Margaret Rayner: Is calculus essential?

Margaret Rayner gave the presentation Is calculus essential? at the Fourth International Congress on Mathematical Education held in Berkeley, California, USA, 10-16 August 1980. The Congress was attended by about 1800 full members and 500 associate members from about 90 countries.

We present below a version of Margaret Rayner's presentation. Before we do this, however, we must explain that the Professor Roberts she refers to is Fred S Roberts of Rutgers University. Roberts gave the lecture Is calculus necessary? which begins with the sentence "Calculus is not necessary" but ends with the conclusion: "It should be clear that my position isn't quite as extreme as my original premise suggested. I just feel that it is very important for us to modify the calculus course so as to expose a large number of students as early as possible to the increasingly broad variety of ideas which make up modern mathematics."

Here is a version of Margaret Rayner presentation.

There are two questions we are discussing today. One is the obvious one about the role of calculus. The second is the implied question as to the feasibility or even desirability of a core syllabus in mathematics courses at universities and other institutions of higher education. Let us take the second issue first.

In 1976, at the International Congress on Mathematical Education at Karlsruhe, Professor van Lint led a very general discussion on mathematics education at university (bachelor's) level and commented that, from the survey he had made, it was apparent that "old-fashioned" calculus courses were being replaced by courses of a more abstract nature; he mentioned abstract integration theory, general topology, metric spaces. He also said that some universities were then moving back from the more abstract courses: I hope that in the subsequent discussion we may hear comments on the present situation in a number of universities. I think we may reasonably suppose that there is less common ground between first year university courses for mathematicians than there used to be (even within universities in the same country) and we should therefore what the implications are. Does it question the whole concept of 'a degree in mathematics'? I believe that it does and it could reduce the possibility of students moving to graduate programmes or employment in other universities and other countries. Such a consequence would be deplorable.

This problem of diverse backgrounds has confronted the universities in Great Britain for an earlier age group - the young people coming from schools into university courses which depend on prior mathematical knowledge. About four years ago, university departments expressed considerable alarm at the decreasing overlap between syllabuses used in schools for the 16-18 year olds and claimed that it was becoming almost impossible to run first year courses economically, i.e. without splitting audiences into small groups to take account of students' previous training. There have been consultations amongst teachers, examining boards and universities and a minimum core syllabus has emerged which is receiving substantial support.

Has the situation yet arisen - or is likely to arise - that some 'core' of mathematics would be advantageous at university level? This is not an occasion when Professor Roberts and I battle to the death, but it is, I hope, an occasion on which teachers from many universities and countries can give information (and so bring the rest of us up-to-date) and also contribute comments on the idea of a core syllabus.

Now let me return to the topic of this debate: is calculus essential? If we accept that some core is required, should calculus be in it? This question must itself be interpreted in an elastic way to include the possibility that calculus may be a pre-requisite.

If any topic is to be included in a core it must satisfy the following requirements:

The topic should introduce to students those ideas which are important to mathematics itself, are of very wide application in mathematics and elsewhere, and give rise to problem which are of interest to professional mathematicians.

I wonder if this criteria is universally acceptable?

A second criterion might be:

The topic should have applications which are of interest to students because it relates to other courses they may be studying at college or because they believe it might be needed in subsequent careers.

The second criterion appears to me to be an obvious one for including a topic within the mathematics course, but I think it is not itself an adequate reason for including a topic in the core. I am increasingly aware of students who enter university courses with no clear idea bout careers or who have firm ideas which subsequently change as graduation approaches. I see therefore, enormous difficulties in finding, as a core subject, a topic which is essentially career related, which interests all students for all of the time. I think therefore we must treat the second criterion less seriously than the first - I will repeat it:

The topic should introduce to students those ideas which are important to mathematics itself, are of very wide application in mathematics and elsewhere, and give rise to problem which are of interest to mathematicians.

On this criterion, I believe that calculus is assured a place in the core.

By calculus, I mean the study of elementary functions of one variable, differentiation, integration and limits, in an intuitive way, with plenty of applications. it is the ideas here that are important, not the proofs. It is true that a few well posed questions can show the need for precision and for proof but these best come after some experience which will give a student confidence and will encourage him to be adventurous..

Dr Roberts has already mentioned many areas of applicability of calculus to problems which arise outside mathematics, but of course it is the starting point for investigations in analysis of many sorts, in differential equations, in the calculus of variations, in geometry, all areas of very great concern to present mathematicians. I think I need not elaborate this point to this audience.

Over the past year I have asked many mathematicians if calculus is essential: most of them looked at me as if I was joking or mad. But just because a topic has been included in syllabuses for any years, it cannot continue to be included by tradition alone. Think of what has happened to geometry over the last fifty years.

I contend that calculus has, as yet, no rival for richness of ideas and applications and it therefore essential in a mathematics course.

There are two main themes in mathematics; the mathematics of finite processes and the mathematics of infinite processes and I think that a graduate of mathematics should have an appreciation of both, even if he becomes an expert in only one. The two themes are like twins, closely related, having a great deal in common, but each having its own existence. But unlike some human twins, they are also of the greatest assistance to each other by exchanging ideas for bypassing difficulties. To have a mathematics course which ignores either finite or infinite processes is missing half the story.

The question is, what topic shall represent infinite processes? Shall it be convergence? From my experience, that is a subject which does not often generate enthusiasm amongst students. Shall it be analysis without a preliminary course in calculus? This appears to be a curious idea because it suggests that students should be introduced to the remedies for pathological cases before they are really familiar with the well-behaved ones. So they have no framework within which to work. Analysis has, undeniably, an important role as an introduction to mathematical rigour, but it is nothing if it does not grow out of the problems raised by calculus. It is a good working principle: Don't prove a theorem until you need it. Analysis also lacks one other feature: scope for intuition. Here I would like to quote from the preface of one of the books of Richard Courant:

The presentation of analysis as a closed system of truths without reference to their origins and purpose has, it is true, an aesthetic charm and satisfies a deep philosophical need. But the attitude of those who consider analysis solely as an abstractly logical, introverted science is not only highly unsuitable for beginners but endangers the future of the subject; for to pursue mathematical analysis while at the same time turning one's back on its applications and on intuition is to condemn it to hopeless atrophy. To me it seems extremely important that the student should be weaned from the very beginning against a smug and presumptuous purism; this is not the least of my purposes in writing this book.

And the book he wrote was Differential and Integral Calculus!

I claim, therefore, that calculus, with its ample fund of graphical illustrations, its concern with familiar concepts like area, volume, curve length provides the most appropriate introduction to infinite processes. I will therefore summarise my case for inclusion of calculus in a core syllabus as follows:

It has applications in many different areas of study outside mathematics: physical science, statistics, numerical solutions, and so forth.
It provides the best starting point for a study of infinite processes because there is great scope for demonstration of its ideas.
Without its techniques, whole areas of mathematics (for example, differential equations) which are of very considerable interest to mathematicians are out of reach.
It is a subject which gives great scope, even at an early stage, for inventiveness and original application.

I submit therefore that calculus is essential in any mathematics core which leads to a degree.

Last Updated April 2020