148 



NATURE 



[November 30, 191 1 



trigonometry, including' the dcrmitioiis and nuiiu-rical 

 work, is not begun until the third year. In arithmetic 

 stress is laid on the importance of a grasp of principles, 

 and reference is made to the harm text-books have 

 done in the way of muUiplying rules and settinjj up 

 mere trivial examples as important types. We do not 

 know the average age of entry to the secondary schools 

 of N.S.W., but we doubt whether pupils in their first 

 year are sufficiently mature in mind for such questions 

 as " Retail and Banker's Discount " and " Balance 

 Sheets." In England arithmetic has suffered seriously 

 by the introduction of the technicalities of commerce 

 and the money market at too early an age. The 

 second-year course is to go deeper into such things ; we 

 should have thought it wiser to postpone most of this 

 work to the third or fourth year, and bring in more 

 numerical trigonometry and easy mechanics to take its 

 place — it is intended to do something of the sort in the 

 second year as "simple problems on the lever, wheel 

 and axle, and inclined plane " are included, but there 

 are no suggestions as to the practical work on which 

 to base these problems. 



The section devoted to algebra begins with a philo- 

 sophical treatment of negative and fractional quanti- 

 ties, but it is not proposed that pupils should be taken 

 through such a treatment "in their first steps in 

 algebra " ; but the course suggested for the first two 

 years is what schools in England have been trying to 

 break away from in the last decade. The whole 

 attitude seems to be too abstract for beginners — we 

 should have liked to see the factors of x^ + aS, long 

 H.C.F., and the solution of two simultaneous equa- 

 tions, "in which only terms of the second degree and 

 constant terms occur," all postponed until after the 

 second year, and indices and logarithms brought back 

 in their place. We feel that it would have been wiser 

 to make the early treatment much less abstract and 

 more numerical, and to let the fourth-year course 

 include a more scientific treatment of the elements of 

 the subject. 



Differential calculus is introduced into the fourth 

 year's algebra, but seems to be intended only for those 

 who matriculate in higher mathematics. We should 

 like to see a short introduction to both differential And 

 integral calculus introduced into secondary schools 

 for all pupils of average ability. The treatment sug- 

 gested in this report for the differential is good, 

 including, as it does, the principles of the subject 

 without elaborating the technique — we suspect that 

 integral calculus is meant to be included as reference 

 is made to the evaluation of areas, &c. 



The report goes on to discuss geometry : — 



'■ !• i> common knowledge that within the last few 

 V. ar- the methods adopted in the teaching of geometry 

 have been greatly altered. No less common is the belief 

 that many of the changes which have been made have 

 hardly justified themselves, that the relative importance of 

 the v'arious parts of the subject has been frequently for- 

 gotten, that much unsatisfactory reasoning is being 

 accepted as logical, and that much unnecessary confusion 

 exists. ... A boy who has made a comparatively close 

 acquaintance with straight lines, angles, circles, triangles, 

 parallelograms, &c., by actual drawing and measurement, 

 knows far more of their properties than one who has learnt 

 by heart long lists of definitions and some of Euclid's 

 propositions. And when, by a carefully graduated series 

 of experiments and drawings, he has discovered for him- 

 self the fundamental theorems regarding congruent 

 triangles, the theory of parallels, the measurement of areas, 

 and the circle, he is ready to proceed to the study of 

 deductive geometry, and should profit by that study in 

 many different ways. 



" This is the first, and one of the most important, of 

 the changes which have been made in the teaching of 

 geometry ; but, with regard to it, some words of warning 

 are still necessary. There is no doubt that the rdle of 



NO. 2196, VOL. 88] 



experiment, careful drawing, exact measurement, an 

 calculation from the figures drawn by the pupils has bee;, 

 exaggerated." 



With all this we agree most heartily, but vmiIi 

 Prof. Carslaw's remedy we cannot agree; this prat 

 tically amounts to taking the theorems of Euclid ' 

 the end of parallels, and setting them up as a stnt 

 order to be followed by all schools. Prof. C;.i 

 seems to feel that such a retrograde step is likely ; 

 meet with opposition, for he says : "Without pr« 

 scribing rigid adherence to the scheme drawn up an 

 embodied in its programme, the Department intend 

 to base its instruction upon it." This seems to giv 

 the teacher the choice the famous Hobson gave to hi 

 customers when they came to hire horses. We ct 

 only express our deep regret that such a retrograd 

 step should be deemed necessary in N.S.W. to remed- 

 an evil which is doubtless the same in all its featur. 

 as that we are experiencing in England. 



The evil may be considered to be due to two distinc 

 causes : (i) some teachers have not shown a prop< 

 appreciation of what constitutes a sound logical proo: 

 and have let their pupils use slipshod arguments 

 (ii) pupils have a break in their careers when they pa*- 

 from the elementary to the secondary schools. Wit 

 regard to (i) it may be pointed out that practically n 

 teachers, both in England and Australia, have ha^ 

 their training in geometry under a cast-iron system, 

 viz. that of Euclid ; some teachers under that system 

 have acquired a splendid appreciation of what is 

 logical, but the very existence of the evil referred to 

 proves that other teachers have not acquired that 

 appreciation — this does not point to finding the remedy 

 in another cast-iron system, particularly one closely 

 following the lines of Euclid. 



Doubtless a rigid system would remedy the difficulty ', 

 of transition from school to school, but that difficulty 

 ought never to exist ; for, as Prof. Carslaw points out, 

 the only trouble due to the lack of a standard order 

 lies in the fundamental theorems about congruence, 

 and parallels and the angle-sum for a triangle; arn 

 that part of geometry, in our opinion, should not b- 

 treated deductively before the age of fourteen with aii 

 but pupils of very exceptional mathematical abilii} 

 and possibly not wich them. It will be rememberei 

 that a couple of years ago the Board of Educatio: 

 published a circular recommending that these funda- 

 mental theorems be arrived at by induction, and then 

 taken as a base on which to build up a logical systen 

 of deductive geometry. Prof. Carslaw says : — 



" With this advice I find myself unable to agree. Oiv 

 of my reasons for disagreeing with their method is" th; 

 I am sure the difficulty that these fundamental congruem 

 theorems offer to the pupil is exaggerated, and that i 

 believe the reasoning, by means of which they are to b 

 proved, can be of value to him. Another ground for ni 

 dissent from the plan of that circular is that the treatmen 

 of parallels which it recommends, by its introduction o 

 the idea of direction as fundamental, and by making tlv 

 angle-sum theorem independent of the theor\' of parallel- 

 includes one of those fallacies with which the long histor> 

 of that theory is crowded." 



We wonder whether Prof. Carslaw has seen th< 

 advice of the Board carefully followed by sympathetic 

 teachers; it is generally found that the results o: 

 teaching on these lines are much better than thos- 

 where a deductive treatment of congruence, &c., i- 

 attempted — there is greater knowledge of geometry 

 greater appreciation of logical geometry, and morf 

 power to tackle new work. We must differ from Prof 

 Carslaw when he says the difficulty of the congruenr* 

 theorems is exaggerated — in a comparatively wid- 

 experience of boys, we have never found the proofs o: 

 these theorems clearlv understood until the boy ha- 



