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Now, let us suppose that the multiplicand, instead of being a whole number, had a dot (.) between the 3 and 6 ; then, of course, we have used it as if it were 1000 times greater than it is ; consequently, our product is 1000 times too large, but can easily, as we have seen in page 42, be made 1000 times smaller, by placing a dot between the two fours, which removes the unit's place from the first to the fourth figure from the right hand. Again : suppose that the multiplier, instead of 842, should have been 8.42 ; then we shall have multiplied by a number 100 times too large ; consequently, our product, being also 100 times too large, will have to be changed once more, by removing the dot two places more to the left, between the 1 and 8. By repeating a similar process and explanation, a few times on the black board, an intelligent child, unprompted, will come to the conclusion, that there must always be as many decimal places in the product as in both factors; and, vice versa, as many in both factors as in the product. Now, multiplication of decimals differs from that of integers in no other respect, than in having a dot to show the place of units.
In division, let it be taken for granted, that the child knows that the divisor and quotient are factors of the dividend. Or, if he does not, he may learn it, by being asked, what two numbers are the factors of the third, in the following example in division :
Now, then, let us suppose, that it is required to divide 24 by 8. The quotient, you observe, is 3. But suppose the number had been 2.4, what would the quotient then be? The numbers 8 and 2.4 are given, and, of course, not to be altered, then what is this 3 ? Recollect, you yourself found out, that there must be as many decimal places in both factors, as in the product. Now, you say, that 2.4 is the product, in which there is one decimal place; and, as there is none in the factor 8, how many must there be in the other factor ? One, you say ; that is, the quotient is 3 tenths.
But suppose our dividend, or product, had been .24 instead of 2.4. Here, then, the product having two decimal places, whilst there is none in the divisor-factor, of course, there must be two in the quotient-factor. But, , as there is only one figure in the quotient, we must place a 0 for the other; and the question is, whether it should be put after it, or before it. If we place it after it, and then place our dot .30, we find it to be useless, because it does not change the value of the 3; therefore, it must go before,.03.
The last possible case of any difficulty is, when the divisor is a decimal, thus .8, and the product, or dividend, a whole number. The rule that we discovered, above, was, that there must be as many decimal places in the product, as in both factors. Now, there being one in the given factor, and none in the product, we must put one in the product, but without changing its value ; thus, 24.0. If we now perform the division, we shall have 30 for the quotient, which is the correct answer.
It will probably appear somewhat extraordinary to a child, that when 24 is divided by .8, the quotient should be larger than the original number, viz. 30; and this would be a proper place to explain the apparent contradiction, which may be done as follows : Dividing one number by another, is finding how many times the one number is contained in the other. How many ls are there in 24 ? Answer, 24. But, as .8 is less than one, of course there must be more of them, viz. 30.
It is probable, that all this may appear a little complicated in the reading ; but, when shown on the blackboard, it will be readily understood by every child who can understand division of integers ; especially if they have been properly trained to habits of attention by the other exercises of the school.
Having thus shown, that, from their great similarity, decimals should be taught simultaneously with integers, or, at all events, alternately ; that is, 1. Numeration of integers ; 2. Numeration of decimals ; 3. Addition of integers ; 4. Addition of decimals, and so on; it may be proper to notice, in this connexion, another error committed in most of our treatises on Arithmetic, leading to similar results; that is, leading students to imagine a difficulty, where there is none in reality. The error, alluded to, consists in making separate articles of addition, subtraction, multiplication, and division, of Federal money, as if they, in any respect, differed from the same processes in decimals. There is, generally, even an article devoted to reduction of Federal money; as if any thing more was necessary, in reducing one sort of coin into another, than a mere change of the unit's place. Thus,
45678 mills may be changed into cents, by placing a dot between the 7 and 8, thereby dividing it by 10, the number of mills in a cent; into dimes, by placing the dot between 6 and 7; into dollars, by placing it between 5 and 6 ; and into eagles, by placing it between 4 and 5; reduction, descending, carrying the dot backwards on the same principle.
