Learning and teaching axiomatic geometry

Summary To sum up, we have examined the various difficulties-mathematical, conceptual, and logical-involved in learning and teaching axiomatic geometry. Taking as a typical axiomatic approach one that couples Emil Artin's treatment of affine geometry with Choquet's handling of the basic order and orthogonality phenomena of Euclidean geometry, we have tried to identify these difficulties in some detail. We have noted that this particular approach isolates the graver difficulties, separating them from one another in such a way that they can be mastered by comparatively simple techniques. For both student and teacher the burden is shifted to the fuller understanding of certain fundamental concepts, such as that of a geometrical transformation, which permit making an early contact with algebra-specifically, with group theory-in the spirit of Descartes and Felix Klein. We have proposed an organization of the school program in geometry from grade K or grade 1 through grade 12 aimed at gradually acquainting the student with these basic concepts and their technical uses as they emerge from the persistent contemplation of the physical, mathematical, and logical facts. We have stated our conviction that by the 10th grade students should be prepared to understand the strategy and the significance of the axiomatic approach, and by the 11th and 12th grades to carry through that approach in adequate detail. We have taken explicitly into account the need to arrange the school geometry program in such a way that students can reach terminal points appropriate to their capacities, needs, and interests. We must emphasize that what has been presented here is no more than a rough outline, to be filled out on the basis of further psychological, mathematical and curricular investigations. I am confident that these suggestions can mature into a school geometry program not only far more significant intellectually and technically than those now commonly offered but also one easier to learn and to teach. This is not just a matter of personal belief, because many elements of the proposed program have actually been tried out in one form or another with success. What is now needed is a grand effort to assemble the tested and untested parts into an experimental whole.