Designing Representations: Reasoning About Functions of Two Variables

One major part of the effort to reform secondary school mathematics is the project of changing the goal of studying school algebra from the mastery of symbolic manipulations to the ability to reason algebraically. Another major component of these reform efforts is the creation of opportunities for students to communicate within and about mathematics. The ability to generalize, especially when the generalization requires a major breakthrough in habits of mind, is one indication of algebraic reasoning. In this article, I describe generalization activity as an opportunity to learn about seventh graders' understanding of functions. A group of students who had studied functions modeled a multivariable situation. Through individual and group work, they designed, described, and discussed various representations for functions of 2 variables. Their modeling efforts allowed them to analyze their understanding of representations of quantities, relationships among quantities, and relationships among the representations of quantities in both single- and multivariable functions.