Geometry as a Course in Reasoning

Abstract

How can geometry be reclaimed from its present position in most of our secondary schools as little more than an unsuccessful course in formal mathematical logic, is a question that has engaged the attention of alert teachers for many years. There is a movement on foot at the present time to overhaul the whole of our mathematical teaching in all schools below the college grade which is doing much to improve the situation. As soon as school authorities throughout the country had decided that geometry could and should be taught in the newly organized junior high schools, the problem was put squarely up to mathematics teachers to reorganize the content and methods of this subject. It was quite evident from the start that the old style presentation of mathematical proofs that we call demonstrative geometry would not be understood by pupils below the ninth grade, and there was a sneaking suspicion in the minds of many that it was not too well understood by most pupils above that grade. Accordingly Euclid's organization of the subject matter of geometry was thrown boldly overboard by mathematical committees appointed to study the situation, and a course of study was recommended which was based on the capacities of childrens’ minds rather than on those of the ancient Greek philosophers. Rigid proofs were eliminated and intuitive, observational, inventional, and numerical geometry was substituted.