The Role of Comparison in Mathematics Learning.

The Role of Comparison in Mathematics Learning Shanta Hattikudur (hattikudur@wisc.edu) Department of Psychology, 1202 W. Johnson Street Madison, WI 53706 USA Martha W. Alibali (mwalibali@wisc.edu) Department of Psychology, 1202 W. Johnson Street Madison, WI 53706 USA Abstract Effects on Procedural and Conceptual Learning To better understand how comparison can be effectively used in mathematics instruction, we reviewed research in psychology and education, with the aim of identifying types of comparison that take place in mathematics learning, and considering the effects of comparison on procedural and conceptual understanding. We identified three types of comparison that are commonly utilized in mathematics instruction and learning: problem-to-problem comparisons, (2) step-to-step comparisons, and (3) item-to- abstraction comparisons. Of these three types, only the effects of problem-to-problem comparisons on learning have been well documented. This paper therefore highlights the need for further research to elucidate the unique contributions of different types of comparison in mathematics learning. Keywords: comparison; mathematics; learning; instruction Students often have difficulty learning both mathematical procedures and their conceptual underpinnings (e.g., Kamii & Dominick, 1997; Pesek & Kirshner, 2000). In order to improve students’ procedural and conceptual understanding, it is important to understand the cognitive processes involved in mathematics learning. In an effort to support student learning, teachers often make connections between problems or concepts by comparing them, but they sometimes fail to provide students with the cognitive support needed to help students benefit from these comparisons (Richland, Zur, & Holyoak, 2007). How can comparison be used to effectively promote mathematics learning? Several techniques utilized in mathematics education involve comparison, but the effects of different types of comparison are not well understood. This paper reviews research in psychology and education in order to (1) identify what types of comparison take place in mathematics learning, and (2) consider the effects of comparison on procedural and conceptual understanding of mathematics. Three Types of Comparison We identified three types of comparison that are commonly utilized in mathematics instruction and learning: (1) problem- to-problem comparison, (2) step-to-step comparison, and (3) item-to-abstraction comparison. Later sections of this paper define each type of comparison and document its effects on learning. The effects of comparison on mathematical learning could be measured in myriad ways. In this paper, we focus on two critical aspects of mathematical knowledge: procedural and conceptual knowledge (Hiebert, 1986). Procedural knowledge refers to the ability to execute action sequences for solving problems (including the ability to adapt procedures for new problems) (Rittle-Johnson & Alibali, 1999) and conceptual knowledge refers to explicit or implicit understanding of principles that govern a domain and of interrelations among aspects of mathematical knowledge (Rittle-Johnson & Alibali, 1999; Tennyson & Cocchiarella, Examining the effects of comparison on procedural and conceptual understanding is a primary goal of this paper. However, in reviewing this literature, it quickly became clear that few studies have sought to determine the unique effects of comparison on procedural and conceptual knowledge. Many studies have investigated how comparison affects procedural knowledge, but few studies have addressed the effects of comparison on conceptual knowledge. In the domain of math, gains in procedural and conceptual knowledge are often difficult to assess separately. The two forms of knowledge are tightly linked, with procedural knowledge affecting conceptual knowledge and conceptual knowledge informing procedures (Gelman & Gallistel, 1978; Siegler & Crowley, 1994; Rittle-Johnson & Alibali, 1999). Many studies measure only procedural gains, which seems to imply that procedural knowledge is the most important measure of learning. This review will highlight the need for studies investigating the effects of comparison on conceptual knowledge, and will emphasize the unique and interrelated importance of both types of knowledge. Inclusion Criteria The studies included in this review were limited to those pertaining to students’ mathematical learning found in the psychology and education literatures. The keywords ―math*‖, ―student‖, and ―learn*‖ were used in combination to search the databases PsycINFO, ERIC, and Web of Knowledge for relevant empirical articles and book chapters. Although this may not be a complete sample of studies, we have tried to include a representative sample of relevant