The secondary school curriculum
Let us organize this discussion around the "In and Out" principle. That is, we will list the topics which should be "In" or strongly emphasized, and the topics which should be "Out" or very much underplayed. We will also be concerned to recommend or castigate, as the case may be, certain teaching strategies and styles. We do not claim that all our recommendations are strictly contemporary, in the sense that they are responses to the current prevailing changes in mathematics and its uses; some, in particular those devoted to questions of teaching practice, are of a lasting nature and should, in my judgment, have been adopted long since. We will present a list of "In" and "Out" items, followed by commentary. We begin with the "Out" category, since this is more likely to claim general attention; and within the "Out" category we first consider pedagogical techniques.
OUT(Secondary Level)
1. TEACHING STRATEGIES
Authoritarianism.
Orthodoxy.
Pointlessness.
Pie-in-the-sky motivation.
2. TOPICS
Tedious hand calculations.
Complicated trigonometry.
Learning geometrical proofs.
Artificial "simplifications".
Logarithms as calculating devices.
There should be no need to say anything further about the evils of authoritarianism and pointlessness in presenting mathematics. They disfigure so many teaching situations and are responsible for the common negative attitudes towards mathematics which regard it as unpleasant and useless. By orthodoxy we intend the magisterial attitude which regards one "answer" as correct and all others as (equally) wrong. Such an attitude has been particularly harmful in the teaching of geometry. Instead of being a wonderful source of ideas and of questions, geometry must appear to the student required to set down a proof according to rigid and immutable rules as a strange sort of theology, with prescribed responses to virtually meaningless propositions.
Pointlessness means unmotivated mathematical process. By "pie-in-thesky" motivation we refer to a form of pseudomotivation in which the student is assured that, at some unspecified future date, it will become clear why the current piece of mathematics warrants learning. Thus we find much algebra done because it will be useful in the future in studying the differential and integral calculus - just as much strange arithmetic done at the elementary level can only be justified by the student's subsequent exposure to algebra. One might perhaps also include here the habit of presenting to the student applications of the mathematics being learnt which could only interest the student at a later level of maturity; obviously, if an application is to motivate a student's study of a mathematical topic, the application must be interesting.
With regard to the expendable topics, tedious hand calculations have obviously been rendered obsolete by the availability of hand-calculators and minicomputers. To retain these appalling travesties of mathematics in the curriculum can be explained only by inertia or sadism on the part of the teacher and curriculum planner. It is important to retain the trigonometric functions (especially as functions of real variables) and their basic identitites, but complicated identities should be eliminated and tedious calculations reduced to a minimum. Understanding geometric proofs is very important; inventing one's own is a splendid experience for the student; but memorizing proofs is a suitable occupation only for one contemplating a monastic life of extreme asceticism. Much time is currently taken up with the student processing a mathematical expression which came from nowhere, involving a combination of parentheses, negatives, and fractions, and reducing the expression to one more socially acceptable. This is absurd; but, of course, the student must learn how to substitute numerical values for the variables appearing in a natural mathematical expression.
Let us now turn to the positive side. Since, as our first recommendation below indicates, we are proposing an integrated approach to the curriculum, the topics we list are rather of the form of modules than full-blown courses.
IN (Secondary Level)
1. TEACHING STRATEGIES
An integrated approach to the curriculum, stressing the
interdependence of the various parts of mathematics.
Simple application.
Historical references.
Flexibility.
Exploitation of computing availability.
2. TOPICS
Geometry and algebra (e.g. linear and quadratic
functions, equations and inequalities).
Probability and statistics.
Approximation and estimation, scientific notation.
Iterative procedures, successive approximation.
Rational numbers, ratios and rates.
Arithmetic mean and geometric mean (and harmonic mean).
Elementary number theory.
Paradoxes.
With respect to teaching strategies, our most significant recommendation is the first. (I do not say it is the most important, but it is the most characteristic of the whole tenor of this article.) Mathematics is a unity, albeit a remarkably subtle one, and we must teach mathematics to stress this. It is not true, as some claim, that all good mathematics - or even all applicable mathematics - has arisen in response to the stimulus of problems coming from outside mathematics; but it is true that all good mathematics has arisen from the then existing mathematics, frequently, of course, under the impulse of a "real world" problem. Thus mathematics is an interrelated and highly articulated discipline, and we do violence to its true nature by separating it- for teaching or research purposes- into artificial watertight compartments. In particular, geometry plays a special role in the history of human thought. It represents man's (and woman's!) primary attempt to reduce the complexity of our three-dimensional ambience to one-dimensional language. It thus reflects our natural interest in the world around us, and its very existence testifies to our curiosity and our search for patterns and
order in apparent chaos. We conclude that geometry is a natural conceptual framework for the formulation of questions and the presentation of results. It is not, however, in itself a method of answering questions and achieving results. This role is preeminently played by algebra. If geometry is a source of questions and algebra a means of answering them, it is plainly ridiculous to separate them. How many students have suffered through algebra courses, learning methods of solution of problems coming from nowhere? The result
of such compartmentalized instruction is, frequently and reasonably, a sense of futility and of pointlessness of mathematics itself.
