398 



NATURE 



[January i8, 



thft lines of the rulen that hnve b«en tauffht. and the«« 

 problems tan b« efTectunlly dealt with only by p«Ti»onii who 

 possoKs «tomr real fjrasp of muthpmatical principles, a« 

 distinct from a mere knowledg** of certain practical rules 

 and methods. Whilst mnintiiinin(< that o student should 

 thoroughly understand thi- grounds upon which the formula- 

 and ruK's which he employs are based. I do not believe 

 that he ought to be expected to commit to memory, and 

 to be able to reproduce at any time, formal proofs of all 

 such formuhc and rules. Much precious time and '•htcv 

 has been unprofitably employed in the past in .n 

 to satisfy the unreasonable demands made by ■ 

 in some branches of mathematics that formal prtw.is vninnii 

 be forthcoming of everything that the candidates are sup- 

 posed to have learned. The burden thus thrown on the 

 memories of the candidates is far too heavy, and much 

 time and energy which should have been employed in an 

 endeavour to grasp and realise principles has thus been 

 diverted to a far less profitable use. 



It appears to me to be eminently desirable that the Hme 

 saved by the diminution, in school work, of the amount 

 of time spent on unessential details and on unnecessarily 

 prolonged drill in the manipulation of symbols should be 

 employed in introducing the pupils to a considerably greater 

 range of mathematical thinking than has hitherto been 

 usual, and in particular in endeavouring to make them 

 acquainted with more of the fundamental and fruitful ideas 

 which make mathematical science what it is. In the 

 higher classes some time might profitably be spent on the 

 principles, as distinct from the practice, of arithmetic. It 

 would be of great educational advantage if the principles 

 which underlie the practice with which all the pupils have 

 become familiar were brought explicitly to their conscious- 

 ness. For example, they should understand the principle 

 of our arithmetic notation, so that they may have an 

 adequate appreciation of its beautiful simplicity, and of the 

 fact that it embodies a great time-saving invention. In 

 order to attain this object it is necessary to deal with the 

 theory of scales of notation and radix-fractions, so that the 

 arbitrary element involved in the adoption of the scale 

 of ten may be clearly appreciated. I do not, of course, 

 contemplate the introduction into such a course of artificial 

 problems on scales of notation ; only the fundamental prin- 

 ciples should be explained, with such quite simple illustra- 

 tions as may be found necessary for their complete 

 elucidation. 



I do not know to what extent some rudimentary and 

 informal treatment of the properties of simple figures in 

 three-dimensional space has at the present time become part 

 of the normal instruction in geometry in our .schools. I 

 am quite sure of the urgent necessity for finding time for 

 a small modicum of study of this part of geometry. I 

 remember, a few years ago, in a paper on mathematics 

 for candidates for a college scholarship in physics, the 

 candidates were asked to construct the shortest distance 

 between two given non-intersecting straight lines. One of 

 the candidates, who showed a considerable knowledge of 

 plane geometry, informed me that two non-intersecting 

 straight lines are necessarily parallels. It is unnecessary 

 to insist upon the importance of an endeavour to uproot 

 ignorance of this kind, due as it is to lack of stimulation 

 of the power of observing simple spatial properties. 



In considering the yariou" directions in which mathe- 

 matical teaching may be made to extend beyond the domain 

 that consists of drill in the employment of processes which 

 up to a certain point is undoubtedly necessary, one ques- 

 tion of great importance arises — that is the very 

 important question as to the possibility of making a rudi- 

 mentary treatment of the ideas and processes of the 

 calculus part of the normal course of mathematics in the 

 higher classes of schools. In the hands of a really skilful 

 teacher, the purely formal element in the treatment of the 

 calculus could be reduced to very small dimensions — all 

 the leading notions and processes could be sufficiently illus- 

 trated by means of functions of the very simplest types. 

 I believe that some of the time saved by lightening the 

 matter in such subjects as algebra might be more profit- 

 ably employed in this manner than in any other. The 

 calculus, as embodying and utilising the fundamental 

 notion of a " limit," is the gate to a mathematical world 

 of incomparably gri>ater dimensions than the on.- in which 



NO. 2203 V)!. 88] 



the student ha« moved during the earlier part of hi« courae. 

