The Teaching of Geometry

by David Eugene Smith

Whichever plan of selection is taken, it is important to introduce a considerable number of one-step exercises immediately, that is, exercises that require only one significant step in the proof. In this way the pupil acquires confidence in his own powers, he finds that geometry is not mere memorizing, and he sees that each proposition makes him the master of a large field. To delay the exercises to the end of each book, or even to delay them for several lessons, is to sow seeds that will result in the attempt to master geometry by the sheer process of memorizing.

As to the nature of these exercises, however, the mistake must not be made of feeling that only those have any value that relate to football or the laying out of a tennis court. Such exercises are valuable, but such exercises alone are one-sided. Moreover, any one who examines the hundreds of suggested exercises that are constantly appearing in various journals, or who, in the preparation of teachers, looks through the thousands of exercises that come to him in the papers of his students, comes very soon to see how hollow is the pretense of most of them. As has already been said, there are relatively few propositions in geometry that have any practical applications, applications that are even honest in their pretense. The principle that the writer has so often laid down in other works, that whatever pretends to be practical should really be so, applies with much force to these exercises. When we can find the genuine application, if it is within reasonable grasp of the pupil, by all means let us use it. But to put before a class of girls some technicality of the steam engine that only a skilled mechanic would be expected to know is not education,—it is mere sham. There is a noble dignity to geometry, a dignity that a large majority of any class comes to appreciate when guided by an earnest teacher; but the best way to destroy this dignity, to take away the appreciation of pure mathematics, and to furnish weaker candidates than now for advance in this field is to deceive our pupils and ourselves into believing that the ultimate purpose of mathematics is to measure things in a way in which no one else measures them or has ever measured them.

In the proof of the early propositions of plane geometry, and again at the beginning of solid geometry, there is a little advantage in using colored crayon to bring out more distinctly the equal parts of two figures, or the lines outside the plane, or to differentiate one plane from another. This device, however, like that of models in solid geometry, can easily be abused, and hence should be used sparingly, and only until the purpose is accomplished. The student of mathematics must learn to grasp the meaning of a figure drawn in black on white paper, or, more rarely, in white on a blackboard, and the sooner he is able to do this the better for him. The same thing may be said of the constructing of models for any considerable number of figures in solid geometry; enough work of this kind to enable a pupil clearly to visualize the solids is valuable, but thereafter the value is usually more than offset by the time consumed and the weakened power to grasp the meaning of a geometric drawing.

There is often a tendency on the part of teachers in their first years of work to overestimate the logical powers of their pupils and to introduce forms of reasoning and technical terms that experience has proved to be unsuited to one who is beginning geometry. Usually but little harm is done, because the enthusiasm of any teacher who would use this work would carry the pupils over the difficulties without much waste of energy on their part. In the long run, however, the attempt is usually abandoned as not worth the effort. Such a term as "contrapositive," such distinctions as that between the logical and the geometric converse, or between perfect and partial geometric conversion, and such pronounced formalism as the "syllogistic method,"—all these are happily unknown to most teachers and might profitably be unknown to all pupils. The modern American textbook in geometry does not begin to be as good a piece of logic as Euclid's "Elements," and yet it is to be observed that none of these terms is found in this classic work, so that they cannot be thought to be necessary to a logical treatment of the subject. We need the word "converse," and some reference to the law of converse is therefore permissible; the meaning of the reductio ad absurdum, of a necessary and sufficient condition, and of the terms "synthesis" and "analysis" may properly form part of the pupil's equipment because of their universal use; but any extended incursion into the domain of logic will be found unprofitable, and it is liable to be positively harmful to a beginner in geometry.

A word should be said as to the lettering of the figures in the early stages of geometry. In general, it is a great aid to the eye if this is carried out with some system, and the following suggestions are given as in accord with the best authors who have given any attention to the subject:

1. In general, letter a figure counterclockwise, for the reason that we read angles in this way in higher mathematics, and it is as easy to form this habit now as to form one that may have to be changed. Where two triangles are congruent, however, but have their sides arranged in opposite order, it is better to letter them so that their corresponding parts appear in the same order, although this makes one read clockwise.

2. For the same reason, read angles counterclockwise. Thus ∠A is read "BAC," the reflex angle on the outside of the triangle being read "CAB." Of course this is not vital, and many authors pay no attention to it; but it is convenient, and if the teacher habitually does it, the pupils will also tend to do it. It is helpful in trigonometry, and it saves confusion in the case of a reflex angle in a polygon. Designate an angle by a single letter if this can conveniently be done.

3. Designate the sides opposite angles A, B, C, in a triangle, by a, b, c, and use these letters in writing proofs.

4. In the case of two congruent triangles use the letters A, B, C and A', B', C', or X, Y, Z, instead of letters chosen at random, like D, K, L. It is easier to follow a proof where some system is shown in lettering the figures. Some teachers insist that a pupil at the blackboard should not use the letters given in the textbook, hoping thereby to avoid memorizing. While the danger is probably exaggerated, it is easy to change with some system, using P, Q, R and P', Q', R', for example.