Robert L. Devaney
Department of Mathematics
Boston, MA 02215
One of the most interesting applications of technology in the mathematics classroom is the fact that it allows teachers to bring many new and exciting topics into the curriculum. In particular, technology lets teachers bring some topics of contemporary interest in research mathematics into both middle school and high school classrooms.
The mathematical topics of chaos and fractals are particularly appropriate in this regard. They are timely---many ideas in these fields were first conceived during the students' lifetimes. They are applicable---fields as diverse as medicine, business, geology, art, and music have adopted ideas from these areas. And they are beautiful---there is something in the gorgeous computer generated images of objects such as the Mandelbrot set, Julia sets, the Koch snowflake, and others that capture students' interest and enthusiasm.
Therein, however, lies the problem. Many research mathematicians cringe at the sight of ``still another fractal.'' Most often, discussions of chaos and fractals degenerate to simple ``pretty picture shows,'' devoid of any mathematical content. As a consequence, students get the idea that modern mathematics is akin to a video game---lots of computer-generated action, but mindless activity at best.
This attitude is both unfortunate and unnecessary. There is mathematics behind the pretty pictures, and moreover, much of it is quite accessible to secondary school students. Furthermore, the mathematics behind the images is often even prettier than the pictures themselves! In this sense it is a tragedy that students come so close to seeing some exciting, contemporary topics in mathematics, yet miss out in the end. Our goal in this note is to help remedy this situation by describing some easy-to-teach topics involving ideas from fractal geometry.