=

In multiplication, as already suggested, we may look at a product of two factors in two ways: For example, 20 feet = 4 feet × 5 = also 5 feet x 4, or five times four times 1 foot four times five times 1 foot-that is, we may use the primary unit of measure "1 foot" with either the four times or the five times. Or, stated in general terms, 6 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 520 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.

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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

are said to be two totally different operations; for, it is alleged, in one case we are searching for the size (the numerical value) of a group-the unit of measurement; in the other for the number of groups. But a little reflection will show that they are not "radically different" operations; they are psychological correlates, if not identities. In counting out fifteen oranges in groups of five there is a count of five, then another count of five, then another count of five, and finally a counting of the number of groups. Psychologically, in counting out five there is a mental sequence of five acts (a partial synthesis); this is repeated three times, and finally the number of these sequences is counted (complete synthesis), and found to be three. In the second case, where the number (five) of groups is given, we begin by putting one orange in each of five places, making, as before, a "count" of five oranges; this operation is repeated till all are counted out; and finally we count the number in each of the five groups. That is, there is a mental sequence of five acts, which is repeated three times, and finally the number of such sequences is counted in counting the number of oranges in a group. It would be hardly too much to say that these two mental processes are so closely correlated as to be identical. Neither the three times in the one question nor the three oranges in the other can be found without counting out the whole quantity in groups of five oranges each (see page 75). There is hardly a difference even in the rhythm of the mental movement. This division by counting is the actual process with things; it is the way of the child and of the savage or the illiterate man; it is exactly symbolized in the "two kinds of division"

-that by a concrete divisor when we are searching for the number of the parts as actual units of measure; and that by an abstract divisor when we are searching for the size of the parts-i. e., for the number of minor units in the actual unit of measure.

With this actual process of counting out the objects the arithmetical operation exactly corresponds. Working by long division as more typical of the general arithmetical operation, we have:

I. Division: 15 oranges 5 oranges; i. e., 15 oranges are to be counted out in groups of 5 oranges; how many groups?

5 oranges 15 oranges | 1°-1st partial multiplier

5 times 15 oranges | 1 orange-1st partial multiplicand

That is, once more, both problems are solved by counting out the whole quantity in groups of five.

(2) Solve the following problems: (a) Find the cost of a town lot of 36 feet frontage at $54 a foot. (b) At the rate of $54 a foot, a town lot was sold for $1944,

find the number of feet frontage. (c) Find the price per foot frontage when 36 feet cost $1944.

On comparing the successive steps in (b) with those in (a) they will be seen to correspond exactly-that is, (b) is the exact inverse of (a). But the steps in (c) do not correspond with those in (a), the operation is not the exact inverse of (a); it is seen to be the exact inverse of the correlative (ii) of (a). This indicates the connection of the operations through the law of commutation; and shows, once more, that either of the correlated measurements (¿) and (ii) may be used in the solution of (c). It should be noted, further, that (c) is a case of so-called partition, yet involves a series of subtractions that is a series of partial dividends (why not partiends?) and partial quotients.

Not Two Kinds of Division.-From the foregoing we see that just as a product of two factors may be interpreted in two ways, so there may be two interpretations of the result of the inverse operation, division. The factor sought may be either the numerical value of the dividing part (" derived unit ") in terms of the pri

mary unit which measures it, or the numerical value of the quantity in terms of the derived unit. But these numerical values may be interchanged at convenience, provided the results are rightly interpreted. There are not two kinds of division; there is one operation leading to one numerical result having two related meanings. It seems therefore unnecessary, either on psychological or practical grounds, to institute two kinds of division-viz., division (why not quotition ?) in the ordinary sense of the word, and "partition "--when the search is for the numerical value of the measuring quantity. When the search is for the numerical value of the measuring unit, is not the pupil likely to become perplexed by a series of parallel definitions-of divisor, dividend, quotient-for the two divisions when he finds that the operations in both cases are exactly alike? If there is confusion in using the term division in two senses, is there not more confusion in using the two terms, divisor and quotient, each with two different meanings Without doubt, the meaning of the result should be grasped; but this can not be done by simply giving two names to exactly the same arithmetical operation. Better give one name to one operation resulting in two correlated meanings than to have two names for one and the same operation. The new name does not help the pupil either in the numerical work or in the interpretation of the result. How is the child to know whether a given problem is a case of division or of partition"? He can not know without an intellectual operation, analysis, by which he grasps what is given and what is wanted in the problem. In other words, he must know the meaning of the problem, must know