THE PRINCIPLES OF MATHEMATICS IN RELATION TO ELEMENTARY TEACHING
(_International Congress of Mathematicians, Cambridge, August,
1912_)
WE are concerned not with the advanced teaching of a few specialist
mathematical students, but with the mathematical education of the
majority of boys in our secondary schools. Again these boys must be
divided into two sections: one section consists of those who desire to
restrict their mathematical education; the other section is made up of
those who will require some mathematical training for their subsequent
professional careers, either in the form of definite mathematical
results or in the form of mathematically trained minds.
I shall call the latter of these two sections the mathematical section,
and the former the non-mathematical section. But I must repeat that by
the mathematical section is meant that large number of boys who desire
to learn more than the minimum amount of mathematics. Furthermore, most
of my remarks about these sections of boys apply also to elementary
classes of our University students.
The subject of this paper is the investigation of the place which
should be occupied by modern investigations respecting mathematical
principles in the education of both of these sections of school-boys.
To find a foothold from which to start the inquiry, let us ask why the
non-mathematical section should be taught any mathematics at all beyond
the barest elements of arithmetic. What are the qualities of mind which
a mathematical training is designed to produce when it is employed as
an element in a liberal education?
My answer, which applies equally to both sections of students, is
that there are two allied forms of mental discipline which should be
acquired by a well-designed mathematical course. These two forms though
closely allied are perfectly distinct.
The first form of discipline is not in its essence logical at all.
It is the power of clearly grasping abstract ideas, and of relating
them to particular circumstances. In other words, the first use of
mathematics is to strengthen the power of abstract thought. I repeat
that in its essence this has nothing to do with logic, though as a
matter of fact a logical discipline is the best method of producing the
desired effect. It is not the philosophical theory of abstract ideas
which is to be acquired, but the habit and the power of using them.
There is one and only way of acquiring the habit and the power of using
anything, that is by the simple common-place method of habitually using
it. There is no other short cut. If in education we desire to produce
a certain conformation of mind, we must day by day, and year by year,
accustom the students' minds to grow into the desired structural shape.
Thus to teach the power of grasping abstract ideas and the habit of
using them, we must select a group of such ideas, which are important
and are also easy to think about because they are clear and definite.
The fundamental mathematical truths concerning geometry, ratio,
quantity, and number, satisfy these conditions as do no others. Hence,
the fundamental universal position held by mathematics as an element of
a liberal education.
But what are the fundamental mathematical truths concerning geometry,
quantity and number?
At this point we come to the great question of the relation between
the modern science of the principles of mathematics and a mathematical
education.
My answer to the question as to these fundamental mathematical truths
is, that in any absolute sense there are none. There is no unique
small body of independent primitive unproved propositions which are
the necessary starting points of all mathematical reasoning on these
subjects. In mathematical reasoning the only absolute necessary
pre-suppositions are those which make logical deduction possible.
Between these absolute logical truths and so-called fundamental truths
concerning geometry, quantity and number, there is a whole new world of
mathematical subjects concerning the logic of propositions, of classes,
and of relations.
But this subject is too abstract to form an elementary training ground
in the difficult art of abstract thought.
It is for this reason that we have to make a compromise and start from
such obvious general ideas as naturally occur to all men when they
perceive objects with their senses.
In geometry, the ideas elaborated by the Greeks and presented by Euclid
are, roughly speaking, those adapted for our purpose, namely, ideas
of volumes, surfaces, lines, of straightness and of curvature, of
intersection and of congruence, of greater and less, of similarity,
shape, and scale. In fact, we use in education those general ideas of
spatial properties which must be habitually present in the mind of any
one who is to observe the world of phenomena with understanding.
Thus we come back to Plato's opinion that for a liberal education,
geometry, as he knew it, is the queen of sciences.
In addition to geometry, there remain the ideas of quantity, ratio,
and of number. This in practice means, elementary algebra. Here the
prominent ideas are those of "any number," in other words, the use
of the familiar _x_, _y_, _z_, and of the dependence of variables on
each other, or otherwise, the idea of functionality. All this is to be
gradually acquired by the continual use of the very simplest functions
which we can devise: of linear functions, graphically represented
by straight lines; of quadratic functions, graphically represented
by parabolas; and of those simple implicit functions, graphically
represented by conic sections. Thence, with good fortune and a willing
class, we can advance to the ideas of rates of increase, still
confining ourselves to the simplest possible cases.
