Technology - Enabled Pedagogy as an Informal Link Between Finite and Infinite Concepts in Secondary Mathematics

This paper is a reflection on activities designed for computer-enhanced in-service training of high school mathematics teachers. The goal of these activities is twofold: to promote advanced mathematical thinking, and to introduce new uses of existing technology tools. The authors suggest using jointly a computer-based graphing calculator, a dynamic geometry program, and a spreadsheet program in exploring linear algebraic equations to bridge finite and infinite mathematics structures. A linear algebraic equation may be introduced in the context of mathematically modeling 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 includes the 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 arrive at the frontiers of mathematical knowledge, and, in developing mature 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.