The Merging of Calculators and Computers: A Look to the Future of Technology Enhanced Teaching and Learning of Mathematics

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 so-called 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 TI-92, a relatively inexpensive hand-held computer with built-in 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 hand-held 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, non-calculus, "black box" procedures. The white-box/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.