The good sense of including applications and, where appropriate, references to the history of mathematics is surely self-evident. Both these recommendations could be included in a broader interpretation of the thrust toward an integrated curriculum. The qualification that the applications should be simple is intended to convey both that the applications should not involve sophisticated scientific ideas not available to the students - this is a frequent defect of traditional "applied mathematics" - and that the applications should be of actual interest to the students, and not merely important. The notion of flexibility with regard to the curriculum is inherent in an integrated approach; it is obviously inherent in the concept of good teaching. Let us admit, however, that it can only be achieved if the teacher is confident in his, or her, mastery of the mathematical content. Finally, we stress as a teaching strategy the use of the hand-calculator, the minicomputer and, where appropriate, the computer, not only to avoid tedious calculations but also in very positive ways. Certainly we include the opportunity thus provided for doing actual numerical examples with real-life data, and the need to re-examine the emphasis we give to various topics in the light of computing availability. We mention here the matter of computer-aided instruction, but we believe that the advantages of this use of the computer depend very much on local circumstances, and are more likely to arise at the elementary level.
With regard to topics, we have already spoken about the link between geometry and algebra, a topic quite large enough to merit a separate article. The next two items must be in the curriculum simply because no member of a modern industrialized society can afford to be ignorant of these subjects, which constitute our principal day-to-day means of bringing quantitative reasoning to bear on the world around us. We point out, in addition, that approximation and estimation techniques are essential for checking and interpreting machine calculations.
It is my belief that much less attention should be paid to general results on the convergence of sequences and series, and much more on questions related to the rapidity of convergence and the stability of the limit. This applies even more to the tertiary level. However, at the secondary level, we should be emphasizing iterative procedures since these are so well adapted to computer programming. Perhaps the most important result - full of interesting applications. It is probable that the whole notion of proof and definition by induction should be recast in "machine" language for today's student. The next recommendation is integrative in nature, yet it refers to a change which is long overdue. Fractions start life as parts of wholes and, at a certain stage, come to represent amounts or measurements and therefore numbers. However, they are not themselves numbers; the numbers they represent are rational numbers. Of course, one comes to speak of them as numbers, but this should only happen when one has earned the right to be sloppy by understanding the precise nature of fractions . If rational numbers are explicitly introduced, then it becomes unnecessary to treat ratios as new and distinct quantities. Rates also may then be understood in the context of ratios and dimensional analysis.
However, there is a further aspect of the notion of rate which it is important to include at the secondary level.
I refer to average rate of change and, in particular, average speed. The principles of grammatical construction suggest that, in order to understand the composite term "average speed" one must understand the constituent
terms "average" and "speed". This is quite false; the term "average speed" is much more elementary than either of the terms "average", "speed", and is not, in fact, their composite. A discussion of the abstractions "average" and "speed" at the secondary level would be valuable in itself and an excellent preparation for the differential and integral calculus. Related to the notion of average is, of course, that of arithmetic mean. I strongly urge that there be, at the secondary level, a very full discussion of the arithmetic, geometric and harmonic means and of the relations between them. The fact that the arithmetic mean of the non-negative quantities a 1 , ~, ... , an is never less than their geometric mean and that equality occurs precisely when a 1 = a2 = ... = an, may be used to obtain many maximum or minimum results which are traditionally treated as applications of the differential calculus of several variables - a point made very effectively in a book by Ivan Niven.
Traditionally, Euclidean geometry has been held to justify its place in the secondary curriculum on the grounds that it teaches the student logical reasoning. This may have been true in some Platonic academy. What we can observe empirically today is that it survives in our curriculum in virtually total isolation from the rest of mathematics; that it is not pursued at the university; and that it instills, in all but the very few, not a flair for logical reasoning but distaste for geometry, a feeling of pointlessness, and a familiarity with failure. Again, it would take a separate article (at the very least) to do justice to the intricate question of the role of synthetic geometry in the curriculum. Here, I wish to propose that its hypothetical role can be assumed by a study of elementary number theory, where the axiomatic system is so much less complex than that of plane Euclidean geometry. Moreover, the integers are very "real" to the student and, potentially, fascinating. Results can be obtained by disciplined thought, in a few lines, that no high speed computer could obtain, without the benefit of human analysis, in the student's lifetime.
Of course, logical reasoning should also enter into other parts of the curriculum; of course, too, synthetic proofs of geometrical propositions should continue to play a part in the teaching of geometry, but not at the expense of the principal role of geometry as a source of intuition and inspiration and as a means of interpreting and understanding algebraic expressions. My final recommendation is also directed to the need for providing stimulus for thought. Here I understand, by a paradox, a result which conflicts with conventional thinking, not a result which is self-contradictory. A consequence of an effective mathematical education should be the inculcation of a healthy scepticism which protects the individual against the blandishments of self-serving propagandists, be they purveyors of perfumes, toothpastes, or politics. In this sense a consideration of paradoxes fully deserves to be classified as applicable mathematics! An example of a paradox would be the following: Students A and B must submit to twenty tests during the school term. Up to half term, student A had submitted to twelve tests and passed three, while student B had submitted to six tests and passed one. Thus, for the first half of the term, A's average was superior to B's. In the second half of the term, A passed all the remaining eight tests, while B passed twelve of the remaining fourteen. Thus, for the second half of the term, A's average was also superior to B's. Over the whole term, A passed eleven tests out of twenty, while B passed thirteen tests out of twenty, giving B a substantially better average than A.
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