THE MATHEMATICAL CURRICULUM
(_Presidential Address to the London Branch of the Mathematical
Association, 1912_)
THE situation in regard to education at the present time cannot find
its parallel without going back for some centuries to the break-up of
the mediæval traditions of learning. Then, as now, the traditional
intellectual outlook, despite the authority which it had justly
acquired from its notable triumphs, had grown to be too narrow for the
interests of mankind. The result of this shifting of human interest
was a demand for a parallel shifting of the basis of education, so
as to fit the pupils for the ideas which later in life would in fact
occupy their minds. Any serious fundamental change in the intellectual
outlook of human society must necessarily be followed by an educational
revolution. It may be delayed for a generation by vested interests or
by the passionate attachment of some leaders of thought to the cycle
of ideas within which they received their own mental stimulus at an
impressionable age. But the law is inexorable that education to be
living and effective must be directed to informing pupils with those
ideas, and to creating for them those capacities which will enable them
to appreciate the current thought of their epoch.
There is no such thing as a successful system of education in a vacuum,
that is to say, a system which is divorced from immediate contact with
the existing intellectual atmosphere. Education which is not modern
shares the fate of all organic things which are kept too long.
But the blessed word "modern" does not really solve our difficulties.
What we mean is, relevant to modern thought, either in the ideas
imparted or in the aptitudes produced. Something found out only
yesterday may not really be modern in this sense. It may belong to
some bygone system of thought prevalent in a previous age, or, what
is very much more likely, it may be too recondite. When we demand
that education should be relevant to modern thought, we are referring
to thoughts broadly spread throughout cultivated society. It is this
question of the unfitness of recondite subjects for use in general
education which I wish to make the keynote of my address this afternoon.
It is in fact rather a delicate subject for us mathematicians.
Outsiders are apt to accuse our subject of being recondite. Let us
grasp the nettle at once and frankly admit that in general opinion it
is the very typical example of reconditeness. By this word I do not
mean difficulty, but that the ideas involved are of highly special
application, and rarely influence thought.
This liability to reconditeness is the characteristic evil which is
apt to destroy the utility of mathematics in liberal education. So
far as it clings to the educational use of the subject, so far we
must acquiesce in a miserably low level of mathematical attainment
among cultivated people in general. I yield to no one in my anxiety to
increase the educational scope of mathematics. The way to achieve this
end is not by a mere blind demand for more mathematics. We must face
the real difficulty which obstructs its extended use.
Is the subject recondite? Now, viewed as a whole, I think it is.
_Securus judicat orbis terrarum_--the general judgment of mankind
is sure.
The subject as it exists in the minds and in the books of students of
mathematics _is_ recondite. It proceeds by deducing innumerable
special results from general ideas, each result more recondite than the
preceding. It is not my task this afternoon to defend mathematics as a
subject for profound study. It can very well take care of itself. What
I want to emphasise is, that the very reasons which make this science
a delight to its students are reasons which obstruct its use as an
educational instrument--namely, the boundless wealth of deductions from
the interplay of general theorems, their complication, their apparent
remoteness from the ideas from which the argument started, the variety
of methods, and their purely abstract character which brings, as its
gift, eternal truth.
Of course, all these characteristics are of priceless value to
students; for ages they have fascinated some of the keenest intellects.
My only remark is that, except for a highly selected class, they are
fatal in education. The pupils are bewildered by a multiplicity of
detail, without apparent relevance either to great ideas or to ordinary
thoughts. The extension of this sort of training in the direction
of acquiring more detail is the last measure to be desired in the
interests of education.
The conclusion at which we arrive is, that mathematics, if it is
to be used in general education, must be subjected to a rigorous
process of selection and adaptation. I do not mean, what is of course
obvious, that however much time we devote to the subject the average
pupil will not get very far. But that, however limited the progress,
certain characteristics of the subject, natural at any stage, must be
rigorously excluded. The science as presented to young pupils must
lose its aspect of reconditeness. It must, on the face of it, deal
directly and simply with a few general ideas of far-reaching importance.
Now, in this matter of the reform of mathematical instruction, the
present generation of teachers may take a very legitimate pride in
its achievements. It has shown immense energy in reform, and has
accomplished more than would have been thought possible in so short a
time. It is not always recognised how difficult is the task of changing
a well-established curriculum entrenched behind public examinations.
But for all that, great progress has been made, and, to put the matter
at its lowest, the old dead tradition has been broken up. I want to
indicate this afternoon the guiding idea which should direct our
efforts at reconstruction. I have already summed it up in a phrase,
namely, we must aim at the elimination of reconditeness from the
educational use of the subject.
