Essays On Mathematical Education

ESSAYS ON M A T H E M A T I C A L EDUCATION

B Y

G. S T . L . C A R S O N

W I T H AN I N T R O D U C T I O N BY

DAVID EUGENE

L O N D O N

A N D

G I N N A N D COMPANY, l

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SMITH

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PUBLISHERS

C O P Y R I G H T , 1913, B Y G I N N A N D ALL RIGHTS RESERVED

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COMPANY

INTRODUCTION I t has always been hard for people to judge with any accuracy the work of their own age, and it is hard for us to do so to-day. I n spite of our optimism and of our certainty that we are pro­ gressing, what we conceive to be an era of great educational awakening may appear to the historian of the future as one in which noble ideals were sacrificed to the democratizing of the school, and the twentieth century may not rank with the sixteenth when the toll is finally taken. I t is, therefore, with some hesitancy that we should assert that we live in a period of remarkable achievement in all that pertains to education. That the period is one of advance is in harmony with the general principle of evolution, but that all that we do is uniformly progressive is not at all in accord with general experience. Certain it is that the present time is one of agita­ tion, of the shattering of idols, and of the setting up of strange gods in their places. Nothing is sacred to the iconoclast, and he is found in the school as he is found in the church, in govern­ ment, and in the social world. Among the objects of attack in this generation is " the science venerable" that has come down to us from Pythagoras and Euclid, from Mohammed ben Musa and Bhaskara, and from Cardan, Descartes, and Newton. A n d yet it does not seem to be mathematics itself that is challenged so much as the way in which it has been presented to the youth in our schools, and to most of us the challenge seems justified. With all the excel­ lence of Euclid, his work is not for the child ; and with all the value of formal algebra, the science needs some other introduction than the arid one until recently accorded to it. iii

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I t is on this account that M r . Carson's work in the English schools and before bodies of English teachers has great value. H e is thoroughly trained as a mathematician, is a product of the college where Newton studied and taught, is a lover of the science in its purest form, and has had an unusual amount of experience in the technical applications of the subject; but he is a teacher by instinct and by profession, and is imbued with the feeling that mathematics can be saved to the school only through an improvement in our methods of teaching and in our selection of material. H e stands for the principle that mathe­ matics must be made to appeal to the learner as interesting and valuable, and he has shown in his own classes that, after this appeal has been successful, pupils need to be held back rather than driven forward in this branch of learning. I t is because of this feeling on the part of M r . Carson that his essays on the teaching of mathematics have peculiar value at this time. They will encourage teachers to continue their advocacy of a worthy form of mathematics, at the same time seeking better lines of approach and endeavouring to relate the subject in a reasonable manner to the various other interests of the pupil. The problem is much the same everywhere, but the ties of a common language, a common spirit of freedom, and a common ancestry make it practically identical in English-speak­ ing lands. On this account we, in the United States, feel that M r . Carson's message is quite as much to us as to his own countrymen, and we shall appreciate it as we have appreciated the noteworthy work that he has already achieved in the teaching of mathematics in England. D A V I D TEACHERS COLUMBIA NEW

COLLEGE UNIVERSITY

YORK

CITY

E U G E N E

S M I T H

CONTENTS PAGE SOME

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INTUITION T H E

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SOME P R I N C I P L E S OF MATHEMATICAL EDUCATION ( R e p r i n t e d from The Mathematical

Gazette,

January, 1 9 1 3 )

SOME

PRINCIPLES OF M A T H E M A T I C A L EDUCATION

O f all the problems w h i c h have perplexed teachers o f mathematics i n this generation, probably none has been more i r r i t a t i n g and insistent than the choice of assumptions w h i c h must be made i n each branch of the science. I n geometry, i n analysis, i n mechanics, one and the same difficulty arises. A r e we to prove that any two sides of a triangle are greater than the t h i r d ? T h a t the l i m i t of the sum of a finite number o f functions is equal to the sum of their limits ? T h a t the total m o m e n t u m of two bodies is uninfluenced by their mutual action ? A n d i n every such case, on what is the proof to depend ? A clear under­ standing of the answers to such questions, or, better still, a clear understanding of principles by w h i c h answers may be found, would go far to co-ordinate and simplify elemen­ tary t e a c h i n g ; the object of this paper is to state such principles and indicate their application. A X I O M , POSTULATE, PROOF

I t is first necessary to lay down definitions, as precise as may be possible, of the terms " a x i o m , " " p o s t u l a t e , " " proof." I t is not i m p l i e d that these definitions should be insisted on, or the terms used, i n elementary t e a c h i n g ; n o t h i n g could be more likely to lead to failure. B u t a full comprehension of each is essential to every teacher of mathematics, and is too often l a c k i n g i n current usage. 3

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A n axiom, or " c o m m o n n o t i o n " i n E u c l i d ' s language, is a statement which is true of all processes of thought, whatever be the subject matter under discussion. T h u s the f o l l o w i n g are a x i o m s : " I f A is identical w i t h B, and C is different f r o m B, then C is different from A . " " I f B is a necessary consequence of A , and also C of B, then C is a necessary consequence of A . " B u t " T w o and two make four " and " T h e straight line is the shortest distance between two points " are not axioms, although they may be considered no less obvious. A statement is not an axiom because i t is obvious, but because i t concerns u n i ­ versal forms of thought, and not a particular subject matter such as arithmetic, geometry, and the l i k e . A postulate is a statement which is assumed concerning a particular subject m a t t e r ; for example, " T h e whole is greater than a p a r t " (subject matter, finite aggregates); " A l l r i g h t angles are equal " (subject matter, Euclidean space). I t is essential to observe that, whereas an axiom is an axiom once for all, a postulate i n one treatment of a science may not be a postulate i n another. I n E u c l i d ' s development o f geometry, the statement that any two sides of a triangle are together greater than the t h i r d side is not a postulate, because i t is deduced f r o m other statements (postulates) w h i c h are avowedly assumed; but i n many current developments i t is adopted at once, without refer­ ence to other statements, and is therefore a postulate i n such cases. T o use an unconventional but expressive t e r m , postulates are " jumping-off places " for the logical explo­ ration of a subject. T h e i r number and nature are i m m a t e r i a l ; they may be readily acceptable, or difficult of credence. T h e i r one function is to supply a basis for reasoning,

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w h i c h is conducted i n accordance w i t h the axioms. Postu­ lates are thus doubly relative : they relate to one particular subject matter (number, space, and so on) and to one par­ ticular method of v i e w i n g that subject matter. A statement w h i c h is deduced, by use of the axioms, f r o m two or more postulates is said to be proved. T h e r e is thus no such t h i n g as absolute proof. Proofs are related to the postulates o n w h i c h they are based, and a demand for a proof must inevitably be met by a counter demand for a place to start f r o m , that is, for some postulates. W h e n a statement is said to have been proved, what is meant is that i t has been shown to be a logical consequence of some other statements w h i c h have been accepted; i f these statements are found to be incorrect, the statement w h i c h is said to be proved can no longer be accepted, t h o u g h the logical character of the proof is i n no way i m p u g n e d . T h u s the type of a proof is, " I f A , then B" ; relentless and final certainty surrounds " then " ; but A , w h i c h is assumed i n the " i f , " may nevertheless be utterly fantastic as viewed i n the l i g h t of experience. T H E T H R E E FUNCTIONS OF M A T H E M A T I C S

T h e first application of mathematics to any domain of knowledge can now be explained. S t a r t i n g f r o m postulates, the t r u t h of w h i c h is no concern of mathematics, sets of deductions are evolved by use of the a x i o m s ; agreement of the results w i t h experience strengthens the evidence i n favour of these postulates. I f this evidence be deemed sufficient, as, for example, i n geometry and mechanics, then deduction yields acceptable results w h i c h could not other­ wise have been predicted or ascertained.

