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The Educational Significance of Algebra and

I

Geometry

BERNARD C. EWER, PH.D., MT. HERMON, MASS.

F some pedagogical Philistine were to ask, "What is the practical value of training in algebra and geometry?" a fair answer would run as follows: These subjects are indispensable to the student of certain branches of pure and applied science, e. g., physics and engineering, since they are necessary to calculation, but for other persons they have little or no direct practical utility; rarely need one have specific recourse to x's and y's or the line AB to solve a concrete problem of daily life. If the questioner persisted, "Why, then, since only a few pupils will enter the regions where algebraic formulas and geometrical diagrams flourish, should we spend so much time upon them in our schools?" the reply would be that while their direct utility is inconsiderable, their indirect utility is important. They are indeed vital in education, and it is the purpose of the following paragraphs to point out some details of this practical signifi

cance.

That education consists not only in the acquisition of truth, but also in the development of acquisitive power, in mental exercise as well as in mental contents, may be taken as a premise. Since, therefore, for most students, the truths of algebra and geometry will never become directly useful, it follows that these subjects must demonstrate their usefulness through the pursuit of them as an active mental process. They are, in a word, helpful studies. By means of them the pupil forms mental habits which are available and efficient in studying other things and in solving daily problems. For it is true of habit that it brings not only ease and rapidity in performing the operation which developed it, but also, though in a lower degree, additional skill in performing similar operations. Acts which employ the same muscles react by facilitating each other; the knotting of one's necktie and the folding of paper train the

fingers in deftness which is of service in applying a bandage. Especially is this principle true of learning as a pursuit. Different branches require, to a greater or a less extent, the same kinds of mental activity. It is remarkable that the great scholar, though perhaps known to the world only through proficiency in a single subject, is usually well versed in several others, in which his attainments have without doubt been assisted by the skill and thoroughness which made him master of the first. Similarly, too, the utility of algebra and geometry is no doubt indirect, yet it is genuine and far-reaching.

Just how does this indirect utility accomplish itself? Simply stated, by forming habits of precision and systematization. In fact, algebra and geometry are in some respects the best training ground for the formation of these habits. Let me explain this more fully.

By precision I mean careful attention to details. At first thought any alleged connection between plus and minus signs, coefficients, exponents, and on the other hand the constituents of language, literature, and inexact science, may seem largely fictitious; and hence my assertion that precision in the former inculcates precision in the latter may appear incorrect. But however unlike these objects of attention may be, the process of attention is much the same in both cases. Physiologically, the mechanism of perception is rendered more accurate by exercise; and psychologically, the practice of dwelling on a particular until its subsumption under a general rule guarantees its reliability, develops a most useful habit of study. For there is a natural tendency, in following a course of reasoning, to pass hastily over a step which seems probable or vaguely familiar, although a moment of deliberation may reveal a serious logical defect or omission. So in geometry a pupil will refer glibly to this or that theorem without really considering whether the given construction satisfies the necessary conditions. Hence by enforcing precision in such cases the teacher will foster a self-critical habit of thought that will be universally beneficial in study. Hasty, unclear thinking may thus be made to stamp itself as uncertain and unsatisfactory. I might add also that there is here an opportunity for training in

intellectual honesty which is likely to be of profound ethical importance in later development.

In the first place, therefore, algebra and geometry may be made to serve in the formation of good mental habits by training the pupil in precision of attention and thus of understanding.

Secondly, systematization. By this I mean the mental habit of arranging details as parts of a whole. For example, I am now conscious of several particular things I wish to say on this subject. They come to my mind in a confused jumble, and if I were to set them down as they appear they would probably baffle the reader and lose much of their significance. A bit of reflection, however, tells me that they may be classified under three heads the definition of systematization, its importance, and its relation to algebra and geometry. The result is at least an orderly progress in the thought, expressed in the succession of paragraphs. Again, the reading of an essay may be the simple perusal of sentence after sentence, each perhaps clear in itself, or it may be the further relating of the thought of the sentence to the subject of the paragraph, and the paragraph to the purpose of the essay. Such systematization involves the twin processes of analysis and synthesis. It is necessary first to find out the fundamental thought of the subject, "to go to the root of the matter;" and second to construct thereupon an arrangement of parts, each of which has a definite place and significance with respect to the fundamental thought. This method of study is usually hard work, for it requires a large amount of abstraction, i. e., the searching for relations between the part and the whole, and unfortunately most writers do not attempt to help the reader in this respect; but the result is sure to be a systematic comprehension of a complex thought as against a confused or uncertain acquaintance with its various details.

