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Abstract
Although multiplicative structures can be modeled by additive structures, they have their own characteristics inherent in their nature, which cannot be explained solely by referring to additive aspects. Thus, this study is about preservice teachers understanding and sense making of representational quantities generated by magnetic color cubes and algebra tiles, the quantitative units (linear vs. areal) inherent in the nature of these quantities, and the quantitative addition and multiplication operations - referent preserving vs. referent transforming compositions (Schwartz, 1988) - acting on these quantities. I devised a set of tasks focusing on identities of the form Sum = Product, which can also be thought of as summation formulas. Data came from videotaped individual interviews during which I asked five (2 middle school and 3 high school mathematics) preservice teachers problems related to six main mathematical ideas: modeling prime and composite numbers; summation of counting numbers, odd numbers, even numbers; and multiplication and factorization of polynomial expressions in x and y. I base my analysis within a framework of unit-coordination with different levels of units (Steffe, 1988, 1994) supported by a theory of quantitative reasoning (Schwartz, 1988; Thompson, 1988, 1993, 1994, 1995). I used a simplified version of Behr, Harel, Post, & Leshs (1994) generalized notation for mathematics of a quantity and Vergnauds (1983, 1988, 1994) theorems and concepts in action formalisms, which helped me describe the preservice teachers understanding of linear and areal quantities and their units, and the quantitative operations taking place; and translate students mathematical performance into a series of terminology based on a simple notation: Relational notation and mapping structures duo (Caglayan, 2007b). There was a pattern, which showed itself in all my findings. Two students constantly relied on an additive interpretation of the context whereas three others were able to distinguish between and when to rely on an additive or a multiplicative interpretation of the context. My results indicate that the identification and coordination of the representational quantities and their units at different categories (multiplicative, additive, pseudo-multiplicative) are critical aspects of quantitative reasoning and need to be emphasized in the teaching-learning process.