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child to grasp.” Yes, if it is treated simply as an accumulation or aggregation of individual separate fixed units. Very few adults can definitely grasp 100 in that sense. The Grube method, proceeding on the basis of the separate individual thing as unit, is quite logical in insisting upon exhausting all the combinations of all the lower numbers. No, if 100 is treated as a natural whole of value, needing to be definitely valued by being measured out into sub-units of value. One dollar is one, we repeat, as much as one block or one pebble, but it is also (which the block and pebble as fixed things are not) two 50's, four 25's, ten 10's, and so on.
It may be well to remind the reader that while we are dealing here only with the theory of the matter, yet the successful dealing with such magnitude as the dozen and the dollar is not a matter of theory alone. Actual results in the schoolroom more than justify all that is here said on grounds of psychology. During the six months in which a child is kept monotonously drilling upon 1 to 5 in their various combinations, he may, as proved by experience, become expert in the combination of higher numbers, as, for example, 1 ten to 5 tens, 1 hundred to 5 hundred, etc. If the action of the mind is judiciously aided by use of objects in the measuring process which gives rise to number, he knows that 4 tens and 2 tens are 6 tens, 4 hundred and 2 hundred are 6 hundred, etc., just as surely as he knows that 4 cents and 2 cents are 6 cents; because he knows that 4 units of measurement of any kind and 2 units of the same kind are 6 units of the same kind. Moreover, this introduction of larger quantities and larger units of measurement saves the child from the chilling effects
of monotony, maintains and even increases his interest in numerical operations through variety and novelty, and through constant appeals to his actual experience.
Many a child who has never seen “four birds sitting on a tree and two more birds come to join them, making in all six birds sitting on the tree,” has heard of one of his father's cows being sold for $40 (4 tens), and another for $20 (2 tens), making in all 6 tens or $60; or of one team of horses being sold for 4 hundred dollars, and another for 2 hundred dollars, in all 6 hundred dollars.
When it is urged that these higher numbers are beyond the child's grasp, what is really meant? If the meaning is that the child can not picture the hundred, cannot visualise it, this is perfectly true; but it is about equally true in the case of the adult. No one can have a perfect mental picture of a hundred units of quantity of any kind. Yet we all have a conception of a hundred such units, and can work with this conception to perfectly certain and valid results. So a child, getting from the rational use of concrete objects as symbols of measuring units the fact that 4 such units and 2 such units are 6 such units, gets a clear enough working conception for any units whatever. The opposite assumption proceeds from the fallacy of the fixed unit method, and from the kindred fallacy that to know a quantity numerically we must mentally image its numerical value; grasp in one act of attention all the measuring parts contained in the quantity. Neither adult nor child, we repeat, can do this. We can not visualise a figure of a thousand sides—perhaps but few of us can "picture" one of even ten sides—but we nevertheless know the
figure, have a definite conception of it, and with certain given conditions can determine accurately the properties of the figure. The objection, in short, proceeds from the fallacy that we know only what we see; that only what is presented to the senses or to the sensuous imagination is known; and that the ideal and universal, the product of the mind's own working upon the materials of sense perception, is not knowledge in any true sense of the word.
To sum up: One method cramps the mind, shutting out spontaneity, variety, and growth, and holding the mind down to the repetition of a few facts. The other expands the mind, demanding the repetition of activities, and taking advantage of dawning interest in every kind of value. One method relies upon sheer memorising, making the “memory” a mere fact-carrier; the other relies upon the formation of habits of action or definite mental powers, and secures memory of facts as a product of spontaneous activity. One method either awakens no interest and therefore stimulates no developing activity; or else appeals to such extrinsic interest as the skilful teacher may be able to induce by continual change of stimulus, leading to a varying activity that produces no unified result either in organising power or in retained knowledge. The other method, in relying on the mind's own activity of parting and wholing-its natural functions-secures a continual support and re-enforcement from an internal interest which is at once the condition and the product of the mind's vigorous action.
THE DEVELOPMENT OF NUMBER ; OR, THE ARITHMETICAL
NUMERICAL OPERATIONS AS EXTERNAL AND AS
INTRINSIC TO NUMBER.
ADDITION, SUBTRACTION, MULTIPLICATION.—As we have already seen, number in the strict sense is the measure of quantity. It definitely measures a given quantity by denoting how many units of measurement make up the quantity. All numerical operations, therefore, are phases of this process of measurement; these operations are bound together by the idea of measurement, and they differ from one another in the extent and accuracy with which they carry out the measuring idea.
As ordinarily treated, the fundamental operationsaddition, subtraction, etc.—are arithmetically connected but psychologically separated. Addition seems to be one operation which we perform with numbers, subtraction another, and so on. This follows from a misconception of the nature of number as a psychical process. Wherever one is regarded as one thing, two as two things, three as three things, and so on, this thought of numerical operations as something externally performed upon or done with existing or ready
made numbers is inevitable. Number is a fixed external something upon which we can operate in various ways: it simply happens that these various ways are addition, subtraction, etc.; they are not intrinsic in the idea of number itself.
But if number is the mode of measuring magnitude -transforming á vague idea of quantity into a definite one—all these operations are internal and intrinsic developments of number; they are the growth, in accuracy and definiteness, of its measuring power. Our present purpose, then, is to show how these operations represent the development of number as the mode of measurement, and to point out the educational bearing of this fact.
The Stages of Measurement.—We have already seen that there are three stages of measurement, differing from one another in accuracy and definiteness. We may measure a quantity (1) by means of a unit which is not itself measured, (2) with a unit which is itself measured in terms of a unit homogeneous with the quantity to be measured, (3) with a unit which is not only defined as in (2), but has also a definite relation to some quantity of a different kind. If, for example, we count out the number of apples in a peck measure, we are using the first type of measurement; there is no minor unit homogeneous with the peck measure by which to define the apple. If we measure the number of pounds of apples, we are using the second type of measurement; each apple may itself be measured and defined as so many ounces, and as therefore capable of exact comparison with the total number of pounds. In this case we have a continuous scale of homogeneous