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Nursery jingles, such as
One, two, buckle my shoe ;
may be conveniently and profitably employed, primarily as aids to the memory. Thus, he will learn addition, subtraction, multiplication, and division, and a part of the multiplication table before he is taught anything but counting.
Those who attempt to teach numbers by means of objects do not sufficiently discriminate between the origin and the nature of number. In practical affairs we have to deal with things. Number is one of the means by which we classify and express our ideas in regard to things. We see a certain object in nature. We give that object a name which shall represent it and call it to mind whenever we hear the name spoken or see it upon the written or printed page. Instead of simply man, tree, rock, we may have man, man ; tree, tree; rock, rock. We must have a name to represent that which is common to these and all other similar groups.
That name is two. Thus, two always means precisely the same thing, whether we say simply two, or two men, or two feet, two units, two tens, or two anything else. That which is common to man, man, man; tree, tree, tree ; rock, rock, rock ; etc., we name three. The general name for one, two, three, etc., is number. Our dealing with objects calls for the theory of numbers. The theory is developed as a pure science, - for it can be nothing else, — and is then applied to objects for practical use.
The naturalness of the method here advocated has, as I trust, been fully demonstrated. Its simplicity must also be already apparent. But there is a still further reason why it must sooner or later be universally adopted. It is the only method by which the various arithmetical operations can be logically and consistently explained. Warren Colburn in his preface says:
It is remarkable that a child, although he is able to perform a variety of examples which involve addition, subtraction, multiplication, and division, recognizes no operation but addition. Indeed, if we analyze these operations when we perform them in our minds, we shall find that they all reduce to addition. They are only different ways of applying the same principle. And it is only when we use an artificial method of performing them that they take a different form.”
In this paragraph Warren Colburn condemns his own method in the strongest possible manner. Neither he nor any of his disciples have ever attempted the task, from their standpoint an impossible one, of explaining division, except in a partial and incomplete manner, in any such way as is here mentioned. According to them all, division is "the process of finding how many times one number is contained in another, or of finding one of the equal parts of a number.” Thus, division is not one thing, but sometimes one thing and sometimes something else. And how do they find one of the equal parts of a number? Here is the method as explained by Fish (“ The Complete Arithmetic,” by Daniel W. Fish, 1875):
"Ex. If 4 pencils cost 20 cents, what is the cost of 1 pencil ?
“Since 5 cents taken 4 times equals 20 cents, it follows that 5 cents is one of the four equal parts of 20 cents (5 + 5 + 5 + 5 = 20), and we say one-fourth of 20 cents is 5 cents."
As a demonstration of the correctness of the result obtained this reasoning is perfectly sound, but as an explanation of division it is a mere juggling with words, for to say that one thing is the reverse of another is to cite what may be but an accidental circumstance. To show the fallacy still more clearly, let the cost be 23 cents instead of 20. Now what we have is this: Since 5 cents taken 4 times equals 23 cents, it follows that 54 cents is one of the four equal parts of 23 cents (54 +58 +53 +54 = 23), and we say one-fourth of 23 cents is 54 cents. As we have been finding equal parts, the i must be either 3 of the 4 equal parts into which 1 is divided, or,
more logically, 1 of the 4 equal parts into which 3 is divided. In either case it is an expression of a performed division, and so we are attempting to explain division by division itself. But if we start with a conception according to which number is or can be anything else than abstract, this result is inevitable, unless - and this most unscientific procedure is the one almost universally adopted — we carefully select our illustrative examples so that the difficulty may not be suggested. We cannot find how many times 4 pencils are contained in 23 cents, and, if cents are to enter into the demonstration, there is nothing left to try except to find one of the 4 equal parts of 23 cents. The conclusion is irresistible, division cannot be explained by means of cents and pencils. Right here, then, is a complete demonstration that numbers are always abstract and that division is not essentially a reversed multiplication.
If 5 men can perform a certain amount of work in 2 days, how many men would it take to do the work in 4 days?
Following the directions in the books, we obtain the result, 24 men. But from their standpoint this answer is wrong. } a man, according to their definitions of a fraction, is 1 of the 2 equal parts into which a man is divided. This piece of a man could do no work, for it would be dead. There would be but 2 men capable of performing work, and 2 men are not enough. So, notwithstanding we have made no mistake in figuring, we have not succeeded in obtaining a correct answer. But if we start with a conception according to which number is or can be anything else than abstract, this result is inevitable. Here, then, is a demonstration that an arithmetical fraction is not a piece or a collection of pieces, and a second demonstration that numbers are always abstract.
What we now call algebra was formerly regarded as part of arithmetic, and the change in the name does not alter the nature of that to which the name is given. The one science
is merely an outgrowth and extension of the other, and, for this reason, any presentation of arithmetical operations which cannot be carried into algebra is illogical and unscientific.
If Charles has 9 apples, and should give 4 of them away, how many apples would he have left ?
" To find how many he would have left, we take 4, a part of 9, away, and, by counting or otherwise, find there are 5 left; thus we know that he would have 5 apples left.”
If this is the way we teach subtraction in arithmetic, what shall we do when we come to algebra, and have to subtract 9 from 4 ?
Numerous other difficulties and inconsistencies might be pointed out, but enough has been said to show the fallacies of existing methods. For the NATURAL method I have but one favor to ask, – that it be judged fairly and impartially upon its merits.
J. F. B. MAY, 1892.