 Any method of presentment which evade« the notion of a 

 *' limit," a» it appear* in the differential coefficient or lit 

 kinematics as a velocity," is much to be •' ' 



The poaseasion of this notion i« the nujst vali 

 Qf .1... ....^.- 1.^..!, f..r^ '-'iMrational and for •• 



p<. ly chosen exani; 



ar:t 11' domains, a y\ . 



up to this (undamviital notion, so that it nuty in tiu- ed' 



become really his own. To this end it i^t wholly m 



necessary that any treatment of tlv ' ' 



employed which would satisfy the i 



fessional mathematician. The import....; ^....,1 



tion with this idea, as with many others, i 



student should really have the notion as part • 



manent mental furniture, and not that he shoui.i l»- abi 



to give a complete description of it, or of its philo*4.|>hv, i 



conceptual language. I do not propose to in<i 



even in outline, a schedule of those parts of 1 



which would be suitable as part of a grr- - ' 



This is a matter which might with mu< 



fully discussed by the association, when tlv 



tical teachers as to the possibilities in this direction woui' 



receive the fullest attention. 



There is a danger which arises in connection with th 

 democratisation of education that less than justice mav t 

 done to the minority who, by natural aptitude, are rnpa^N 

 of making much more rapid progress than the rar'. .; <i 

 file. The danger is probably not so great in oui 

 country as in some others ; with us, the old leaven 

 impels teachers to make the most of their more 

 pupils still works strongly enough, and the questiun.jb. 

 stimulus provided by scholarship examinations and oth' 

 competitions exercises an influence in the same di: 

 which is very powerful, and perhaps, indeed, too p<. 

 In some countries the rigid system by which every p„,... 

 a school is taken in a general class in a certain numb, 

 of years through prescribed portions of a subject act- 

 detrimentally upon those pupils who are capable of learn- 

 ing much more rapidly than the average. In America I 

 was told that it would be regarded as undemocratic t 

 make any special provision in a school for the more rap 

 advance of gifted pupils. This view seems about ;. 

 reasonable as it would be to prescribe, as a thoroughl 

 democratic arrangement, that all the pupils should t 

 supplied with boots of the same size. The general gcx 

 demands that, so far as possible, equality of opportunii 

 should be afforded to all for their mental development i 

 accordance with their enormously varying abilities; 

 does not demand a mechanical equality of treatment, r^ 

 presented by forcing all students to move at the pace > 

 the less gifted or of the average. .Although, however, th 

 danger may be a real one in some quarters in this countr\ 

 the opposite fault, of sacrificing to some extent the nee<l 

 of the average to those of the abler students, is probab! 

 still the more prevalent one. 



The movement which I have spoken of as tli 

 democratisation of mathematical education is a progVessiv 

 development. Something not inconsiderable has be. 

 accomplished in our time; very much more remains to t- 

 done. The difficulties which arise in this connection ar 

 largely those of finding the true coordination between tli 

 practical and the theoretical sides of the subject. An undu 

 emphasis placed on either side is apt to have di>.' " 

 results. The perfect mean is in all such cases pr 

 an unattainable ideal: a certain degree of compi ... 

 depending upon a variet>' of circumstances, is usually ti 

 practicable course; but the most earnest endeavours shoul 

 be made to prevent such compromise going too far. ^^'hil- 

 recognising to the full the importance of the practical si<l 

 of mathematics, both as affording the right approach t 

 the subject, in view of sound psychological principles, an 

 also on account of its importance as an equipment fc 

 various departments of practical life, let us never lo- 

 sight of the paramount importance of mathematics as pari 

 of a real education of the intellect. Such education is in- 

 complete unless a few, at least, of the many illuminating 

 notions which our race has achieved in its long struggle 

 to attain clearness in the domain of mathematical think- 

 ing aiP made the common property of our iiilcUtu-tuat 

 (l.'Diocracv. 



