These studies also suggest that prior cognitive science research on comparison as a basic learning mechanism may be generalizable to new domains (algebra and estimation), a new age group (school-aged children), and a new setting (the classroom).
These findings were strengthened by our use of random assignment of students to condition within their regular classroom context, along with maintenance of a fairly typical classroom environment. Further, rather than comparing our intervention to standard classroom practice, which differs from our intervention on many dimensions, we compared it to a control condition which was matched on as many dimensions as possible. This allowed us to evaluate a specific component of effective teaching and learning.
The current studies are an important first step in providing experimental evidence for the benefits of comparing alternative solution methods, but much is yet to be done. In particular, it is important to evaluate when and how comparison facilitates learning. We are presently conducting several studies exploring the effectiveness of different types of comparison, including comparing solution strategies (the same problem solved in two different ways), comparing problem types (two different problems, solved using the same strategy), and comparing isomorphs (two similar problems, solved using the same strategy). Our preliminary analyses suggest that the type of comparison that is most effective appears to depend on prior knowledge and ability.
1. Students in the compare condition made greater gains in procedural knowledge.
2. Students in the compare condition made greater gains in flexibility.
Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93, 373-397.
Blume, G. W., & Heckman, D. S. (1997). What do students know about algebra and functions? In P. A. Kenney & E. A. Silver (Eds.), Results From the Sixth Mathematics Assessment (pp. 225-277). Reston, VA: National Council of Teachers of Mathematics.
Case, R., & Sowder, J. T. (1990). The development of computational estimation: A neo-Piagetian analysis. Cognition and Instruction, 7, 79-104.
Fraivillig, J. L., Murphy, L. A., & Fuson, K. (1999). Advancing children's mathematical thinking in Everyday Mathematics classrooms. Journal for Research in Mathematics Education, 30, 148-170.
Gentner, D., Loewenstein, J., & Thompson, L. (2003). Learning and transfer: A general role for analogical encoding. Journal of Educational Psychology, 95(2), 393-405.
Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 390-419). New York: Simon & Schuster.
Kilpatrick, J., Swafford, J. O., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington DC: National Academy Press.
Lindquist, M. M. (Ed.). (1989). Results from the fourth mathematics assessment of the National Assessment of Educational Progress. Reston, VA: National Council of Teachers of Mathematics.
Reys, R. W., Bestgen, B., Rybolt, J. F., & Wyatt, J. W. (1980). Identification and characterization of computational estimation processes used by in-school pupils and out-of-school adults (No. ED 197963). Washington, DC: National Institute of Education.
Rittle-Johnson, B. & Star, J. (in press). Does comparing solution methods improve conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology.
Schmidt, W. H., McKnight, C. C., Cogan, L. S., Jakwerth, P. M., & Houang, R. T. (1999). Facing the consequences: Using TIMMS for a closer look at U.S. mathematics and science education. Dordrecht: Kluwer.
Sowder, J. T., & Wheeler, M. M. (1989). The development of concepts and strategies used in computational estimation. Journal for Research in Mathematics Education, 20, 130-146.
Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16(4), 475-522.
Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24, 287-301.
Star, J.R., & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31, 280-300.
Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world's teachers for improving education in the classroom. New York: Free Press.
3. Compare and sequential students achieved similar and modest gains in conceptual knowledge.
Samples of Assessment Items
We compared learning from studying contrasting cases (comparegroup) to learning from studying sequentially presented solutions (sequential, or control, group) in the domains of multi-step linear equations (Study 1; Rittle-Johnson & Star (in press)) and computational estimation (Study 2).
Participants, Study 1: Seventy (36 female) 7th graders and their teacher
Participants, Study 2: Sixty-nine (32 female) 5th graders and their teacher
Procedure: We randomly paired students and assigned them to condition. Pairs studied worked examples of other students’ solutions and answered questions about the solutions during a three-day intervention in their intact math classes. Both conditions were introduced to the same solution methods and received mini-lectures from the teacher during the intervention.