Our courses of instruction should be planned to illustrate simply a
succession of ideas of obvious importance. All pretty divagations
should be rigorously excluded. The goal to be aimed at is that the
pupil should acquire familiarity with abstract thought, should realise
how it applies to particular concrete circumstances, and should know
how to apply general methods to its logical investigation. With this
educational ideal nothing can be worse than the aimless accretion
of theorems in our text-books, which acquire their position merely
because the children can be made to learn them and examiners can set
neat questions on them. The bookwork to be learnt should all be very
important as illustrating ideas. The examples set--and let there
be as many examples as teachers find necessary--should be direct
illustrations of the theorems, either by way of abstract particular
cases or by way of application to concrete phenomena. Here it is worth
remarking that it is quite useless to simplify the bookwork, if the
examples set in examinations in fact require an extended knowledge of
recondite details. There is a mistaken idea that problems test ability
and genius, and that bookwork tests cram. This is not my experience.
Only boys who have been specially crammed for scholarships can ever
do a problem paper successfully. Bookwork properly set, not in mere
snippets according to the usual bad plan, is a far better test of
ability, provided that it is supplemented by direct examples. But this
is a digression on the bad influence of examinations on teaching.
The main ideas which lie at the base of mathematics are not at all
recondite. They are abstract. But one of the main objects of the
inclusion of mathematics in a liberal education is to train the pupils
to handle abstract ideas. The science constitutes the first large
group of abstract ideas which naturally occur to the mind in any
precise form. For the purposes of education, mathematics consists of
the relations of number, the relations of quantity, and the relations
of space. This is not a general definition of mathematics, which, in
my opinion, is a much more general science. But we are now discussing
the use of mathematics in education. These three groups of relations,
concerning number, quantity, and space, are interconnected.
Now, in education we proceed from the particular to the general.
Accordingly, children should be taught the use of these ideas by
practice among simple examples. My point is this: The goal should be,
not an aimless accumulation of special mathematical theorems, but the
final recognition that the preceding years of work have illustrated
those relations of number, and of quantity, and of space, which are of
fundamental importance. Such a training should lie at the base of all
philosophical thought. In fact elementary mathematics rightly conceived
would give just that philosophical discipline of which the ordinary
mind is capable. But what at all costs we ought to avoid, is the
pointless accumulation of details. As many examples as you like; let
the children work at them for terms, or for years. But these examples
should be direct illustrations of the main ideas. In this way, and
this only, can the fatal reconditeness be avoided.
I am not now speaking in particular of those who are to be professional
mathematicians, or of those who for professional reasons require a
knowledge of certain mathematical details. We are considering the
liberal education of all students, including these two classes. This
general use of mathematics should be the simple study of a few general
truths, well illustrated by practical examples. This study should
be conceived by itself, and completely separated in idea from the
professional study mentioned above, for which it would make a most
excellent preparation. Its final stage should be the recognition of
the general truths which the work done has illustrated. As far as I
can make out, at present the final stage is the proof of some property
of circles connected with triangles. Such properties are immensely
interesting to mathematicians. But are they not rather recondite,
and what is the precise relation of such theorems to the ideal of
a liberal education? The end of all the grammatical studies of the
student in classics is to read Virgil and Horace--the greatest thoughts
of the greatest men. Are we content, when pleading for the adequate
representation in education of our own science, to say that the end of
a mathematical training is that the student should know the properties
of the nine-point circle? I ask you frankly, is it not rather a "come
down"?
This generation of mathematical teachers has done so much strenuous
work in the way of reorganising mathematical instruction that there is
no need to despair of its being able to elaborate a curriculum which
shall leave in the minds of the pupils something even nobler than "the
ambiguous case."
Let us think how this final review, closing the elementary course,
might be conducted for the more intelligent pupils. Partly no doubt
it requires a general oversight of the whole work done, considered
without undue detail so as to emphasise the general ideas used, and
their possibilities of importance when subjected to further study. Also
the analytical and geometrical ideas find immediate application in the
physical laboratory where a course of simple experimental mechanics
should have been worked through. Here the point of view is twofold, the
physical ideas and the mathematical ideas illustrate each other.
The mathematical ideas are essential to the precise formulation of the
mechanical laws. The idea of a precise law of nature, the extent to
which such laws are in fact verified in our experience, and the rôle of
abstract thought in their formulation, then become practically apparent
to the pupil. The whole topic of course requires detailed development
with full particular illustration, and is not suggested as requiring
merely a few bare abstract statements.
