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unity, instead of simply a measuring part-a means of measuring a magnitude.

The Unit itself Measured.--As necessary to the growth of the true conception of unit as a measuring part, the idea of the unit as a unity of measured parts must be clearly brought out. The given quantity is measured by a certain unit; this unit itself is a quantity, and so is made up of measuring parts. This idea must be used from the beginning; it is absolutely essential to the clear idea of the unit, and of number as measurement of quantity. Beginning with a group of 12 objects requiring measurement, or with counters representing such objects, we have them counted off into two equal parts, noting the relation of the parts to one another and to the whole; then each of these two units (half of the given whole) is counted off into two equal parts, and the relation of these minor parts to each other and to the whole they compose is noticed ; then each of the first units of measurement (halves of the given whole) is counted off into three equal parts, and their relation to one another and to the whole which they make is carefully observed ; and so on, with similar exercises in parting and wholing. Such constructive exercises help in the growth of the true idea of the unit as a measuring part, which is or may be itself measured by other units. But the true idea of the essential property of the unit—its measuring function can be fully developed only by exercises belonging to the second stage of measurement, in which exact and equal units are used for precise measurement. These measurements of groups of like things (apples, oranges, etc.) by groups which are themselves measured by still

smaller groups, must be supplemented by the use of exactly measured quantities-quantities defined by equal units, which in turn are measured by other equal units. Without such exercises there can be no adequate conception of measurement, or of number as the tool of measurement, or of the real meaning of multiplication and division, and especially of fractions, the full and precise statement of the measuring process. To free arithmetic from the tyranny of irrational method, an indispensable step is the emancipation of the unit from the cast-iron fetters which have paralyzed its measuring function. 3. The Idea of Times.—

With the intelligent use of these constructive exercises to make clear the idea of the unit, there is necessarily growth towards recognition of the times of repetition of the unit to make up some magnitude towards, that is, the true idea of number. To discuss the evolution of this idea would be to repeat in the main what has been laid down in the preceding paragraphs. It is therefore necessary only to state explicitly the chief things to be considered as bearing upon the natural growth of the idea of times -i. e., of number in the strict sense of the word.

(a) The preliminary operations as already illustrated - dealing with groups of like things, and so leading to a working idea of the unit as measuring part--are to be supplemented by constructive acts with exactly measured quantities. The measured whole must be analyzed into its measured units, and again built up from these parts. For example, exercises such as the quantity 12 apples measured by the unit 4 apples, by the unit 3 apples, etc., must be supplemented by exercises such

as the quantity 12 inches measured by the unit 4 inches, by the unit 3 inches, etc.; or the quantity 20 cents measured by the unit 10 cents, by the unit 5 cents, etc. The movement towards the real number idea began in operations with undefined units, and is strengthened by these supplementary exercises with exactly measured quantity; there is a more rapid growth towards the numerical discriminating and unifying power. (6) Count by ones, but not necessarily by single things ; in fact, to avoid the fixed unit error, do not begin with counting single things. The 12 things in the

group have been measured off, for example, into four groups, or into three groups; these are units, are ones, and in counting there is a first one, a second one, a third one that is, in all “three timesone; and so with the four ones when the quantity is divided into four equal parts. Proceed similarly with exactly measured quantities: the four 3-inch ones or the six 2-inch ones making up the linear foot, or other exactly measured quantity. As before said, the child first of all sees related things, and with the repetition of the exercises—parting and wholing—begins to feel the relations of things, and in due time consciously recognises these relations, and the goal is at last reached-a definite idea of number.

(c) Use the Actual Units.- In these constructive processes let the child at first use-as before suggested—the actual concrete units to make up or equal the measured quantity; then apply the single concrete unit the requisite number of “times.” In the first case, in measuring, for example, a length of 12 feet, four actual units of measure (3 feet) are put together to equal the 12 feet; in the second case, one unit is applied, laid down, and


taken up three times. This application of the single unit so many times is an important step in the process of numerical abstraction and generalization; it is from the less abstract and more concrete to the more abstract and less concrete. It may be noted, also, that the other senses, especially the sense of hearing, may be made to co-operate with sight in the evolution of the times idea. Appeal by a variety of examples to the trusty eye, but appeal also to the trusty ear-strokes on a bell, taps on the desk, uttered syllables, etc. Here, as in all other cases, we do not confine ourselves to single bellstrokes or syllables; we count the number of double strokes, triple strokes; of double and triple syllables, as, for example, oh, oh, oh, oh, oh, oh-i. e., 3 counts of two sounds each, etc.

Counting and Measuring.-In the separating and combining processes referred to, counting goes on. This is at first chiefly mechanical, and care must be taken in the interest of the number idea to make it become rational. Through practice in parting and wholing the idea of the function of the unit is gradually formed; it is the concrete, spatial thing used to measure quantity. The point is, not to neglect either the spatial element or the other essential factor in number, the counting, the actual relating process.

In the method of number teaching usually followed, counting is the prominent thing, to the almost total exclusion of the measuring idea ; the emphasis is apon the how many, with but little attention to the how much. But the counting is largely mechanical. There is a repetition of names without definite meaning. The child is groping his way towards the light. He can not help

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 countingone, two, three, etc.—is now'a rational process.

So much ; 80 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

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