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,

=

who will make him take account of every possible circumstance and condition, and work his own way as far as possible, and then he will be likely to appreciate the circumstances and conditions when he comes that way again. In other words, he will the better remember what has been taught him.

Every one will doubtless agree that it is of vital importance that a pupil should acquire the habit of An illustration thinking clearly, and as far as posfrom geography

sible profoundly, regarding the world of people and objects with which he must come into relations, either immediately or remotely. The study of geography should develop in pupils this ability to think effectively in respect to certain aspects of the world about them. To illustrate prevalent methods of presenting this subject in the schools, we may glance at the plan pursued in a series of lessons recently observed in a fifth-grade class. On the first day the lesson related to the elevation of the land masses out of the water, and the action of various agents, atmospheric and otherwise, upon rock formations, leading to the disintegration of the surface of the rocks. This brought up a discussion of erosion, and the making and movement of detritus. The children in the class averaged about nine years of age. They had had but little previous work in geography. All they had gained was ac

quired from an elementary text-book, which they had learned memoriter. They were now studying the advanced geography, which treated the subject in a logical, technical way, using terms which many adults even could not readily pronounce, to say nothing about understanding them. The writer, after listening to this recitation, asked several intelligent grown people what detritus was, and they all threw up their hands in utter helplessness. There were in this lesson a number of such expressions as atmospheric agents and the like, which the majority of the children could not pronounce, even after some instructions by the teacher.

The pupils had been required to "study" the lesson at their seats. When they came to the recitation, it was evident that the majority of them had not assimilated so much as one clear idea from their struggle with the text. The whole thing, content as well as terminology, was quite beyond them, largely, though not wholly, on account of the technical character of the terms used, and the general abstract treatment of the subject. The man who had prepared the geography stated his facts and principles in the briefest way he could, viewing them through his adult experience. His statements were, of course, intelligible to himself, and he naïvely inferred that a child in the fifth grade ought to under