For numerical ideas involve the simplest forms of this higher process of mental elaboration; they enter into all human activity; they are essential to the proper interpretation of the physical world; they are a necessary condition of man's emancipation from the merely sensuous; they are a powerful instrument in his reaction against his environment; in a word, number and numerical ideas are an indispensable condition of the development of the individual and the progress of the race. It would therefore seem to be contrary to the "beautiful economy of Nature" if the mind had to be forced to the acquisition of that knowledge and power which are essential to individual and racial development; in other words, if the conditions of progress involved other conditions which tended to retard progress.

The position here taken on theoretical grounds, that the normal activity of the mind in constructing number is full of interest, is confirmed by actual experience and observation of the facts in child life. There are but few children who do not at first delight in number. Counting (the fundamental process of arithmetic) is a thing of joy to them. It is the promise and potency of higher things. The one, two, three of the "six-years' darling of a pygmy size" is the expression of a higher energy struggling for complete utterance. It is a proof of his gradual emergence from a merely sensuous state to that higher stage in which he begins to assert his mastery over the physical world. We have seen a firstyear class—the whole class—just out of the kindergarten, become so thoroughly interested in arithmetic under a sympathetic and competent teacher, as to prefer an

exercise in arithmetic to a kindergarten song or a romp

in the playground.

Arrested Development. Since, then, the natural action of the child's mind in gaining his first ideas of number is attended with interest, it seems clear that when under the formal teaching of number that interest, instead of being quickened and strengthened, actually dies out, the method of teaching must be seriously at fault. The method must lack the essentials of true method. It does not stimulate and co-operate with the rhythmic movement of the mind, but rather impedes and probably distorts it. The natural instinct of number, which is present in every one, is not guided by proper methods till effective development is reached. The native aptitude for number is continually baffled, and an artificial activity, opposed to all rational development of numerical ideas, is forced upon the mind. From this irrational process an arrested development of the number function ensues. An actual distaste for number is created; the child is adjudged to have no interest in number and no taste for mathematics; and to nature is ascribed an incapacity which is solely due to irrational instruction. It is perhaps not too much to say that nine tenths of those who dislike arithmetic, or who at least feel that they have no aptitude for mathematics, owe this misfortune to wrong teaching at first; to a method which, instead of working in harmony with the number instinct and so making every stage of development a preparation for the next, actually thwarts the natural movement of the mind, and substitutes for its spontaneous and free activity a forced and mechanical action accompanied with no vital interest, and lead

ing neither to acquired knowledge nor developed power.

Characteristics of this defective method have been frequently pointed out in the preceding pages, and it is unnecessary to notice them here further than to caution the teacher against a few of them, which it is especially necessary to avoid.

Avoid what has been called the "fixed-unit" meth od. No greater mistake can be made than to begin with a single thing and to proceed by aggregating such independent wholes. The method works by fixed and isolated unities towards an undefined limit; that is, it attempts to develop accurate ideas of quantity without the presence of that which is the essence of quantity— namely, the idea of limit. It does not promote, but actually warps, the natural action of the mind in its construction of number; it leaves the fundamental numerical operations meaningless, and fractions a frowning hill of difficulty. No amount of questioning upon one thing in the vain attempt to develop the idea of "one," no amount of drill on two such things or three such things, no amount of artificial analysis on the numbers from one to five, can make good the ineradicable defects of a beginning which actually obstructs the primary mental functions, and all but stifles the number instinct.

Avoid, then, excessive analysis, the necessary consequence of this "rigid unit" method. This analysis, making appeals to an undeveloped power of numerical abstraction, becomes as dull and mechanical and quite as mischievous in its effects as the "figure system," which is considered but little better than a mere jugglery with number symbols.

Avoid the error of assuming that there are exact numerical ideas in the mind as the result of a number of things before the senses. This ignores the fact that number is not a thing, not a property nor a perception of things, but the result of the mind's action in dealing with quantity. Avoid treating numbers as a series of separate and independent entities, each of which is to be thoroughly mastered before the next is taken up. Too much thoroughness in primary number work is as harmful as too little thoroughness in advanced work.

Avoid on the one hand the simultaneous teaching of the fundamental operations, and on the other hand the teaching which fails to recognise their logical and psychological connection.

Avoid the error which makes the "how many' alone constitute number, and leaves out of account the other co-ordinate factor, "how much." The measuring idea must always be prominent in developing number and numerical operations. Without this idea of measurement no clear conception of number can be developed, and the real meaning of the various operations as simply phases in the development of the measuring idea will never be grasped.

Avoid the fallacy of assuming that the child, to know a number, must be able to picture all the numbered units that make up a given quantity.

Avoid the interest-killing monotony of the Grube grind on the three hundred and odd combinations of half a dozen numbers, which thus substitutes sheer mechanical action for the spontaneous activity that simultaneously develops numerical ideas and the power to retain them.

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Rational Method. The defects which have been enumerated as marking the "fixed-unit" method suggest the chief features of the psychological or rational method. This method pursues a diametrically opposite course. It does not introduce one object, then another closely observed" object, and so on, multiplying interesting questions in the attempt to develop the number one from an accurate observation of a single object. It does as Nature prompts the child to do: it begins with a quantity—a group of things which may be measured-and makes school instruction a continuation of the process by which the child has already acquired vague numerical ideas. Under Nature's teaching the child does not attempt to develop the number one by close observation of a single thing, for this observation, however close, will not yield the number one. He develops the idea of one, and all other numerical ideas, through the measuring activity; he counts, and thus measures, apples, oranges, bananas, marbles, and any other things in which he feels some interest. Nature does not set him upon an impossible task-i. e., the getting of an idea under conditions which preclude its acquisition. She does not demand numerical abstraction and generalization when there is nothing before him for this activity to work upon. Let the actual work of the schoolroom, therefore, be consistent with the method under which by Nature's teaching the child has already secured some development of the number activity.

In all psychical activity every stage in the development of an instinct prepares the way for the next stage. The child's number instinct begins to show itself in its