Understanding the calculus

A number of significant changes have have occurred recently that give us a golden opportunity to review the teaching of calculus. The most obvious is the arrival of the microcomputer in the mathematics classroom, allowing graphic demonstrations and individual investigations into the mathematical ideas. But equally potent are new insights into mathematics and mathematics education that suggest new ways of approaching the subject. In this article I shall consider some of the difficulties encountered studying the calculus and outline a viable alternative approach suitable for specialist and non-specialist mathematics students alike

School algebra and the computer

How are we to use the computer in the teaching and learning of algebra? In the longterm the new technology is introducing new possibilities that may radically change the algebra curriculum. However, in the short-term we already have the National Curriculum placing its template on the development of algebra in school. The recent regrouping of topics into five attainment targets has integrated number pattern with the development of algebraic symbolism. It seems natural to build from expressing patterns in words to expressing them in a shorthand algebraic notation, but, although this proves a sound tactic for the more able, there are subtle difficulties for the majority of children. Instead I shall advocate introducing algebraic symbolism by using it as a language of communication with the computer, through programming in a suitable computer language. This has two distinct benefits – it develops a meaningful algebraic language which can be used to describe number patterns, and it gives a foundation for traditional algebra and its manipulation

A versatile approach to calculus and numerical methods

Traditionally the calculus is the study of the symbolic algorithms for differentiation and integration, the relationship between them, and their use in solving problems. Only at the end of the course, when all else fails, are numerical methods introduced, such as the Newton-Raphson method of solving equations, or Simpson’s rule for calculating areas. The problem with such an approach is that it often produces students who are very well versed in the algorithms and can solve the most fiendish of symbolic problems, yet have no understanding of the meaning of what they are doing. Given the arrival of computer software which can carry out these algorithms mechanically, the question arises as to what parts of calculus need to be studied in the curriculum of the future. It is my contention that such a study can use the computer technology to produce a far more versatile approach to the subject, in which the numerical and graphical representations may be used from the outset to produce insights into the fundamental meanings, in which a wider understanding of the processes of change and growth will be possible than the narrow band of problems that can be solved by traditional symbolic methods of the calculus

The notion of infinite measuring number and its relevance in the intuition of infinity

In this paper a concept of infinity is described which extrapolates themeasuring properties of number rather thancounting aspects (which lead to cardinal number theory). Infinite measuring numbers are part of a coherent number system extending the real numbers, including both infinitely large and infinitely small quantities. A suitable extension is the superreal number system described here; an alternative extension is the hyperreal number field used in non-standard analysis which is also mentioned. Various theorems are proved in careful detail to illustrate that certain properties of infinity which might be considered false in a cardinal sense are true in a measuring sense. Thus cardinal infinity is now only one of a choice of possible extensions of the number concept to the infinite case. It is therefore inappropriate to judge the correctness of intuitions of infinity within a cardinal framework alone, especially those intuitions which relate to measurement rather than one-one correspondence. The same comments apply in general to the analysis of naive intuitions within an extrapolated formal framework