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I t is here that the prevailing concept of the power of mathematics ends ; but such a concept presents a view of the subject so l i m i t e d and distorted as to be almost gro­ tesque. T h e process just described may be regarded as an upward development; a downward research is also possible, and no less valuable. I t consists of a logical review of the set of postulates w h i c h have been adopted ; i n the result, either it is shown that some must be rejected, or the evidence i n favour of all may be considerably enhanced. T h i s review consists of two processes, w h i c h w i l l be described i n t u r n . I t is first necessary to ascertain whether the set of pos­ tulates is consistent; that is, whether some a m o n g t h e m may not be logically contradictory of others. F o r example, E u c l i d defines parallel straight lines as coplanar lines w h i c h do not intersect, and proves i n his twenty-seventh proposi­ t i o n that such lines can be drawn ; for this purpose he uses his fourth postulate, w h i c h makes no allusion to parallels. I f he had included a m o n g his postulates another, stating that every pair of coplanar lines intersect i f produced suffi­ ciently far, and had o m i t t e d his definition of parallel lines, his postulates would not have been consistent; for the twenty-seventh proposition proves that i f the fourth postu­ late be granted, then the existence of non-intersecting co­ planar lines must be admitted also. I t is essential to realize that the contradiction i m p l i e d i n the t e r m " i n c o n s i s t e n t " is based on logic, not on experience ; assumptions w h i c h are contrary to all experience are not thereby inconsistent. T h e r e is n o t h i n g i n logic to veto the assumption that, for certain types of matter, weight and mass are inversely propor­ tional ; or that life may exist where there is no atmosphere, as on the m o o n . Such assumptions are not inconsistent

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w i t h the other postulates of mechanics or b i o l o g y ; they are merely contrary to all experience gained up to the present time. H e r e , then, is the second function of the mathematician — the investigation of the consistence of a set of postulates. A n d the task is not superfluous. Physical measurements are perforce inaccurate, and a set of inconsistent assump­ tions m i g h t well appear to be consistent w i t h actual obser­ vations. M o r e accurate measurements must, of course, expose the discrepancy, but these may for ever remain beyond our powers ; logic renders t h e m superfluous by demonstrating the consistence or otherwise of each set considered. T h e next investigation concerns the redundance of a set of postulates. Such a set is said to be redundant i f some of its members are logical consequences of others. F o r example, any ordinary adult w i l l accept without difficulty the properties of congruent figures, the angle properties of parallel lines, and the properties of similar figures, as i n maps or plans, regarding t h e m as " i n the nature of t h i n g s . " A n d electricians may, by experiment, convince themselves first, that Coulomb's law of attraction is very approximately t r u e ; and secondly, that w i t h i n the limits of observation there is no electric force i n the interior of a closed con­ ductor. I n neither the one case nor the other need there be the least suspicion that the statements are logically connected, so that they must stand or fall together. Y e t so i t is, and the fact is expressed by the statement that the assumptions are redundant. T h e investigation of redundance, and the demonstration that sets of postulates are free therefrom, forms the t h i r d

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function of the mathematician. I t s value, i n connection w i t h any subject matter to w h i c h it may be applied, may not at once be evident. I t is based on the fact that all experiments, necessary and inevitable t h o u g h they be, are nevertheless sources of uncertainty; i t reduces this uncer­ tainty to a m i n i m u m by r e m o v i n g the redundant assump­ tions into the category of propositions, and exposing the science i n question as based on a m i n i m u m of assumption. A n d m o r e ; it can offer several alternative sets of assump­ tions for choice, that one being taken w h i c h is most nearly capable of verification. T h e labours of Faraday resulted i n the offer of such a choice to electricians ; either, they were told, you can base electrostatics {inter alia) on Cou­ lomb's experiment, or on the absence of electric force i n ­ side a closed conductor; i t is logically immaterial w h i c h course you adopt. T h e latter experiment, being far more capable of accurate demonstration i n the laboratory, is chosen as the p r i m a r y basis for faith i n the deductions of electrostatics — a faith w h i c h is, of course, very m u c h strengthened as such deductions are found to accord w i t h our experience. B u t these considerations are for the physi­ cist ; the task of the mathematician is ended when he has put forward, for choice by the physicist, alternative sets of assumptions w h i c h are at once consistent w i t h each other and free f r o m redundance. I n this way does he free the physicist, so far as may be, from the uncertainties of assumption, and assure h i m that no further increase of such freedom can be attained. 1

1

T h e antithesis between mathematician and p h y s i c i s t does not imply

that the functions are of n e c e s s i t y p e r f o r m e d by different i n d i v i d u a l s ; it is u s e d m e r e l y to enforce the argument.

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Such is the range of application of mathematics to other sciences. W h e n complete i t reveals each science as a firmly k n i t structure of logical reasoning, based on assump­ tions whose number and nature are clearly exposed; of these assumptions i t can be asserted that no one is incon­ sistent w i t h the others, and that each is independent of the others. T h e r e is thus no fear that contradictions may i n t i m e emerge, and no false hope that one assumption may i n t i m e be shown to be a logical consequence of the others. F i n a l i t y has been reached. T h e acute critic may, of course, ask the mathematician whether his o w n house is i n order. W h a t is the precise statement of the axioms w h i c h are the basis of his science, and can they be shown to depend on a set of consistent mental postulates, free f r o m redundance ? H e r e i t need only be said that the labours of the last generation have done m u c h to answer these questions, and that their com­ plete solution is certainly possible, i f not actually achieved; to go further would be beyond the limits of this paper. THE

D I D A C T I C PROBLEM

T h e complete application of mathematics to any branch of knowledge b e i n g thus exhibited, the didactic problem can now be stated i n explicit terms. I n any g i v e n science— geometry, mechanics, and so o n — what is the r i g h t point of entry to the structure, and i n what order should its exploration be made ? W h a t results should be regarded as postulates, and should their consistence and possible inter­ dependence one o n another be investigated before upward deduction is undertaken ? Should the m i n i m u m number be chosen o n the g r o u n d that the p u p i l should at once be

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placed i n possession of the ultimate point of view ? O r should some larger number be taken, and i f so, on what principles should they be chosen ? B e a r i n g i n m i n d that the pupils concerned are not presumed to be adults, i t is easy to indicate principles f r o m w h i c h answers to such questions may be deduced. One of the few really certain facts 'about the juvenile m i n d is that i t revels i n exploration of the u n k n o w n , but loathes analysis of the k n o w n . I t is often said that boys and girls are indifferent to, and cannot appreciate, exact l o g i c ; that i t is unwise to force detailed reasoning upon t h e m . Few statements are farther f r o m the t r u t h . L o g i c , provided that i t leads to a comprehensible goal, is not only appreciated, but demanded, by pupils whose i n ­ stincts are n o r m a l . B u t the goal must be comprehensible ; it must not be a result as easily perceived as the assump­ tions o n w h i c h the proof is based. L e t any one w i t h expe­ rience i n e x a m i n i n g consider the types of answer given to two p r o b l e m s ; one, an " obvious " rider on congruence, i n v o l v i n g possibly the pitfall of the ambiguous case; the other, some simple but not obvious construction or rider concerning areas or circles. I n the former, paper after paper exhibits f u m b l i n g uncertainty or bad l o g i c ; i n the latter, there is usually success or silence, and more usually success; bad logic is hardly ever f o u n d . T h e phenomenon is too universal to be comfortably accounted for by abuse of the teachers; the abuse must be transferred to the crass methods w h i c h enforce the premature application of logic to analysis of the k n o w n , rather than to exploration of the u n k n o w n . T h e natural order of exploration should now be evident. L e t the leading results of the science under consideration

PRINCIPLES OF E D U C A T I O N

I I

be divided into two groups : one, those w h i c h are accept­ able, or can be rendered acceptable by simple illustration, to the pupils under consideration ; the other, those which would never be suspected and whose verification by exper­ iment would at once produce an unreal and artificial atmos­ phere. L e t the former group — w h i c h i n geometry would include many o f Euclid's propositions — be adopted as postulates, and let deductions be made f r o m t h e m w i t h full rigour. W h e r e v e r possible, let the results of such deduc­ tions be tested by experiment, so as to give the utmost feeling of confidence i n the whole structure. Later, when speculation becomes more natural, let i t be suggested that gratuitous assumption is perhaps inadvisable, and let the meanings of the consistence and redundance of the set of postulates be explained. Finally, i f i t prove possible, let the postulates be analysed, their consistence and independ­ ence be demonstrated, and the science exposed i n its ultimate f o r m . These second and t h i r d stages are even more essential to a " liberal education " than the first, for they exhibit scientific method and human knowledge i n t h e i r true aspect. I t is not suggested that they can be dealt w i t h i n schools, except perhaps tentatively i n the last year of a l o n g course. B u t i t is definitely asserted that the general ideas involved should f o r m part of the compulsory element of every U n i ­ versity course, even t h o u g h details be excluded, for they are of the very essence of the spirit of mathematics. T h e method of developing such ideas remains to be considered. I t may be presumed that the pupils concerned have some knowledge of arithmetic, geometry, the calculus, and mechanics, each subject h a v i n g been developed f r o m a

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redundant set of postulates. I n which, then, of these four branches is i t most natural to suggest the analysis of these assumptions ? Since analysis of the k n o w n may still be presumed to have its dangers, the branch chosen must be that one i n which the investigation bears this aspect least p r o m i n e n t l y . N o w the m a i n ideas of arithmetic, geometry, and the cal­ culus are so firmly held by boys and girls, that any attempt to discuss t h e m i n detail produces revolt or boredom. Such attempts account for m u c h ; the writer can well remember his feelings o n first seeing a formal proof that the sum of a definite number of continuous functions is itself a con­ tinuous f u n c t i o n ; and at the same t i m e he realized to the full that the proposition m i g h t well be untrue i f the n u m ­ ber of functions were not finite. G r o u n d such as this is unfavourable for the development of this new analysis. T h e same is by no means true of mechanics. H e r e the postulates, acceptable t h o u g h they be, have been elucidated w i t h i n the memory of the pupils, and they may reasonably be asked to examine the facility w i t h w h i c h these assump­ tions were made, and to consider whether the evidence can i n any way be strengthened. T h i s being done, the ideas of consistence and redundance can be developed, and some idea of the structure of a science imparted. E v e n then i t may probably be wise to lay little stress o n analysis of the geometrical postulates ; i f the ideas are realized i n connec­ tion w i t h mechanics, we may w e l l leave the seed to mature i n minds to which i t is congenial. I n the view o f mathematics here taken, its various branches are regarded as structures w i t h many possible entrances, and the discussion has been concerned w i t h the

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choice of entrance and the route to be taken t h r o u g h the edifice. W e cannot hope that our pupils w i l l ever k n o w more than the outline of each structure. E v e n we w h o are the guides cannot k n o w each detail of any one ; the laby­ r i n t h is too vast. B u t the best guide to a structure is he who knows its m a i n outlines most completely, and a teacher who has clear ideas of mathematical principles can do much, i n leading his pupils t h r o u g h such avenues o f the structure as they can attain, to give t h e m a view of the whole. O f the i m p o r t and beauty of this view more need not be said.