It would, however, be a grave error to put too much emphasis on the
mere process of direct explanation of the previous work by way of
final review. My point is, that the latter end of the course should
be so selected that in fact the general ideas underlying all the
previous mathematical work should be brought into prominence. This may
well be done by apparently entering on a new subject. For example,
the ideas of quantity and the ideas of number are fundamental to
all precise thought. In the previous stages they will not have been
sharply separated; and children are, rightly enough, pushed on to
algebra without too much bother and quantity. But the more intelligent
among them at the end of their curriculum would gain immensely by a
careful consideration of those fundamental properties of quantity in
general which lead to the introduction of numerical measurement. This
is a topic which also has the advantage that the necessary books are
actually to hand. Euclid's fifth book is regarded by those qualified
to judge as one of the triumphs of Greek mathematics. It deals with
this very point. Nothing can be more characteristic of the hopelessly
illiberal character of the traditional mathematical education than the
fact that this book has always been omitted. It deals with ideas,
and therefore was ostracised. Of course a careful selection of the
more important propositions and a careful revision of the argument are
required. This also is to hand in the publications of my immediate
predecessor in the office of president, Prof. Hill. Furthermore, in Sir
T. L. Heath's complete edition of Euclid, there is a full commentary
embodying most of what has been said and thought on the point. Thus
it is perfectly easy for teachers to inform themselves generally
on the topic. The whole book would not be wanted, but just the few
propositions which embody the fundamental ideas. The subject is not
fit for backward pupils; but certainly it could be made interesting to
the more advanced class. There would be great scope for interesting
discussion as to the nature of quantity, and the tests which we should
apply to ascertain when we are dealing with quantities. The work
would not be at all in the air, but would be illustrated at every
stage by reference to actual examples of cases where the quantitative
character is absent, or obscure, or doubtful, or evident. Temperature,
heat, electricity, pleasure and pain, mass and distance could all be
considered.
Another idea which requires illustration is that of functionality.
A function in analysis is the counterpart of a law in the physical
universe, and of a curve in geometry. Children have studied the
relations of functions to curves from the first beginning of their
study of algebra, namely in drawing graphs. Of recent years there has
been a great reform in respect to graphs. But at its present stage it
has either gone too far or not far enough. It is not enough merely
to draw a graph. The idea behind the graph--like the man behind the
gun--is essential in order to make it effective. At present there is
some tendency merely to set the children to draw curves, and there to
leave the whole question.
In the study of simple algebraic functions and of trigonometrical
functions we are initiating the study of the precise expression of
physical laws. Curves are another way of representing these laws. The
simple fundamental laws--such as the inverse square and the direct
distance--should be passed under review, and the applications of the
simple functions to express important concrete cases of physical laws
considered. I cannot help thinking that the final review of this topic
might well take the form of a study of some of the main ideas of the
differential calculus applied to simple curves. There is nothing
particularly difficult about the conception of a rate of change; and
the differentiation of a few powers of _x_, such as _x_^2,
_x_^3, etc., could easily be effected; perhaps by the aid of
geometry even sin _x_ and cos _x_ could be differentiated.
If we once abandon our fatal habit of cramming the children with
theorems which they do not understand, and will never use, there will
be plenty of time to concentrate their attention on really important
topics. We can give them familiarity with conceptions which really
influence thought.
Before leaving this topic of physical laws and mathematical functions,
there are other points to be noticed. The fact that the precise law is
never really verified by observation in its full precision is capable
of easy illustration and of affording excellent examples. Again,
statistical laws, namely laws which are only satisfied on the average
by large numbers, can easily be studied and illustrated. In fact a
slight study of statistical methods and their application to social
phenomena affords one of the simplest examples of the application of
algebraic ideas.
Another way in which the students' ideas can be generalised is by the
use of the History of Mathematics, conceived not as a mere assemblage
of the dates and names of men, but as an exposition of the general
current of thought which occasioned the subjects to be objects of
interest at the time of their first elaboration. The use of the History
of Mathematics is to be considered at a later stage of our proceedings
this afternoon. Accordingly I merely draw attention to it now, to
point out that perhaps it is the very subject which may best obtain the
results for which I am pleading.
We have indicated two main topics, namely general ideas of quantity
and of laws of nature, which should be an object of study in the
mathematical curriculum of a liberal education. But there is another
side to mathematics which must not be overlooked. It is the chief
instrument for discipline in logical method.
Now, what is logical method, and how can any one be trained in it?
