Jtjnb 10, 1887.] 



sciujsrcu. 



571 



of conducting a large class. My own personal 

 pi'edilections are in favor of a Socratic system of 

 teaching, by asking questions, and so drawing 

 out, — educating, — the mind of the pupil. I do 

 not regard it as good to tell the pupil every thing. 

 It is our object to train him to exercise his own 

 powers. A child who is always carried will never 

 learn to walk. But a child who can walk cannot 

 get over a stile, and requires a lift now and then. 

 It is a matter of tact to decide, in any particular 

 case, whether the assistance is to be given or with- 

 held. I do not feel competent to lay down gen- 

 eral rules. With a pupil who can listen and 

 ■speak, understanding these words as I have ex- 

 plained them, there is little difficulty in ascer- 

 taining whether the supposed inability to proceed 

 arises from want of power or from laziness. It 

 very often arises from want of will, not exactly a 

 downright shirking of work, but a certain defi- 

 ciency in determination. In such cases a guiding 

 hand is better than a lift. 



"That method of teaching is best which most 

 stimulates the mental activity of the pupil, and 

 that is the reason why methods after a time cease 

 to be good : it is just because they are methods, 

 and become mechanical, and so fail to stimulate 

 activity. 



" Algebra should be taught as if to an intelligent 

 person. Unfortunately, all the i^upils in a class 

 are not equally intelligent. Still, people turn out 

 very much as you treat them : draw out the germ 

 of intelligence, and it will grow. A style of 

 teaching that is based on the supposition that the 

 class is unintelligent, is apt to end in making them 

 so. To this end no slovenly work should be al- 

 lowed. It is a mistake to look only at the answers 

 of a set of exercises, and not to care about the 

 orderly setting forth of the argument that leads 

 to the answer. This is a practical detail that re- 

 quires some skill to adjust : the mode of adjust- 

 ment depends on the size and character of the 

 class. Too much of the teacher's energy is in 

 danger of being absorbed in examining exercises. 

 The benefit of the exercise consists chiefly in doing 

 it, and in so doing it that it needs no subsequent 

 alteration : consequently the cori'ectness of the 

 answer is a most important point. But a practice 

 of merely looking at the answer allows the boys 

 to fall into slovenly habits, and may lead them 

 into the unsound habit of working up to an an- 

 swer. 



■' Considerable difference of opinion is expressed 

 as to just how the first steps in algebra should be 

 taken. It may be taken by using letters as gen- 

 eral symbols for numbers, treating algebra as a 

 generalized arithmetic ; and there is much to be 

 said in favor of this. In this way algebra pre- 



sents itself as a language, and this is a view of 

 algebra that ought to be put before the student 

 at an early period. Some of the most instructive 

 of the early exercises in algebra consist In trans- 

 lating general arithmetical statements into sym- 

 bolic language, and in forming the equations 

 which are the algebraical statement of problems. 

 Simple equation problems can hardly be begun 

 too early. 



" On the other hand, the notion of the negative 

 number can be acquired without the use of any 

 fresh apparatus of symbols beyond those that the 

 student has been accustomed to in arithmetic : 

 and, as this is one of the gi'eatest of the early dif- 

 ficulties of algebra, I have sometimes thought it 

 wise to begin with it, so that the difficulty of the 

 negative quantity may be mastered without the 

 comi^lication which the use of letters seems to 

 give to the matter. I think myself that it is more 

 logical to begin with the letters, but that it is, on 

 the whole, easier for the student to begin with 

 the negative quantity. To talk about and explain 

 5 — 8 is simpler to a beginner than the use of 

 a and &. 



"But, whatever sequence of the parts of the 

 subject is adopted in teaching, there should be no 

 departure from a logical development. Algebra 

 is built up on certain few axioms, and certain not 

 very numerous conventions, A pupil should be 

 led to see from the first the distinction between 

 what is axiomatic and what is conventional ; 

 though, in the latter case, he may be unable, at 

 an early stage, to see the convenience of the con- 

 vention : he is not a sufficient judge of this, in 

 many cases, till his studies have proceeded much 

 further. But he should be encouraged to see for 

 himself that the propositions of the science are 

 correctly deduced by means of the axioms of 

 which he admits the truth, and no matter should 

 be taught which cannot thus be put before him. 



" The gradual extension of meaning which such 

 a term as 'multiplication' receives — first in 

 arithmetic, when it is extended to a fractional 

 multiplier ; then in algebra, when the multiplier 

 is likely to be negative ; and finally in applied 

 mathematics, when we contemplate a concrete 

 multiplier — is a matter which should form part 

 of the teaching of algebra to all, who should thus 

 be led to see that in mathematics ' impossible ' is 

 a word of only temporary significance. A stu- 

 dent who knows only arithmetic is justified in 

 saying that 5—8 is impossible ; but the impossi- 

 bility is a stile to be gotten over. 



" In looking over exercises, it is often more im- 

 portant to look over those that are wrong than 

 those that are right. When an example has been 

 done right, correct in reasoning and accurate in 



