Partitive division : a mathematical and semantic analysis
Kietzmann, Gina M.
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The purpose of this thesis was to describe a manipulative-based partitive division model which has a number domain of the rational numbers. The partitive division model has two purposes. The first purpose is as a research tool for investigators studying students understanding of division. The second purpose of the partitive division model is as a tool for teaching division in the classroom. In order to understand all the factors that are involved in the partitive division model an extensive review of the literature begins the thesis. The liter ature review includes information about the variables that affect the solutions of division problems, current models of division, partitioning behaviors and the concept of rational number. The description of the partitive division model has two main components, a numerical categorization of division problems and a physical model for solving division problems. The purpose of the numerical categorization of division problems is to provide prototypical division problems in terms of the numerical variables identified in the literature review. The systematic manipulation of these variables resulted in 298 different categories of division problems. The numerical categorization of division problems serves as the basis for the physical partitive division model. Three distinct phases of solution are identified in the partitive division model: Representation, Action and Calculation. The Representation phase involves physically depicting the quantities in the division problem. The Action phase consists of partitioning and distributing procedures that are involved in the division problems. In the Calculation phase the answer to the division problem is identified in the representation of the problem. Each of the three phases are subdivided according to the procedures used to solve the division problems from the numerical categorization. The partitive division model is theoretical and therefore should be the center of a research study before the model can be successfully implemented in the classroom. The worthiness of this endeavor and the possible research questions to be answered are found in the final section of this thesis.