INTUITION address delivered to the Mathematical A s s o c i a t i o n , and reprinted from The Mathematical

Gazette,

March, 1 9 1 3 )

INTUITION I f there be one duty more incumbent than any other upon mathematicians, i t is to have a clear and c o m m o n understanding of every t e r m w h i c h they use. I do not say a formal definition, though that is most advisable i f and when i t can be obtained ; but a class of entities must be k n o w n and recognised before i t can be defined, and no t e r m should be used unless i t at least gives rise to definite, recognisable, and identical images i n the minds of the speaker and listener. I t cannot fairly be said that mathe­ maticians are at fault i n this respect, when dealing w i t h their own special subjects ; but I fear they cannot so easily be acquitted when discussing the didactic side of their w o r k . Concrete and utilitarian, axiom and postulate, intu­ ition and assumption; how many of us have definite meanings for these terms, and can feel certain that they represent the same meanings to others ? T h e t e r m w h i c h I have chosen as the title of this paper is one of the most commonly used and, as i t seems to me, most often misun­ derstood ; at the same time, the ideas and processes for which i t stands lie at the root of all elementary teaching. I have therefore thought it w o r t h while to discuss its meaning and to show the bearing of the process on math­ ematical education. T h e r e is, I t h i n k , little doubt that to most of those w h o use the t e r m " i n t u i t i o n , " i t connotes some peculiar quality of material certainty. T a k e , for example, the equality of 17

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all r i g h t angles, or the angle properties of parallel lines, and ask one who understands these statements w i t h what degree of certainty he asserts their t r u t h . I t w i l l be found almost invariably that he regards t h e m as far more certain than statements such as " the sun w i l l rise to-morrow m o r n i n g " or " a l l m e n are mortal " ; these, he admits, m i g h t be upset by some perversion of the order which he has regarded as customary, but the geometrical statements appear to be of the essential nature of things, eternal and invariable verities. So m u c h , indeed, is this the case that the very idea of practical tests is grotesque. W h o has ever experimented to ascertain whether, i f two pieces of paper are folded, and the folds doubled again on themselves, the corners so f o r m e d are superposable ? I f the individual under examination be questioned as to the basis for this faith, he can only reply that i t is the nature of things, or that he knows i t i n t u i t i v e l y ; o f the degree o f his faith there is no doubt. I t is to statements asserted i n this manner that the t e r m " i n t u i t i o n " is c o m m o n l y applied ; other facts, such as the mortality of all men, w h i c h are justified by the fact that all h u m a n experience points to them, are not classified under this heading nor, as I have said, are they accepted w i t h the same faith. These alleged certainties can of course be dissipated by purely philosophical considerations concerning the relations and differences between concepts and percepts; but " an ounce of practice is w o r t h a ton of theory," and I propose here to show, mainly f r o m historical considerations, that there is no g r o u n d for absolute faith i n certain intuitions, however tenaciously they may be held. T a k e first the idea, still held by many, that a body i n m o t i o n must be urged

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on by some external agent i f its velocity is to be main­ tained. U n t i l the time of Galileo this belief was held uni­ versally, even men o f eminence who had considered the subject being convinced of its t r u t h . N o w this faith was of just such a k i n d , and just as strongly held, as the faith i n geometrical statements w h i c h I have m e n t i o n e d ; i t was, and still is by many, regarded as i n the nature of things that a body should stop m o v i n g unless i t is propelled by some external agency. A n d yet others, of w h o m Galileo was the forerunner, see the nature of things i n a l i g h t wholly different. T h e y regard i t as utterly certain that a body can o f itself neither increase nor retard its own mo­ t i o n . A s k a clever boy who has learnt some mechanics, or even a graduate who has not thought overmuch on the foundations of the subject, w h i c h he regards as more un­ likely : that an isolated body should, contrary to N e w t o n ' s first law, set itself i n m o t i o n , or that the secret of i m m o r ­ tality should be discovered. H e w i l l tell you that the second m i g h t happen, t h o u g h personally he does not be­ lieve that i t ever w i l l ; but that a body can never begin to move unless i t has some other body " to lever against." W e thus see two contradictory intuitions i n existence, each held w i t h equal strength. C o m i n g to more recent history, let me r e m i n d you of the development of the theory of parallels, and the rise of non-Euclidean geometries. U n t i l the last century i t may fairly be said that no one had ventured to doubt the so-called t r u t h of the parallel postulate, t h o u g h many emi­ nent mathematicians had endeavoured to deduce i t f r o m the other postulates of geometry. T h e genius of Bolyai and Lobachewsky, however, put the matter i n quite another

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light. T h e y showed that a completely different theory of parallels was just as m u c h i n accord w i t h the nature of things as that hitherto h e l d ; and that, to beings w i t h more extended experience or finer perceptions than ours, this different theory m i g h t appear to correspond with ob­ servation while the current belief failed to do so. I n other words, they showed that there are several ways of account­ i n g for such space observations as we can make w i t h our restricted o p p o r t u n i t i e s ; just as i t was then well k n o w n that there were two theories w h i c h fitted the observations of astronomers, of w h i c h Newton's was the more simple and self-consistent. I t thus becomes clear that intuitions are no more than w o r k i n g hypotheses or assumptions ; they are on the same footing as the p r i m a r y assumptions concerning gravitation, electrostatics, or any other branch of knowledge based on sensation. T h e y differ f r o m these i n that they are formed unconsciously, as a result of universal experience rather than conscious e x p e r i m e n t ; and they are so formed i n regard to those experiences — space and motion — w h i c h are forced on all of us i n virtue of our existence. I t is not i m p l i e d that their possessor is even fully conscious of t h e m ; ask some comparatively untrained adult how to test rulers for straightness, and he may be at a loss or give some i n ­ effective r e p l y ; but suggest placing t h e m back to back and then reversing one, and he at once assents. H e re­ gards this not as new information, but as something so simple and obvious that i t had not occurred to h i m . I t is to h i m the essential nature of t h i n g s ; he has held this view f r o m so early an age, and it has remained so entirely free f r o m challenge, that he revolts at the suggestion that

INTUITION

21

things, viewed from another standpoint, may appear to have a different nature. T h e formation of such w o r k i n g hypotheses is the normal method by w h i c h the m i n d investigates natural phenom­ ena. A f t e r observation of a certain set of events, a theory is formed to fit them, the simplest being chosen i f more than one be found to fit the facts equally well. T h i s theory is developed, and its consequences compared w i t h the results of further observations; so l o n g as these are i n accord, and so l o n g as no simpler theory is found to ac­ commodate the fact, the first theory holds the field. B u t , should either of these events occur, i t is abandoned r u t h ­ lessly i n favour of some better description of the recorded observations. T h e r e are famous historical cases of each e v e n t ; Newton's corpuscular theory of light yielded de­ ductions i n actual disaccord w i t h observation, and was therefore abandoned. T h e ancient theory of astronomy, wherein the stars were imagined to be fixed on a crystal sphere on w h i c h the planets travelled i n epicycles, was abandoned i n favour of the modern theory, not because i t could not be modified to accord w i t h observation, but be­ cause of its greater complexity. I n every such case the question of absolute t r u t h is irrelevant and beyond our reach ; the problem is to find the simplest theory i n accord w i t h all the facts, abandoning i n the quest each theory as a successor is found w h i c h better fulfils these requirements. Shortly, then, we may say that intuitions are merely a particular class of assumptions or postulates, such as f o r m the basis of every science. T h e y are distinguished f r o m other postulates first, i n that they, w i t h their subject matter — for example, space or m o t i o n — are c o m m o n

22

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EDUCATION

f r o m an early age to every h u m a n being endowed w i t h the ordinary senses; and secondly, i n that no other assump­ tions fitting the sensations concerned ever occur to those who make t h e m . T h e i r formation is forgotten, and they are therefore regarded as e t e r n a l ; they h o l d the field un­ challenged, and are therefore regarded as inviolable. Before passing to the consideration o f the bearing of intuition o n the teaching of mathematics, i t may be well to illustrate what has been said by the consideration of a few particular cases. First, suppose that one sees a jar o n a shelf, and puts his hand up to find out whether it is empty. I s the act based on an i n t u i t i o n from the appearance of the jar ? T h i s is not the case ; i f asked before the act, one would not express any final certainty that the hand could enter the j a r ; i t m i g h t have a l i d or be a d u m m y . T h e i n ­ dividual can make more than one assumption which corre­ sponds to the sight sensation; the first assumption made — that the hand can enter the jar — is merely the most likely as j u d g e d by experience. N e x t suppose that a knock is heard i n a r o o m . T h e natural exclamation " W h a t is that ? " is based on i n t u i t i o n , for i t expresses the now universal conviction that such a noise is an invariable accompaniment of some happening which, given opportunity, w i l l also appeal to the other senses. A c c u m u l a t i o n of h u m a n experience has led to the belief that such is invariably the case ; but belief i t is, and not certainty. I f the reply were, " I t is n o t h i n g ; under no circumstances could you have correlated any sight or feel­ i n g w i t h that sound," i t would be received w i t h complete incredulity.

