idea of number as a mode of measurement is followed, it will be practically impossible to work in any other way. Even while working explicitly with addition and subtraction-inches, feet, ounces, pounds, dollars, cents, etc.—the process of ratio is constantly being introduced. The child can not help feeling that 1 inch is one third of 3 inches, 10 cents (1 dime) one tenth of a dollar, etc.; and this natural growth towards the definite conception of ratio is only checked, not forwarded, by compelling a premature conscious recognition of the nature of the process.

ADDITION AND SUBTRACTION.-The general nature of these operations as concerned with measurement through the process of aggregating minor units or parts has already been dealt with. Two or three points may, however, be considered in more detail.

1. Work from and within a Whole.-Here, as everywhere, the idea of a magnitude—a whole of quantity— corresponding to some one unified activity should be present from the first. Some vague quantity or whole, which is to be measured by the putting together of a number of parts, alone gives any reason for performing the operation and sets any limit to it. The process of breaking up the whole into parts and then putting together these parts into a whole, measures or defines what was originally a vague magnitude and gives it precise numerical value. In dealing, say, with 6, we may begin with a figure like this* This is a unity or whole-it is one. But its value is indefinite. The

* This may, of course, be constructed out of splints, or whatever is convenient.

counting off of the various sticks changes the vague unity into a measured unity, but these parts always fall within the original unity. There is always a sense of the whole connecting them together. If the square has already been mastered, the figure will be recognised as one 4+ one 2. Or, if one of the diagonals is changed thus, it will be recognised as two triangles—that is, as one 3+ one 3. Or, of course, it may be taken all to pieces and put together again and recognised as 6 parts of the value of 1 each. Or, the pupil may be told to make "pickets" or "tents" of the figure, and, arranging them as follows, ^ ^ ^, see that there are three groups of the value of 2 each.

The principle kept in mind in this instance is that of the equation and its rhythmic construction. (a) According to the prevalent method, six, when reached, would be simply six ones, six separate unities, that is-—not, as in the foregoing illustration, six parts of unit value each. No matter how much the teacher is urged to have the pupils recognise six at a glance, and not count up the various unities in it separately, still the fact remains that it can not, by that method, be grasped as a whole; while by the psychological method it can not be grasped in any other way. (b) It is also, upon the psychological method, regarded as having a value equal to (measured by) its constituent minor wholes. We are always working within a value, simply making it more clear and definite, not blindly or vaguely from fixed unities to their accidental sum-accidental, that is, so far as the action of mind is concerned. As a result, the psychological method appeals directly to the power of breaking up a larger whole into minor wholes, and putting

these together to make a larger whole. It appeals to the constructive rhythmic interest, never to mere memorizing. It gives the maximum opportunity for the exercise of power; it leaves the minimum for mere mechanical drill. Because, dealing with wholes, intuition may be used; the rationality of the principle-the construction of a complete whole by means of partial wholes may be objectively seen and clearly appreciated.

It may be laid down, then, in the most emphatic terms, that the value of any device for teaching addition depends upon whether or not it begins with a whole which may be intuitively presented, and whether or not it proceeds by the rhythmic partition of this original whole into minor wholes, and their recombination.

2. Use of Subtraction as Inverse Operation.-Upon this basis the process of subtraction is always used simultaneously with addition. In beginning with a fixed unity, or an aggregate of such unities, the "method" may tell us to teach addition and subtraction together (or, what is really meant, one immediately after the other), but they can not be employed at the same time. If 1 is one thing, 2 two things, 3 three things, and so on, it requires one mental act to unite two or more such things, and notice the resulting sum; and another act to remove one or more, and note the resulting difference. But in beginning with and noting that it is made up of or 4, and X, or 2, the synthesis (recognition of the whole of parts) and analysis (recognising the parts in the whole) are absolutely simultaneous. It is one and the same act (6=4+2), which becomes in outward statement addition or subtraction, according as the emphasis is directed upon both of the parts equally, or upon the whole and

one of the parts. If, for example, in the above in stance the and the X are both equally familiar, then the construction would probably appeal to the child as addition, putting together the more familiar to make the more unfamiliar. But if the alone is very 'familiar, he might rather notice that the difference between the square and the original whole, namely, X, or 2 units.

3. The Conscious Process of Subtraction slightly more Complex.-The conscious recognition of subtraction, however, is a slightly more complex processmakes more demand upon attention-than the conscious interpretation of addition. In addition, the whole emphasis is upon the result; it is not necessary to keep the parts separate at all. The sum of 5 and 4, e. g., is first of all supplied by intuition, and where the association is complete the mind merely touches, as it were, the symbols, and the sum appears in consciousness. If, for example, we know that James and John and Peter have a certain amount of money-the undefined whole-of which James has 6 and John 8 and Peter 12 cents, we instantly merge or absorb each preceding quantity in the next greater-6, 14, 26. As soon as the two parts are added they are dropped as separate parts, the resulting whole is alone kept in mind. But in subtraction it is necessary to note both the whole and the given part, and the relation between them. If we say that of the total amount* James has 6 cents, John

* While it is not necessary always to introduce the idea of the total first in words, it should be done even verbally until we are sure that the child's mind always supplies the idea of a whole from and within which he is constantly working.

2 more than James, and Peter 4 more than John, then the addition problem requires the same attention to the two terms separately and to the result as is required in subtraction. There is the idea of definiteness or relative moreness, and not merely the idea of an aggregate more ness. Here, as in subtraction, we are approaching nearer to ratio.

MULTIPLICATION: GENESIS OF THE FACTOR IDEA.

We have seen that, though multiplication is not identical with addition (even with the special case of addition where the addends are all equal), it has its genesis in addition, taking its rise in counting, which is the fundamental numerical operation. Counting is the relating process in the mental activity which transforms an indefinite whole of quantity into a definite whole. It begins with discrete quantity, and is first of all largely mechanical-an operation with things. The child in his first countings does not consciously relate the things; his act is not one of rational counting. He is apt to think that the number-names are the names of things; that three, e. g., is not the third of three related things, but the name of the third thing; and on being asked to take up three he will fix upon the single thing which in counting was called three.

But starting with groups of objects and repeating the operations of parting and wholing, he soon begins to feel that the objects are related to one another and to the whole. This is a growth towards the true idea of number, but the idea is not yet developed. There is a relating, but not the relating which constitutes number. In the process of counting one, two, etc., getting