It is little more than a blind manipulation of number symbols. The child simply takes, for example, the figures 3 and 12, and performs certain "operations with them, which are dignified by the names addition, subtraction, multiplication, etc.; he knows very little of what the figures signify, and less of the meaning of the operations. The second method, the simple perception or observation method, depends almost wholly upon physical operations with things. Objects of various kinds-beans, shoe-pegs, splints, chairs, blocks—are separated and combined in various ways, and true ideas of number and of numerical operations are supposed necessarily to arise. Both of these methods are vitiated by the same fundamental psychological error; they do not take account of the fact that number arises in and through the activity of mind in dealing with objects. The first method leaves out the objects entirely, or at least makes no reflective and systematic use of them; it lays the emphasis on symbols, never showing clearly what they symbolize, but leaving it to the chances of future experience to put some meaning into empty abstractions. The second method brings in the objects, but so far as it emphasizes the objects to the neglect of the mental activity which uses them, it also makes number meaningless; it subordinates thought (i. e., mathematical abstraction) to things. Practically it may be considered an improvement on the first method, because it is not possible to suppress entirely the activity which uses the things for the realization of some end; but whenever this activity is made incidental and not important, the method comes far short of the intelligence and skill that should be had from instruction based on psychological principles. While the method of symbols is still far too widely used in practice, no educationist defends it; all condemn it. It is not, then, necessary to dwell upon it longer than to point out in the light of the previous discussion why it should be condemned. It treats number as an independent entity-as something apart from the mental activity which produces it; the natural genesis and use of number are ignored, and, as a result, the method is mechanical and artificial. It subordinates sense to symbol. The method of things-of observing objects and taking vague percepts for definite numerical conceptstreats number as if it were an inherent property of things in themselves, simply waiting for the mind to grasp it, to "abstract" it from the things. But we have seen that number is in reality a mode of measuring value, and that it does not belong to things in themselves, but arises in the economical adaptation of things to some use or purpose. Number is not (psychologically) got from things, it is put into them. It is then almost equally absurd to attempt to teach numerical ideas and process without things, and to teach them simply by things. Numerical ideas can be normally acquired, and numerical operations fully mastered only by arrangements of things-that is, by certain acts of mental construction, which are aided, of course, by acts of physical construction; it is not the mere perception of the things which gives us the idea, but the employing of the things in a constructive way. The method of symbols supposes that number arises - wholly as a matter of abstract reasoning; the method of objects supposes that it arises from mere observation by the senses that it is a property of things, an external energy just waiting for a chance to seize upon consciousness. In reality, it arises from constructive (psychical) activity, from the actual use of certain things in reaching a certain end. This method of constructive use unites in itself the principles of both abstract reasoning and of definite sense observation. If, to help the mental process, small cubical blocks are used to build a large cube with, there is necessarily continual and close observation of the various things in their quantitative aspects; if splints are used to inclose a surface with, the particular splints must be noted. Indeed, this observation is likely to be closer and more accurate than that in which the mere observation is an end in itself. In the latter case there is no interest, no purpose, and attention is laboured and wandering; there is no aim to guide and direct the observation. The observation which goes with constructive activity is a part of the activity; it has all the intensity, the depth of excitation of the activity; it shares in the interest of and is directed by the activity. In the case where the observation is made the whole thing, distinctions have to be separately noted and separately memorized. There is nothing intrinsic by which to carry the facts noted; that the two blocks here and the two there make four is an external fact to be carried by itself in memory. But when the two sets are so used as to construct a whole of a certain value, the fact is internal; it is part of the mind's way of acting, of seeing a definite whole through seeing its definite parts. Repetition in one case means simply learning by rote; in the other case, it means repetition of activity and formation of an intelligent habit. The rational factor is found in the fact that the constructive activity proceeds upon a principle; the construction follows a certain regular or orderly method. The method of action, the way of combining the means to reach the end, the parts to make the whole, is relation; acting according to this relation is rational, and prepares for the definite recognition of reason, for consciously grasping the nature of the operations. Rational action will pass over of itself when the time is ripe into abstract reasoning. The habit of abstracting and generalizing of analysis and synthesis grows into definite control of thinking. THE FACTORS IN RATIONAL METHOD.-In more detail, dealing with number by itself, as represented by symbols, introduces the child at an early stage to abstractions without showing how they arise, or what they stand for; and makes clear no reason, no necessity, for the various operations performed, which are all reducible to (a) synthesis-addition, multiplication, involution; and (b) analysis-subtraction, division, evolution. The object or observation method shows the relation of number to things, but does not make evident why it has this relation; does not bring out its value or measuring use, and leaves the operations performed purely external manipulations of number, or rather with things which may be numbered, not internal developments of its measuring power. The method which develops numerical ideas in connection with the construction of some definite thing, brings out clearly (a) the natural unity, the limit (the magnitude) to which all number refers; (b) the unit of measurement (the particular thing) which helps to construct the whole; and (c) the process of measuring, by which the second of these factors is used to make up or define the first-thus de. termining its numerical value. (a) Only this method presents naturally the idea of a magnitude from which to set out. The end to be reached, the object to be measured, supplies this idea of a given quantity, and thus gives a natural basis for the development and use of ideas of number. In numbers simply as objects, or in things simply as observed things, there is no principle of unity, no basis for natural generalization. Only using the various things for a certain end brings them together into one; we count and measure some quantitative whole. (b) While every object is a whole in itself, a unity in so far as it represents one single act, no object simply as an observed object is a unit. Objects which we recognise as three in number may be before the child's senses, and yet there may be no consciousness of them as three different units, or of the sum three. Some writers tell us that each object is one, and so gives the natural basis for the evolution of number; that the starting point is one object, to which another object is "added," then a third, etc. But this overlooks the fact that each object is one, not a unit but one whole, differing from and exclusive of every other whole. That is, to take it as an observed object is to centre attention wholly upon the thing itself; attention would discriminate and unify the qualities which make the thing what it is a qualitative whole; but there |