this, in spite of its maxim to teach all processes simultaneously:* first, a process involving all the numerical operations, then, as the power of attention and interpretation ripens, making the process already performed an object of attention to bring out what is involved, is the psychological law.

4. The method which neglects to recognise number as measurement (or definition of the numerical value of a given magnitude), and considers it simply as a plurality of fixed units, necessarily leads to exhausting and meaningless mechanical drill. The psychological account shows that the natural beginning of number is a whole needing measurement; the Grube method (with many other methods in all but name identical with the Grube) says that some one thing is the natural beginning from which we proceed to two things, then to three things, and so on. Two, three, etc., being fixed, it becomes necessary to master each before going on to the next. Unless four is exhaustively mastered, five can not be understood. The conclusion that six months or a year should be spent in studying numbers from 1 to 5, or from 1 to 10, the learner exhausting all the combinations in each lower number before proceeding to the higher, follows quite logically from the premises. Yet no one can deny that, however much it is sought to add interest to this study (by the introduction of various objects, counting eyes, ears, etc., dividing the children into groups, etc.), the process is essentially one

* Which, of course, it never does. It only teaches all of them about one "number" before it goes on to another, each number being an entity in itself-which it ought not to do. This matter of the various operations is discussed in the next chapter.

of mechanical drill. The interest afforded by the objects remains, after all, external and adventitious to the numbers themselves. In the number, as number, there is no variety, but simply the ever-recurring monotony of ringing the changes on one and two and three, etc. Moreover, the appeal is constantly made simply to the memorising power. These combinations are facts to be learned. All the emphasis is laid upon the products, upon the accumulation of the information that 2 plus 2 equal 4, 2×3 +1=7, 18 x 5 = 7, etc.; as a result, the "numbers" remain something external to the mind's own activity; something impressed upon it, and carried by it, not something growing out of its own action and coming to be a normal habit of intrinsic mental working.

Contrast with the True Method.-For the sake of indicating more clearly the defects of this method, let us follow out the contrast with the true or psychological method:

(a) The emphasis is all the time upon the performance of a certain mental process; the product, the particular fact or item of information to be grasped, is simply the outcome of this process. There is a given whole to be counted off into minor wholes; a group of objects to be marked off into sub-groups; a given magnitude of surface to be cut up into equal minor units of surface; a weight to be measured through equalising it with a number of sub-units of weight, etc. Then the number of sub-groups, minor unities or parts, has to be counted up in order to find the numerical value of the original whole. The entire interest is in the actual process of distinguishing the whole into its parts, and

combining the parts so as to make up the value of the whole. Wherever there is a break with the mind's own activity, there the facts or principles learned are external, and interest must be partial and defective. The operation becomes mechanical, and the operator a mere machine; or else it is maintained only by a series of artificial stimulations, which keep the mind in a condition of strain-an effort which has its sole source in the need of covering the gap between the intrinsic mental activity and the abnormal action which is forced upon the mind. Wherever there is intrinsic mental activity there is interest; interest is nothing but the consciousness arising from normal activity.* Besides, this activity of parting and wholing, of measuring off into minor units of value, and summing up these minor units into the one whole, can not be performed without the mind's getting the information needed, that, e. g., 1+1+1+1, or 1+1+2, etc.,= 4.

(b) The appeal, according to the psychological method (number as mode of measurement), is not to memory or memorising, but is a training of attention and judgment; and this training, which forms the habit of definite analysis and synthesis, forms the habit of the rhythmic balancing of parts against one another in a whole, and the habit of the rhythmic or orderly breaking up of a whole into its definite parts. So far as this habit is formed the

* Wherever we have to appeal to external stimulus to make a subject interesting, it indicates, of course, that the activity if left to itself would cease, that the mind would wander or become listless. This means that there is no intrinsic interest, no spontaneous movement, no self-developing energy in the mind. Wherever there is this intrinsic activity, the subject is interesting of necessity and does not have to be made so.

memory will take care of itself. The facts do not need to be seized and carried by sheer effort of memory, but are reproduced, whenever needed, out of the mind's own power. The learning of facts, the preservation and retention of information, is an outcome of the formation of habit, of the attainment of power. The method which neglects the measuring function of number can not possibly lead to a definite habit; it can result only in the ability to remember.

There is no question here about the need of drill, of discipline, in all instruction. But there is every question about the true nature of drill, of discipline. The sole conception of drill and of discipline which can be afforded by the rigid unit method is that of ability to hold the mind fixed upon something external, and of ability to carry facts by sheer force of memory. By the psychological method of treating the unit as means to an end, a basis of measurement, the discipline consists in the orderly and effective direction of power already struggling for expression or utterance. is the drill of a slave to fit him for a task which he himself does not understand, and which he does not care for in itself. The other is the discipline of the free man in fitting him to be an efficient agent in the realization of his own aims.

One

(c) Finally, the fixed unit method deadens interest and mechanizes the mind in not allowing free play to its tendencies to variety, to continual new development. As already said, according to the Grube method, the fact that 2 + 2 = 4 always remains precisely the same, no matter how much its monotony is disguised by permutations with blocks, shoe pegs, pictures of birds, etc.

According to the measuring method, the habit or general direction of action remains the same, but is constantly differentiated through application to new facts. According to the Grube method unity is one thing, and that is the end of it. According to the measuring method unity may be 12 (the dozen oranges as measured by the particular orange, the day as measured by the hour, the foot as measured by the inch, the year as measured by the month, etc.), or it may be 100—e. g., the dollar as measured by the cent.

Instead of relying upon a minute and exhaustive drill in numbers from 1 to 5, allowing next to no spontaneity, severing nearly all connection with the child's actual experience, ruling out all variety as diametrically opposed to its method, it can lay hold of and give free play to any and every interest in a whole which comes up in the child's life. Unity as 12, as a dozen, is likely to be indefinitely more familiar and interesting to a child than 7; the desire to be able to tell time comes to be an internal demand, etc. But the Grube method must rule out 12. Twenty-five as a unity (of money, the quarterdollar), 50 (as the half-dollar), 100 (as the dollar), are continual and lively interests in the child's own activities. Each of these is just as much one as is one eye or one block, and is arithmetically a very much better type of unit than the block by itself, because it is capa ble of definite measurement or rhythmic analysis into sub-units, thus involving division, multiplication, fractions, etc.-operations which are entirely external and irrelevant to the fixed unit.

Some will probably say, "But 100, or even 12, is altogether too complex and difficult a number for a