The Merging of Calculators
and Computers: A Look to the Future
of Technology Enhanced Teaching
and Learning of Mathematics
Bert K. Waits
The Ohio State University
waitsb@math.ohiostate.edu
Abstract
Fifteen years
ago desk top computers and calculators
were viewed as quite different.
Computers were powerful, expensive,
and ran sophisticated software.
Calculators were inexpensive and
did only elementary numerical computations.
Scientific calculators are now very
inexpensive ($10 to 20 US) and have
significantly changed some of the
mathematics curriculum taught in
most countries. For many years desktop
computers have remained expensive
and thus still are not used nearly
as widely as they should be in the
teaching and learning of mathematics
in colleges and universities. Ten
years ago calculators took a giant
evolutionary step and added new
software functionality in ROM found
only desktop PC computers. These
were the socalled graphing calculators,
first invented by Casio in 1985.
Graphing calculators started a revolution
in the teaching and learning of
mathematics in the United States
and in many other countries as well.
Before graphing calculators, professors
had to rely exclusively on expensive
computers (usually housed in a separate
computer laboratory) to deliver
computer enhanced visualization
in mathematics teaching and learning.
Only a few elite colleges and universities
could provide such an experience
to all mathematics students on a
regular basis. A CAS (computer algebra
system), available usually only
on expensive PC's, generally consists
of three main software packages
 symbol manipulating software,
numerical solvers, and computer
graphers. 1995.
In late 1995
Texas Instruments introduced the
TI92, a relatively inexpensive
handheld computer with builtin
computer symbolic algebra system
(using powerful Derive^{TM }
algorithms) and computer interactive
geometry (an almost complete version
of Cabri II^{TM}). It was
about 2 times the cost of a graphing
calculator but probably 25 times
more powerful! It was the first
of a no doubt new generation of
powerful handheld computers for
mathematics education representing
the merging of calculators and
computers. It is clear to us
that inexpensive CAS technology
will change the nature of the current
style of "computing" in
the teaching and learning of mathematics
from an almost exclusive paper and
pencil symbol manipulation approach
to a more balanced approach.
CAS allows
new pedagogical methods. For example,
calculus procedure are presented
as "white box" procedures
where we allow student use of some
algebraic, noncalculus, "black
box" procedures. The whitebox/black
box principal was first introduced
by Professor Bruno Buchberger from
the Research Institute for Symbolic
Computation in Linz, Austria. We
now need to be more specific and
explicit about a controversial issue.
We can no longer spend out time
in the mathematics classroom doing
everything we did in the past paper
and pencil era and adding
on the many topics and methods our
students need for the technological
intensive future they face. We have
much to learn about our future mathematics
curriculum and the details of how
we will get there.