INTUITION

23

Consider again the statement that, given a sufficient number of weights, no matter how small, one can w i t h t h e m balance a single weight, however large. N o one would doubt this or treat i t as a n y t h i n g but the most obvi­ ous of truisms, and yet i t is a pure assumption, formed unconsciously as the result of general experience ; i t an­ swers i n every respect to our definition of an i n t u i t i o n . I t may be thought by some that the statement can be proved arithmetically, but i n every such alleged proof the assump­ tion itself w i l l be found somewhere concealed. W e have, i n fact, no warrant for assuming that the phenomenon called weight retains the same character, or even exists, for portions of matter w h i c h are so small as to be beyond our powers of subdivision. Finally, consider the statement, " I knew intuitively that you would come to-day." I n what respect do those who use i t regard i t as differing from, " I thought it was almost certain that you would come t o - d a y " ? I t may fairly be said that the former expresses less basis of knowledge but more feeling of certainty than the l a t t e r ; it means, " I don't i n the least k n o w how I knew i t , but I d i d k n o w beyond all doubt that you would come." Such ideas, w i t h or without the use of the actual term " i n t u i t i o n , " are com­ m o n enough. T h e y are here quoted to justify the statement, made above, that the t e r m connotes to many of those w h o use it some peculiar degree of certainty. Such statements are not intuitions ; they are mere superstitions, and those who are subject to t h e m fail to realise how often they are unjustified by the event. Belief i n the absolute t r u t h of the angle properties of parallels or of the Laws of M o t i o n is equally a superstition, though these are, u n t i l now, justified

24

MATHEMATICAL

EDUCATION

by the event. T h e t r u t h is that they can never receive this absolute justification, for no material observation is beyond the possibility of error, nor can i t be certain that some simpler theory w i l l not be formed, accounting equally well for the observations; i t is the belief i n this impossible finality w h i c h constitutes the superstition. T u r n i n g now to the more educational 'aspect of the subject, the first problem w h i c h confronts us is t h i s : children, when they commence mathematics, have formed many intuitions concerning space and m o t i o n ; are they to be adopted and used as postulates w i t h o u t question, to be tacitly ignored, or to be attacked ? H i t h e r t o teaching methods have tended to ignore or attack such i n t u i t i o n s ; instances of their adoption are almost non-existent. T h i s statement may cause surprise, but I propose to justify i t by classifying methods w h i c h have been used under one or other of the two first heads, and I shall urge that com­ plete adoption is the only method proper to a first course in mathematics. Consider first the treatment of formal geometry, either that of E u c l i d or of almost any of his modern r i v a l s ; i n every case i n t u i t i o n is ignored to a greater or less extent. E u c l i d , of set purpose, pushes this policy to an e x t r e m e ; but all his competitors have adopted i t i n some degree at least. Deductions of certain statements still persist, al­ though they at once command acceptance when expressed in non-technical f o r m . F o r example, i t is still shown i n elementary text-books that every chord of a circle perpen­ dicular to a diameter is bisected by that diameter. D r a w a circle on a wall, then draw the horizontal diameter, mark a point on it, and ask any one you please whether he w i l l

INTUITION

25

get to the circle more quickly by g o i n g straight up or straight down from this point. Is there any doubt as to the a n s w e r ? A n d are not those who deduce the propo­ sition just quoted, f r o m statements no more acceptable, i g n o r i n g the intuition w h i c h is exposed i n the immediate answer to the question ? A l l that we do i n u s i n g such methods is to make a chary use of i n t u i t i o n i n order to reduce the detailed reasoning of E u c l i d ' s scheme; our attitude is that statements w h i c h are accepted intuitively should nevertheless be deduced f r o m others of the same class, unless the proofs are too involved for the juvenile m i n d . W e oscillate to and fro between the Scylla of ac­ ceptance and the Charybdis of proof, according as the one is more revolting t o ourselves or the other to our pupils. 1

A t this point I wish to suggest that a distinction should be drawn between the terms " d e d u c t i o n " and " p r o o f . " T h e r e is no doubt that proof implies access of material conviction, while deduction implies a purely logical process i n w h i c h premisses and conclusion may be possible or impossible of acceptance. A proof is thus a particular k i n d of deduction, wherein the premisses are acceptable (intuitions, for exam­ ple), and the conclusion is not acceptable u n t i l the proof carries conviction, i n virtue of the premisses on w h i c h i t is based. F o r example, E u c l i d deduces the already accept­ able statement that any two sides of a triangle are together greater than the t h i r d side f r o m the premiss {inter alia) that all r i g h t angles are equal to one another ; but he proves that triangles o n the same base and between the same 1

T h e r e is often apparent doubt;

but it will usually be found that

this is due to an attempt to estimate the want of truth of the c i r c l e as drawn.

26

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EDUCATION

parallels are equal i n area, starting from acceptable prem­ isses concerning congruent figures and converging lines. T h e distinction has didactic importance, because pupils can appreciate and obtain proofs l o n g before they can understand the value of deductions; and it has scientific importance, because the functions of proof and deduction are entirely different. Proofs are used i n the erection of the superstructure of a science, deductions i n an analysis of its foundations, undertaken i n order to ascertain the number and nature of independent assumptions involved therein. I f two intuitions or assumptions, A and B, have been adopted, and i f we find that B can be deduced from A , and A f r o m B, then only one assumption is involved, and we have so m u c h the more faith i n the bases of the science. H e r e i n lies the value of deducing one accepted statement from another ; the element of doubt involved i n each acceptance is thereby reduced. N e x t , to justify the statement that intuition has been attacked. B o t h E u c l i d and his modern rivals knew well enough that their schemes must be based on some set of assumptions; they differed only i n the choice. Each agrees that i n t u i t i v e assumptions are undesirable, but the m o d e r n school regards the extreme logic entailed by E u c l i d ' s principle of the m i n i m u m of assumption as impossible for y o u n g pupils. T h e r e is, however, a t h i r d school w h i c h pursues a different course; i t professes to replace i n t u i t i o n by experimental demonstration. Pupils are directed to draw pairs of intersecting lines,- measure the vertically opposite angles, and state what they observe; to perform similar processes for isosceles triangles, parallel lines, and so o n . Instead of being asked, " D o you t h i n k that, i f these lines

INTUITION

27

were really straight, and you cut out the shaded pieces, the corners would fit ? " they are told to find out, by a clumsy method, a belief w h i c h they had previously held, t h o u g h i t had never, perhaps, entered definitely into t h e i r consciousness. T h e question suggested is, i n these homely terms, just sufficient to b r i n g the idea before t h e m , and i t is at once recognised as according w i t h the child's previous notions; he does not regard i t as new, but merely as some­ t h i n g of w h i c h he had not before t h o u g h t so definitely. I t is this type of exercise i n d r a w i n g and measurement w h i c h I regard as an attack u p o n i n t u i t i o n . I t replaces this natural and inevitable process by hasty generalisation f r o m experiments of the crudest type. Some advocates o f these exercises defend t h e m o n the g r o u n d that they lead to the formation of intuitions, and that the pupils were not previously cognisant of the facts involved. B u t i n the first place, a conscious induction f r o m deliberate experiments is not an i n t u i t i o n ; i t lacks each o f the special elements connoted by the t e r m . A n d as to the alleged ignorance o f the elementary idea o f space, i t appears to me to be a mis­ taken impression, based o n undoubted ignorance of mathe­ matical terminology. I f you say to a c h i l d o f twelve, " A r e these angles equal?" he has to stop to t h i n k first, what an angle is, and next, when angles are e q u a l ; by the t i m e he has done this his m i n d is incapable o f grasping the pecul­ iar relations of the angles i n question, and he is labelled as ignorant o f the answer. T h e real difficulty, and i t is not a small one, is to lead the child to express familiar facts i n precise mathematical t e r m i n o l o g y ; to say "angles e q u a l " rather than " c o r n e r s fit." U n t i l this terminology is thoroughly familiar, the effort of using i t must absorb

28

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EDUCATION

a large part of the child's attention, leaving little available for the matter i n hand. T h i s paper is not concerned w i t h the methods or practice of teaching, but I would strongly urge all those who are concerned w i t h y o u n g children to guard against this danger, by constant transition to and fro between c o m m o n and technical phraseology, appealing at once to the former at the least sign o f doubt or hesita­ t i o n . T h e l e a r n i n g of technical terms should not appear as part of the definite w o r k , or it w i l l inevitably be regarded as the major p a r t ; i t should come incidentally and by gradual transition, as I have suggested. 1

T h e only alternative to this evasion or suppression of i n t u i t i o n is to accept i t f r o m the commencement as the natural basis for p r i m a r y education. B u t to be of any avail, the acceptance must be unquestioned and complete ; every i n t u i t i o n w h i c h can be f o r m e d by the pupils must, without suggestion of doubt, be adopted as a postulate, none being deduced f r o m others w h i c h are themselves no more easy of acceptance. Such a course leads, i t need hardly be said, to considerable simplification i n the early treatment of any subject. F o r example, i n geometry the angle properties of parallel lines, properties of figures evident f r o m symmetry, and the theory o f similar figures (excluding areas) appear as postulates; i n the calculus i t is not proved that the differential coefficient of the sum of a finite number of functions is equal to the sum of their differential coeffi­ cients ; the statement is illustrated by, say, consideration 1

I t is no good to say, " C o m e now, what is an angle ? " A p p e a l first

to the tangible fact i n the child's m i n d by s a y i n g , " C a n n o t you see that those c o r n e r s must

fit ? " a n d t h e n r e m i n d h i m that " equal a n g l e s "

merely m e a n s the same thing.

