Caging the Capybara: Understanding Functions through Modeling

How do advances in technology expand and improve the ways we can teach about mathematical functions? In addition to indispensable tools for dealing with stochastics, technology gives us modeling tools: tools that enhance data analysis by letting students graph functions with their data, create new computed variables, and control model functions dynamically by varying the functions’ parameters. In this paper, we will use an extended example—an optimization task we will call the “Capybara Problem”—to show how we can use these tools to address common difficulties students have with functions. This paper describes seven different approaches to this problem, beginning at the concrete end of the spectrum— using physical materials to represent the problem and its constraints—and then gradually introducing abstraction in the form of variables and functions. Technology supports students throughout this process, helping them understand the nature of variables, and helping them learn to construct symbolic functions and to meaning in their forms and parameters.