The Teaching of Geometry

by David Eugene Smith

The statement that a given individual has received a sound geometrical training implies that he has segregated from the whole of his sense impressions a certain set of these impressions, that he has then eliminated from their consideration all irrelevant impressions (in other words, acquired a subjective command of these impressions), that he has developed on the basis of these impressions an ordered and continuous system of logical deduction, and finally that he is capable of expressing the nature of these impressions and his deductions therefrom in terms simple and free from ambiguity. Now the slightest consideration will convince any one not already conversant with the idea, that the same sequence of mental processes underlies the whole career of any individual in any walk of life if only he is not concerned entirely with manual labor; consequently a full training in the performance of such sequences must be regarded as forming an essential part of any education worthy of the name. Moreover, the full appreciation of such processes has a higher value than is contained in the mental training involved, great though this be, for it induces an appreciation of intellectual unity and beauty which plays for the mind that part which the appreciation of schemes of shape and color plays for the artistic faculties; or, again, that part which the appreciation of a body of religious doctrine plays for the ethical aspirations. Now geometry is not the sole possible basis for inculcating this appreciation. Logic is an alternative for adults, provided that the individual is possessed of sufficient wide, though rough, experience on which to base his reasoning. Geometry is, however, highly desirable in that the objective bases are so simple and precise that they can be grasped at an early age, that the amount of training for the imagination is very large, that the deductive processes are not beyond the scope of ordinary boys, and finally that it affords a better basis for exercise in the art of simple and exact expression than any other possible subject of a school course.

Are these results really secured by teachers, however, or are they merely imagined by the pedagogue as a justification for his existence? Do teachers have any such appreciation of geometry as has been suggested, and even if they have it, do they impart it to their pupils? In reply it may be said, probably with perfect safety, that teachers of geometry appreciate their subject and lead their pupils to appreciate it to quite as great a degree as obtains in any other branch of education. What teacher appreciates fully the beauties of "In Memoriam," or of "Hamlet," or of "Paradise Lost," and what one inspires his pupils with all the nobility of these world classics? What teacher sees in biology all the grandeur of the evolution of the race, or imparts to his pupils the noble lessons of life that the study of this subject should suggest? What teacher of Latin brings his pupils to read the ancient letters with full appreciation of the dignity of style and the nobility of thought that they contain? And what teacher of French succeeds in bringing a pupil to carry on a conversation, to read a French magazine, to see the history imbedded in the words that are used, to realize the charm and power of the language, or to appreciate to the full a single classic? In other words, none of us fully appreciates his subject, and none of us can hope to bring his pupils to the ideal attitude toward any part of it. But it is probable that the teacher of geometry succeeds relatively better than the teacher of other subjects, because the science has reached a relatively higher state of perfection. The body of truth in geometry has been more clearly marked out, it has been more successfully fitted together, its lesson is more patent, and the experience of centuries has brought it into a shape that is more usable in the school. While, therefore, we have all kinds of teaching in all kinds of subjects, the very nature of the case leads to the belief that the class in geometry receives quite as much from the teacher and the subject as the class in any other branch in the school curriculum.

But is this not mere conjecture? What are the results of scientific investigation of the teaching of geometry? Unfortunately there is little hope from the results of such an inquiry, either here or in other fields. We cannot first weigh a pupil in an intellectual or moral balance, then feed him geometry, and then weigh him again, and then set back his clock of time and begin all over again with the same individual. There is no "before taking" and "after taking" of a subject that extends over a year or two of a pupil's life. We can weigh utilities roughly, we can estimate the pleasure of a subject relatively, but we cannot say that geometry is worth so many dollars, and history so many, and so on through the curriculum. The best we can do is to ask ourselves what the various subjects, with teachers of fairly equal merit, have done for us, and to inquire what has been the experience of other persons. Such an investigation results in showing that, with few exceptions, people who have studied geometry received as much of pleasure, of inspiration, of satisfaction, of what they call training from geometry as from any other subject of study,—given teachers of equal merit,—and that they would not willingly give up the something which geometry brought to them. If this were not the feeling, and if humanity believed that geometry is what Mr. Locke's words would seem to indicate, it would long ago have banished it from the schools, since upon this ground rather than upon the ground of utility the subject has always stood.

These seem to be the great reasons for the study of geometry, and to search for others would tend to weaken the argument. At first sight they may not seem to justify the expenditure of time that geometry demands, and they may seem unduly to neglect the argument that geometry is a stepping-stone to higher mathematics. Each of these points, however, has been neglected purposely. A pupil has a number of school years at his disposal; to what shall they be devoted? To literature? What claim has letters that is such as to justify the exclusion of geometry? To music, or natural science, or language? These are all valuable, and all should be studied by one seeking a liberal education; but for the same reason geometry should have its place. What subject, in fine, can supply exactly what geometry does? And if none, then how can the pupil's time be better expended than in the study of this science?[14] As to the second point, that a claim should be set forth that geometry is a sine qua non to higher mathematics, this belief is considerably exaggerated because there are relatively few who proceed from geometry to a higher branch of mathematics. This argument would justify its status as an elective rather than as a required subject.

Let us then stand upon the ground already marked out, holding that the pleasure, the culture, the mental poise, the habits of exact reasoning that geometry brings,

and the general experience of mankind upon the subject are sufficient to justify us in demanding for it a reasonable amount of time in the framing of a curriculum. Let us be fair in our appreciation of all other branches, but let us urge that every student may have an opportunity to know of real geometry, say for a single year, thereafter pursuing it or not, according as we succeed in making its value apparent, or fail in our attempt to present worthily an ancient and noble science to the mind confided to our instruction.

The shortsightedness of a narrow education, of an education that teaches only machines to a prospective mechanic, and agriculture to a prospective farmer, and cooking and dressmaking to the girl, and that would exclude all mathematics that is not utilitarian in the narrow sense, cannot endure.

The community has found out that such schemes may be well fitted to give the children a good time in school, but lead them to a bad time afterward. Life is hard work, and if they have never learned in school to give their concentrated attention to that which does not appeal to them and which does not interest them immediately, they have missed the most valuable lesson of their school years. The little practical information they could have learned at any time; the energy of attention and concentration can no longer be learned if the early years are wasted. However narrow and commercial the standpoint which is chosen may be, it can always be found that it is the general education which pays best, and the more the period of cultural work can be expanded the more efficient will be the services of the school for the practical services of the nation.[15]

Of course no one should construe these remarks as opposing in the slightest degree the laudable efforts that are constantly being put forth to make geometry more