A Microgenetic Study of the Conceptual Development of Inversion on Multiplication/Division Inversion Problems - eScholarship

A Microgenetic Study of the Conceptual Development of Inversion on Multiplication/Division Inversion Problems Katherine M. Robinson (katherine.robinson@uregina.ca) Department of Psychology, Campion College, University of Regina 3737 W ascana Parkway Regina, SK S4S 0A2 Abstract The purpose of this study was to conduct a microgenetic study of the development of the concept of inversion as it applies to multiplication and division inversion problems. The study was modelled on Siegler and Stern’s (1998) study in which Grade 2 participants solved addition and subtraction inversion problems (a + b - b) for 6 weekly sessions. In session 7, modified inversion problems (b + a - b) as well as lure problems (b - a + b) were also included. In the current study, Grade 6 participants solved multiplication and division inversion problems (d x e ÷ e) during 6 weekly sessions. Previous research has shown that this latter type of inversion problems is more difficult than the former type. The present results indicate that there are differences in how frequently participants discover and apply the inversion concept compared to Siegler and Stern’s (1998) work. The findings add to the recent body of knowledge indicating that the concept of inversion as it applies to multiplication and division is significantly more difficult than it is for addition and subtraction. Keyw ords: arithmetic; inversion; conceptual knowledge; microgenetic. Introduction In the domain of mathematical cognition, it has been historically difficult to assess conceptual knowledge, and in particular the development of conceptual knowledge (Bisanz & LeFevre, 1990). Conceptual knowledge is the understanding of the underlying structures of mathematics (Bisanz & LeFevre, 1990). Recently, there has been an increasing interest in the development of conceptual knowledge (e.g., Rittle-Johnson, Siegler, & Alibali, 2001) and yet it is often difficult to directly assess children’s understanding of the underlying structures of mathematics. One type of task that has been successfully used in the past, however, is the inversion problem (Starkey & Gelman, 1982). These problems are of the form a + b - b and can be used to assess whether participants understand the inversion concept. Because addition and subtraction are inverse operations, no calculations are required to solve an inversion problem as the answer to the problem is simply the first number, a. This solution approach is called the inversion- based shortcut. There are a number of advantages to using the inversion problem to assess conceptual knowledge or understanding. First, for participants who do not yet know a written numerical system (e.g., preschoolers), inversion problems can be demonstrated using manipulatives (Klein & Bisanz, 2000). Second, unlike many tasks that assess conceptual understanding, participants do not have to have the verbal abilities to explain their understanding but can instead demonstrate their understanding through problem solving. Finally, supporting evidence that participants are indeed using the inversion concept, as demonstrated via stating that the answer is the first number, can be obtained by using standard problems. If a participant is simply stating that the answer is the first number, a, then the answer will be incorrect on standard problems of the form a + b - c. Using inversion and standard problems, researchers have found that children, even preschoolers, can make use of the inversion concept to solve the inversion problems without any calculation and that inversion shortcut use increases across age (Bisanz & LeFevre, 1990; Bryant, Christie, & Rendu, 1999; Rasmussen, Ho, & Bisanz, 2003; Stern, 1992; Vilette, 2002). Siegler and Stern (1998) further extended the research on inversion shortcut use by examining the implicit and explicit components of conceptual understanding. They used a microgenetic design and found that by the end of the study Grade 2 participants were using the inversion shortcut over 90% of the time to solve inversion problems and all of the participants discovered the shortcut during the course of the study. In more recent research, a new type of inversion problem has been investigated (Robinson & Ninowski, 2003; Robinson, Ninowski, & Gray, in press). T his type of inversion problem makes use of the inverse relationship between multiplication and division and takes the form of d x e ÷ e. The same inversion-based shortcut can be used. Robinson and Ninowski (2003) compared adult performance on both types of inversion problems: Addition/Subtraction inversion problems and Multiplication/Division inversion problems. Adults used the inversion shortcut on both types of inversion problems but more frequently on the Addition/Subtraction problems (94% vs 85% for Addition/Subtraction and M ultiplication/Division problems, respectively). In a following study, performance of Grade 6 and 8 students on both types of inversion problems was examined (Robinson, et al., in press). Once again, inversion shortcut use was much higher on Addition/Subtraction problems than on Multiplication/Division problems (44% vs. 19% in Grade 6 and 60% and 39% in Grade 8). Overall, the