570 



SCIEN'CE. 



LVOL. IX., No. 227 



fied with a single word as an answer, and who 

 habitually put their questions so as to admit of 

 such an answer. This does not encourage the art 

 of speaWng in the pupil ; in fact, it destroys it, 

 and is not to be commended. 



" A third preliminary is the power of reading : 

 this is far more difiScult, and far more usually ab- 

 sent, than the preceding. Many a boy who can 

 listen and speak has no idea of reading. He can, 

 it is true, form the sounds approj)riate to the 

 words he sees, but he has not the habit of using a 

 laook as a mine of information, of reading in order 

 to get the sense : his main idea too frequently is 

 that of learning the sound of the words, like a 

 parrot. 



"There are few more valuable lessons that can 

 be given to a boy than to teach him to read a book, 

 and extract the sense out of it. This is what young 

 children naturally do with their fairy-tales; but 

 when they become school-boys and school-girls, 

 their natural reading seems somehow to give place 

 to a mechanical lesson-reading. 



" Now, mathematical reading differs from most 

 other reading in this : that it requires writing. 

 This is the fourth essential preliminary. It is pos- 

 sible, no doubt, for a great genius to carry on all 

 the steps of a piece of algebraical reasoning in his 

 head. The ordinary school-boy cannot do this, 

 •cannot pass from one statement of the book to the 

 next without inserting an intermediate step. The 

 boy who has learnt to write, who always, while 

 leading, has a piece of paper and pencil at hand 

 to work out details as they arise, will learn alge- 

 bra : the one who tries merely Co remember the 

 words and symbols of the book will make no real 

 progress. 



"These preliminaries of listening, speaking, 

 reading, writing, do not properly come under the 

 head of teaching algebra : they are so obviously 

 essential, that I scarcely need have mentioned 

 them, but in so many cases absent, that I implore 

 those who have the early training of children not 

 to lose sight of them in the vain hope that with- 

 out them any progress in higher studies is pos- 

 sible. 



"Another essential preliminary more distinctly 

 bears on the subject. The teaching of algebra 

 must be based on, and naturally arises out of, a 

 sound knowledge of the principles of arithmetic. 

 In return, the knowledge of algebra will enable a 

 student concisely to express these principles, and 

 to understand them more clearly. On this ac- 

 count, it is necessary that those who undertake 

 the teaching of arithmetic should have a sufficient 

 knowledge of algebra. This is another lower rea- 

 ■son for studying algebra ; namely, in order to be 

 able to teach arithmetic. 



" It is a mistake to teach a pupil any thing that 

 he has subsequently to unlearn ; the persistence of 

 first impressions is notorious, therefore arithmetic 

 should not be taught in such a way that it needs 

 correction when algebra is studied. The two are 

 naturally and historically connected ; and one who 

 is wholly ignorant of either is apt, also, to be un- 

 familiar with the other. The teacher should be 

 above his subject, not in the sense of despising it, 

 but as one who looks from a height upon a plain 

 can see the topography of the country more dis- 

 tinctly than one on the lower land. 



"Therefore, in the interest of algebra, I protest 

 against the practice of despising arithmetic, of 

 setting it to be taught in schools by persons ig- 

 norant of algebra, and, it may be, contemptuous 

 of the subject they have to teach. A teacher of 

 algebra ought to find the ground prepared for him 

 by a sound knowledge of arithmetic ; and it would 

 be better, therefore, that the mathematical masters 

 should undertake arithmetic. 



" This leads to the next question, Who are to 

 teach algebra ? It may, perhaps, be thought by 

 some that a teacher requires to be very little ahead 

 of his pupil, and that one who has slight knowl- 

 edge is good enough to teach a beginner. On the 

 contrary^ the proper teaching of the elements of 

 any subject requires a teacher who has a knowl- 

 edge considerably in advance. I do not hesitate 

 to say that it would be well that a teacher of al- 

 gebra should know something — and that some- 

 thing soundly — of the method of co-ordinate 

 geometry, of trigonometry, and of the differen- 

 tial calculus. Teaching should be anticipatory. 

 The algebra taught should be such as to prepare 

 for these higher subjects, and this can only be 

 effectually done by one who is acquainted with 

 them. Moreover, the elementary teaching re- 

 quires more care and more knowledge than more 

 advanced. Nothing is worse than to lay founda- 

 tions imperfectly. A necessary qualification for 

 a teacher of algebra is, therefore, a sound knowl- 

 edge of mathematics considerably in advance of 

 the subject he is teaching. 



"Next let us ask. How is algebra to be taught? 

 It is fashionable nowadays, and I do not say it is 

 a bad fashion, to attach importance to the train- 

 ing of teachers in methods of teaching. But I 

 think too much importance can be attached to 

 method. Methods that seem good, and are good 

 when first introduced, seem to lose their virtue 

 after a few years. An energetic teacher will be 

 constantly changing his methods, and adapting 

 them to the various characters of his pupils. 

 Freshness and vigor are far more important quali- 

 ties. Nevertheless, an unmethodical teachei', who 

 would do very well for a single pupil, is incapable 



