A Distributed Computing Approach to Mathematics in Education
The Fractal Microscope is an
interactive tool designed by the Education Group at the National Center
for Supercomputing Applications (NCSA) for exploring the Mandelbrot set
and other fractal patterns. By combining supercomputing and networks with
the simple interface of a Macintosh or X-Windows workstation, students and
teachers from all grade levels can engage in discovery-based exploration.
The program is designed to run in conjunction with NCSA imaging tools such
as DataScope and Collage. With this program students can enjoy the art
of mathematics as they master the science of mathematics. This
focus can help one address a wide variety of topics in the K-12 curriculum
including scientific notation, coordinate systems and graphing, number
systems, convergence, divergence, and self-similarity.
Why Fractals?
Many people are immediately drawn to the bizarrely beautiful images known as fractals. Extending beyond the typical perception of mathematics as a body of sterile formulas, fractal geometry mixes art with mathematics to demonstrate that equations are more than just a collection of numbers. With fractal geometry we can visually model much of what we witness in nature, the most recognized being coastlines and mountains. Fractals are used to model soil erosion and to analyze seismic patterns as well. But beyond potential applications for describing complex natural patterns, with their visual beauty fractals can help alter students' beliefs that mathematics is dry and inaccessible and may help to motivate mathematical discovery in the classroom.A popular representation of fractal geometry lies within the Mandelbrot set, named after its creator Benoit B. Mandelbrot who coined the name "fractal" in 1975 from the Latin fractus or "to break" (Jürgens et al., 1990). The Mandelbrot set (figure 1) is the set of all points that remain bounded for every iteration of z = z*z + c on the complex plane, where the initial value of z is 0 and c is a constant (Jürgens et al., 1992).
Figure 1. The Mandelbrot set visualized and shaded in blue.
But we can appreciate the beauty of the fractals encompassed in the Mandelbrot set without the specific mathematics behind it. With the help of an NCSA supercomputer and two programs written by Michael South and Dr. Robert M. Panoff working with the Education Group at NCSA, it is possible to explore many common elementary mathematical principles while examining the Mandelbrot set. In fact, some students from Wiley Elementary School in Urbana, Illinois have done just that. One program, the Fractal Microscope, allows anyone to zoom in and out of the Mandelbrot set quickly (in a few seconds, as opposed to a few hours with most home computers) and easily by simply pointing and clicking within the Macintosh environment. The other program, Starstruck, visualizes the path produced through the Mandlebrot set by each iteration.
Information provided by: http://archive.ncsa.uiuc.edu