THE CURRENT MATHEMATICS CURRICULUM: TRADITIONS AND CONCERNS

For many years, a crucial place in the mathematics curriculum of the last year of secondary school or the first year of university studies has been occupied by the differential and integral calculus. The calculus can be seen both as the culmination of the secondary school mathematics curriculum and as the beginning of the serious study of mathematics in the university. In some sense, the study of calculus has become synonymous with the serious study of mathematics. The central and essential position occupied by calculus can be traced to at least two interrelated causes.

For mathematicians, calculus represents the methodology and techniques needed for the study of functions, first defined on the real line, then on higher-dimensional Euclidean spaces, and finally on the complex plane. Thus, the study of the calculus allows students for the first time to acquire the formal and abstract tools that are essential for the further study of higher mathematics.

On the other hand, calculus provides the foundation for the application of mathematics to the physical sciences and engineering. These applications date back to Newton's original development of the calculus in the seventeenth century, and since that time they have been wildly successful across a vast collection of disciplines, even including (in recent years), the biological sciences and economics. All of the calculus-based applications are based on mathematical models that can be regarded as being continuous; that is, the quantities being modeled are real numbers (or elements of some Euclidean space Rn).