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difference is in favour of the making-up method, as in. volving less demand upon conscious attention. However this may be, it is certainly known from actual knowledge of school practice that pupils who have been instructed under psychological methods have had but little difficulty in comprehending the making-up method, and have quickly acquired skill in the application of it.
Fundamental Principles of Addition and Subtraction.— When a quantity is expressed by means of several terms connected by the signs + and —, the expression is called an aggregate ; and when the several operations are performed the result is the total or sum of the aggregate. Some of the fundamental principles connecting the operations of addition and subtraction are:
(1) If equals be added to equals, the wholes are equal.
(2) If equals be subtracted from equals, the remainders are equal.
(3) Adding or subtracting zero from any quantity leaves the quantity unchanged.
(4) Changing the order of performing the additions and subtractions in any aggregate does not change the total or sum of the aggregate.
The pupil can use these principles, and abstract recognition of them will come in good time.
MULTIPLICATION AND DIVISION.
Multiplication. From the preceding discussion (see especially page 109 et seq.) of multiplication as a stage in the development of number, it is clear that certain points are to be kept steadily in view, if the process is to be made really intelligible to the pupil.
1. It is not simply addition of a special kind. It means development and conscious use of the idea of number—that is, of the ratio of the measured quantity to the unit of measure, whatever the magnitude of the unit may be in terms of minor units. In counting with a 1-unit measure, one, two, three, ... nine, the number is known when the unit it names is recognised as the ninth in a series of nine units constituting a whole when, that is, the defined quantity is grasped as nine times the unit of measure.
2. In the development of the measuring process (as in the exact stage of measurement) there is the explicit recognition that the measuring unit is itself measured off into a definite number of minor units. This gives rise to the process of multiplication, and of course to a inore definite and adequate idea of number as denoting times of repetition of the unit to make up or equal the magnitude. Nine times one is nine is understood in its full significance. 15
3. A quantity expressed in terms of a given unit of measure is, by multiplication, expressed in terms of the minor units in the given unit of measure; in other words, for the number of derived units in the quantity is substituted the number of primary units in the quantity. If we buy 7 barrels of flour at $5 a barrel, the measured cost is $5 x 7; seven units of $5 each. By multiplication this is changed to $35—i. e., $1 x 35. This product, as it is called, this new measurement, is not seven fives. It denotes the same quantity under a different though related measurement; it is thirty-five ones. In one of these measurements the number is seven, in the other it is thirty-five.
4. The multiplicand must always be regarded as a unit of measure—a measure made up of primary units; and the operation looked upon as simply making the quantity more definite by expressing it in a better known or more convenient unit of measure.
5. While the multiplicand as multiplicand must always be interpreted to mean measured quantity, we can take either factor as multiplier or multiplicand. This idea must be used from the first, even in the primary stage. In finding the number of primary units (dollars) in 12 yards of velvet at $5 a yard, there is no known law that decrees 12 as unchangeably the multiplier, and $5 as the only multiplicand. On the contrary, by a necessary law of mind, every measuring process has two phases, and so the measurement $5 x 12 carries with it the measurement $12 x 5. Only a total misconception of number and the measuring process could prompt the question, How can 12 yards become $12 ? The proposition $5 x 12 = $12 x 5, is not a proposition about things; it is a proposition concerning a psychical process—the mind's mode of defining and interpreting a certain quantity. This principle of measurementof interchange of times and parts—is essential to the proper understanding of numerical operations, and can from the beginning be intelligently used. Intelligent use leads to perfect mastery. The problem of multiplication then is : Given the number of unit-groups in a ineasured quantity, and the number of minor units in each unit-group, to determine, from these related factors, the number of minor units in the quantity.
The Formal Process of Multiplication.-It may be well to consider the logical steps in learning the process :
(1) The multiplication of a quantity by powers of ten. Beginning with some ultimate or primary unit of measure, we conceive a measured quantity as making up ten such units—that is, we multiply the unit by ten ; we may further conceive this 10 unit quantity used as a unit of measure, and repeated ten times to make up a larger quantity—that is, the 10-unit quantity is multiplied by ten to express this larger quantity in terms of the minor unit, it is 100 of them, etc. It has already been shown how the notation corresponds with this process. The 1-unit multiplied by 10 becomes 10, the 10-unit multiplied by 10 becomes 100; in other words, the 1 increases 10 times with every removal to the left of the decimal point. So the product of 5 ones is 10 fives or 5 tens—i. e., 50; the product of 5 tens by 10 is 50 tens or 500—i. e., 5 multiplied by 100, etc.
(2) We may find the total product which measures a quantity by finding the sum of partial products. If a given quantity is measured by 4 feet x 28, we may multiply the 4 feet by 20 and by 8, and the sum of the partial products will be the total product-in all 28 times the multiplicand. This is the basis of the work in long multiplication.
(3) We may multiply by the factors of the multiplier. This is using the relation between parts and times. If we have, e. g., a quantity expressed by $2 x 20, it is expressed equally by $2 x 5 x 4-i. e., by $10 x 4; in other words, we have made the measuring unit 5 times as large, and the number of them 5 times as small.
In the following, e. g., we take the multiplicand 5 times, getting the first partial product; in multiplying by 4, we have in fact taken the multiplicand 10 times, and this product 4 times, obtaining the second partial product 11,120.
45 1390 ... 5 times multiplicand. 1112 ... 40 (i. e., 4 times 10 times multiplicand). 12510 = 45 times multiplicand.
Special Processes.—Special processes may be used in many cases. These afford good practice for mental work, and give better ideas of number and numerical operations, as well as preparation for subsequent work. A few of these processes may be noticed.
1. When the multiplier is any of the numbers 11 to 19, the product can be obtained in one line, thus :