Beyond being “whatever mathematicians do”, mathematics is the study of abstract structures: integers or points in space, for instance, or sets, and sets of sets. The beauty of mathematics is revealed when we develop a deeper understanding of such structures. To cope with mathematical abstractions, people have used practical representations as a way to visualize mathematical ideas ever since they first realized they had ten fingers.
Tracing the history of technology in mathematics, we see the invention and use of the abacus, the quipu, Napier’s bones, and the slide rule, all of which were improved ways of manipulating numbers. Drawings have always been used as a tool to express geometric ideas; Archimedes was slain as he pondered a drawing in the sand. The ruler and the compass were more than just drawing tools to the ancient Greeks. Their use inspired deep mathematical questions as to what could and could not be drawn with them–questions that eventually led to the development of modern algebra.
Recently, with the advent of digital computers, machines have given us even greater insight into mathematical problems. Simulations and computer graphics make it possible to visualize complex ideas and systems. Computers have been used to provide evidence for and identify counter-examples to conjectures in number theory, solving problems that had stood open and unchallenged for hundreds of years. Certain artificial-intelligence programs attempt to find theorems by heuristic methods, and mechanical theorem provers can find proofs for certain theorems, given a conjecture and some axioms.
But perhaps the most substantial impact computers have had on mathematics has been delivered by software systems such as Macsyma, Maple, and Mathematics. Programs like these let users manipulate symbols the way calculators let them manipulate numbers. They are capable of handling many of the unpleasant tasks associated with algebra and calculus in better ways than unassisted people can.
Computer algebra systems have been available to researchers since the introduction of Macsyma in the 1960s. Because they consumed huge amounts of memory, they resided on large mainframes and were consulted as oracles when difficult problems arose. User-unfriendliness, combined with massive resource demands, kept them out of the hands of all but a few mathematicians, scientists, and engineers.
Time and technology have brought about notable changes in this arrangement, however. Computer mathematics systems that can be used on much more modest machines are becoming widely available. Mathematica, for one, runs well on a Macintosh, a 386-based machine, or a Next workstation (on which it is a part of the system software). Since machines capable of running computer mathematics systems (which demand a few MIPS of performance and several megabytes of memory) should become relatively inexpensive and widely available over the next five years, it is reasonable to expect that they will soon be available to the masses. This development could lead to dramatic changes in education, science, and engineering.
What is a computer mathematics system” First, it is a tool for doing numerical mathematics which has capabilites that exceed those of a calculator. Beyond the manipulation of numbers, it is also used as a tool for doing symbolic mathematics. It can do mathematically oriented graphics that are much more interesting than pie charts. Let’s look at some examples of one such mathematics system, Mathematica, in action.
Suppose you have to work with big numbers, bigger than will fit on a calculator, such as
And you need to compute this exactly–scientific notation won’t cut it. In Mathematica, which uses arbitrary precision arithmetic, we simply type “Binomial [5906,2953]”, as in Figure 1, and the digits come spilling out.
Or remember yourself back in high-school algebra, laboriously expanding polynomials by the FOIL (first, outer, inner, last) rule. Careless errors constantly foiled your work, but had you had access to Mathematica, you would have been a key-press away from the solution shown in Figure 2a.
A third example concerns the inverse operation of expanding polynomials, that of factoring them. Factorization is well defined mathematically, but simplifying to find the nicest closed form is largely an aesthetic problem. The correct answer, produced by Mathematica, is shown in Figure 2b.
Generations of engineers have lugged around huge tables of integrals, with the hope that the identity they needed was somewhere in the scriptures. Computer mathematics systems make those tables obsolete, as demonstrated in Figure 3.
As powerful as the mind’s eye is, if unaided it can have considerable difficulty in visualizing mathematical surfaces in three dimensions. Computer graphics, however, can make complicated trigonometric functions considerably easier to understand (see Figure 4).
A final example involves the use of series expansions, one of the time-consuming tasks encountered in introductory calculus. I always felt they were taught as a test of character, and didn’t realize until years later that they were useful to approximate functions. In Figure 5 we’ve painlessly taken the first six terms of the power series expansion of a function around x=0.
If you remember the efforts of learning how to do these kinds of operations, you will realize what a boon to mathematics access to systems such as these can be. They have the potential to change the way math is taught and used.
=When portable calculators became widely available, there was a long, loud debate on how they would affect the teaching of mathematics. As the dust settled and it became clear that calculators were here to stay, their existence forced a re-evaluation of what it is students actually need to learn. Most people realized that while it was still useful to know how to add and subtract, extracting square roots by hand, or using a slide rule or logarithm tables, were skills of the past. And yet, there still exist school systems where these things are taught. Imagine the howling when we tell parents and the educational establishment that much of what is taught in algebra and calculus also must change.
It seems fair to assume, however, that within five years many high-school students will have the same sort of access to a computer mathematics system as they now have to a good encyclopedia. Even if mom and dad can’t afford a computer to run it on, there will be one in the school library. This changes the game of what should be taught in school. When students have a program that can manipulate algebraic expressions and do integrals better than a math major, it is clear that obtaining fluency in these manipulations is less important than it used to be. Thus less class time and effort should be devoted to them, freeing the schedule to allow for study of remaining subjects in greater depth or the addition of other subjects to the curriculum.
How should the mathematics curriculum change? It is fairly clear that counting, simple arithmetic, and other basic skills will remain important. The first significant changes will occur in high-school algebra. Much of the year is spent practicing how to manipulate algebraic formulae, which is certainly tedious and error-prone. Once you learn how to use a computer mathematics system, you will never attempt to factor [x.sup.4 + 5x.sup.3 + 9x.sup.2 + 8x + 2 by hand] again. Thus, a significant portion of the year can be freed up, leaving enough time, for example, to explain how computer algebra systems do their thing.
Calculus is usually taught in colleges and universities in a four-semester sequence. The first semester introduces the theory of limits, rates of change, and derivatives. The second semester traditionally concentrates on integration, with most of the time spent on tricks such as integration by parts and a variety of substitutions which are valuable only when you take integrals by hand. The third semester covers multidemensional integration–which is fairly straightforward but tedious–and the fourth semester differential equations, among other topics. With a program to do the formal integration and differentiation, a semester can be cut from this sequence with no reduction of its intellectual content. Since using a computer mathematics system requires practice and experience, these courses can be restructured to teach students how to enlist mechancial aid effectively in solving mathematics problems and what the inherent limitations of such systems are.