INTUITION

29

of some expanding rods placed end to end, and at once commands acceptance. H e r e the question of terminology again arises ; I have often been struck, i n teaching school­ boys and students, by their slowness to accept this and similar results i n the calculus; the clue was given to me by a boy who remarked that i t was t a k i n g h i m all his t i m e to remember what a rate of increase was, and he could not manage any more at the m o m e n t . Since that t i m e I have avoided many seeming difficulties w i t h elementary and advanced pupils by appeal f r o m technical to familiar terms, always of course rephrasing the result i n the proper f o r m before leaving the matter i n hand. I t w i l l , I know, be t h o u g h t by many that this adoption of all natural intuitions involves an appalling lack of rigour. B u t I would ask those who are of this opinion to do one t h i n g before passing j u d g m e n t , and that is, to define and ex­ emplify w i t h some care the meaning of the t e r m " r i g o u r . " W h e n they have done this, I t h i n k they may be disposed to agree w i t h the answer to their accusation w h i c h I am now g o i n g to put forward. I t is that the scheme suggested is perfectly rigorous, provided that every deduction made f r o m the postulates adopted is logically sound; on the other hand, i t is admitted that the mathematical t r a i n i n g thus imparted is not complete, because no attempt has been made to analyse these i n t u i t i v e postulates into their com­ ponent parts, s h o w i n g how many must perforce be adopted i n the most complete system of deduction. I n other words, we may be rigorous i n regard to logical reasoning, or i n regard to lessening the number of assumptions w h i c h f o r m the basis of a science. T h e view for w h i c h I contend is, that i n a l l stages of mathematical education, deductions

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EDUCATION

f r o m the assumptions made should be r i g o r o u s ; but that i n the earlier stages every acceptable statement or i n t u i t i o n should be taken as an assumption, the analysis of these, to show on how small an amount of assumption the science can be based, being deferred. T o avert misapprehension, let me say again that I pro­ pose that, w h e n all intuitions are accepted as postulates, this should be done w i t h o u t question or discussion other than that necessary to give t h e m some precision. T o em­ bark o n a discussion of t h e i r nature, or to appear to cast doubt u p o n t h e m , would be fatal, as fatal as has been the apparently futile process of deducing one accepted state­ m e n t f r o m another. T h e p u p i l is already i n possession of a body o f accepted t r u t h ; let us b u i l d o n that and defer its analysis, or a n y t h i n g that pertains thereto, u n t i l he is sufficiently mature to appreciate the motive. T h e first course of mathematics would, then, range from arithmetic and analysis t h r o u g h geometry to mechan­ ics. I n this last subject there is little scope for i n t u i t i o n . Most of the mechanical intuitions formed by the race as a whole have been mistaken, and i t is just this fact w h i c h gives some indication of the proper commencement for the second course, i n w h i c h the intuitive postulates are to be analysed and reduced as far as possible. L e t the student learn s o m e t h i n g of the history of mechanics, realising that ideas w h i c h he regards as impossible and absurd were held, by m e n o f great eminence, w i t h faith just as strong as that w h i c h he places i n his geometrical postulates. T h e n let i t be suggested to h i m that this renders care i n regard to assumption o f vital importance, and so commence an analysis of the mechanical postulates, h i t h e r t o redundant,

INTUITION

31

obtaining deductions of one f r o m another to show their inter-connection. T h i s completed, and the task is not a large one, i t is natural to suggest that the postulates of geometry deserve some examination, and so, according to the t i m e available and the ability of the p u p i l , we may pass backward t h r o u g h a review of the foundations of geometry to an examination of the foundations of analysis and arith­ metic. I t is not, o f course, i m p l i e d that every student of mathematics can reach this g o a l ; few can ever get beyond some consideration of the foundations of geometry, w i t h a clear understanding of the end to be attained i n its general application to all sciences. B u t I do wish to put forward, w i t h such emphasis as I can, this general scheme o f math­ ematical education; namely, an upward progress, based o n i n t u i t i o n , f r o m arithmetic t h r o u g h geometry to mechanics, followed by consideration i n the reverse order o f the founda­ tions of each branch, the upward progress constituting the first course, and the downward review the second course. I t would, I believe, give an intelligible unity to the whole subject, a n d would do s o m e t h i n g to restore that purely intellectual appreciation w h i c h has so largely declined d u r i n g the past generation. Mathematics is a useful tool, but i t is also something far greater, for i t presents i n unsullied outline that model after w h i c h all scientific thought must be cast. I have endeavoured to show how this outline may be developed, starting f r o m those intuitions w h i c h are c o m m o n to us all, and e n d i n g i n an analysis demonstrating their true nature. T h e concrete illustrations, so necessary and i l l u m i n a t i n g i n elementary teaching, are so many draperies, fashioned to render this outline visible to those who cannot otherwise

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appreciate i t . E v e n the several branches—analysis, geom­ etry, mechanics—serve

the same e n d ; behind t h e m all

is the one pure structure of mathematical thought. T h e y who most appreciate the structure w i l l best fashion the draperies, and so render i t most clearly visible to those w h o m they instruct.

T H E USEFUL A N D T H E REAL

THE

USEFUL AND THE

REAL

A m o n g the many changes i n mathematical education d u r i n g the last twenty years, and a m o n g the many and often conflicting ideals w h i c h have directed these changes, one element at least appears t h r o u g h o u t ; a desire to relate the subject to reality, to exhibit i t as a l i v i n g body of thought w h i c h can and does influence h u m a n life at a m u l t i t u d e of points. T h e old scholastic ideal of development i n the most abstract way, the realities b e i n g allowed to take care of themselves, is exploded for this as for most branches of education ; i t is recognised that the separated mediaeval worlds of t h o u g h t and action must be replaced by a single w o r l d wherein each exerts profound influences on the other. O u r c h i l d r e n must learn to t h i n k , and to t h i n k about the w o r l d as i t now is and the manner of its evolution. Some few there may be w h o can w i t h profit to us all devote themselves to one or other side of this w o r l d of t h o u g h t and action, but the mass of m e n must be fitted to play their part between the two. So far all are agreed, but c o m m u n i t y of pious opinion has before now been k n o w n to result i n discord, and dis­ cord none the less acute because due to diversity of policy alone. Such has been the case w i t h mathematical educa­ tion ; the c o m m u n i t y of ideals just described has not resulted i n c o m m u n i t y of action ; i t is more nearly true that each man is a law unto himself i n his method of for­ warding t h e m . L i k e most disorganised armies we have 35

36

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EDUCATION

our shibboleths, and a m o n g the most p r o m i n e n t are " r e a l , " " u s e f u l , " " c o n c r e t e . " A n examination of what these do and should represent may not be without profit. Starting f r o m agreement that the w o r l d of thought is to be related to the w o r l d of action or reality (not thereby dependent u p o n or l i m i t e d by that world), the natural course is to attempt to f o r m some concept of the particular w o r l d of reality w i t h w h i c h we are concerned. Suppose that one desires to explain the principles o f the calculus to an assemblage of doctors; tables of population, mortality, and the l i k e f o r m one obvious w o r l d of reality f r o m w h i c h thought can be developed; i f to an assemblage of mer­ chants, statistics of trade and finance w o u l d f o r m such a world, and so o n for other avocations. B u t suppose that the assembly consisted o f m e n engaged i n no one p u r s u i t ; the difficulty would be greatly enhanced, for there would be no obvious w o r l d o f reality c o m m o n and familiar to them a l l . So also i n his dealings w i t h y o u n g c h i l d r e n must the teacher of mathematics determine fitting worlds of reality and develop his instruction for t h e m . B u t what is reality ? W h a t considerations determine the entities w h i c h have this attribute ? F o r me, m y hands, m y furniture, this town, E n g l a n d , Cromwell, Macbeth, the binomial theorem, are all real ; but Cromwell's hands, the furniture i n a strange house, Hepscott ( I take the name at random f r o m a gazetteer), F a n n i n g Island, B e n Jonson, H e d d a Gabler, n i t r a t e s — a l l lack reality. Each one o f the first is related to some definite recognisable sensation or concept of m y own, but each of the second is (for me) a mere name w h i c h bears no relation to any such sensation or concept; I k n o w n o t h i n g of F a n n i n g Island, nor sufficient

