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In this article, we present a mathematical analysis that distinguishes two distinct quantitative perspectives on ratios and proportional relationships: variable number of fixed quantities and fixed numbers of variable parts. This parallels the distinction between measurement and partitive meanings for division and between two meanings for multiplication—one rooted in counting equal-sized groups, the other in scaling the size of the groups. We argue that (a) the distinction in perspectives is independent from other distinctions in the literature on proportional relationships, including the within measure space versus between measure space ratio distinction; (b) the psychological roots for multiplication suggest the accessibility of the two perspectives to learners; and (c) the fixed numbers of variable parts perspective, though largely overlooked in past research, may provide an important foundation for central topics that build on proportional relationships. We also suggest directions for future empirical research.

Teaching experiments have generated several hypotheses concerning the construction of fraction schemes and operations and relationships among them. In particular, researchers have hypothesized that children's construction of splitting operations is crucial to their construction of more advanced fractions concepts (Steffe, 2002). The authors propose that splitting constitutes a psychological structure similar to that of a mathematical group (Piaget, 1970b): a structure that introduces mutual reversibility of students' partitioning and iterating operations that the authors refer to as the splitting loope. Data consisted of 66 sixth–grade students' written performance on 20 tasks designed to provoke responses that would indicate particular fractions schemes and operations. Findings are consistent with hypotheses from related teaching experiments. In particular, they demonstrate–consistent with the notion of the splitting loope—that equipartitioning and the partitive unit fraction scheme mediate the construction of splitting from partitioning and iterating operations.

Prospective teachers' knowledge of division was investigated through an open-response written instrument and through individual interviews. Problems were designed to focus on two aspects of understanding division: connectedness within and between procedural and conceptual knowledge and knowledge of units. Results indicated that the prospective teachers' conceptual knowledge was weak in a number of areas including the conceptual underpinnings of familiar algorithms, the relationship between partitive and quotitive division, the relationship between symbolic division and real-world problems, and identification of the units of quantities encountered in division computations. The research also characterized aspects of individual conceptual differences. The research results suggest conceptual areas of emphasis for the mathematical preparation of elementary teachers.

One hundred twenty-eight children in Grades 1, 2, and 3 were given two multiplication and four division word problems that differed in semantic structure. The children's solution strategies were classified by degree of abstraction into direct representation, double counting, transitional counting, additive or subtractive, or recalled number fact. Strategies were also classified by use of physical objects: use as representations of individual elements, use as tallies or repeated references, or no use. Analyses of the solution strategies indicated the existence of a two-step intuitive model for multiplication, two common intuitive models for partitive division and measurement division each of which was related to an intuitive model for subtraction, and a third model for partitive division.

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. The present study was inspired by previous findings on the difficulties children encounter when faced with verbal problems in multiplication and division. Bell, Swan, and Taylor (1981) have shown that when children are presented with a series of problems with the same content, they may change their minds about the operation needed to solve the problem, depending on the specific numerical data that are given. For instance, 12- to 15-year-old