on paper, the candy may so absorb attention that the two dots do not present themselves in consciousness at all. This is undoubtedly one reason why the mathematical attainments of savages are so meagre; they are so given up to one absorbing thing-which is to them what the candy is to the child-that the rest of the universe, however much it may affect their senses, does not become an object of attention.

Or, again, the child may be conscious of the dots as well as of the candy, and yet not be able to recognise that these various objects are connected or make one whole. The qualitative unlikeness of the objects may be so great as to make it difficult or even impossible for the child's mind to relate them, to view them all from a common standpoint as forming one group. The candy is one thing and the dots are another and entirely different thing. Here, again, rational counting is out of the question.

Nor, finally, is it to be concluded that from the mere presentation of three like objects the idea of three will be secured. There must be enough qualitative unlikeness-if only of position in space or sequence in timeto mark off the individual objects, to keep them from fusing or running into one vague whole. Part of the difficulty of performing the abstraction which is required to get the idea of number is, accordingly, that this abstraction is complex, involving two factors: the difference which makes the individuality of each object must be noted, and yet the different individuals must be grasped as one whole-a sum. It requires, then, considerable power of intellectual abstraction even to count three. Unlike objects, in spite of differences in

quality, must be recognised as forming one group; while a group of like objects, in spite of their similarities in quality, is to be recognised as made up of separate things. Three differently coloured cubes, for example, must be apprehended as one group, while a group of three cubes exactly alike must be apprehended as three individuals. In other words, the objects counted, whatever be their physical resemblances or differences, are numerically alike in this: they are parts of one whole they are units constituting a defined unity.

The delight which a child four or five years old often manifests in the apparently mechanical operation of counting chairs, books, slate-marks, playthings, or even in simply saying over the names of number symbols is really delight in his newly acquired or rapidly growing power of abstraction and generalization. There is abstraction because the child now knows, in a definite, objective way, that one chair, although a different chair from every other, is, nevertheless, in some particular identical with every other—it is a chair. He is able to neglect all that sensuous qualitative difference which previously so claimed his attention as to prevent his conscious or objective recognition of the common quality or use through which the things may be classed as one whole. Now, ability to neglect certain features of things in view of another considered more important, is of course of the essence of abstraction in its highest as well as in this rudimentary form. Generalization, on the other hand, is simply the obverse of abstraction; they are correlative phases of one activity. In leaving out of account the qualities now seen to be unimportant to the end in view, though sensuously they may be very

prominent and attractive, the mind grasps in one whole the objects that have a common quality or use, though the objects are decidedly unlike as regards other qualities or uses. If from a collection of objects of different colours a child is required to select all the red ones, he not only neglects all that are not red; he neglects also all the other qualities—shape, size, material, etc.—of the red objects themselves; and when this abstraction is completed, there is the conception of the group of red things as the result of the other side of the mental process-viz., generalization.

The manifestation of the conscious tendency in a child to count coincides, then, with the awakening in his mind of conscious power to abstract and generalize. This power can show itself only when there is ability to resist the immediate solicitations of colour, sound, etc., ability to hold the mind from being absorbed in the delight of mere seeing, hearing, handling; and this means power of abstraction. But this very power to resist the stimulus of some sense qualities and to attend to others means also the power to group the different objects together on the basis of some principle not directly apprehended by the senses-some use or function which all the different objects have—and this is, again, generalization.

DISCRIMINATION AND RELATION.-This power to form a whole out of different objects may be studied in somewhat more detail. It includes the two correlative powers of discrimination and relation.

1. Discrimination.-As adults we are constantly deceiving ourselves in regard to the nature and genesis of our mental experiences. Because an object presents

a certain quality directly to us, we are apt to assume that the quality is inherent in the object itself, and is presented to everybody quite apart from any intellectual operation. We forget that the objects now have certain qualities for us simply because of analyses previously performed. We see in an object just what we have learned to see in it. The contents of the concept resulting from an elaborate process of analysis-synthesis are at last given in the percept. An expert geometrician's percept of a triangle is quite a different thing, from that of a mere tyro in geometry. A man may become such a chemist as never to see water without being conscious that it is composed of oxygen and hydrogen; or such a botanist that a passing glance at a flower instantly recalls the name orchid, or ranunculus, and all the differential qualities which belong to this class of plant life. In like manner all of us have become sufficiently familiar with numerical ideas to know at a glance that a tree has a great many leaves, a chair a certain number of parts, a cube a definite number of faces. Although this knowledge is now direct and "intuitive," it is the result of past discriminations. We may be perfectly sure that they are not "intuitions" to the child; to him the tree, the house, the cube, the blackboard, the group of six objects, is one undefined whole, not a whole of parts. The recognition of separate or distinct parts always implies an act of analysis or discrimination definitely performed at some period; and such definite analysis has always been preceded by a vague synthesis-that is, the idea of a whole of as yet undistinguished parts.

There is perhaps no point at which the teacher is

more likely to go astray than in assuming that objects have for a child the definiteness or concreteness of qualities which they have for us. In the application of the pedagogical maxim "from the concrete to the abstract," he is very apt to overlook the necessity of making sure that the "concrete" is really present to the child's mind. He too easily assumes as already existing in the consciousness of the learner what can really exist only as the product of the mind's own activity in the process of definition-of discriminating and relating. It is a grave error to suppose that a triangle, a circle, a written word, a collection of five objects, are concrete wholes, that is, definitely grasped mental wholes to the child, simply because there are certain physical wholes present to his senses. Definite

ideas are thus assumed as the basis of later work when there is absolutely nothing corresponding to them in the child's mind, in which, indeed, there is only a panorama of vague shifting imagery, with a penumbra of all sorts of irrelevant emotions and ideas. Thus, this noted maxim, when translated to mean concrete things before the senses, therefore concrete knowledge in the mind, becomes really a mischievous fallacy.

Or, again, the teacher, mislead by the formula-first, the isolated definite particular; second, the interconnection; third, the organic whole-introduces distinction and definition where normally the child should deal only with wholes in vague outline; and thus substitutes for the poetic and spontaneous character of mental action a forced mechanical analysis all out of harmony with his existing stage of development. Of this we have an example in the prevailing methods of primary