THE USEFUL AND T H E REAL

37

of B e n Jonson to distinguish h i m f r o m other writers of his t i m e ; I have not read Ibsen, and I k n o w little chemistry. T h e essence o f reality is thus found i n definite recognisable percepts or concepts, and is therefore a func­ t i o n of the individual and the t i m e ; what is real to me is not necessarily real to another, and m u c h that was real to me i n childhood is no longer so. I t is for the teacher to determine the realities of his pupils and exemplify mathe­ matical principles by as many as are suitable for the purpose. H e w i l l also find i t necessary to enlarge their spheres of reality, but he must avoid confusion between a name and a t h i n g ; he must, for example, make sure that his pupils k n o w what a parallelogram is before they use the name. I t is at this p o i n t that various policies have arisen, des­ pite general agreement o n ideals. O n e of these confuses the many worlds o f reality, different for each individual, w i t h some absolute w o r l d of reality supposed to be com­ m o n to a l l . T h i s absolute w o r l d is usually based o n those applications of mathematics which have some commercial or scientific utility, such utility being considered to involve reality for the p u p i l . T h e result of this confusion o f the useful w i t h the real is seen i n problems w h i c h deal w i t h such mysteries as resistance i n pounds per t o n weight, the extension of helical springs, efficiency and load, ton-inches of t w i s t i n g moment. T o all children (and many adults) these phrases are as meaningless as the symbols of the purely scholastic algebra of t h i r t y years ago ; they are merely a cumbrous way of w r i t i n g the x and y of that alge­ bra and i m p l y as little to those for w h o m they are intended. B u t their use may t e n d to impart an idea that realities are being dealt w i t h — an idea thoroughly vicious i n that i t

38

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replaces entities by words. W e may name entities w h i c h are direct sensations, and we may name entities w h i c h are pure creations of the i m a g i n a t i o n ; but to imagine that a name w h i c h is co-related to neither sensation nor imagina­ t i o n possesses any sort of reality is the grossest of errors. T o o many teachers are content to use words for w h i c h they have no definite meanings, and to allow their pupils to imagine that they have acquired something i n learning such words ; but we need not go out of our w ay to spread this error, the more so as we are concerned w i t h the one subject w h i c h should suppress i t most completely. 7

I t is not, of course, suggested that any existing courses of mathematics are l i m i t e d to such applications alone. B u t there is an obvious tendency t o judge applications by such standards, a t t r i b u t i n g more and more importance to those w h i c h accord w i t h t h e m . A n excellent instance of such judgments is the condemnation of the traditional problems dealing w i t h tanks w h i c h are emptied and filled simultane­ ously by different pipes. I t is argued that no adult ever deals w i t h a cistern i n this way, and that the problems should therefore be replaced by others h a v i n g more reality. T h e t e r m " r e a l i t y " begs the whole question, for i t has no absolute m e a n i n g for all people at all ages and must be defined by those who use i t . A n d i t is here confused w i t h utility, a very different attribute. T h e essence of the con­ tention is that no application should be used i n education unless i t is of actual use i n some branch of science or walk of life. T h i s is a far cry f r o m the pupil's w o r l d of reality ; the formalist attempted to transport h i m to a w o r l d of ab­ stract thought wherein the entities are typified by letters x and but our utilitarian proposes to l i m i t the play of

T H E USEFUL AND T H E REAL

39

his imagination to matters used by adults, no matter how far these may be removed from his cognisance or interest. T h e r e is at bottom little difference between the two, but the formalist is the more open i n that he does not cloak his meaning under a mass of words w h i c h are f u l l of sound and signify n o t h i n g . A variant o f this school may reply that they are being unjustly accused; that they are i n entire accord w i t h the rejection of matters such as voltage and twisting m o m e n t on the g r o u n d that they have no reality for the pupils con­ cerned, but that there are plenty of applications w h i c h are real and also useful. T h i s may be so, t h o u g h examination of modern text-books hardly supports the c l a i m ; but i n any case we cannot o n such lines develop mathematical t h o u g h t f r o m any large portion of the pupil's w o r l d of reality ; i t is related to those parts only of that w o r l d w h i c h coincide w i t h the worlds o f various adults, and these may well be neither the most interesting nor the most familiar portions of his own w o r l d . A second policy, exemplified for the most part i n con­ nection w i t h geometry, interprets the child's world o f reality as the w o r l d of his senses, and more particularly the senses of sight and touch, and so is allied w i t h the concrete rather than the useful. I t endeavours to develop t h o u g h t f r o m manipulations and measurements performed by the p u p i l himself, and is thus l i m i t e d to the perceptory or concrete portion of his realities. I n itself and so far as i t goes this is an entire and most valuable gain as compared w i t h the practice of t h i r t y years ago ; the pupils feel that they are dealing w i t h matters w i t h i n their own personal cognisance instead of abstractions w h i c h are evidently familiar only

4°

MATHEMATICAL

EDUCATION

to m e n w i t h w h o m , intellectually, they have and w i l l have little i n c o m m o n . Unfortunately, however, there has been a strong tend­ ency to l i m i t the w o r k to this concrete domain, refusing any part to that world of imagination which is, especially i n children, just as real and a great deal more v i v i d . I n t r o ­ ductory courses of geometry consist of the construction and measurement by the p u p i l of figures whose dimensions are prescribed. T h e y develop a detailed knowledge of perceptory space but make no use of that m u c h larger and more i m p o r t a n t conceptual space wherein the creations of his imagination move and have their being. Travels, ad­ ventures, romances, history, and the hundred and one utterly useless but apparently practical things which interest a boy are situated i n this space, and here, as well as i n the smaller concrete space of t h e senses, should the w o r l d of thought be exemplified, for these t h i n g s also are realities for h i m . T h r e e distinct policies have now been discussed : the first rejects all applications and insists throughout o n devel­ opment i n abstract t e r m s ; the second insists that illustra­ tions must be drawn f r o m applications w h i c h are relevant to some branch of science, industry, or commerce; and the t h i r d insists that development must originate i n the i m m e ­ diate evidence of the senses. O f course, no m a n or body of m e n holds one of these views to the entire exclusion of the other two, nor is the w o r l d of imagination entirely ignored i n current practice. B u t most text-books and writ­ ings on mathematical education are influenced mainly by some one of them, and may be placed i n a class w h i c h holds that particular policy as paramount. T h e r e is most i n

T H E USEFUL AND T H E REAL

41

common between those who hold the second or t h i r d view, for they give a c o m m o n allegiance to the use o f reality and differ only i n the scope of the t e r m . M a n y teachers are, indeed, influenced by considerations of utility i n algebra and considerations of concrete reality i n geometry, their utterances on one subject often contradicting those on the other. N o w among the many uncertainties and conflicts w h i c h surround these (as all) questions of education, two state­ ments at least stand out as certain beyond dispute. T h e first is that the operations and processes o f mathematics are i n practice concerned at least as m u c h w i t h creations of the imagination as w i t h the evidences of the senses; i t is enough to m e n t i o n points, complex numbers, ether, electric charges, to make this plain. T h e second is that the purpose of mathematical education is to put the p u p i l i n a " mathematical way " ; to permeate his whole being w i t h the elementary principles of the science so that he w i l l apply t h e m spontaneously i n considering any matter to w h i c h they may be relevant. T h e formalists held that i f principles were imparted i n their utmost generality, each individual could and would make such applications as he m i g h t require, a statement not justified by experience and not i n accord w i t h such knowledge of the m i n d as we possess; the moderns believe that principles can only be seen by their exemplification throughout the w o r l d of reality of the p u p i l . T h e formalists thus seek u n i t y of treatment for a class i n generality of presentation, the moderns seek it among the experiences and concepts of the various pupils. Fortunately for education i n general, this modern search is certain to prove successful as regards children, because

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their experiences and imaginations r u n i n grooves more or less alike ; they are interested i n puzzles, hidden treasures, travels, railways, ships, and the like, and problems concerned w i t h these entities are real to t h e m no matter how absurd they appear from the standpoint of practical life. T h e i r educational utility is not to be measured by their commercial or scientific value, but by their degree o f reality for the pupils under instruction. P u t t i n g the matter i n more or less mathematical phrase­ ology, we may say that the mathematical instruction of a beginner must be exemplified by a m a x i m u m number of his realities i n order that the principles may permeate his whole b e i n g ; i n dealing w i t h a class we must therefore find the greatest c o m m o n measure of their realities and w o r k f r o m that. I f the class is composed of adults h a v i n g v a r y i n g antecedents, this c o m m o n measure may be small compared w i t h the realities of any one member; but i f the pupils are children, i t is large i n comparison w i t h their individual realities, and the task of the teacher is corre­ spondingly simplified. L e a v i n g generalities w h i c h may have appeared somewhat vague, we may now consider a few problems w h i c h are real for children but not directly useful to t h e m or any one else. F i r s t take the type already mentioned, which deals w i t h the e m p t y i n g and filling o f a tank. T h e r e is no doubt that this is sufficiently real for any c h i l d ; he can visualise the whole process, and its value is increased because the entities are imagined and not perceived t h r o u g h the senses. T h e purpose served is the exemplification of the m e t h o d o f adding or comparing several rates by reducing all to a common unit, an idea sufficiently important i n after life.

