feeling, as he counts his units, that one, two, three is not so much-because it is not so many-as one, two, three, four. These first vague ideas must be made clear and definite; the natural movement of the mind is aided by the proper presentation of right material; the initial mechanical operation of naming the units in order gives place to an intelligent relating of the units to one another, and finally to a conscious grasp of the relation of each to the unified whole; the counting— one, two, three, etc.-is now a rational process.

So much; so many.-In the development of this rational process there must be no divorce between the how much and the how many, between the measuring process and the results of measurement. The so much is determined only by the so many, and the so many has significance only from its relation to the so much. These are co-ordinate factors of the idea of number as measurement. Now, the development of countingdetermining the how many that defines the how much -is aided by symmetrical arrangements of the units of measure (see page 34). The child at first counts the units one, two, six, with only the faintest idea of

the relations of the units in the numbers named. Both the analytic and relating activities are greatly aided by the rhythmic grouping of the units of measure, or of the counters used to represent them; the mastery of the number relations (of both addition and multiplication) as so many units making up a quantity, becomes much easier and more complete. Thus, when exercises in parting and wholing (accompanied with counting) a quantity, say a length of 12 inches, have given rise to even imperfect ideas of unit of measurement and times

of repetition, the symmetric forms may be used with great advantage; indeed, they may be used in the exercises from the first. We have counted, according to the unit of measure used, one part, two parts; one part, two parts, three parts, etc. Both the times and the unit values are more easily grasped through the number forms; for example, six, one of the two measuring units, may be shown as a whole of related units (threes, twos, ones) as in the arrangement,; and so on with the whole quantity and all its minor parts (addition) and repeated units. Real meaning is given to the operation of counting when, instead of using unarranged units, we have the rhythmic arrangement :

::|

Six,

five,

four,

three, two, one. The actual values of the measuring units, and the meaning of counting-necessarily related processes are fully brought out. Six is at last perceived as six without the necessity of counting.

CHAPTER IX.

ON PRIMARY NUMBER TEACHING.

Relation between Times and Parts.-With the growth of the idea of the unit as a measure itself measured by minor units, and of number as indicating times of repetition of a unit of measure, there is gradually developed a clear idea of the relation between the value of the actual measuring unit (as made up of minor units) and the number of them in the given quantity; in other words, of the relation between the number of derived units in the quantity and the number of primary units in the derived unit. A common error, as has been often pointed out, is that of making too broad a distinction between these related factors in the measuring process. They are said to be totally different conceptions. It has been shown that they are absolutely inseparable. They are, in any and every case, two aspects of the same measurement. The direct unit in a given measurement is not wholly concrete; it is a quantity measured by a number of other units; and so it involves, as every measured quantity involves, the space element in the single concrete (minor) unit, and the abstract element in the number applied to the unit. When we speak of the "size" of the numbered parts (derived units) composing a given quantity, we mean the number of minor units

of which one part is composed. When, for instance, we conceive of $15 as measured by the unit $3, we get the number five; when we are required to divide $15 into five equal parts, we are searching for the "size" ($3) of the measuring unit-i. e., for the numerical value of the unit in terms of the minor unit ($1) by which it is measured.

The relation, then, between "times and parts" is the relation between the number of derived units in the measured quantity and the number of primary units in the derived unit. It is clear that the rational processes of parting and wholing that ultimately give clear ideas of unit and number, must also bring out clearly the relation between these two factors in measured quantity: the smaller the unit the larger the number; or, the number of the measuring units in the quantity varies inversely as the number of primary units in the derived unit. Measuring a length of one foot by a 6-inch unit, by a 3-inch unit, and by a 1-inch unit, the numbers are respectively 2, 4, and 12; measuring a length of one decimetre by 10, 20, 30, 40, 50 centimetres, the numbers are respectively 10, 5, 3, 24, 2; measuring $20 by the $1 unit, $2 unit, $4 unit, the numbers are 20, 10, 5, etc. In the constructive exercises already described, attention to measuring unit and its times of repetition must lead to the conscious recognition of this principle, which is fundamental in number as measurement. It has already been given in the complete statement in "fractional" form of the process of measurement : Any measured quantity may be expressed in the form 1 2 3 4

1' 2' 3' 4'

n This principle is, of course, the basal

n

principle in the treatment of fractions (because it is the primary principle of number, and "fractions" are numbers)—namely, both terms of a fraction may be multiplied or divided by the same number without altering the value of the fraction.

The Law of Commutation.-The development of this principle through the rational use of its ideathat is through the use of the facts supplied by senseperception in the rational use of things—is the development of the psychological law of commutation which is primary and essential in all mathematics. The method that ignores this necessary relation between times and parts, or regards them as totally different things, never leads to a clear conception of this important principle. It is, as a consequence, always finding difficulties where, for rational method, none really exist. The principle is difficult for the child only when the method is wrong. With right presentation of material he can have no difficulty in seeing that the larger the units the smaller their number in a given quantity. When he counts out a collection of 24 objects into piles of 3 each, and into piles of 6 each, can he fail to see that the respective numbers differ? And with rightly directed attention to the concrete processes, may he not be led slowly, perhaps, but surely, to a clear thought of how and why they differ? The learner can not help seeing, for example, the difference between the 2-inch unit and the 6-inch unit, and the corresponding difference between six times and three times. To see clearly is to think clearly; there is a rationality in rationally presented facts, and this rationality leads with certainty to a complete recognition of the meaning-the law-of the facts.