I wish here emphatically to remind you that both in geometry and in
algebra a clear grasp of these general ideas is not what the pupil
starts from, it is the goal at which he is to arrive. The method of
progression is continual practice in the consideration of the simplest
particular cases, and the goal is not philosophical analysis but the
power of use.
But how is he to practise himself in their use? He cannot simply sit
down and think of the relation _y_ = _x_ + 1, he must employ
it in some simple obvious way.
This brings us to the second power of mind which is to be produced by
a mathematical training, namely, the power of logical reasoning. Here
again, it is not the knowledge of the philosophy of logic which it is
essential to teach, but the habit of thinking logically. By logic, I
mean deductive logic.
Deductive logic is the science of certain relations, such as
implication, etc., between general ideas. When logic begins, definite
particular individual things have been banished. I cannot relate
logically this thing to that thing, for example this pen to that pen,
except by the indirect way of relating some general idea which applies
to this pen to some general idea which applies to that pen. And the
individualities of the two pens are quite irrelevant to the logical
process. This process is entirely concerned with the two general ideas.
Thus the practice of logic is a certain way of employing the mind
in the consideration of such ideas; and an elementary mathematical
training is in fact nothing else but the logical use of the general
ideas respecting geometry and algebra which we have enumerated above.
It has therefore, as I started this paper by stating, a double
advantage. It makes the mind capable of abstract thought, and it
achieves this object by training the mind in the most important kind of
abstract thought, namely, deductive logic.
I may remind you that other choices of a type of abstract thought
might be made. We might train the children to contemplate directly the
beauty of abstract moral ideas, in the hope of making them religious
mystics. The general practice of education has decided in favour of
logic, as exemplified in elementary mathematics.
We have now to answer the further question, what is the rôle of
logical precision in the teaching of mathematics? Our general answer
to the implied question is obvious: logical precision is one of the
two objects of the teaching of mathematics, and it is the only weapon
by which the teaching of mathematics can achieve its other object. To
teach mathematics is to teach logical precision. A mathematical teacher
who has not taught that has taught nothing.
But having stated this thesis in this unqualified way, its meaning must
be carefully explained; for otherwise its real bearing on the problem
will be entirely misunderstood.
Logical precision is the faculty to be acquired. It is the quality of
mind which it is the object of the teaching to impart. Thus the habit
of reading great literature is the goal at which a literary education
aims. But we do not expect a child to start its first lesson by reading
for itself Shakespeare. We recognise that reading is impossible till
the pupil has learnt its alphabet and can spell, and then we start it
with books of one syllable.
In the same way, a mathematical education should grow in logical
precision. It is folly to expect the same careful logical analysis at
the commencement of the training as would be appropriate at the end.
It is an entire misconception of my thesis to construe it as meaning
that a mathematical training should assume in the pupil a power of
concentrated logical thought. My thesis is in fact the exact opposite,
namely, that this power cannot be assumed, and has got to be acquired,
and that a mathematical training is nothing else than the process of
acquiring it. My whole groundwork of assumption is that this power does
not initially exist in a fully developed state. Of course like every
other power which is acquired, it must be developed gradually.
The various stages of development must be guided by the judgment and
the genius of the teacher. But what is essential is, that the teacher
should keep clearly in his mind that it is just this power of logical
precise reasoning which is the whole object of his efforts. If his
pupils have in any measure gained this, they have gained all.
We have not yet, however, fully considered this part of our subject.
Logical precision is the full realisation of the steps of the argument.
But what are the steps of the argument? The full statement of all the
steps is far too elaborate and difficult an operation to be introduced
into the mathematical reasoning of an educational curriculum. Such a
statement involves the introduction of abstract logical ideas which
are very difficult to grasp, because there is so rarely any need to
make them explicit in ordinary thought. They are therefore not a fit
subject-ground for an elementary education.
I do not think that it is possible to draw any theoretical line
between those logical steps which form a theoretically full logical
investigation, and those which are full enough for most practical
purposes, including that of education. The question is one of
psychology, to be solved by a process of experiment. The object to be
attained is to gain that amount of logical alertness which will enable
its possessors to detect fallacy and to know the types of sound logical
deduction. The objects of going further are partly philosophical, and
also partly to lay bare abstract ideas whose investigation is in itself
important. But both these objects are foreign to education.