THE

USEFUL AND T H E REAL

43

Those who attack such an illustration must find others w h i c h w i l l serve the same purpose and satisfy their test of utility, and i n d o i n g this they w i l l i n all probability pass beyond the limits of reality. T h e r e is no doubt that the e m p t y i n g of cisterns, the coincidence of clock hands, and other seeming trivialities do exemplify the h a n d l i n g of rates i n ways w h i c h are more real to y o u n g children than others w h i c h have more actual utility, and they are therefore to be welcomed rather than condemned. T h e mistake i n their treatment, and as gross a mistake as could well be made, has been their g r o u p i n g by subject matter instead of p r i n ­ ciple. A l l questions w h i c h deal w i t h one principle should be grouped together and the subject matter varied continually. N e x t consider the Progressions, which have of late been attacked on the score that they are comparatively useless i n mathematics or anywhere else. T h i s is true, and ad­ vocates of their retention have done their cause no good by saying vaguely that they have their uses and t h e n fail­ i n g to give specific instances. T h e y do, however, provide a number of problems w h i c h have reality for children, and they exemplify three most important matters : the concept of a series, the value of w h i c h extends far beyond math­ ematics ; the insight w h i c h can be gained by a proper g r o u p i n g of various entities; and the construction of a formula or law to cover any number of discrete cases. Consider again the well-known problem i n the calculus of a man who is on a common and wishes to reach a point on a straight road, along which he can walk more quickly than on open country, as soon as possible. I f such a problem ever has practical utility, i t is not for one man i n ten thou­ sand, and to regard it as i n any way generally useful is

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obviously grotesque ; but again i t is real for those who study it, and i t exemplifies the comparison of different modes of transition f r o m one state to another, and the selection of the most suitable. A n o t h e r illustration is provided by the use of statistical graphs i n the introduction of the calculus. Such graphs are of service i n exemplifying the meaning of a differential coefficient and a definite integral by means w h i c h possess reality for the students, and their whole function is described i n this statement. N o w i t may well happen that a set of statistics w h i c h have no practical use of any k i n d , or are even i n actual disaccord w i t h the results o f some branch of science or industry, may serve these purposes better than others which have some direct use or are i n accord w i t h experience. F o r example, excellent problems can be made concerning the consumption of coal by locomotives, but they would never occur i n the practice of any engineer, nor would the numbers w h i c h happen to give good graphs occur i n the w o r k i n g of any conceivable locomotive. B u t this is i n no way to the detriment of the problem for the purposes of instruction. T h e inaccuracy of the information contained i n the figures is surely immaterial i f students are told that actual numbers can be found i n any handbook for engineers should they ever chance to need them, and no other objection seems relevant to the purpose of the problem. I t exemplifies principles t h r o u g h illustrations w h i c h are real for the particular students, and thus fulfils its a i m . 1

These examples exhibit the tests by w h i c h applications should be judged. T h e y must exemplify those leading ideas 1

T h e y would not be real for a class of locomotive engineers, a n d the

example would not be u s e d for s u c h a class.

T H E USEFUL AND T H E REAL

45

which it is desired to impart, and they must do so t h r o u g h media w h i c h are real to those under instruction. T h e reality is f o u n d i n the students, the utility i n their acquisition of principles. T h e outcome of our discussion is, then, that illustrations must above all be r e a l ; they must be useful as well, i f that be possible, and particularly w i t h reference to other branches of study such as physics; but reality is the crucial test. A n d reality is a function of the individual and the time, so that no absolute schedule of the more and the less real can be devised ; but there is sufficient c o m m u n i t y between children o f the same age to handle t h e m i n groups, while adults m i g h t , o n the other hand, require classification i n regard to their realities before they could receive efficient instruction i n groups. M a n y problems w h i c h interest and even excite children are to t h e m hope­ lessly banal, and others must be used more i n touch w i t h their particular spheres of reality. Finally, each principle must be exemplified i n as many ways as possible so that unity may be perceived i n principle rather than subject matter. W e have travelled far from the useful applications of mathematics i n our quest for fitting illustrations ; we have been led to consider reality as the proper criterion, and to recognise that the t e r m is essentially relative. B u t so also is "useful " a relative t e r m ; what is useful for one purpose is useless for another, and i t may well be said that many applications of mathematics which are grotesquely useless in any branch of science or commerce are of the utmost use i n education for their v i v i d illustration of ideas so abstract as to be otherwise vague or invisible.

SOME U N R E A L I S E D P O S S I B I L I T I E S OF M A T H E M A T I C A L E D U C A T I O N ( A n address delivered to the Mathematical A s s o c i a t i o n , and reprinted from The Mathematical

Gazette,

M a r c h , 1912)

SOME UNREALISED POSSIBILITIES OF MATHEMATICAL EDUCATION T h e last half-century has seen a great and significant change i n the popular estimation of mathematics. F o r m e r l y the subject was regarded as utterly unpractical and there­ fore useless i n the narrow sense of this term, t h o u g h i t was recognised as p r o v i d i n g a t r a i n i n g unique i n its char­ acter, i n logical thought and i n accurate expression. N o w it is regarded, and correctly regarded, as having enormous practical importance i n science and engineering. Most, i f not all, of those discoveries and inventions w h i c h are so profoundly m o d i f y i n g civic and national life have found their origin, or development, or both, i n the labours of mathematicians, and this fact is widely k n o w n . T h e mathematician is no longer regarded as a dreamer of dreams; he is classed w i t h the doctor, the engineer, the chemist, and all those whose specialised labours have had immense i m p o r t for the human race. B u t simultaneously a change of no less magnitude has taken place i n the mathematical w o r l d . T h e type of i n ­ vestigation which bore such fruit i n the hands of Faraday, Clerk M a x w e l l , K e l v i n , and many others no longer occupies the attention of those who are i n the forefront of mathe­ matical investigation. T h e theories of pure number, of space, of functions, and such names as D e d e k i n d , Cantor, Grassmann, K l e i n , and, i n our own country, Hobson, W h i t e h e a d , and Russell, have little or no connotation for 49

SO

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EDUCATION

the outer w o r l d . I n so far as this outer w o r l d is cognisant of their existence, these theories, and the men to w h o m they are due, appear as chimerical and unpractical as would the labours of Clerk M a x w e l l have appeared to the L a n ­ cashire cotton spinner of 1850. A n d I fear that this view is too often shared, consciously or unconsciously, by mathe­ maticians themselves, and especially by those who teach the subject. Here, they say or t h i n k , is a type of thought or investigation of great interest to those who can appre­ ciate it, but it is utterly and permanently out of touch w i t h the w o r l d at large. I t can have no relevance or i m p o r t for the ordinary boys and girls who learn mathematics at school, and can i n no way assist t h e m to become efficient citizens. B u t is this really the case ? H e would be a bold man who would say w i t h certainty that any branch of scientific investigation must be regarded, once for all, as h a v i n g no bearing on the development of the individual or race. Is not the better answer that the practical i m p o r t of these investigations has not yet been perceived ; that i t behoves all mathematicians, but especially those w h o are engaged i n teaching, and therefore have some knowledge of the youthful m i n d , to do what they can to correlate this w o r k w i t h the outer world, and to examine to what extent i t can now influence the manner or matter of teaching i n our schools ? T h e question w i l l probably receive an affirmative answer f r o m each one of you, but you may perhaps add that I am w a l k i n g i n the mists w h i c h hide from us the development of future centuries; that sufficient unto the day is the vision thereof; and that the ground to which I invite you is a morass w h i c h may conceivably be made firm by our great-grandchildren.

