Graphic Insight into Calculus and Differential Equations

Summary

The human brain is powerfully equipped to process visual information. By using computer graphics it is possible to tap this power to help students gain a greater understanding of many mathematical concepts. Furthermore, dynamic representations of mathematical processes furnish a degree of psychological reality that enables the mind to manipulate them in a far more fruitful way than could ever be achieved starting from static text and pictures in a book. Add to this the possibility of student exploration using prepared software and the sum total is a potent new force in the mathematics curriculum.

In this paper we report on the development of interactive high resolution graphics approaches at different levels of teaching calculus and differential equations. The first author has been concentrating on the calculus in the U.K. [Tall 1985] and the second is working with John H. Hubbard in the U.S.A. on differential equations [Hubbard & West 1985], (later referred to as [T] and [H&W] respectively). We are particularly grateful to Professor Hubbard for his assistance in the preparation of this article.

Others have pioneered a computer approach to these topics, particularly [Artigue & Gautheron 1983] who used computer graphics to build up pictures of solutions of autonomous systems of differential equations and [Sanchez et al. 1983] who emphasized a qualitative approach to the theory. A suitable qualitative approach can lean to an insightful understanding of the formal quantitative theory. The major advance in our work is the interactive nature of the prepared software, enabling students to explore the ideas and develop their own conceptualizations.