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implies division; the definite unit of measurement which constitutes the multiplicand is always a certain exact (equal) portion of some whole. Hence multiplication always implies ratio; the whole magnitude bears to the unit of measure a ratio which is expressed in the number of times (represented by the multiplier) the unit has to be taken to measure that magnitude—to give it accurate numerical value. In fact, the process is simply one of changing the number which measures a magnitude by changing the unit of measure-i. e., by substituting for the given unit of measure the primary unit from which it was derived.*

2. In multiplication, then, as in addition, we are not performing a purposeless operation, or one with unrelated parts and isolated units; rather, we begin and end with some magnitude requiring measurement, keeping in mind that what distinguishes multiplication is the kind of measurement it uses-that, namely, in which a unit itself measured off by other units is taken a certain number of times.

3. The psychology of number, therefore, imperatively demands that the quantity which is to be finally expressed by the "product" should first be suggested, just as in addition the quantity given by "sum," within which and towards which we are working, is kept in

* In such instances as multiply 7 apples by 4, the idea of exact division or ratio is not so evident, but the 7 apples must be taken as one of four equal portions-i. e., as having the ratio to the whole quantity. The fact, however, that the idea of an exact unit of measurement is not so clearly present, is a strong reason for using fewer examples of this sort, and more of those involving standard units of measure.

mind from the first. If the child sees, e. g., that there is a certain field of given dimensions whose area is to be ascertained, or a piece of cloth of given length and price per yard, of which the cost is to be determined, the mind has something to rest upon, a clearly defined purpose to accomplish. Beginning with a more or less definite image of the thing to be reached, the subsequent steps have a meaning, and the entire process is rational and consequently interesting. But when he is asked how much is 4 times 8 feet, or 9 times 32 cents, there is no intrinsic reason for performing the operation; psychologically it is senseless, because there is no motive, no demand for its performance. The sole interest which attaches to it is external, as arising from the mere manipulation of figures. Under an interested teacher, indeed, even the pure" figuring" work may be interesting; but this interest is re-enforced, transformed, when the mechanical work is felt to be the means by which the mind spontaneously moves by definite steps towards a definite end. This does not mean, we may once more remark, that examples like 8 feet x 4, or even 8 × 4, are to be excluded, but only that the habit of regarding number as measuring quantity should be permanently formed. The pupil should be so trained that all addends, sums, minuends, products, multiplicands, dividends, quotients, could be instantly interpreted in their nature and function as connected with the process of measurement. For example: A farmer has 8 bushels of potatoes to sell, and the market price is 55 cents a bushel: how much can he get for them?

This and similar examples are often presented in such a way that when the pupil gets the product, $4.40,

his mind stops short with the mere idea of the product as a series of figures. This is irrational; $4.40 in itself is not a product; no quantity or value is ever in itself a product; but as a product it measures more definitely the value of some quantity. In other words, the product must always be interpreted; it must be recognised as the accomplished measurement of a measured quantity in terms of more familiar or convenient units of

measure.

4. The multiplicand must always be seen to be a unit in itself, no matter how large it is as expressed in minor units. It signifies the known value of the unit with which one sets out to measure; it is the measuring rod, as it were, which is none the less (rather the more) a unit because it is defined by a scale of parts. A foot is none the less one because it may be written as 12 inches or as 192 sixteenths; nor is a mile any the less a unit because it is written as 320 rods or as 5280 feet. The ineradicable defect of the Grube method, or any method which conceives of a unit as one thing instead of as a standard of measuring, is that it can never give the idea of a multiplicand as just one unit-a part used to measure a whole.

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5. It is important so to teach from the beginning that a clear and definite conception of the relation between parts and times may be developed. Of course, nothing is said till the time is ripe about the law of " tation"; but the idea should be present, and should be freely used. If a quantity of 12 units is measured by 3 units repeated four times, the child can be led to see will probably discover for himself that this measurement is identical with the measurement

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4 units repeated three times. Rationally using this idea of commutation in repeated operations, the child will soon get possession of a principle by which he can easily interpret both processes and results in numerical work.

CHAPTER VII.

NUMERICAL OPERATIONS AS EXTERNAL AND AS INTRINSIC TO NUMBER.

DIVISION AND FRACTIONS.

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DIVISION. AS multiplication has its genesis in addition, but is not identical with it, so division has its genesis in subtraction, but is not identical with it. Just as multiplication comes from the explicit association of the number of equal addends with their sum, and the substitution of the factor idea (ratio) for the part idea, so division comes, in the last analysis, from the explicit association of the number of equal subtrahends from the same sum (dividend), and the substitution of the factor idea for the part idea. In other words, division is the inverse of multiplication, just as subtraction is the inverse of addition. Further, as in multiplication, both factors are the expression of a measured quantity and are interchangeable, so in division either of the factors (divisor and quotient) which produced the dividend can be commuted with the other. In multiplication, for example, we have 4 feet x 55 feet x 4 = 20 feet; and the inverse problem in division is, given the 20 feet, and either of the factors, to find the other factor. We solve the problem not by subtraction, but by the use of the factor, or ratio, idea.

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