SOME U N R E A L I S E D POSSIBILITIES

51

Nevertheless, I am g o i n g to ask you to bear w i t h me while I endeavour to convince you that we can now com­ mence to bridge the morass. I admit that i t is one. H e s i t a t i n g and imperfect our endeavours may be, but I am honestly convinced that the t i m e is ripe for a com­ mencement, and that the future of mathematics as a universal subject i n the curricula of schools depends, i n some part at least, on this commencement being made at once. M y ground for this conviction is best stated tersely. I believe that the modern theories of pure mathematics are destined to i l l u m i n e our understanding of the human m i n d and of cities and nations, just as the pure mathe­ matics of fifty years ago has already i l l u m i n e d the previ­ ously dark and chaotic field of physical science; that modern mathematics is or w i l l be to psychology, history, sociology, and economics as has been the older mathe­ matics to electricity, heat, light, and other branches of physical science. F o r example, i t may well be that the theory of sets of points or the theory of groups w i l l find fruitful application i n economics. Y o u w i l l see that I am suggesting that the range of applied mathematics may be widened far beyond its present scope. I t was asserted recently at a meeting of head-masters that the reign of pure mathematics was closed. W o u l d i t not be more accurate to say that pure mathematics has of late extended and co-ordinated its dominions to an amazing extent, and that corresponding extensions of applied mathematics have yet to be found ? I f I am right, then our subject has an irresistible claim. W e may trust our lives to engineers and scientists, just as we entrust our bodies to doctors and surgeons ; but each member of a human society should,

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so far as he may, be competent to analyse and estimate for himself the w o r k i n g s of his own m i n d and the devel­ opment of the society of w h i c h he is a unit. I n the more detailed remarks w h i c h I am about to make, I w i l l ask you to bear i n m i n d that their m a i n inspiration and justi­ fication lies i n what I have just said — that they represent an individual attempt to relate mathematical education to h u m a n thought and social development. Mathematics has been defined by Russell as the class of propositions, " I f A , then and is applied to classes of entities concerning which certain propositions A are assumed; the t r u t h of these is no concern of the subject. T h e entities f o r m the universe of discourse. T h e y can be ordered i n respect of each of the attributes w h i c h charac­ terise their class. T h i s universe of discourse may be of any number of dimensions from one u p w a r d s ; i n arith­ metic i t is one-dimensional, and i n geometry i t should be three-dimensional, but is more often two-dimensional. I may remark i n passing that some attempt to estimate the number of dimensions, that is, of quantities required for exact specification, of the entities discussed i n such subjects as economics would often t h r o w considerable l i g h t o n these subjects. T h e abstract idea of entities and their dimen­ sions is too often w a n t i n g . Hence arithmetic forms the basis of mathematics, since i t explores the properties o f onedimensional fields. A n y treatment of arithmetic which fails to explore the whole domain of such fields is ipso facto incomplete, and its v i c t i m is i n possession of an imper­ fect instrument which cripples h i m alike i n concrete and abstract applications. M y first plea is, therefore, for a mathe­ matical treatment of arithmetic from the earliest stages.

SOME U N R E A L I S E D POSSIBILITIES

53

T h e r e is m u c h w h i c h m i g h t be said concerning integers and fractions, and i n particular scales of notation. M y omission of these subjects is only to be interpreted as an admission that decimals and the theory of exact measure­ m e n t are of more immediate importance, and must occupy such t i m e as I can devote to arithmetic. T o m y t h i n k i n g , young children are h u r r i e d on to fractions far too soon. T h e r e are many unexplored fields of concrete problems, possessing real interest for y o u n g pupils, the study of w h i c h would give a m u c h firmer basis for future develop­ ments than is now obtained. A n d the proofs of such simple rules as " casting out the nines " may provide easy exercises i n deduction, not w i t h o u t value. T o commence, then, w i t h measurement. W h e n , i n actual practice, one measures a length, there are three distinct objects, any one of w h i c h may be i n view. T h e purpose may be either ( i ) to state a l e n g t h greater than that of the given object, but as little greater as may be, or (2) to state a l e n g t h less than that of the given object, but as little less as may be, or (3) to state two lengths as close together as may be, between which the given object lies. I venture to suggest that t r a i n i n g i n measurement can only become of any value (other than manipulative) if i t proceeds on these lines, phrases such as " nearly " and " e x a c t l y " b e i n g abolished as inexact, and therefore unscientific. " Nearly " is useless u n t i l we are told how near or w i t h i n what nearness, and " exactly " only means " as nearly as I can see." B y the use of a vernier — the theory of w h i c h should be included i n every course of arithmetic — children should learn how nearly they can see, and then say, for example, 13-40111. w i t h i n 0-2 m m .

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W e should thus sweep away all the loose statements w h i c h are, I honestly believe, responsible for m u c h of that lack of accurate thought which is the subject of present com­ plaint, and replace t h e m by a t r a i n i n g i n the exact expres­ sion of practical measurements, the final f o r m being of the type " between 7-38 and 7-39 c m . " T h e g r o u n d is now prepared for the extension of the idea of number, this being done, probably, i n connection w i t h mensuration, that is, by questions such as " F i n d the length of the side of a square whose area is 2 sq. i n . " Few trials are necessary i n order to ensure conviction of the fact that the number of inches is not a fraction, and systematic approximation f r o m above and below is at­ tempted. B y actual trial, u s i n g multiplication only, i t is found that the following pairs of numbers are respectively smaller and greater than the number r e q u i r e d : ( 1 , 2), (1-4, 1-5), (1-41, 1-42), and so on. So far n o t h i n g more appears than can be realised by measurement, but i t is at once seen that (1) this process can be continued indefi­ nitely, given t i m e and energy; and (2) that there is no l i m i t to the closeness of the approximation. T h e human m i n d , by this systematic approach, has thus ridden rough­ shod over the imperfections of physical measurement. T h e latter leaves, and must always leave, an unexplored gap w h i c h cannot be diminished, but the method of suc­ cessive approximation enables us to d i m i n i s h the gap below any l i m i t , however small. N o w this process, i f carefully developed, is not beyond the comprehension of y o u n g pupils, and i t may fairly be said to contain the germ of any proper study of functions and the calculus, whether this be undertaken on a graphical

SOME U N R E A L I S E D POSSIBILITIES

55

or analytical basis. I n either case this method of inclu­ sion between converging pairs is essential to any exact comprehension of the subject. A n d beyond this i t develops the theory of pure number so far as to give the pupils — however unconsciously — an early example of a perfect mental structure, fashioned by extension from concrete experience, and i t gives t h e m the only true ideal for the exact estimation of any set of phenomena. Shortly, then, I suggest the continuous development of the idea of a cut or ScJudtt o f the rational numbers, commencing i t at an early age i n connection w i t h a scientific treatment of simple measurements, the purpose being to give a true concept of number i n its relation to measurement. I next make some reference to algebra, stating first that I am not to be taken as i m p l y i n g that the subject should be taught before geometry. O n the contrary, I am convinced f r o m actual experience that geometry should have been studied for two years at least before algebra is commenced. A t the risk of appearing to raise needlessly large issues, I must ask the question, W h a t is an algebra for our present purpose, and what educational purpose may be served by its study ? T o m y m i n d there are two essential steps i n the development of an algebra: the first is the development of a symbolism w h i c h is usually suggested by certain combinations of entities, for example, a + b = b + a, ab—ba; and the second is the extension of this symbol­ ism to cases w h i c h bear no interpretation i n terms of these entities, and its subsequent application to other classes of entities. B y this I mean the interpretation of symbols such

56

MATHEMATICAL

a s

m

e

a

c

n

EDUCATION w

m

3 —• S> 3 + ^~ 7> ° f c h the entities origin­ ally considered are found to f o r m part of a larger class. I propose to allude shortly to each of these steps. A s regards the first I have little to say, for the unrealised possibilities w i t h which I am concerned are here not con­ spicuous. B u t I do feel that the laws of algebra have re­ ceived far too little attention i n current and past teaching, i n that their interpretation is so exclusively confined to the domain of pure number. A n y ordinary boy or g i r l of 1 5 is able to realise that a + b = b + a and a + (b + c) = a + b + c are true w h e n a, b, c are vectors, and to make simple deductions therefrom, as, for example, the proof of the median properties of a triangle. Such work, even i f only a little t i m e be devoted to it, gives a larger and truer view of algebra as a language w i t h more than one inter­ pretation. A n d i t gives the idea of an algebra relevant to any field of human thought, an idea far more s t i m u l a t i n g and fruitful for the ordinary man or woman than the nar­ rower view of one absolute algebra, w h i c h is too often the only result o f our teaching. B u t , w h e n all is said and done, this first part of the subject only presents itself as the formation on methodical lines of a shorthand language; every step i n the solution of equations, factorisation, or what you w i l l , can be expressed i n words whether the entities be numbers or vectors, and no new methods are involved. B u t now take the second step, the interpretation of algeP q

braic symbols such as x or V — y w h i c h have at first no meaning. T h e process involved is, or should be, purely logical. W e assume that such laws of combination as x x x =x and (x ) = x must also hold i n cases f

m

n

m + n

m n

mn

SOME UNREALISED POSSIBILITIES

57

which already bear interpretation, and then find that the p 9

p

one interpretation x = ~^lx is consistent with each of these laws. I t is too often assumed without proof that, because the one law x x x" = x leads to this interpretation, the other laws, such as (x ) = x , must also be true in this case. I do not believe that complex exercises in the manipulation of fractional and negative indices can be of any profit, but I am convinced that a complete and logi­ cal interpretation of these indices, if only in particular numerical cases, can and should form part of every course in algebra. I t is one of the best examples of constructive logic to be found in elementary mathematics, and it gives a sense of new methods for the discovery of hidden fields of entities which is hardly to be found elsewhere. Passing now to imaginary expressions, I would suggest that the geometrical interpretation of these is not beyond the capacity of pupils of seventeen or eighteen years of age, and, further, that it provides a valuable link between the symbolism thus far developed and geometry of two dimensions. Not much knowledge of trigonometry is re­ quired in order to understand the expressions a + bi, (a + bi) (c + di), nor is it necessary to plunge into useless elaborations. The pupils have ample scope for exercise in written descriptions of the processes ; for example, in show­ ing that this interpretation satisfies the laws r , =