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would be little room for the abstracting and relating action involved in all number. A numerical unit is not merely a whole, a unity in itself, but is, as we have seen, a unity employed as a means for constructing or measuring some larger whole. Only this use, then, transforms the object from a qualitative unity into a numerical unit. The sequence therefore is: first the vague unity or whole, then discriminated parts, then the recognition of these parts as measuring the whole, which is now a defined unity-a sum. Or, briefly, the undefined whole; the parts; the related parts (now units); the sum.

(c) Beginning with the numbers in themselves, as represented by mere symbols, or with perceived objects in themselves, there is no intrinsic reason, no reason in the mind itself, for performing the operations of putting together parts to make a whole (using the unit to measure the magnitude), or of breaking up a whole into units-discovering the standard of reference for measuring a given unity. These operations,* from either of these standpoints, are purely arbitrary; we may, if we wish, do something with number, or rather with number symbols: the operations are not something that we must do from the very nature of number itself. From the point of view of the constructive (or psychical) use of objects, this is reversed. These processes are simply phases of the act of construction. Moreover, the operations of addition, multiplication, division, etc., in the method of perceived objects, have to be regarded as

*It will be shown in a later chapter that all numerical operations grow out of this fundamental process.

physical heaping up, physical increase, physical partition; while in that of number by itself they are purely mental and abstract. From the standpoint of the psychological use of the things, these processes are not performed upon physical things, but with reference to establishing definite values; while each process is itself concrete and actual. It is not something to be grasped by abstract thought, it is something done.

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Finally, to teach symbols instead of number as the instrument of measurement is to cut across all the existing activities, whether impulsive or habitual. To teach number as a property of observed things is to cut it off from all other activities. To teach it through the close adjustment of things to a given end is to re-enforce it by all the deepest activities.

All the deepest instinctive and acquired tendencies are towards the constant use of means to realize ends; this is the law of all action. All that the teaching of number has to do, when based upon the principle of rationally using things, is to make this tendency more definite and accurate. It simply directs and adjusts this process, so that we notice its various factors and measure them in their relation to one another. More

The complications introduced in schools-e. g., that you can not multiply by a fraction, nor increase a number by division, etc., because multiplication means increase, etc.—result from conceiving the operations as physical aggregation or separation instead of synthesis and analysis of values-mental processes. To multiply $10 by one third is absurd if multiplication means a physical increase; if it means a measurement of value, taking a numerical value of $10 (a measured quantity) in a certain way to find the resulting numerical value, it is perfectly rational.

over, it relies constantly upon the principle of rhythm, the regular breaking up and putting together of minor activities into a whole; a natural principle, and the basis of all easy, graceful, and satisfactory activity.

CHAPTER V.

THE DEFINITION, ASPECTS, AND FACTORS OF NUMERICAL IDEAS.

WE may sum up the steps already taken as follows: (1) The limitation of an energy (or quality) transforms it into quantity, giving it a certain undefined muchness or magnitude, as illustrated by size, bulk, weight, etc. (2) This indefinite whole of quantity is transformed into definite numerical value through the process of measurement. (3) This measuring takes place through the use of units of magnitude, by putting them together till they make up an equivalent value. (4) Only when this unit of magnitude has been itself measured (has itself a definite numerical value) is the measurement of the whole magnitude or construction of the entire numerical value adequate. Forty feet denotes an adequately measured quantity, because the unit is itself defined; forty eggs denotes an inadequately measured quantity, because the unit of measure is not definite. Were eggs to become worth, say, twenty times as much as they are now worth, they would be weighed out by the pound-that is, inexact measurement would give way to exact measurement. Having before us, then, the psychological process which constitutes measured quantity, we may define number.

DEFINITION OF NUMBER.-The simplest expression of quantity in numerical terms involves two components:

1. A Standard Unit; a Unit of Reference.-This is itself a magnitude necessarily of the same kind as the quantity to be measured. Or, as it may be otherwise expressed, the unity of quantity to be measured and the unit of quantity which measures it are homogeneous quantities. Thus, inch and foot (measuring unit and measured unity), pound and ton, minute and hour, dime and dollar are pairs of homogeneous quantities.

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2. Numerical Value.-This expresses how the standard units make up, or construct, the quantity needing measurement. Examples of numerical value are the yard of cloth costs seventeen cents; the box will hold thirty-six cubic inches; the purse contains eight ten-dollar pieces. The seventeen, thirty-six, eight represent just so many units of measurement, the cent, the cubic inch, the ten-dollar piece; they express the numerical values of the quantities; they are pure numbers, the results of a purely mental process. The numerical value alone represents the relative value or ratio of the measured quantity to the unit of measurement. The numerical value and the unit of measurement taken together express the absolute value (or magnitude) of the measured quantity.

In the teaching of arithmetic much confusion arises from the mistake of identifying numerical value with absolute magnitude—that is, number, the instrument of measurement with measured quantity. Number is the product of the mere repetition of a unit of measurement; it simply indicates how many there are; it is purely abstract, denoting the series of acts by which

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