The teaching of elementary mathematics in Scotland in the 19th Century

Abstract

In the title we have used the term "Elementary Mathematics" a somewhat vague phrase which demands a word of explanation. Hardy's "Pure Mathematics" a wellknown work studied by advanced students at our Universities is described by its author as "being really elementary" that is, dealing with the fundamental ideas which are the starting points for chains of deductive reasoning. On the other hand, in common speech, by association of ideas, 'elementary' is often used as synonymous with the initial stages of a study and even with the easy parts of the study. In the teaching of Latin it is customary to study the 'elements' of the language before proceeding, to read the authors. As the study progresses less stress is laid on the 'elements'. Mathematics however is somewhat unusual in that the novice and the don both study its elements and perhaps the don gives far more attention to them. Leslié wrote that it was "the nature of mathematical science to advance in continual progression. Each step carries it to others still higher." Mathematics starts from certain hypotheses which may or may not be true and from these develops trains of reasoning leading to conclusions which are true if the hypotheses are true. In this sense Mathematics may be held to 'advance'. But it does not advance in only one direction. Mathematicians are constantly examining their hypotheses to see if some more general hypothesis could be found of which these are particular cases. Thus Mathematicians try to extend their knowledge in the reverse direction. Indeed to use a metaphor from its own language, Mathematics is like a continuum, the origin can be chosen anywhere and progress made in either the positive or the negative direction. Or a rotation of the axis can be made and the whole investigation given a new direction altogether. Mathematics is most definitely a science in which a study of the 'elements' is not confined to the initial stages but is taking place all the time. Thus if our subject had been "The Teaching of the Elements of Mathematics" its scope would have included the whole field of mathematics. However it is "The Teaching of Elementary Mathematics ", a change in phrase which permits of quite a different interpretation. "Elementary Mathematics" will be taken as referring to those parts of mathematics which are normally studied as the elements or basis of mathematical education not as the basis of mathematics itself. The phrase 'normally studied' is used advisedly for we have known a student who had never studied trigonometry make excellent progress in the calculus and learn trigonometry as a by- product of his main study.