Introduction
My intention in this series of articles is to study, grosso modo, the dominant trends in present-day mathematics, and to draw from this study principles that should govern the choice of content and style in the teaching of mathematics at the secondary and elementary levels. Some of these principles will be time-independent, in the sense that they should always have been applied to the teaching of mathematics; others will be of special application to the needs of today's, and tomorrow's, students and will be, in that sense, new.
The principles will be illustrated by examples in order to avoid the sort of frustrating vagueness which often accompanies even the most respectable recommendations (thus, "problem solving [should] be the focus of school mathematics in the 1980's" [1]). However, before embarking on a intended as a contribution to the discussion of how to achieve a successful mathematical education, it would be as well to make plain what are our criteria of success. Indeed, it would be as well to be clear what we understand by successful education, since we would then be able to derive the indicated criteria by specialization. Let us begin by agreeing that a successful education is one which conduces to a successful life. However, there is a popular, persistent and paltry view of the successful life which we must immediately repudiate. This is the view that success in life is measured by affluence and is manifested by power and influence over others. It is very relevant to my theme to recall that, when Queen Elizabeth was the guest of President and Mrs Reagan in California, the "successes" who were gathered together to greet her were not Nobel prize-winners, of which California may boast remarkably many, but stars of screen and television. As the London Times described the occasion, "Queen dines with celluloid royalty". It was apparently assumed that the company of Frank Sinatra, embodying the concept of success against which I am inveighing, would be obviously preferable to that of, say, Linus Pauling.
The Reaganist-Sinatrist view of success contributes a real threat to the integrity of education; for education should certainly never be expected to conduce to that kind of success. At worst, this view leads to a complete distortion of the educational process; at the very least, it allies education far too closely to specific career objectives, an alliance which unfortunately has the support of many parents naturally anxious for their children's success. We would replace the view we are rejecting by one which emphasizes the kind of activity in which an individual indulges, and the motivation for so indulging, rather than his, or her, accomplishment in that activity.
The realization of the individual's potential is surely a mark of success in life. Contrasting our view with that which we are attacking, we should seek power over ourselves, not over other people; we should seek the knowledge and understanding to give us power and control over things, not people. We should want to be rich but in spiritual rather than material resources. We should want to influence people, but by the persuasive force of our argument and example, and not by the pressure we can exert by our control of their lives and, even more sinisterly, of their thoughts. It is absolutely obvious that education can, and should, lead to a successful life, so defined. Moreover, mathematical education is a particularly significant component of such an education. This is true for two reasons. On the one hand, I would state dogmatically that mathematics is one of the human activities, like art, literature, music, or the making of good shoes, which is intrinsically worthwhile. On the other hand, mathematics is a key element in science and technology and thus vital to the understanding, control and development of the resources of the world around us. These two aspects of mathematics, often referred to as pure mathematics and applied mathematics, should both be present in a well-balanced, successful mathematics education.
Let me end these introductory remarks by referring to a particular aspect of the understanding and control to which mathematics can contribute so much. Through our education we hope to gain knowledge. We can only be said to really know something if we know that we know it. A sound education should enable us to distinguish between what we know and what we do not know; and it is a deplorable fact that so many people today, including large numbers of pseudo-successes but also, let us admit, many members of our own
academic community, seem not to be able to make the distinction. It is of the essence of genuine mathematical education that it leads to understanding and skill; short cuts to the acquisition of skill, without understanding, are often favored by self-confident pundits of mathematical education, and the results of taking such short cuts are singularly unfortunate for the young traveler. The victims, even if "successful", are left precisely in the position of not knowing mathematics and not knowing they know no mathematics. For most, however, the skill evaporates or, if it does not, it becomes out-dated. No real ability to apply quantitative reasoning to a changing world has been learned, and the most frequent and natural result is the behavior pattern known as "mathematics avoidance". Thus does it transpire that so many prominent citizens exhibit both mathematics avoidance and unawareness of ignorance.
This then is my case for the vital role of a sound mathematical education, and from these speculations I derive my criteria of success.
No comments:
Post a Comment