Technology-enhanced Discourse on a Uniform Movement as a Window on Limits, Cycles, and Chaos

Sergei Abramovich
abramovs@potsdam.edu
Teacher Education
State University of New York at Potsdam
44 Pierrepont Avenue, NY 13676-2294
U.S.A

Anderson Norton
anorton@coe.uga.edu
Mathematics Education
105 Aderhold Hall
University of Georgia
Athens, GA 30602-7124
U.S.A 

Abstract

This paper reflects on activities designed for computer-enhanced in-service training of high school mathematics teachers. The goal of these activities is two-fold: to promote advanced mathematical thinking and to introduce newer tools of technology. The authors suggest using jointly a graphing calculator, a dynamic geometry program, and spreadsheet in bridging finite and infinite mathematics structures by exploring linear algebraic equations in this setting. A linear algebraic equation may arise in a holistic content as a mathematical model of a uniform movement. In turn, in the technology-rich environment solving a linear algebraic equation can be introduced through the method of iterations that ultimately leads to the discussion of infinite processes. This opens a window on the complexity of infinite structures, which include convergent, divergent and cyclic behavior of iterative sequences. Computer-enhanced representations of infinite processes include bisector-bounded staircases and cobweb diagrams, animated pencils of straight lines, and iterations of sequences both in numeric and graphic notations. Finally, by exploring a piece-wise linear recursion one can be just a key press from the frontiers of mathematical knowledge and, along with the concepts of convergence, divergence and cycles, experience how chaos — a remarkable phenomena of modern mathematics — can arise in dynamic systems of a surprisingly simple form.