Fifth Graders' Representations and Reasoning on Constant Growth Function Problems: Connections between Problem Representations, Student Work and Ability to Generalize

Abstract

Student difficulties learning algebra are well documented. Many mathematics education researchers (e.g., Bednarz&Janvier, 1996; Davis, 1985, 1989; Vergnaud, 1988) argued that the difficulties students encounter in algebra arose when students were expected to shift suddenly from arithmetic to algebraic reasoning and that the solution to the problem was to integrate opportunities for elementary school students to simultaneously develop both arithmetic and algebraic reasoning. The process of generalization, or describing the overall pattern underlying a set of mathematical data, emerged as a focal point for extending beyond arithmetic reasoning to algebraic reasoning (Kaput, 1998; Mason, 1996). Given the critical importance for students to have opportunities to develop understanding of the fundamental algebraic concepts of variable and relationship, one could argue that providing opportunities to explore linear functions, the first function studied in depth in a formal algebra course, should be a priority for elementary students in grades 4-5. This study informs this debate by providing data about connections between different representations of constant growth functions and student algebraic reasoning in a context open to individual construction of representations and reasoning approaches. Participants included 9 fifth graders from the same elementary class. Data shows that students can generate representations which are effective reasoning tools for finding particular cases of the function and generalizing the function but that this depends on features of the problem representation, most importantly the representation of the additive constant. I identified four categories of algebraic reasoning on the task to find the tenth term and found that only students who used reasoning approaches with the additive constant separate and functional reasoning to find the variable component were able to generalize the function. These instances occurred on a story problem and two geometric pattern problems. None of the students used such a reasoning approach or were able to generalize on the numeric sequence problem which did not represent the additive constant separately. Implications for future research and for teaching for conceptual understanding of variable and relationship are discussed.