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Method Introduction It Pays so that Compare! The Benefits of Contrasting Cases on Students? Learning of Mathematics Jon R. Star1, Bethany Rittle-Johnson2, Kosze Lee3, Jennifer Samson2, in addition to Kuo-Liang Chang3 1Harvard University, 2Vanderbilt University, 3Michigan State University
DePauw University, IN has reference to this Academic Journal, It Pays so that Compare! The Benefits of Contrasting Cases on Students? Learning of Mathematics Jon R. Star1, Bethany Rittle-Johnson2, Kosze Lee3, Jennifer Samson2, in addition to Kuo-Liang Chang3 1Harvard University, 2Vanderbilt University, 3Michigan 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, in addition to 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 in addition to Hong Kong often have students produce in addition to discuss multiple solution methods (Stigler & Hiebert, 1999). While these in addition to other studies provide evidence that sharing in addition to comparing solution methods is an important feature of expert mathematics teaching, existing studies do not directly link this teaching practice so that 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 in consideration of the benefits of comparing contrasting examples in consideration of learning in other domains, mostly in laboratory settings (e.g., Gentner, Loewenstein, & Thompson, 2003; Schwartz & Bransford, 1998). For example, college students who were prompted so that compare two business cases by reflecting on their similarities were much more likely so that transfer the solution strategy so that a new case than were students who read in addition to reflected on the cases independently (Gentner et al., 2003). Thus, identifying similarities in addition to differences in multiple examples may be a critical in addition to fundamental pathway so that flexible, transferable knowledge. However, this research has not been done in mathematics, alongside 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 in addition to 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 in addition to novel problems, (2) conceptual knowledge of the target domain, in addition to (3) procedural flexibility, which includes the ability so that generate more than one way so that solve a problem in addition to evaluate the relative benefits of different procedures. Algebra equation solving. The transition from arithmetic so that algebra is a notoriously difficult one, in addition to improvements in algebra instruction are greatly needed (Kilpatrick et al., 2001). Algebra historically has represented students? first sustained exposure so that the abstraction in addition to symbolism that makes mathematics powerful (Kieran, 1992). Regrettably, students? difficulties in algebra have been well documented in national in addition to international assessments (Blume & Heckman, 1997; Schmidt et al., 1999). Current mathematics curricula typically focus on standard procedures in consideration of solving equations, rather than on flexible in addition to meaningful solving of equations (Kieran, 1992). In contrast, prompting students so that solve problems in multiple ways leads them so that 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 so that problems (e.g., Case & Sowder, 1990; Reys, Bestgen, Rybolt, & Wyatt, 1980). This disuse or inability so that use mental math or estimation is a significant barrier so that using mathematics in everyday life. In addition so that being a fundamental, real-world skill, the ability so that quickly in addition to accurately perform mental computations in addition to estimations has two additional benefits: 1) It allows students so that check the reasonableness of their answers found through other means, in addition to 2) it can help students develop a better understanding of place value, mathematical operations, in addition to general number sense (Kilpatrick et al, 2001). Method We compared learning from studying contrasting cases (compare group) so that 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)) in addition to computational estimation (Study 2). Participants, Study 1: Seventy (36 female) 7th graders in addition to their teacher Participants, Study 2: Sixty-nine (32 female) 5th graders in addition to their teacher Procedure: We randomly paired students in addition to assigned them so that condition. Pairs studied worked examples of other students? solutions in addition to answered questions about the solutions during a three-day intervention in their intact math classes. Both conditions were introduced so that the same solution methods in addition to received mini-lectures from the teacher during the intervention. Samples of Intervention Materials Samples of Assessment Items Results 2. Students in the compare condition made greater gains in flexibility. 3. Compare in addition to sequential students achieved similar in addition to modest gains in conceptual knowledge. 1. Students in the compare condition made greater gains in procedural knowledge. Discussion Comparing in addition to contrasting alternative solution methods led so that greater gains in procedural knowledge in addition to flexibility, in addition to comparable gains in conceptual knowledge, compared so that studying multiple methods sequentially. These findings provide direct empirical support in consideration of 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 so that new domains (algebra in addition to estimation), a new age group (school-aged children), in addition to a new setting (the classroom). These findings were strengthened by our use of random assignment of students so that condition within their regular classroom context, along alongside maintenance of a fairly typical classroom environment. Further, rather than comparing our intervention so that standard classroom practice, which differs from our intervention on many dimensions, we compared it so that a control condition which was matched on as many dimensions as possible. This allowed us so that evaluate a specific component of effective teaching in addition to learning. The current studies are an important first step in providing experimental evidence in consideration of the benefits of comparing alternative solution methods, but much is yet so that be done. In particular, it is important so that evaluate when in addition to 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), in addition to comparing isomorphs (two similar problems, solved using the same strategy). Our preliminary analyses suggest that the type of comparison that is most effective appears so that depend on prior knowledge in addition to ability. References 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 in addition to 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 in addition to Instruction, 7, 79-104. Fraivillig, J. L., Murphy, L. A., & Fuson, K. (1999). Advancing children’s mathematical thinking in Everyday Mathematics classrooms. Journal in consideration of Research in Mathematics Education, 30, 148-170. Gentner, D., Loewenstein, J., & Thompson, L. (2003). Learning in addition to transfer: A general role in consideration of analogical encoding. Journal of Educational Psychology, 95(2), 393-405. Kieran, C. (1992). The learning in addition to teaching of school algebra. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching in addition to 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 in addition to characterization of computational estimation processes used by in-school pupils in addition to 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 in addition to procedural knowledge? An experimental study on learning so that 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 in consideration of a closer look at U.S. mathematics in addition to science education. Dordrecht: Kluwer. Sowder, J. T., & Wheeler, M. M. (1989). The development of concepts in addition to strategies used in computational estimation. Journal in consideration of Research in Mathematics Education, 20, 130-146. Schwartz, D. L., & Bransford, J. D. (1998). A time in consideration of telling. Cognition in addition to Instruction, 16(4), 475-522. Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. (2005). Moving from rhetoric so that praxis: Issues faced by teachers in having students consider multiple solutions in consideration of 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 in consideration of improving education in the classroom. New York: Free Press.
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Informed Choice- Do We Need It? Introduction Law in addition to Ethics Various Approaches Law in addition to Ethics Revisited The Basics The Elements of Choice The Professional-Client Model The Professional-Client Model Difficult Issues Informed Refusal Patient Chooses- Provider Does Not Agree Rules so that Avoid Abandonment Provider-Patient Disagreement The Limits of Autonomy 1 The Limits of Autonomy 2 The Limits of Autonomy 3 The Limits of Autonomy 4 The Limits of Autonomy 5 The Limits of Autonomy 6 The Limits of Autonomy 7 Autonomy-Not a Trump Card ? A Note About Ethics Minors Parental Refusal 1 Parental Refusal 2 Parental Refusal 3 Incompetent Patients Research in addition to Consent Stubborn Patients ?Life is not Worth Living? Patients Testing in consideration of HIV Uniform Anatomical Gift Act Exceptions so that Need in consideration of Consent Documenting the Decision Consent-Do we Need It? Case Study #1 Case Study #2 Case Study #3 Case Study #4 Case Study #5 When Policies Clash Conclusion Remember?
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