Demonstrative Algebra

Abstract

High School or college algebra, in comparison with high school geometry, is commonly recognized as a loosely organized subject. In algebra, the usual emphasis is on generalization of the concepts and rules of arithmetic, and on the use of a more powerful symbolism to solve numerical problems, but not on the systematic, logical development of the subject from a set of postulates and undefined terms. Standard terminology distinguishes between informal geometry of the junior high school and demonstrative geometry of the senior high school, but in the field of algebra we generally have only such designations as elementary, intermediate, advanced, or college algebra. These various courses certainly differ in the complexity of topics considered, and in the later courses somewhat more attention is probably paid to such fundamental postulates as commutative and associative laws, and more proofs of specific thorems may be given. But it can hardly be said that under usual conditions the logical structure of the subject, as a whole, is significantly much more advanced in the later than in the earlier courses. Somehow, we do not hear of courses entitled “demonstrative algebra.”