There is, also, a deficiency, when treating of Federal money, in not explaining the etymology of the names of the coins, which, when understood, makes the subject perfectly plain. It might be done thus : a dollar is considered the unit; the meaning of dime, is tenth ; of cent, hundredth ; of mill, thousandth; consequently,
44.368 dollars may be read, either 44 dollars, 3 tenths, (or dimes,) 6 hundredths, (or cents,) and 8 thousandths, (or mills ;) or 44 dollars, 36 hundredths, (or cents,) and 8 thousandths, or (mills ;) or 44 dollars, and 368 thousandths, (or mills ;) or, finally, as 10 dollars make an eagle, (so called, from an eagle being the ensign, or arms of the United States, the 44 dollars may be considered as 4 eagles and 4 dollars.
Before leaving the subject of decimals, it may not be improper, here, to enter a protest against the novelty introduced by some authors, of using a comma instead of a dot, to indicate the place of units ; that character being already appropriated to a different purpose, viz., the division of large numbers into series of threes, for the convenience of reading them readily. Thus, when the comma is used for both, it is impossible to say whether
346,789 stands for 346 thousand and 789 ; or 346 and 789 thousandths. Other authors, again, have, very improperly, taken the liberty of leaving out the separating commas, altogether, in large numbers ; thus rendering them exceedingly difficult to be read. The eye can readily distinguish four figures, at a glance. But, when the number consists of more than four, they should always be separated, by commas, into series of threes.
But the chief impediment to the thorough acquisition of arithmetic lies in the multiplicity of Rules, with which our popular treatises are burdened. Perhaps it would be well, if these books consisted merely of a collection of questions, systematically arranged, so as to lead the students into a knowledge of the principles of the science, somewhat on the plan of Colburn's Mental Arithmetic.' Every teacher ought to be able to give a clear demonstration of the whole subject on the blackboard, on simple, philosophical principles. If this were properly done, children might be completely versed in arithmetic, by the time they reached the age of eight or nine. A specimen will be given, in another part of the treatise, of the manner of conveying this sort of practical instruction on the black board, and, at the same time, a more simple and natural classification of the whole subject will be offered. We shall attempt to supply, also, a very important omission on the part both of books and teachers, viz., the different modes in which the working-out of questions may be shortened by cancelling, division, &c. By such abbreviations, not only a vast amount of time and labor is saved, but several of the most important principles of arithmetic are brought out and elucidated.
The last error to be noticed, on the subject of written arithmetic, is the prevailing custom of furnishing readymade tables of addition and multiplication for the student. This is, probably, one of the most injurious species of labor-saving machinery. The habit of relying on readyformed tables, instead of forming them for one's self, may be classed with that of depending on mechanical rules, and the practice of getting the teacher to work out the difficult problems, instead of simplifying the subject by a few leading questions. These are perversions, which tend to convert a science, that is probably one of the best adapted to lead the mind into trains of reasoning, and habits of patient investigation, into a means of leading youth into implicit, blind reliance on the dicta of books and teachers : an admirable method of preparing the community to become the ready dupes of demagogues, quacks, and fanatics.
Mental Arithmetic. This branch of the science of numbers, properly taught, has a tendency to produce the most valuable results. Colburn's little work, on this subject, is by far the best that has yet appeared. It is nothing but a mere collection of questions, but so admirably arranged, that the student can proceed onward, step by step, from the simple question of how many fingers are on the hand, to the most complicated and abstruse problems, almost without the aid of a teacher, and all the while forming his own rules. the acquisition of arithmetic is one of the smallest advantages that may be derived from the proper use of this work. If the child is not allowed to study it ; if he is never allowed to see the questions at all ; and if they are never read to him till his class is called upon to recite ; and if he is regularly required to state the manner in which he worked them, and his reasons for the adoption of such a course, he will thus acquire habits of strict attention to the person by whom he is addressed ; of patient investi