The Role of Implicit Models in Solving Verbal Problems in Multiplication and Division

Abstract

Arithmetical operations were assumed to remain attached to primitive behavioral models that influence tacitly the choice of an operation even after the learner has had solid formal-algorithmic training. The model for multiplication was conjectured to be repeated addition, and two primitive models (partitive and quotative) were seen as linked to division. A total of 623 pupils enrolled in 13 Italian schools (Grades 5, 7, and 9) were asked to choose the operation needed to solve 26 multiplication and division word problems. Violations by the numerical data of the constraints imposed by the assumed tacit models (for instance, when the operator was a decimal number) constituted particular sources of difficulty at all three grade levels. The findings seemed to confirm the impact of the repeated addition model on multiplication and of the partitive model on division. The quotative division model influenced the pupils' choices only at the ninth-grade level.

Strategies of Pre-Service Early Childhood Teachers for Solving Multi-Digit Division Problems

“…For instance, [29] revealed that English primary students find integer quotitive problems significantly easier to solve than partitive ones. By contrast, [31] discovered that Italians achieve better results on whole-number partitive problems than on quotitive ones; [32] concluded similar results to those of [31] with a sample of American pre-service primary teachers.…”

“…Three division problems were interspersed with three nondivision problems "to reduce the likelihood that correct answers would result from guessing" [32] (p. 96). This strategy was employed previously in studies of this topic [31,32]. The three division problems were multi-digit, involving integer numbers with various digits, or decimal numbers.…”

“…When the dividend is a decimal number smaller than an integer divisor, students attain a correct expression easier than when problems comprise integer numbers only. This is because solvers try to avoid having a decimal divisor [31,32]: "the faulty rule that the divisor should be a whole number seems to have a greater strength than the faulty rule about the relative sizes of divisor and dividend" [32] (p. 100).…”

“…It represents a partitive model where the dividend is a decimal number (7.2), the divisor an integer (8), and the remainder is zero. This problem violates the partitive division rule because the dividend is lower than the divisor [31]. Problem 3 (chewing gum) is a one-step division problem in which both the dividend (4.5) and divisor (0.9) are decimal numbers.…”

Security of Attachment to Spouses in Late Life

“…Analysis of social networks gives priority to social behaviors over subjective experience of relationship quality, and reports of feeling lonely or connected do not typically account for the social context in which the social exchange takes place. A third window on human relationships that bridges the domains of subjectivity and behavior is that of mental models (Craik, 1943; Johnson-Laird, 1983)—internal representations of external reality that shape a broad range of experiences and behaviors, from how we assess probable outcomes (Tversky & Kahneman, 1983), to how we make political judgments (Westen, Blagov, Harenski, Kilts, & Hamann, 2006), to how we solve math problems (Fischbein, Deri, Nello, & Marino, 1985). Mental models typically have both explicit features that are conscious and implicit features that operate outside of awareness (Johnson-Laird, 1983).…”