My opinion is, that, on the whole, the type of logical precision handed
down to us by the Greek mathematicians is, roughly speaking, what we
want. In geometry, this means the sort of precision which we find in
Euclid. I do not mean that we should use his famous _Elements_ as
a text-book, nor that here and there a certain compression in his mode
of exposition is not advisable. All this is mere detail. What I do mean
is, that the sort of logical transition which he made explicit, we
should make explicit, and that the sort of transition which he omits,
we should omit.
I doubt, however, whether it is desirable to plunge the student into
the full rigour of euclidean geometry without some mitigation. It is
for this reason that the modern habit, at least in England, of laying
great stress in the initial stages on training the pupil in simple
constructions from numerical data is to be praised. It means that after
a slight amount of reasoning on the euclidean basis of accuracy, the
mind of the learner is relieved by doing the things in various special
cases, and noting by rough measurements that the desired results are
actually attained. It is important, however, that the measurements
be not mistaken for the proofs. Their object is to make the beginner
apprehend what the abstract ideas really mean.
Again in algebra, the notation and the practical use of the symbols
should be acquired in the simplest cases, and the more theoretical
treatment of the symbolism reserved to a suitable later stage. My
rule would be initially to learn the meaning of the ideas by a crude
practice in simple ways, and to refine the logical procedure in
preparation for an advance to greater generality. In fact the thesis of
my paper can be put in another way thus, the object of a mathematical
education is, to acquire the powers of analysis, of generalisation,
and of reasoning. The two processes of analysis and generalisation were
in my previous statement put together as the power of grasping abstract
ideas.
But in order to analyse and to generalise, we must commence with the
crude material of ideas which are to be analysed and generalised.
Accordingly it is a positive error in education to start with the
ultimate products of this process, namely the ideas in their refined
analysed and generalised forms. We are thereby skipping an important
part of the training, which is to take the ideas as they actually exist
in the child's mind, and to exercise the child in the difficult art of
civilising them and clothing them.
The schoolmaster is in fact a missionary, the savages are the ideas in
the child's mind; and the missionary shirks his main task if he refuse
to risk his body among the cannibals.
At this point I should like to turn your attention to those pupils
forming the mathematical section. There is an idea, widely prevalent,
that it is possible to teach mathematics of a relatively advanced
type--such as differential calculus, for instance--in a way useful to
physicists and engineers without any attention to its logic or its
theory.
This seems to me to be a profound mistake. It implies that a merely
mechanical knowledge without understanding of ways of arriving at
mathematical results is useful in applied science. It is of no use
whatever. The results themselves can all be found stated in the
appropriate pocket books and in other elementary works of reference. No
one when applying a result need bother himself as to why it is true. He
accepts it and applies it. What is of supreme importance in physics and
in engineering is a mathematically trained mind, and such a mind can
only be acquired by a proper mathematical discipline.
I fully admit that the proper way to start such a subject as the
differential calculus is to plunge quickly into the use of the notation
in a few absurdly simple cases, with a crude explanation of the idea
of rates of increase. The notation as thus known can then be used by
the lecturers in the Physical and Engineering Laboratories. But the
mathematical training of the applied scientists consists in making
these ideas precise and the proofs accurate.
I hope that the thesis of this paper respecting the position of logical
precision in the teaching of mathematics has been rendered plain. The
habit of logical precision with its necessary concentration of thought
upon abstract ideas is not wholly possible in the initial stages of
learning. It is the ideal at which the teacher should aim. Also logical
precision, in the sense of logical explicitness, is not an absolute
thing: it is an affair of more or less. Accordingly the quantity of
explicitness to be introduced at each stage of progress must depend
upon the practical judgment of the teacher. Lastly, in a sense, the
instructed mind is less explicit; for it travels more quickly over a
well-remembered path, and may save the trouble of putting into words
trains of thought which are very obvious to it. But on the other
hand it atones for this rapidity by a concentration on every subtle
point where a fallacy can lurk. The habit of logical precision is the
instinct for the subtle difficulty.