Applets for Mathematical Learning

Introduction

It is common knowledge that abstraction is hard to understand if the learner is not equipped with the proper background. Many years of teaching engineering students have led me to conclude that mathematics students have this problem with partial differential equations and Fourier series, because in many cases they have little mechanical experience. For years when teaching the heat equation, for example, I described the heating and cooling of iron bars at the blacksmith's forge, and the changing of the colors with temperature, and this did not help the students very much. I could talk about something closer to their experience, such as the cooling of pizza. This seemed to be more effective. With the wave equation it is possible to discuss the vibrations of a musical instrument in order to introduce Fourier series. This topic is very real to the string musicians in the class who have experienced the higher harmonics with their own eyes and ears. Even so, this discussion does not usually convey to all students the mental pictures which are needed for understanding the mathematics. For this reason, writing some dynamic software to help students experiment with the ideas of these equations in a visual manner seemed imminent. These are interactive java applets, and the main purpose is to let students explore the mathematics visually as a supplement to whatever textbook is being used. Very little background is required of the student. This chapter describes these applets. We show pictures from some of them, and discuss the mathematical context. Of course, one cannot see the full power of the dynamical environment unless the applets are actually utilized.