SUGGESTIONS FOR THE USE OF COLORED

PLATES.

IN beginning the study of solid geometry, a new difficulty is encountered, the difficulty of seeing the figures correctly. Through plane geometry, the pupil has acquired the habit of looking at the figures simply as lines making different angles and running in varying directions, but always limited to one plane. To the untrained eye, the line figures in solid geometry do not look essentially different. The teacher sees, the pupil does not, and, worse than all, too frequently the teacher fails to realize that what represents to him a solid figure is, to the pupil, a number of lines similar to those in plane geometry, only hopelessly compli cated in arrangement.

The first thing necessary, then, is to train a class to visualize correctly, to see in imagination, not a seeming confusion of lines, but the solids outlined by those lines.

In the hope of accomplishing this, various aids have been offered by text-books in the way of graphic representation, but all of them, while attractive at first, have, when tried, fallen short of expectation in teaching value.

Work with actual models is accurate and helpful, but photographic reproductions of these models make nearly the same demands upon the untrained imagination that the line figures do. Shaded figures have been used, but the similarity in tone of grays and blacks is confusing to the uneducated eye.

With the color scheme here presented, the confusion

vanishes, and the pupil not only may see, but must see, the planes in their true relations to each other. If his first glimpse of figures for solids is right, he is ready then to look for depth, distance, and three dimensions, in all succeeding figures.

The few colored figures here presented are valuable in the beginning, to show the pupil the kind of thing that he is to look for what he is expected to see. Take, for instance, the figure on page 236. To the beginner there is little sug gestion of various planes intersecting, disappearing behind each other, and reappearing, but by Plate I all this is instantly revealed. The correct visual impression here gained will then be transferred naturally to the line figure.

Another objection to the aids thus far presented lies in the fact that the text-book does all the work, leaving the pupil only an observer. If the work stops with looking at the figures and studying from them, their greatest teaching value is lost. It is comparatively easy, with a figure that has been carefully drawn and effectively shaded or colored, to grasp for the moment the general idea indicated, but the impression will be neither complete nor lasting.

Purposely only a few suggestive figures are here presented in color, it being the plan that the pupil, from the figures given in the text, and from the accompanying demonstration, shall interpret in color the solids indicated. When he is compelled thus to fix the limitations of the planes, he is led to definite knowledge that is otherwise impossible. Here is represented all the vast distance that lies between looking at a picture done by some one else, and reproducing that picture yourself, all the difference between observing and doing.

The plan here offered is capable of practical application in several ways.

Send the class to the board to draw the figures for the day with colored crayons; the result will reveal their understanding or misunderstanding of the proposition under con

sideration. With the color in their own hands, pupils are compelled to decide where plane intersects plane, where one disappears behind another, and many other things that escaped their observation in studying from the book. Take, for instance, the figure on page 236. It probably never occurred to the pupil in studying to observe into how many planes the argument is carried. When he colors it he must know.

This first work should be done rapidly, with no attempt at finished drawings. Sometimes it is well to have a class draw entirely free-hand, laying in the color rapidly, attempting only to bring out the geometric idea. At first strongly contrasting colors should be used, and, as the work is not permanent, they may be even crude, if only striking.

Following this, certain figures should be put into permanent form. Such drawing should be done carefully, with as close mathematical accuracy as board and chalk will allow. Here attention should be given to the color scheme and the result made restful to the eye. There is actual teaching value in having these difficult figures long before the attention.

With the figures thus before the class, it is easy in the few spare moments that occasionally come at the close of a recitation, to give a quick review that would not be possible if time had to be consumed in drawing.

Blackboard work is strengthened by outlining all planes. in strong white lines, just as in the book they are outlined in black.

Another useful expedient in the use of color is the careful preparation of plates outside of class. The most interesting and effective figures should be selected, and all members of the class required to execute a certain number, this number varying according to the ability of different classes. It is usually well to suggest a uniform size of paper; seven by nine inches, approximately, is desirable. One figure only should be placed on a sheet. As to size of figure, it is better not to dictate. When the first drawings are brought

together, it will not take a class long to decide which is most effective. From this they will modify their scale, approximating the best, but retaining perfect individuality.

One additional direction should be given, applying equally to board work and to work on paper. Leave only such lines as would be visible if the planes were opaque.

A glance at the colored plates here given will show that by omitting the dotted lines the figure is more effective, and the solidarity greatly emphasized. The geometric value of the lines is not lost, for the eye naturally carries them along behind the plane, and joins the parts correctly if they reappear. outlining the figure at first, these lines, of course, should be drawn, for by them is frequently determined where the planes intersect.

In

As to the medium used, that is a matter of taste and equipment. Colored pencils are easiest for the untrained hand, and good effects can be obtained with them. If any one in the class handles water colors, he should be encouraged to use them, for they make stronger figures, and the influence of even one or two working in them will elevate the entire standard.

Some or all of these expedients may be used as the conditions of individual classes indicate, but let it always be insisted that the class do the work. The most carefully executed drawing of teacher or text-book is worth less than the poorest attempt of the poorest pupil.

Finally, the method here presented is offered only as a practical suggestion for clearer teaching, not as an integral part of geometry, and may be used or not as teachers desire. Like everything else, it is capable of abuse and perversion, and whoever uses it should be ever watchful lest it overstep its proper limitations. Its purpose is not to produce a fine set of drawings, but to assist in teaching geometry. It is a means, not an end; an expedient, not a science.