## 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.