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 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 x 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 we have in all"; or the teacher might ask, at the end of every problem, "What were we counting up or measuring that time?" letting each one interpret as he pleased. Just how far this is carried is a matter of detail; what is not a matter of detail is that the habit of interpretation be formed by continually referring the numbers to some quantity. 2. The unit is never to be taught as a fixed thing (e. g., as in the Grube method), but always as a unit of measurement. One is never one thing simply, but always that one thing used as a basis for counting off and thus measuring some whole or quantity. Absolutely everything and anything which we attend to is one; is made one by the very act of attending. If we could take in the whole system of things in one observation, that would be one; if we could isolate an atom and look at that, it would equally be one. The forest is one when we view it as a whole; the tree, the branch, the stem, the leaf, the cell in the leaf, is equally one when it becomes the object or whole with which we are occupied. But this oneness, this unity possessed by every object of attention, has nothing but the name in common with the numerical unit. In itself it is not quantitative at all; it is mere unity of quality, of meaning. It becomes a quantitative unity (a quantity or magnitude) only when considered as limited (page 36), and as an end to be reached by the use of certain means. It becomes a unit only when used as one of the means to construct a value equivalent to a certain other value. The assumption that some one object is the natural unit of quantity, which is then increased by bringing in other objects, is the very opposite of the truth; number does |
