Very recent advances in computer technology provide the power of mainframe systems in relatively compact and inexpensive personal computers; soon the computing power of even a supercomputer will be available on a desktop at a price comparable to today's personal computers. Over the next decade this tremendous computing power can and probably will become available in schools throughout the world. Here we discuss the possibility of harnessing this new technological resource as a teaching tool for specific topics in mathematics and science, focusing on random processes in nature and their deep connection to concepts in probability and fractal geometry. Such natural phenomena as the growth of snowflakes via random aggregation and the disordered geometric configurations of polymer chains demonstrate that fundamentally random microscopic processes can give rise to predictable macroscopic behaviors. They also give rise to random fractal structures of inherent interest and great beauty. Because it is impossible to view the underlying processes directly, computer simulation and visualization is an indispensable tool for understanding and studying these phenomena. In the process of “doing science” with both hands-on experiments and computer simulations, students would learn abstract mathematical concepts in a context which is at once concrete and inherently motivating. Furthermore, the techniques they could employ would mirror in most respects those in current use by researchers, thus forging an unprecedented link between this curriculum and the professional worlds of science and mathematics.
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