3i8 



NA TURE 



[August 2, 1900 



I have studied these phenomena very carefully, and 

 I affirm that they are directly traceable to the absurd 

 thing called mathematical teaching in schools and 

 colleges. 



The framers of educational methods took in their 

 youth to abstract reasoning as a duck takes to water, 

 and of course they assume that a boy who cannot in one 

 year understand a little Euclid must be stupid. In truth, 

 it is a very exceptional mind, and not, perhaps, a very 

 healthy mind, which can learn things or train itself through 

 abstract reasoning ; nor, indeed, is much ever learnt in 

 this way. Do we philosophise about swimming before 

 we know how to swim ; or about walking or jumping, or 

 cycling or riding a horse, or planing wood, or chipping 

 or filing metals, or about playing billiards or cricket ? 

 Is it through philosophy that we learn a game of cards, 

 or to read or to write ? No ; we first learn by actual 

 trial ; we practice as our mind lets us ; we philosophise 

 afterwards— perhaps long afterwards. Then if we are 

 too clever or stupid, we insist on teaching a pupil from 

 the point of view which we have at the end of our 

 studies, and we refuse to look at things from the pupil's 

 point of view. 



What a natural but ghastly statement the boy made 

 who said: "Yes, Euclid and Xenophon, the beasts, 

 wrote books for the third and fourth forms " ! It is even 

 a ghastlier notion that the jokesomeness of a philosopher, 

 the unessential fringe of a subject, often becomes the 

 soul-destroying, weary, worrying study of a schoolboy. 



In a short article I shall not attempt to put forward my 

 views as to how mathematics ought to be taught ; I have 

 published some of them in a summary of lectures on 

 "Practical Mathematics," published by the Science and 

 Art Department, and in my " Calculus for Engineers." 



We let a Board School boy jump over all the ancient 

 philosophy of arithmetic with its twenty-seven independent 

 Greek characters (for our ten figures), the study of which 

 required a lifetime, so that only old men could do multi- 

 plication, and they not only needed many hours to do one 

 easy bit of multiplication, but declared that if the art 

 were not practised every day it could not be remembered. 

 Why not also let a boy jump over all the Euclidian 

 philosophy of geometry, and assume even the forty- 

 seventh proposition of the first book of Euclid to be 

 true ? Why not let him replace the second and fifth 

 books of Euclid by a page of simple algebra, and give 

 him much of the sixth book as axiomatic ? If you must 

 insist on abstract reasoning, you had better remember 

 that nothing is really axiomatic ; but any well-established 

 truths may be looked upon as fundamental or axiomatic, 

 and a system of abstract reasoning maybe founded upon 

 them. At present, a man at Cambridge finishes just 

 where the really interesting and useful part of mathe- 

 matics begins. There would not indeed be much diffi- 

 culty in framing a course in which he would begin by 

 studies where the studies of good mathematicians now 

 end. This has been tried and proved successful. The 

 present rules of the game are really a little too absurd. 

 A difficult vector subject like geometry must be studied 

 before algebra. Simple exercises on squared paper, well 

 within the capacity of even illiterate persons, must not 

 be approached until one has wasted years on higher 

 algebra and trigonometry and geometrical conies, because 

 they belong to the subject of co-ordinate geometry. It 

 is assumed that it is not until after co-ordinate geometry 

 is thoroughly studied that a man can take in the idea 

 which underlies the calculus, an idea which is possessed 

 by every young boy with absolute accuracy, and by every 

 healthy mind. 



Some friends of mine assert that no boy or man 

 ought to be allowed to use logarithms until he knows 

 how to calculate logarithms. They say this, knowing 

 that the calculation is a branch of what is called higher 

 mathematics, and that the average schoolboy, after six 



NO. 1605, VOL. 62] 



years at mathematics, finds it hopeless to even begin the 

 study of the exponential theorem. It is a hard saying I 

 It is exactly like saying that a boy must not wear a 

 watch or a pair of trousers until he is able to make a 

 watch or a pair of trousers. I am an advocate for the 

 use by all students of all appliances which may be useful 

 to them, whether made by tailors, or watchmakers, or 

 instrument makers, or builders, or pure mathematicians. 

 We need not believe a craftsman when he tells us that 

 we cannot utilise his results without practising his trade. 

 Nevertheless, it is good to be able to do some things for 

 one's self, such as sewing on buttons, or using the lathe or 

 a blowpipe, or the development of a little mathematics. 

 If readers will refer to the above-mentioned summa?-/ 

 they will see that I consider a good system of mathe- 

 rnatics teaching of fundamental importance in the educa- 

 tion of all men. 



I must not dwell any longer on the imperfections of the 

 existing system, but I hope that even readers who do not 

 quite agree with me that much of the sixth book of 

 Euclid ought to be regarded as axiomatic, will agree 

 that what we usually call arithmetic is useless. For 

 races not troubled with our abominable system of 

 weights and measures, the whole of arithmetic consists 

 merely of multiplication and division. To them a 

 decimal is no more difficult to understand than an 

 ordinary number. It is supposed that an English boy 

 understands at once the meaning of 4,590,000 or 4590 

 or 459, but that such a number as 45*9 or "459 or '00459 

 is beyond his comprehension. I say that this is a 

 difficulty artificially maintained by our stupid methods of 

 teaching. Like the rest of our stupid methods, it is due 

 to our unscientific ways of thinking. Because the embryo 

 passes through all the stages of development of its 

 ancestors, a boy of the nineteenth century must be 

 taught according to all the systems ever in use and in 

 the same order of time. The decimal system of stating 

 numbers is 700 years old in Europe, but it was not till 

 280 years ago that Napier invented the use of decimals 

 and the decimal point. Think of compelling all emigrants 

 to pass to America through Cuba, because Cuba was dis- 

 covered first. Think of making boys learn Latin and 

 Greek before they can write English, because Latin and 

 Greek were the only languages in which there was a 

 literature known to Englishmen 450 years ago. 



Again, the ingenious teachers of last century incor- 

 porated every kind of arithmetical example in a book 

 and called each kind by a slang name— practice, 

 interest, discount, tare and tret, alligation, position, &c., 

 and we must teach exactly as they did. I do not mind 

 retaining the buttons at the back of my coat ; many 

 useless ancient ceremonies may still be practised, and I 

 shall not object. I can even admire them, but the un- 

 scientific waste of the valuable youth of millions of our 

 people, now that we are face to face with nations who 

 are determined to destroy England through commerce 

 and war, is so abhorent to me that I cannot think of it 

 with patience. It is not merely in arithmetic and 

 geometry, but in the higher parts of mathematics that 

 this waste goes on. Newton employed geometrical 

 conies in his astronomical studies, and mechanics was 

 developed ; and therefore it is that every young engineer 

 must study mechanics through astronomy, and he dare 

 not think of the differential calculus till he has finished 

 geometrical conies. The young applier of physics, the 

 engineer, needs a teaching of mathematics which will 

 make his mathematical knowledge part of his mental 

 machinery, which he shall use as readily and certainly as 

 a bird uses its wings ; and we teach him in such a way 

 that he hates the sight of a mathematical symbol all his 

 life after. 



It is just as in classics. Ask the average man if he 

 ever reads anything now in Latin or Greek ; ask him 

 about anything to which he devoted ten years of his 



