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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.
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 ineasure 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.
THE DEFINITION, ASPECTS, AND FACTORS OF NUMERICAL
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.
2. Numerical Value. This expresses how many of 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 the mind constructs defined parts into a unified and definite whole. Absolute value (quantity numerically defined) is represented by the application of this how many to magnitude, to quantity—that is, to limited quality. To take an example of the confusion referred to: we are told that division is dividing a (1) number into a (2) number of equal (3) numbers. This definition as it stands has absolutely no meaning ; there is confusion of number with measured quantity. Doubtless the definition is intended to mean : division is dividing a certain definite quantity into a number of definite quantities equal to one another. Only in (2), in the definition as quoted, is the term number correctly used ; in both (1) and (3) it means a measured magnitude. A measured or numbered quantity may be divided into a numnber of parts, or taken a number of times; but no number can be multiplied or divided into parts. Number simply as number always signifies how many times one “so much,” the unit of measurement, is taken to make up another “so much,” the magnitude to be measured. It is, as already said, due to the fundamental activities of mind, discrimination, and relation, working upon a qualitative whole; and we might as well talk of multiplying hardness and redness, or of dividing them into hard and red things, as to talk of multiplying a number or of dividing it into parts.
It may be observed that the problems constantly used in our arithmetics, multiply 2 by 4, divide 8 by 4, are legitimate enough provided they are properly interpreted, if not orally at least mentally, but taken literally are absurd. The first expression means, of course, that a quantity having a value of two units of a certain kind