1. It Pays to Compare! The Benefits of Contrasting Cases on Students’ Learning of Mathematics Jon R. Star 1, Bethany Rittle-Johnson 2, Kosze Lee 3, Jennifer Samson 2, and Kuo-Liang Chang 3 1 Harvard University, 2 Vanderbilt University, 3 Michigan State University Introduction 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). Method We compared learning from studying contrasting cases (compare group) 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. Samples of Intervention Materials next page Sequential condition Algebra (Study 1) Compare condition Sequential condition Estimation (Study 2) Compare condition Samples of Assessment Items Algebra (Study 1)Estimation (Study 2) Procedural Knowledge (familiar) 1. Solve: -1/4 (x – 3) = 10 2. Solve: 5(y – 12) = 3(y – 12) + 20 1. Estimate: 12 * 24 2. Estimate: 37 * 17 Procedural Knowledge (transfer) 3. Solve: 0.25 (t + 3) = 0.5 4. Solve -3(x + 5 + 3x) – 5(x + 5 + 3x) = 24 3. Estimate: 1.92 * 5.08 4. Estimate: 148 ÷ 11 Flexibility1. Solve 4(x + 2) = 12 in two different ways. 2. For the equation 2(x + 1) + 4 = 12, identify all possible steps (among 4 given choices) that could be done next. 3. A student ’ s first step for solving the equation 3(x + 2) = 12 was x + 2 = 4. What step did the student use to get from the first step to the second step? Do you think that this way of starting this problem is (a) a very good way; (b) OK to do, but not a very good way; (c) Not OK to do? Explain your reasoning. 1. Estimate 12 * 36 in three different ways. 2. Leo and Steven are estimating 31 * 73. Leo rounds both numbers and multiplies 30 * 70. Steven multiplies the tens digits, 3█ * 7█ and adds two zeros. Without finding the exact answer, which estimate is closer to the exact value? 3. Luther and Riley are estimating 172 * 234. Luther rounds both numbers and multiplies 170 * 230. Riley multiplies the hundreds digits 1█ █ * 2█ █ and adds four zeros. Which way to estimate is easier? Conceptual Knowledge1. If m is a positive number, which of these is equivalent to (the same as) m + m + m + m? (Responses are: 4m; m 4 ; 4(m + 1); m + 4) 2. For the two equations 213x + 476 = 984 and 213x + 476 + 4 = 984 + 4, without solving either equation, what can you say about the answers to these equations? Explain your answer. 1. What does “ estimate ” mean? 2. Mark and Lakema were asked to estimate 9 * 24. Mark estimated by multiplying 10 * 20 = 200. Lakema estimated by multiplying 10 * 25 = 250. Did Mark use an OK way to estimate the answer? Did Lakema use an OK way to estimate the answer? (from Sowder & Wheeler, 1989) Results 2. Students in the compare condition made greater gains in flexibility. 3. Compare and sequential students achieved similar and modest gains in conceptual knowledge. 1. Students in the compare condition made greater gains in procedural knowledge. Discussion 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. References Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. 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