Advanced Mathematics from an Elementary Viewpoint: Chaos, Fractal Geometry, and Nonlinear Systems

Abstract

We are conducting exploratory research to investigate the instructional issues and educational benefits from introducing both a new paradigm and a new area of applied mathematics into the high school curriculum. The new paradigm is experimental mathematics and the new area is mathematical chaos. By experimental mathematics we mean computer modeling of mathematical processes to gain insight into their structure and behavior so as to inform and guide mathematical inquiry. Mathematical chaos is the study of orderly and chaotic behavior in nonlinear processes and in the real world systems modelled by them. Both depend fundamentally on the use of computers and interactive graphics technology.

School curricula often present the standard subjects in an intellectually impoverished and uncompelling way, teaching modes of thinking and doing that are distinctly different from those used by practitioners. School math is not a model of real mathematics and school science is not genuine science. Education should be directed to grounding knowledge in experience and in contexts of use. Our thesis is that the introduction of experimental mathematics and mathematical chaos will help accomplish this by creating highly motivating computational environments that foster exploration and discovery and bridge the gulf between schoolwork and real mathematics and science.