factors of product, correlated processes of division, are all seen in their true logical and psychological interrelation. These things are organically connected by necessary laws of thought; the method which is rationalized by this idea makes arithmetic a delight to the pupil and a powerful educating instrument. A method which violates this necessary law of mind in dealing with quantity-constantly obstructing the original action of the mind-makes arithmetic a thing of rule and routine, uninteresting to the teacher, and probably detested by the learner.

Make Haste slowly.-As already suggested, time is necessary for the completion of the idea of number. Under sound instruction a working conception suitable for a primary stage of development may be readily acquired, and may be used for higher development. But a perfectly clear and definite conception of number is a product of growth by slow degrees. The power of numerical abstraction and generalization can not be imparted at will even by the most painstaking teacher. Hence the absurdity of making minute mechanical analysis a substitute for nature's sure but patient way. It seems to be thought that mechanical drill upon a few numbers a drill which, if rational, would really use ratio and proportion-will in some unexplained way "impart" the idea of number to the child apart from the self-activity of which alone it is the product. And this fallacious idea is strengthened by the fluent chatter of the child-the apt repeater of mere sense factsabout the "equal numbers in a number," the "equal numbers that make a number," and all the rest of it. This routine analysis and parrotlike expression of it are

a direct violation of the psychology of number. The idea of number can not be got from this forcing process. The conscious grasp of the idea, we repeat, must come from rational use of the idea, and is all but impossible by monotonous analyses with a few "simple" numbers; it is absolutely impossible in the immature state of the child-mind in which it is attempted. The child must, once more, freely and rationally use the ideas; must operate with many things, using many numbers, before the idea of number can possibly be developed. We are omitting the things the child can do-rationally use numerical ideas-and forcing upon him things that he can not do-form at once a complete conception of number and numerical relations. It is high time to change all this: to omit the things he can not do, and interest him in the things he can do. In the comparatively formal and mechanical stage there must be a certain amount of mechanical drill-mechanical, yet in no small degree disciplinary, because it works with ideas which, though imperfect, are adequate to the stage of development attained, and through rational use become in due time accurate scientific conceptions. Besides valuable discipline, the child gets possession of facts and principles of elementary knowledge, it may be said—which are essential in his progress towards scientific concepts and organized knowledge. It seems absurd, or worse than absurd, to insist on thoroughness, on perfect number concepts, at a time when perfection is impossible, and to ignore the conditions under which alone perfect concepts, can arise the wise working with imperfect ideas till in good time, under the law connecting idea and action, facile doing may result in per

fect knowing. Following the nonpsychological method hinders the natural action of the mind, and fails to prepare the child for subsequent and higher work in arithmetic. The rational method, promoting the natural action of the mind by constructive processes which use number, leads surely and economically to clear and definite ideas of number, and thoroughly prepares for real and rapid progress in the higher work.

The Starting Point.-It is commonly assumed that the child is familiar with a few of the smaller numbers —with at least the number three. He has undoubtedly acquired some vague ideas of number, because he has been acting under the number instinct; he has been counting and measuring. But he does not, because he can not, know the number. He knows 3 things, and 5 or more things, when he sees them; he knows that 5 apples are more than 3 apples, and 3 apples less than 5 apples. But he does not know three in the mathematical or psychological sense as denoting measurement of quantity-the repetition of a unit of measure to equal or make up a magnitude-the ratio of the magnitude to the unit of measure. If he does know the number 3, in the strict sense, it is positively cruel to keep him drilling for months and months upon the number five and "all that can be done with it," and years upon the number twenty.

The Number Two.-There can be little doubt that among the early and imperfect ideas of number the idea of two is first to appear. From the first vague feeling of a this and a that through all stages of growth to the complete mathematical idea of two, his sense experiences are rich in twos: two eyes, two ears, two

hands, this side and that, up and down, right and left, etc. The whole structure of things, so to speak, seems to abound in twos. But it is not to be supposed that this common experience has given him the number two as expressing order or relation of measuring units. The two things which he knows are qualitative ones, not units. Two is not recognised as expressing the same relation, however the units may vary in quality or magnitude; it is not yet one + one, or one taken two times -two apples, two 5-apples, two 10-apples, two 100-apples; or two 1-inch, two 5-inch, two 10-inch, two 100-inch units; in short, one unit of measure of any quantity and any value taken two times. But his large experience with pairs of things, and the imperfect idea of two that necessarily comes first, prepare him for the ready use of the idea, and the comparatively easy development of it. There must be a test of how far, or to what extent, he knows the number two. This is supplied by constructive exercises with things in which the idea of two is prominent. The child separates a lot of beans (say 8) into two equal parts, and names the number of the parts two; separates each part into two equal parts, and names the number of the parts two; separates each of these parts into two equal parts, and names the number of the single things two. Or, arranging in perceptive forms, how many ones in ? How many twos in ? How many pairs of twos in

? Similar exercises

and questions may be given with splints formed into two squares, and into two groups of two pickets each; with 12 splints formed into two squares with diagonals (see page 106); then each square (group) formed into

two triangles; how many squares? how many triangles (unit groups)? how many pairs of triangles? Exact measurements are to accompany such exercises: the 12-inch length measured by the 6-inch, this again by the 3-inch; how many 6-inch units in the whole? 3-inch units in the 6-inch units? pairs of 3-inch units in the whole? Put 1-inch units together to make the 2-inch unit, the 2 inch units to make the 4-inch unit, two of these to make 8 inches, etc. It is not meant that these exercises are to be continued till the number two is thoroughly mastered; they carry with them notions of higher numbers, without which a conception of two can not be reached. Beware of "thoroughness" at a stage when thoroughness (in the sense of complete mastery) is impossible.

The Number Three.-The number three is a much more difficult idea for the child. As in the case of two, he knows three objects as more than two and less than four; three units in exact measurement as more than two, etc. But he does not know three in the strictly numerical sense. He may know two fairly well—as a working notion-without having a clear idea of the ordered or related ones making a whole. In three, the ordering or relating idea must be consciously present. It is not enough to see the three discriminated ones; they must at the same time be related, unified-a first one, a second one, a third one, three ones, a one of three. Three must be three units-measuring parts of a qualitative whole-units made up, it may be, of 2, or 3, or 4. . . or n minor units. If 12 objects are counted off in unit-groups of 4 each, 15 objects into unit-groups of 5 each, 18 objects into unit-groups of 6 each, 30 ob