Logical method is more than the mere knowledge of valid types of
reasoning and practice in the concentration of mind necessary to
follow them. If it were only this, it would still be very important;
for the human mind was not evolved in the bygone ages for the sake of
reasoning, but merely to enable mankind with more art to hunt between
meals for fresh food supplies. Accordingly few people can follow close
reasoning without considerable practice.
More than this is wanted to make a good reasoner, or even to enlighten
ordinary people with knowledge of what constitutes the essence of the
art. The art of reasoning consists in getting hold of the subject at
the right end, of seizing on the few general ideas which illuminate
the whole, and of persistently marshalling all subsidiary facts round
them. Nobody can be a good reasoner unless by constant practice he
has realised the importance of getting hold of the big ideas and of
hanging on to them like grim death. For this sort of training geometry
is, I think, better than algebra. The field of thought of algebra is
rather obscure, whereas space is an obvious insistent thing evident to
all. Then the process of simplification, or abstraction, by which all
irrelevant properties of matter, such as colour, taste, and weight, are
put aside is an education in itself. Again, the definitions and the
propositions assumed without proof illustrate the necessity of forming
clear notions of the fundamental facts of the subject-matter and of the
relations between them. All this belongs to the mere prolegomena of the
subject. When we come to its development, its excellence increases. The
learner is not initially confronted with any symbolism which bothers
the memory by its rules, however simple they may be. Also, from the
very beginning the reasoning, if properly conducted, is dominated by
well-marked ideas which guide each stage of development. Accordingly
the essence of logical method receives immediate exemplification.
Let us now put aside for the moment the limitations introduced by
the dullness of average pupils and the pressure on time due to other
subjects, and consider what geometry has to offer in the way of a
liberal education. I will indicate some stages in the subject,
without meaning that necessarily they are to be studied in this
exclusive order. The first stage is the study of _congruence_.
Our perception of congruence is in practice dependent on our judgments
of the invariability of the intrinsic properties of bodies when their
external circumstances are varying. But however it arises, congruence
is in essence the correlation of two regions of space, point by point,
so that all homologous distances and all homologous angles are equal.
It is to be noticed that the definition of the equality of lengths and
angles is their congruence, and all tests of equality, such as the use
of the yard measure, are merely devices for making immediate judgments
of congruence easy. I make these remarks to suggest that apart from the
reasoning connected with it, congruence, both as an example of a larger
and very far-reaching idea and also for its own sake, is well worthy
of attentive consideration. The propositions concerning it elucidate
the elementary properties of the triangle, the parallelogram, and the
circle, and of the relations of two planes to each other. It is very
desirable to restrict the proved propositions of this part within the
narrowest bounds, partly by assuming redundant axiomatic propositions,
and partly by introducing only those propositions of absolutely
fundamental importance.
The second stage is the study of similarity. This can be reduced to
three or four fundamental propositions. Similarity is an enlargement of
the idea of congruence, and, like that idea, is another example of a
one-to-one correlation of points of spaces. Any extension of study of
this subject might well be in the direction of the investigation of one
or two simple properties of similar and similarly situated rectilinear
figures. The whole subject receives its immediate applications in plans
and maps. It is important, however, to remember that trigonometry is
really the method by which the main theorems are made available for use.
The third stage is the study of the elements of trigonometry. This is
the study of the periodicity introduced by rotation and of properties
preserved in a correlation of similar figures. Here for the first time
we introduce a slight use of the algebraic analysis founded on the
study of number and quantity. The importance of the periodic character
of the functions requires full illustration. The simplest properties of
the functions are the only ones required for the solution of triangles,
and the consequent applications to surveying. The wealth of formulæ,
often important in themselves, but entirely useless for this type of
study, which crowd our books should be rigorously excluded, except
so far as they are capable of being proved by the pupils as direct
examples of the bookwork.
This question of the exclusion of formulæ is best illustrated by
considering this example of Trigonometry, though of course I may well
have hit on an unfortunate case in which my judgment is at fault. A
great part of the educational advantage of the subject can be obtained
by confining study to Trigonometry of one angle and by exclusion of the
addition formulæ for the sine and cosine of the sum of two angles. The
functions can be graphed, and the solution of triangles effected. Thus
the aspects of the science as (1) embodying analytically the immediate
results of some of the theorems deduced from congruence and similarity,
(2) as a solution of the main problem of surveying, (3) as a study of
the fundamental functions required to express periodicity and wave
motion, will all be impressed on the pupils' minds both by bookwork and
example.
