Students' construction of algebraic symbol systems
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The purpose of this study was to understand how eighth grade students used their symbolizing activity in interaction with a teacher-researcher to begin constructing an algebraic symbol system. Symbolizing activity notation, diagrams, and natural language that the students generated in the process of solving quantitative problem situationsÑwas taken as the basis for studying the construction of an algebraic symbol system. So, rather than take an algebraic symbol system as a given (e.g., start with problems that involved conventional algebraic notation like 3x= 75), the goal of the study was to understand students' generation of notation, diagrams, and natural language for their operations, schemes, and concepts. A central reason for this approach was to base definitions of algebraic symbol systems in students' generation of notation, diagrams, and natural language. As a teacher-researcher, I taught three eighth graders at a rural middle school in Georgia in a constructivist teaching experiment from October 2005 to May 2006. All teaching episodes were videotaped with two camerasÑone to capture student work and one to capture student teacher interaction. During the year, I posed quantitative problem situations that, from my perspective, involved multiplicative combinations, binomial reasoning, quadratic equations, and linear and quadratic functions. The analysis presented in my dissertation pertains to the problems I posed that involved multiplicative combinations and led to the students finding the sum of the first so many whole numbers (e.g., 1 + 2 + É + 15). In retrospective analysis of the videotapes, I constructed second-order models that accounted for changes that students made in their mathematical way of operating and accounted for how they used their notation, diagrams, and natural language to symbolize their activity. Students' multiplicative structures were significant resources in how they solved problems involving multiplicative combinations, how they produced the sum of the first so many whole numbers, and whether they engaged in recursive multiplicative reasoning. The data suggest that using one's notation to externalize the functioning of one's scheme and operating on this notation with operations that are internal or external to the scheme is one important aspect of constructing an algebraic symbol system.