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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 × 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 commutation"; 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

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.

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 420 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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In multiplication, as already suggested, we may look at a product of two factors in two ways: For example, 20 feet = 4 feet x 5 = also 5 feet x 4, or five times four times 1 foot = four times five times 1 footthat is, we may use the primary unit of measure with either the four times or the five times. Or, stated in general terms, b times a times the primary unit of measure is identical with a times b times this primary unit—that is, we may interchange at pleasure the numerical value of the measuring unit (the derived unit as made up of primary units) with the numerical value of the whole quantity as made up of these derived units. This is important as interpreting the process and result in division. If we have 20 feet and the factor 4 feet given to find the other factor, we use the measurement 4 feet x 5 = 20 feet. If, on the other hand, we have the 20 feet and the number 5 given to find the other factor, we may use either measurement; we may divide directly by the number 5, or we may "concrete" the 5 (consider it as denoting 5 feet), and get the other factor 4 (times); for we know that 4 times 5 feet is identical with 5 times 4 feet, and the conditions of the question require the latter interpretation. In other words, we first of all determine what the problem demands, times or parts, then operate with the pure number symbols, and interpret the result according to the conditions of the problem.

Illustrations of Division.-Let us take a few illustrations of these inverse operations: (1) We count out fifteen oranges, by groups of five, and the number of groups is three. We count them out in five groups, and the number of oranges in each group is three. These

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