It has long been recognized that the development of technology should effect the curriculum of mathematics [3] [2]. A key point of Young's book is that these effects should not be merely pedagogical, but substantive. The speed of computers has fundamentally changed the kinds of computations that are possible and this changes the kinds of mathematics people do. Since the advent of computers, the ability to do numerical calculations has increased astronomically. In response, fields of numerical mathematics have blossomed in recent years and the use of numerical methods in the sciences and engineering has become ubiquitous.
The invention of symbolic manipulators in the mid 1980's added a whole new dimension to the interaction between mathematics and computers. For example, students can now find the integral of a function without learning lots of ``tricks", which were a big part of traditional calculus courses. This example only scratches the surface; both students and researchers can now do many abstract operations on the computer, something that a generation ago was only a dream of a handful of people.
Precisely because the new CS is so remarkable, to use it properly requires training. Consider the now familiar example of a calculator. Even though a calculator alleviates the need for most people to do long division, a calculator is useless to the person who does not understand when to add and when to divide. Take this example and multiply its complexity be several thousand. CS is to a calculator as chess is to tic-tac-toe.4 The user of CS must understand a number of things to take advantage of it. First, since both numerical and symbolic operations are possible, the user must understand the advantages and disadvantages of each. Not only must the user understand what she wants to calculate, but she must also have some understanding of how the computer is going about it's work. Moreover, with the new software it is quite possible to believe that the computer is doing what is desired, but in fact it is doing something completely different. This is because the new software does things for which fail-safe algorithms do not exist. Results of these computations can be misleading or even wrong. Fortunately, most problems of this nature can be avoided if students master a few basic mathematical principals about the software. To not make use of the software as an educational tool is bad enough in itself. However, it would be an even more serious problem if our graduates are ill-prepared to use computational software when they inevitably encounter it in the workplace.