relating parts in a definite whole. On the other hand, there is altogether too much definition, definition carried to the point of isolation, when, in number teaching, a start is made with one thing-endless changes being rung with single objects in order "to develop the number one "-then another object is introduced, then another, and so on. Here the preliminary activity that resolves a whole into parts is omitted, as well as the connecting link that makes a whole of all the parts.

2. Relation or Rational Counting.-This involves the putting of units (parts) in a certain ordered relation to one another, as well as marking them off or discriminating them. If, when the child discriminates one thing from another, he loses sight of the identity, the link which connects them, he gains no idea of a group, and hence there is no counting. There is, to him, simply a lot of unrelated things. When we reach "two" in counting, we must still keep in mind "one"; if we do not we have not "two," but merely another one. Two things may be before us, and the word "two" may be uttered but the concept two is absent. The concept two involves the act of putting together and holding together the two discriminated ones. It is this tension between opposites which is largely the basis of the childish delight in counting. Number is a continued paradox, a continued reconciliation of contradictions. If two things are simply fused in each other, forming a sort of vague oneness, or if they are simply kept apart from each other, there is no counting, no "two." It is the correlative differentiation and identification, the holding apart and at the same time bringing

together, which imparts to the operation of counting its fascination. This activity is simply the normal exercise of what are always the fundamental rational functions; and thus it gives to the child the same sense of power, of ease and mastery in mental movement, that an adult may realize from some magnificent generalization through which a vast, disorderly field of experience is reduced to unity and system. In the simple one, two, three, four of the child, as he counts the familiar objects around him, there is presented the form of the highest operations of discrimination and identification.

EDUCATIONAL SUMMARY.-The idea of number is not impressed upon the mind by objects even when these are presented under the most favourable circumstances. Number is a product of the way in which the mind deals with objects in the operation of making a vague whole definite. This operation involves (a) discrimination or the recognition of the objects as distinct individuals (units); (b) generalization, this latter activity involving two subprocesses; (1) abstraction, the neglecting of all characteristic qualities save just enough to limit each object as one; and (2) grouping, the gathering together the like objects (units) into a whole or class, the sum. Hence :

1. Number can not be taught by the mere presentation of things, but only by such presentation as will stimulate and aid the mental movement of discriminating, abstracting, and grouping which leads to definite

numerical ideas.

2. In this process there must be sufficient qualitative difference among the objects used to facilitate the recognition of individuals as distinct, but not enough to resist

the power of grouping all the individuals, of grasping them as parts of one whole or sum.

The application of this principle will depend largely upon circumstances (sensory aptitudes, etc.) and the tact of the teacher. In some cases it may be well at the outset to use differently coloured cubes, the different colours serving to individualize each object or group of objects as a unit, while the common cubical quality facilitates relation. In other cases the difference in colour might divert attention from the relating process, and hinder the grasping of the different units as one sum; the mere difference of position in space would be enough for the necessary discrimination.

3. In any case the aim must be to enable the pupil to get along with the minimum of actual sense difference, and thus further the power of mathematical abstraction and relation. For discrimination must operate just enough for the recognition of the individuality or singleness of each object or part, and no further. The end is the facile recognition of groups as groups, the individuals, the single, component parts being considered not for their own sake, but simply as giving definite value to the group. That is to say, the recognition, for instance, of three, or four, or five, must be as nearly as possible an intuition; a perception of the parts in the whole or a whole of parts, and not a conscious recognition of each part by itself, and then a conscious uniting it to other parts separately recognised.

4. It is clear that to promote the natural action of the mind in constructing number, the starting point should be not a single thing or an unmeasured whole, but a group of things or a measured whole. Attention

fixed upon a single unmeasured object will discriminate and unify the qualities which make the thing a qualitative whole, but can not discriminate and relate the parts which make the thing a definite quantitative whole. It is equally clear that with groups of things the movement in numerical abstracting and relating may be greatly assisted by the arrangement of the things in analytical forms, as is the case, e. g., with the points on dominoes.

CHAPTER III.

THE ORIGIN OF NUMBER: DEPENDENCE OF NUMBER ON MEASUREMENT, AND OF MEASUREMENT ON ADJUSTMENT OF ACTIVITY.

ADMITTING, then, the psychical nature of number, we are now prepared to deal with its psychological origin. It does not arise, as we have seen, from mere sense perception, but from certain rational processes in construing, in defining and relating the material of sense perception. But we are not to suppose that these processes-numerical abstraction and generalizationaccount for themselves. They give rise to number, but there is some reason why we perform them. This reason we must now discover, for it lies at the root of the problem of the origin of number.

THE IDEA OF LIMIT.-If every human being could use at his pleasure all the land he wanted, it is probable that no one would ever measure land with mathematical exactness. There might be, of course-Crusoe-likea crude estimate of the quantity required for a given purpose; but there would be no definite numerical valuation in acres, rods, yards, feet. There would be no need for such accuracy. If food could be had without trouble or care, and in sufficiency for everybody, we should never put our berries in quart measures, count