The representations of the arithmetic operations include functional relationships

Abstract

Current theories of mathematical problem solving propose that people select a mathematical operation as the solution to a problem on the basis of a structure mapping between their problem representation and the representation of the mathematical operations. The structure-mapping hypothesis requires that the problem and the mathematical representations contain analogous relations. Past research has demonstrated that the problem representation consists of functional relationships, or principles. The present study tested whether people represent analogous principles for each arithmetic operation (i.e., addition, subtraction, multiplication, and division). For each operation, college (Experiments 1 and 2) and 8th grade (Experiment 2) participants were asked to rate the degree to which a set of completed problems was a good attempt at the operation. The pattern of presented answers either violated one of four principles or did not violate any principles. The distance of the presented answers from the correct answers was independently manipulated. Consistent with the hypothesis that people represent the principles, (1) violations of the principles were rated as poorer attempts at the operation, (2) operations that are learned first (e.g., addition) had more extensive principle representations than did operations learned later (multiplication), and (3) principles that are more frequently in evidence developed more quickly. In Experiment 3, college participants rated the degree to which statements were indicative of each operation. The statements were either consistent or inconsistent with one of two principles. The participants' ratings showed that operations with longer developmental histories had strong principle representations. The implications for a structure-mapping approach to mathematical problem solving are discussed.

Stability and change in children’s division strategies

“…Recent research has also highlighted other diYculties that adults and children have with division. Dixon, Deets, and Bangert (2001) found that eighth-graders' and adults' understanding of the concept of division is signiWcantly weaker than that of the other operations, and Squire and Bryant (2003) highlighted the diYculties that younger children have with concrete representations of division.…”

Children’s understanding of the arithmetic concepts of inversion and associativity

“…Less time may be spent teaching, learning, and exploring this operation than is the case with the other operations. Dixon, Deets, and Bangert (2001) found that operations learned later tend to have weaker principle representations. Most children and many adults solve simple division problems (e.g., 48 ‚ 6) often by using related multiplication facts and rarely by retrieving division facts directly (Robinson, Arbuthnott, & Gibbons, 2002;Robinson et al, in press).…”

Following the standard form: Effects of equation format on algebraic modeling

“…In the context of numerical problem solving, mathematical modeling entails choosing an arithmetic operation that corresponds to the situation described in the problem text. College students have extensive experience with this modeling task, and are highly fluent in selecting arithmetic operations that are structurally analogous to the quantitative relations in word problems (e.g., Bassok, Chase, & Martin, 1998;Bassok, Wu, and Olseth, 1995;Dixon, 2005;Dixon, Deets & Bangert, 2001;Martin & Bassok, 2005). When students transition to algebra and learn that algebraic models have a standard (multiplication) form, some of them may be unsure whether and, if so, how algebraic modeling involves selecting an appropriate arithmetic operation.…”