Teaching the concept of a variable with meaning and purpose: Connecting contextual mathematical thought to the abstract symbols and operations of Algebra

The concept of an algebraic variable is both important in its own right and foundational for higher levels of math, but many students struggle to comprehend its meaning and purpose, demonstrating a variety of misconceptions about the interpretation of a variable and algebra’s relation to arithmetic. Common educational practices fail to support a substantial portion of students in connecting their intuitive cognitive capabilities to the formal external representations (i.e., symbolic notation) of algebra, depriving these students of understanding how and why variables are used, as well as their relevance in solving real-world problems. Previous attempts at improving students’ understanding of variables have focused on schematic induction across varied concrete examples or the generalization of relational thinking from arithmetic. While these efforts are important, the approaches do not fully elucidate the purpose of using formal symbols (e.g., letters) to represent unknown numbers. I posit that the clearest way to demonstrate the purpose of symbolic variables is through students’ formulation and attempted solution of mathematical problems where multiple unknowns must be represented (and distinguished from each other), such as in a system of equations word problem. Guided by principles from cognitive psychology and educational research, I formulate a framework for encouraging and supporting students’ intuitive discovery of the concept of variable using purpose-driven contrast comparisons, active learning techniques such as constructive struggling with intuitive hints, and contextual facilitation of students’ natural problem solving for meaningful, concrete tasks. Through this process, variables representations are introduced progressively, first by using more interpretable word equations and later by abbreviating word phrases into letter symbols. I implemented this framework into novel multimedia educational materials, which were iteratively piloted and revised, and then experimentally tested with middle and high school students against a more traditionally structured control version of the materials and a baseline condition. The results from this experimental testing suggest that students who were encouraged to infer the purpose of a variable before its formal representation was introduced went on to provide more correct answers to analogous problems on a post-test given 1-3 weeks later