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.