The value of such a habit of study is exceeding great. Facts are much more easily remembered as parts of a system than as isolated facts. Psychologically, indeed, this is the art of" associative memory" in its most perfect form. Historical dates, for example, which have long eluded us with the most

persistent perversity, are easily pigeon-holed when their occurrences are grasped as items of a period of development, i. e., as details of a historical system. Likewise the peculiar usages of Latin, perplexing in translation and baffling in composition, are rendered much more familiar if learned in relation to certain broad principles of the language. Furthermore, it often happens that our subsequent use of a fact regards just this systematic relationship. Economic statistics, for most of us, are a burden which should be reduced to a minimum; but the outlines of economic structure and development are of great importance for our thinking. Hence for at least two good reasons it is desirable to cultivate the habit of learning things systematically.

That the study of algebra and geometry confirm this habit is certain. An algebraic exercise or problem, or a geometrical demonstration, is in itself a small system, an essential feature of which is orderliness; proper arrangement of parts is necessary to success. In the problem the central thought to be reached by analysis is the equation; secondarily, if there are several unknown quantities, their relations must also be analytically determined. Construction of the equation by synthesis of known and unknown quantities is, in point of difficulty, nine tenths of the performance; it remains only to complete the solution by following definite rules of simplification. In elementary algebra the solution of problems is by far the best illustration of what I have called systematization, although factoring, simultaneous equations, and other minor subjects. may be turned to account in developing the same mental habit. A bit of analysis discovers that 4x6 — 9(y + z) is the difference between the squares of 2x8 and 3(y + z); this is the fundamental thought. Synthesis, by rule, of these latter quantities gives the desired factors, viz., 2x + 3(y + z)1 and 3(y+z). Likewise a pair of simultaneous equations stand in some relation, to be ascertained by analysis, such that a certain process, let us say division, will eliminate an unknown quantity, or otherwise simplify the solution.

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A geometrical theorem is a fertile field for systematic procedure. The analytic question is, What previous theorems or

relationships that may be turned to account are suggested by what is here given? Synthesis, in logical form, of elements thus obtained constitutes the demonstration. For example, analysis of the proposition "The sum of the angles of any triangle is equal to two right angles" reminds us that the "sum" of angles implies arrangement side by side around a common vertex. Once this idea is grasped, the further insight that such an arrangement corresponds to certain theorems about parallel lines cut by a transversal follows by simple association of geometrical ideas. These elements are then easily arranged in a logical system which proves the proposition. It is indeed unfortunately true that most text-books of geometry minimize the necessity for analysis both by offering synthetic demonstrations without explaining or even mentioning their important analytic foundations, and also by accompanying "original" exercises with appropriate "hints" which discourage and frustrate originality. Nevertheless the teacher can inculcate the habit of systematization in at least some members of a class by discussing theorems in the way suggested.

Such a habit, I repeat, becomes useful in all further study. So also with the habit of precision. Let me guard, however, against certain possible objections. It does not follow that good students must possess mathematical proficiency, or have had extraordinary mathematical training. Neither is a mathematical habit of mind in itself a reliable mental equipment for dealing with practical problems. Nor, indeed, are the beneficial results of such habit always distinctly obvious. Our mental life is vastly complex; our theoretical beliefs and practical decisions are due to all sorts of conflicting causes, many of them emotional and irrational. Mathematical habit can be at best but a strictly limited function of life. Yet the importance in education of elementary mathematical study is none the less; indeed, its very abstractness seems to give it a superior general utility. The fact that good students of mathematics are usually good students of other subjects also lends authority to this opinion. Algebra and geometry are therefore valuable features of school training.

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