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perceived in the following, where the dots symbolize both times and units of quantity:

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Measuring by the 3-feet unit we count it off four times —that is, the quantity is expressed by 3 feet taken four times. This is represented by the four vertical columns of three minor units each. But this measuring process necessarily involves the correlated process which is expressed by 4 feet taken three times. For, in measuring by three feet, and finding that it is repeated four times, we perceive that each of its three parts is repeated four times, giving the three horizontal rows a, b, c-that is to say, a is one whole of 4 feet, b a second whole of 4 feet, and c a third whole of 4 feet; or, in all, 4 feet taken three times. Briefly, 1 foot four times is one whole of 4 feet; this is true of every foot of the original measure, 3 feet; and therefore 3 feet four times is 4 feet three times.

It is clear that the two questions, (a) in 12 feet how many counts of 4 feet each, and (b) how many feet in each of 4 counts making 12 feet, are solved in exactly the same way; neither the three counts (times) in the first case nor the three feet in the second case can be found without counting the twelve feet off in groups of four feet each.

This necessary correlation, in the measurement of quantity, between " parts" and "times "-numerical value of the measuring unit and numerical value of the measured quantity-gives the psychology of the fun

damental principle in multiplication known as the law of commutation: the product of factors is the same in whatever order they may be taken-i. e., in the case of two factors, for example, either may be multiplicand or multiplier; a times b is identical with b times a.

It is asserted by some writers that this commutative law does not hold when the multiplicand is concrete; "for," we are told, "though there is meaning in requiring $4 to be taken three times, there is no sense in proposing that the number 3 be taken four-dollars times"--which is perfectly true. Nevertheless, the objection seems to be founded on a misconception of the psychical nature of number and the psychological basis of the law of commutation. Psychologically speaking, can the multiplicand ever be a pure number? If the foregoing account of the nature of number is correct, the multiplicand, however written, must always be understood to express measured quantity; it is always concrete. As already said, 4 × 3 must mean 4 units of measurement taken three times. If number in itself is purely mental, a result of the mind's fundamental process of analysis-synthesis-what is the meaning of 3 x 4 where both symbols represent pure numbers, and where, it is said, the law of commutation does hold? There is no sense, indeed, in proposing to multiply three by four dollars; but equally meaningless is the proposition to multiply one pure number by another-to take an abstraction a number of times.

Thus, if the commutative law "does not hold when the multiplicand is concrete"-indicating a measured quantity-it does not hold at all; there is no such law. But if the psychological explanation of number as aris

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ing from measurement is true, there is a law of commutation. We measure, for example, a quantity of 20 pounds weight by a 4-pound weight, and the result is expressed by 4 pounds × 5, but the psychological correlate is 5 pounds x 4. Here we have true commutation of the factors, inasmuch as there is an interchange of both character and function; the symbol which denotes measured quantity in the one expression denotes pure number in the other, and vice versa. If the 4 pounds in the one expression remained 4 pounds in the commuted expression, would there be commutation?

We have referred to the fallacy of identifying actual measuring parts with numerical value; it may now be said that, on the other hand, failure to note their necessary connection-their law of commutation-is often a source of perplexity. To say nothing at present of the mystery of "Division," witness the discussions upon the rules for the reduction of compound quantities and of mixed numbers to fractions. To reduce 41 yards to feet we are, according to some of the rules, to multiply 41 by 3. According to others, this is wrong, giving 123 yards for product; and we ought to multiply 3 feet by 41, thus getting the true result, 123 feet. Some rulemakers tell us that though the former rule is wrong it may be followed, because it always brings the same numerical result as the correct rule, and in practice is generally more convenient. It seems curious that the rule should be always wrong yet always bring the right results. With the relation between parts and times before us the difficulty vanishes. The expression 41 yards denotes a measured quantity; 41 expresses the numerical value of it, and one yard the measuring unit; our con

ception of the quantity is therefore, primarily, 41 parts of 3 feet each, and we multiply 3 feet by 41; but this conception involves its correlate, 3 parts of 41 feet each; and so, if it is more convenient, we may multiply 41 feet by 3.

A similar explanation is applicable to the reduction, e. g., of $3 to an improper fraction. The denominator of the fraction indicates what is, in this case, the direct unit of measure, one of the four equal parts of the dollar; and so we conceive the $3 as denoting 3 parts of 4 units (quarter dollars) each, and multiply 4 by 3; or, as denoting 4 parts of 3 units each, and multiply 3 by 4.

EDUCATIONAL APPLICATIONS.

1. Every numerical operation involves three factors, and can be naturally and completely apprehended only when those three factors are introduced. This does not mean that they must be always formulated. On the contrary, the formulation, at the outset, would be confusing; it would be too great a tax on attention. But the three factors must be there and must be used.

Every problem and operation should (1) proceed upon the basis of a total magnitude-a unity having a certain numerical value, should (2)-have a certain unit which measures this whole, and should (3) have number-the ratio of one of these to the other. Suppose it is a simple case of addition. John has $2, James $3, Alfred $4. How much have they altogether? (1) The total magnitude, the amount (muchness) altogether, is here the thing sought for. There will be meaning to the problem, then, just in so far as the child feels this amount altogether as the whole of the various parts.

(2) The unit of measurement is the one dollar. (3) The number is the measuring of how many of these units there are in all, namely, nine. When discovered it defines or measures the how much of the magnitude which at first is but vaguely conceived. In other words, it must be borne in mind that the thought of some inclusive magnitude must, psychologically, precede the operation, if its real meaning is to be apprehended. The conclusion simply defines or states exactly how much is that magnitude which, at the outset, is grasped only vaguely as mere magnitude.

Are we never, then, to introduce problems dealing with simple numbers, with numbers not attached to magnitude, not measuring values of some kind; are we not to add 4, 5, 7, 8, etc.? Must it always be 4 apples, or dollars, or feet, or some other concrete magnitude? No, not necessarily as matter of practice in getting facility in handling numbers. Number is the tool of measurement, and it requires considerable practice with the tool, as a tool, to handle it with ease and accuracy. But this drill or practice-work in "number" should never be introduced until after work based upon definite magnitudes; it should be introduced only as there is formed the mental habit of continually referring number to the magnitude which it measures. Even in the case of practice, it would be safer for the teacher to call attention to his reference of number to concrete values in every case than to go to the other extreme, and neglect to call attention to its use in defining quantity. For example, when adding "numbers," the teacher might say, "Now, this time we have piles of apples, or we have inches, etc., and we want to see how much

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