CHAPTER V

TEACHING PUPILS TO THINK- -CONCLUDED

WE may here glance at the relation which exists between clear thinking and a good memory. An

Clear thinking and a good memory was a dull class.

earnest teacher was recently observed instructing what she said The pupils were taking their first lessons in fractions, and they were progressing very, very slowly. Indeed, after twenty-five minutes of struggle and tension, it was not easy to discover that they had learned much, if anything, concerning the topic being taught. The teacher felt discouraged, and her state of mind was expressed in her tone of voice, her features, and even in her bodily attitudes. She was irritated over what she thought was wilful stupidity. She felt her pupils could grasp the simple relations she was trying to teach them, if only they would make an effort so to do. During the entire period she was chiding them, upbraiding them for their lack of application, and charging them with carelessness and indifference. It

was a disagreeable hour, alike for the pupils, for the teacher, and for the visitor.

One of the phrases which the teacher used most freely in the attempt to quicken the mental processes of her pupils was, "I told you that yesterday; why can't you remember it to-day?" This phrase is heard very frequently in the class-room, and it always comes readily from the tongue of almost any teacher. When one has told a pupil a fact, it would seem that he ought to retain it for a day at least. The teacher can easily retain it himself, and the pupil could do so, "if he was only in earnest about the matter". But is this sound psychology? Can a novice remember any fact as readily as one who is already familiar with it? The very question sounds absurd; and yet it is an entirely reasonable one, considering the attitude of most teachers toward a learner who forgets what has been told him. It is likely to seem so simple to the instructor that he can not easily forgive any pupil who fails to retain it when it is presented to him.

Let us take a concrete instance. A teacher is endeavoring to lead her pupils to discover what is the A concrete instance result of multiplying one-fourth of obscure teaching by one-fifth. They sit on the recitation bench while she talks about multiplying the numerators together for a new numerator, and

the denominators for a new denominator. When she uses the term "numerator" she does not indicate it explicitly, or have her pupils come to the board and indicate it, simply because it is so familiar to her that she thinks by calling attention to it once or twice in a general way her pupils will grasp it correctly. In the same manner, when she uses the term "denominator" she does not make it entirely clear what "denominator" means. It is so perfectly obvious to herself that she thinks it is a waste of time, and even throwing a sop to stupidity, to keep dwelling on it. The inevitable result is that as she talks to her pupils there is confusion in the minds of most of them; and when the lesson is over, no clear, definite impression has been established. How then can they remember what was developed when the original perception was so obscure?

Suppose that instead of merely talking to her pupils about this process, she had caused each one Attacking the prob- of them to work the whole lem in another way thing out for himself, and to describe the operation in his own words, based exactly upon what he had done. Suppose she had taken forty splints, we shall say, or similar objects, and had asked her pupils what was meant by taking onefourth of one-fifth of them. Before this problem could be taken up, the pupils would, of course, have

had experience in finding fractional parts of a unity, or a group of objects. They could now find onefifth of these forty splints, and then they could easily find one-fourth of this one-fifth. Then readily they could determine what part of forty was the group of two splints, which was gained by taking onefourth of one-fifth of forty. Next, they could look at the original statement, and note how one-twentieth could be obtained by simply performing the required operations on the figures themselves.

In order to fix the principle the teacher could give problems like these: Find two-fourths of one-fifth of forty; three-fourths of one-fifth of forty; threefourths of two-fifths of forty; and so on through a large number of processes. The outcome of this work would be that, through having actually carried a statement out into its concrete results, the pupils would become so impressed with it on account of handling objects in executing the relations stated in the problems, that they would be likely to remember their experience. But when they have no experience, except working with mere figures, nothing can induce them to remember a process except incessant repetition.

Perhaps a better way to proceed than to use splints would be to have each pupil draw a circle, and then perform upon it the operations which his

problem requires; and the teacher can propose many problems involving the same principle, which will tend to fix it securely. These operations can be performed on a line, or better still on a square or oblong, where the pupil can see, and especially where he can feel through actual execution, what it means to take a part of a part, or, as we teach it, to multiply a fraction by a fraction. Even with this sort of experience, of course, he may go astray in the first stages of his work, and I, as his teacher, may feel that he is stupid, because the thing seems so familiar to me. But we must not forget that a novice is always likely to make mistakes in dealing with any situation which is new to him. The reason for this is that just exactly the sequence of things necessary to think through any problem accurately has not become established in his mind as a result of constant repetition. Such a sequence has become established in the expert's mind, because he has gone through it so frequently, and he has blazed a trail, as it were, which he can follow without difficulty whenever he starts upon it. But when one is new in any situation, he can not recognize a trail, and he is apt to wander here and there without knowing precisely what is the right direction. The only possible way by which he can discover this right direction is to go over the route frequently with a good guide,