transportation of commodities, fixing political boundaries, etc.-it acquires a geographical significance.

In other words, the unity of geography is a certain unity of human action, a certain human interest. Unless, therefore, geographical data are presented in such a way as to appeal to this interest, the method of teaching geography is uncertain, vacillating, confusing; it throws the movement of the child's mind into lines of great, rather than of least, resistance, and leaves him with a mass of disconnected facts and a feeling of unreality in presence of which his interest dies out. All method means adaptation of means to a certain end; if the end is not grasped, there is no rational principle for the selection of means; the method is haphazard and empirical—a chance selection from a bundle of expedients. But the elaboration of this interest, the discovery of the concrete ways in which the mind realizes it, is unquestionably the province of psychology. There are certain definite modes in which the mind images to itself the relation of environment and human activity in production and exchange; there is a certain order of growth in this imagery; to know this is psychology; and, once more, to know this is to be able to direct the teaching of geography rationally and fruitfully, and to secure the best results, both in culture and discipline, that can be had from the study of the subject.

Application to Arithmetic.—In the following pages an attempt has been made to present the psychology of number from this point of view. Number represents a certain interest, a certain psychical demand; it is not a bare property of facts, but is a certain way of interpreting and arranging them-a certain method of constru

ing them.

activity develop?

What is the interest, the demand, which gives rise to the psychical activity by which objects are taken as numbered or measured? And how does this In so far as we can answer these questions we have a sure guide to methods of instruction in dealing with number. We have a positive basis for testing and criticising various proposed methods and devices; we have only to ask whether they are true to this specific activity, whether they build upon it and further it. In addition to this we have a standard at our disposal for setting forth correct methods; we have but to translate the theory of mental activity in this direction-the psychical nature of number and the problem of its origin—over into its practical meaning. Knowing the nature and origin of number and numerical properties as psychological facts, the teacher knows how the mind works in the construction of number, and is prepared to help the child to think number; is prepared to use a method, helpful to the normal movement of the mind. In other words, rational method in arithmetic must be based on the psychology of number.

CHAPTER II.

THE PSYCHICAL NATURE OF NUMBER.

WHY do we ask with respect to any magnitude, "how many," "how much," and set about counting and measuring till we can say "so many," "so much"? Why do we not take our sense experience just as it comes to us, making no attempt to give it these exact quantitative characteristics? If we can find out the psychological reason, the mental necessity, which induces us to put our experience so far as we can into terms of exact measurement, we shall have a principle which will guide us to sound conclusions regarding the nature and origin of number and its rational treatment as a school study. We have here tacitly assumed that number is a psychical product, and has a psychical reason for its origin. Before dealing with the problem of the origin of number, let us put the assumption of its psychical nature on a firmer basis.

NUMBER IS A RATIONAL PROCESS, NOT A SENSE FACT. -The mere fact that there is a multiplicity of things in existence, or that this multiplicity is present to the eye and ear, does not account for a consciousness of number. There are hundreds of leaves on the tree in which the bird builds its nest, but it does not follow that the bird can count.

So hundreds of noises strike the ear, and countless objects appeal to the eye of a child a few weeks old; but he is not conscious of the noises or the objects as quantitative; he does not number or measure them. More than this, sense facts may be even attended to without giving the idea of number. To put, say, five objects before an older child, to call his mind away from all other things and get his attention fixed upon these objects, is not to give him the idea of the number five. Number is not a property of the objects which can be realized through the mere use of the senses, or impressed upon the mind by so-called external energies or attributes. Objects (and measured things) aid the mind in its work of constructing numerical ideas, but the objects are not number. Nor does the bare perception of them constitute number. A child, or an adult, may perceive a collection of balls or cubes, or dots on paper, or a bunch of bananas, or a pile of silver coins, without an idea of their number; there may be clear and adequate percepts of the things quite unaccompanied by definite numerical concepts. No such concepts, no clearly defined numerical ideas, can enter into consciousness till the mind orders the objects—that is, compares and relates them in a certain way.

FACTORS OF THE INTELLECTUAL PROCESS.-In the simple recognition, for example, of three things as three the following intellectual operations are involved: The recognition of the three objects as forming one connected whole or group-that is, there must be a recognition of the three things as individuals, and of the one, the unity, the whole, made up of the three things. If one of the objects is a piece of candy, and the other two are dots

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 time— to 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