Reaching the Unreachable: Technology and the Semantics of Asymptotes

This work is part of a larger attempt to explore the nature of symbolic understanding involving graphic technology. This study describes learning advanced mathematics that occurs through constructing qualitative reasoning methods using graphic technology. Data was gathered from a precalculus class who, for a few weeks, investigated and explored asymptotic behavior of rational functions. The analysis is based on observations of group discussions and written works. Learning about asymptotes using software which serves as tool box for numerical evaluation and graphic representation amplifies epistemological complexities related to the infinity concept. Using the software to watch examples of rational functions, generated by symbolic and graphic operations between polynomial functions, enabled students to leave their own traces on the formalization of asymptotes – on its definition, its symbolic structure, and its computational procedures. The discrepancy between technology as a support for visual perception but a tool that cannot support the conception of approaching infinities, makes the study of asymptotes an intriguing domain for investigating what is being manipulated with software and how. The study of the asymptotic behavior of rational functions is a standard part of both precalculus and calculus courses at the high school level. The study of asymptotes and the investigation of rational functions are based on symbolic work of two types – algebraic manipulations of expressions and computations of limits. Typically, the investigation is simplified for the student by introducing “recipes” for the three types of line asymptotes – finding a vertical asymptote by finding the value for x that zeros the denominator of a rational function; finding a horizontal asymptote through an algebraic exercise known as “dividing all terms by the largest power of x”; and, finding the inclined asymptotes through an algorithm involves limit’s computation. The three procedures are described in Figure 1. The recipes provide procedure (or algorithm) to identify asymptotic behavior of rational functions and works appropriately for practicing analysis of rational functions. However, syntactically correct actions with such recipes are not sufficient for facilitating reasoning and argumentation involved in the analysis of rational functions. This kind of actions involves understanding of concepts such as discontinuity and limits, which appear to be principle cognitive problems in the learning of calculus (Artigue, 1991; Cornu, 1991; Orton, 1983). The complexity of these concepts lies International Journal of Computers for Mathematical Learning 2: 1–25, 1997. c 1997 Kluwer Academic Publishers. Printed in the Netherlands.