If it be desired to extend this course, the addition formulæ should
be added. But great care should be taken to exclude specialising
the pupils in the wealth of formulæ which comes in their train. By
"exclude" is meant that the pupils should not have spent time or energy
in acquiring any facility in their deduction. The teacher may find
it interesting to work a few such examples before a class. But such
results are not among those which learners need retain. Also, I would
exclude the whole subject of circumscribed and inscribed circles both
from Trigonometry and from the previous geometrical courses. It is
all very pretty, but I do not understand what its function is in an
elementary non-professional curriculum.
Accordingly, the actual bookwork of the subject is reduced to very
manageable proportions. I was told the other day of an American college
where the students are expected to know by heart ninety formulæ or
results in Trigonometry alone. We are not quite so bad as that. In
fact, in Trigonometry we have nearly approached the ideal here sketched
out as far as our elementary courses are concerned.
The fourth stage introduces Analytical Geometry. The study of graphs in
algebra has already employed the fundamental notions, and all that is
now required is a rigorously pruned course on the straight line, the
circle, and the three types of conic sections, defined by the forms of
their equations. At this point there are two remarks to be made. It is
often desirable to give our pupils mathematical information which we do
not prove. For example, in co-ordinate geometry, the reduction of the
general equation of the second degree is probably beyond the capacities
of most of the type of students whom we are considering. But that need
not prevent us from explaining the fundamental position of conics, as
exhausting the possible types of such curves.
The second remark is to advocate the entire sweeping away of
geometrical conics as a separate subject. Naturally, on suitable
occasions the analysis of analytical geometry will be lightened by
the use of direct deduction from some simple figure. But geometrical
conics, as developed from the definition of a conic section by the
focus and directrix property, suffers from glaring defects. It is
hopelessly recondite. The fundamental definition of a conic, _SP_ = _e_
· _PM_, usual in this subject at this stage, is thoroughly bad. It is
very recondite, and has no obvious importance. Why should such curves
be studied at all, any more than those defined by an indefinite number
of other formulæ? But when we have commenced the study of the Cartesian
methods, the equations of the first and second degrees are naturally
the first things to think about.
In this ideal course of Geometry, the fifth stage is occupied with
the elements of Projective Geometry. The general ideas of cross ratio
and of projection are here fundamental. Projection is yet a more
general instance of that one-to-one correlation which we have already
considered under congruence and similarity. Here again we must avoid
the danger of being led into a bewildering wealth of detail.
The intellectual idea which projective geometry is to illustrate is
the importance in reasoning of the correlation of all cases which
can be proved to possess in common certain identical properties. The
preservation of the projective properties in projection is the one
important educational idea of the subject. Cross ratio only enters
as the fundamental metrical property which is preserved. The few
propositions considered are selected to illustrate the two allied
processes which are made possible by this procedure. One is proof
by simplification. Here the simplification is psychological and not
logical--for the general case is logically the simplest. What is meant
is: Proof by considering the case which is in fact the most familiar to
us, or the easiest to think about. The other procedure is the deduction
of particular cases from known general truths, as soon as we have a
means of discovering such cases or a criterion for testing them.
The projective definition of conic sections and the identity of the
results obtained with the curves derived from the general equation
of the second degree are capable of simple exposition, but lie on
the border-line of the subject. It is the sort of topic on which
information can be given, and the proofs suppressed.
The course of geometry as here conceived in its complete ideal--and
ideals can never be realised--is not a long one. The actual amount
of mathematical deduction at each stage in the form of bookwork is
very slight. But much more explanation would be given, the importance
of each proposition being illustrated by examples, either worked out
or for students to work, so selected as to indicate the fields of
thought to which it applies. By such a course the student would gain an
analysis of the leading properties of space, and of the chief methods
by which they are investigated.
The study of the elements of mathematics, conceived in this spirit,
would constitute a training in logical method together with an
acquisition of the precise ideas which lie at the base of the
scientific and philosophical investigations of the universe. Would it
be easy to continue the excellent reforms in mathematical instruction
which this generation has already achieved, so as to include in the
curriculum this wider and more philosophic spirit? Frankly, I think
that this result would be very hard to achieve as the result of
single individual efforts. For reasons which I have already briefly
indicated, all reforms in education are very difficult to effect. But
the continued pressure of combined effort, provided that the ideal is
really present in the minds of the mass of teachers, can do much, and
effects in the end surprising modification. Gradually the requisite
books get written, still more gradually the examinations are reformed
so as to give weight to the less technical aspects of the subject, and
then all recent experience has shown that the majority of teachers are
only too ready to welcome any practicable means of rescuing the subject
from the reproach of being a mechanical discipline.