Relevance.

Turner, Paul

It is said that teachers need to make the mathematics they teach in the classroom relevant to the lives of the students. Otherwise, the students will become disengaged and lost to the whole critically important enterprise of science, technology, engineering and mathematics. Furthermore, the economy of the nation will suffer because young people will lack the proper skill set for emerging twenty-first century occupations.

Yet, as I have sometimes said to my students, the thing I like most about mathematics is its irrelevance. On the face of it, this puts me at odds with the conventional wisdom. So, I am looking for a synthesis of these opposing points-of-view.

Back in 1640, Pierre de Fermat claimed that if p is a prime number that does not divide an integer a, then when [a.sub.p-1] is divided by p, the remainder is always 1. This could not have had any relevance to Fermat's work as a lawyer and government official. Indeed, for over 300 years no practical, let alone commercial, application of the result was known.

It is commercially relevant now, thanks to the work of Ronald Rivest, Adi Shamir and Leonard Adleman in the field of cryptography, but Fermat could not have predicted that outcome. Fermat used it as something with which to impress his circle of mathematical contacts who also tended to be engaged in apparently irrelevant pursuits: finding the areas under certain curves and worrying about tangents, for example.

Staying with Fermat, his other famous conjecture--the one about there being no integer solutions in x, y and z to the equation [x.sup.n] + [y.sup.n] = [z.sup.n] for any n > 2, has had no practical application of which I am aware other than to inspire generations of mathematicians to look for a proof, and in the process to invent a large quantity of new mathematics. Somehow, to all these mathematical enthusiasts and to the young Andrew Wiles, who eventually confirmed the result, the search for the missing proof must have seemed highly relevant.

Wiles's achievement is legendary. Yet, I can imagine sceptics questioning its relevance. The rejoinder 'Relevance to what?' springs to mind. Perhaps there are non-utilitarian values that those imaginary sceptics are overlooking.

On the other side of the question, students finishing a college level general mathematics class that I taught recently were asked one-by-one about their perceptions of the course. They commented on the method of delivery of the content as well as its relevance.

Several expressed an emotion bordering on gratitude due to the fact that the course had taught them things about budgeting, living costs, rent, mortgages, taxation and similar matters that they felt they would soon need to know about in order to live independently.

Not everyone in the class was quite so enthusiastic but, overall, the course writers and the teachers felt that they could justifiably give themselves a satisfactory grade for relevance.

This was not the whole story. Others in the class were not expecting to move out of home in the near future and, therefore, were not experiencing any sense of urgency about finding out about the affordability of rents or the cost of utilities. At best, these students went through the motions of completing the coursework. For them, the course was less personally relevant and, while it is hard to be certain, it is probable that less real learning occurred and the experience was under stimulating.

This anecdote illustrates my contention that while a course might have all the trappings and appearances of relevance, in the sense of applicability of the skills learnt in some place called the real-world, it can still very easily fail to capture the imagination of the students. This is because students are a wonderfully heterogeneous group. Their various needs and proclivities are not likely to be uniformly satisfied by any externally selected subset of course content, however worthy it may appear to be. What appeals to some will not necessarily arouse the interest of the rest.

The intellectual needs of developing human beings do not always align well with the demands of outside bodies and pressure groups. Perhaps the key lies at the tail end of my opening mantra: 'Teachers need to make the mathematics they teach in the classroom relevant to the lives of the students'.

If young people are valued primarily as feedstock for industry, along the lines of Aldous Huxley's Brave New World, in which humans are deliberately bred in a range of types to fill certain predetermined roles in society, then the utilitarian model of relevance is apt. The broad expanse of education is reduced to training for the particular. But, nobody really thinks like that, do they?

While it is obviously true that practical skills taught in the classroom are often relevant to the lives of students, there is more. Junior sections of public libraries contain many books written with a view to helping very young children realise that the cardinality of a set is a property that is independent of the nature of the elements of the set. The books do not express it like that, but that is what it is nonetheless. The human act of abstraction starts with the very young.

This, to me, is what mathematics and mathematics education is all about. The dichotomy with which I began fades if I think of 'the lives of the students' not so much as a multitude of circumstances external to themselves, but rather as the web of their personal ideas about the so called real world outside. Thus, according to this model, when given a set of initially unrelated ideas that a student is at least willing to contemplate, the student may make an abstract connection between them. If so, this outcome is pleasurable to the student who then naturally perceives the newly acquired concept as relevant.

A student whose interest is aroused by the juxtaposition of the following apparently unrelated statements, for example, might be willing to think about the problem of comparing proportions.

   Eight out of nine dentists prefer brand 'A'; and yet between 1852
   and 1889 there were 40 721 arrivals and 36 049 departures of
   Chinese immigrants due to the gold rush in this country.
   Furthermore, a guitar string sounding a note 'G' vibrates at the
   rate of 392 Hz while the 'F' two frets lower vibrates at 349 Hz.

In this way, the idea of percentage or of common denominator might become relevant. Ideas in mathematics become relevant, meaning that they gain a place in an individual's construction of the world, when they make sense of some of that individual's otherwise disparate notions and previously unrelated observations about things assumed to have a mathematical essence. Relevance is thus an almost entirely subjective attribute.

Paul Turner

Australian Catholic University

<pturner@gmail.com>