The calculus curriculum in the microcomputer age: A Working Group Report from the Ware Conference

Abstract

The arrival of the computer will cause a number of profound changes in the calculus curriculum:(1) Fast numerical methods will be available on the computer to give numerical solutions of problems previously handled formally by the calculus.(2) These numerical techniques can usually be programmed in simple algorithms. The understanding of the process of the algorithm may be aided by the students carrying out their own programming.(3) The graphic facilities of computers are providing dynamic ways of viewing the concepts, enabling them to be understood with more profound insight by students and mathematicians at all levels.(4) Symbolic mathematical manipulators are becoming available that are able to produce the formulae for differentiating and integrating functions. This may allow more time on theoretical insight and less on specific tricks of integration.(5) Both graphical and symbolic modes of operation are becoming available in interactive modes that enable the user to explore the concepts concerned.(6) The methods of teaching may be modified, with exposition and exercises being enhanced by exploration, conjecture and testing by the pupils.(7) Applications may be concerned with differential equations which may not have solutions in closed formulae. Graphic and numerical techniques used in tandem give the possibility of investigating both qualitative and quantitative aspects of the solution.(8) The development of insights into the processes will lead to alternative approaches to the subject, e.g. numerical differentiation before symbolic differentiation, allowing the topic to be started without an initial discussion on limits. This will require a rethink of the balance between calculus theory and numerical practice and lead to modifications of the syllabus.(9) Pressures from areas of applications such as computing, business studies, data processing, the experimental and social sciences, are demanding new techniques loosely termed “discrete mathematics”. The time for their study may reduce the time available for the calculus.