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Methods of explaining Subtraction.

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borrowed, why it is borrowed, or what sort of morality that is which permits you to borrow ten' in one direction, and pretends to compensate by 'paying back one' in another, are points which are left in obscurity. Language like this, which simulates explanation and is yet utterly unintelligible, is an insult to the understanding of a child; it would be far better to tell him at once that the process is a mystery, than to employ words which profess to account for it, and which yet explain nothing.

There are two ways in which, with a little pains, the Method of reason of this rule may be made clear even to the youngest

decomposi

tion. class. Thus:--

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“9 cannot be taken from 3; so borrow one of the tens from the 50 (in other words, resolve 53 into 40 and 13). 9 from 13 leaves 4. Set down 4 in the units' place.

7 tens cannot be taken from 4 tens; so borrow i from the 8 hundreds (in other words, resolve 8 hundreds and 4 tens into 7 hundreds and 14 tens). 7 tens from 14 tens leave 7 tens. Set down 7 in the tens' place.

4 hundreds from 7 hundreds leave 3 hundreds. Set down 3 in the hundreds place.”

Now here you will observe that the word “borrowing" is not inappropriate. But there is no paying back; for you have only borrowed from one part of your minuend 853 to another, and dealt with its parts in a slightly different order from that indicated by the figures. You have simply resolved 800 + 50 + 3, for your own convenience, into the form 700 + 140 + 13; and have left the

subtrahend 479 untouched. I do not say this is the best method of working, but it is, at least, easy to explain ; and the language you employ is self-consistent

throughout. Method of The second method is a little harder to explain, but equal addi- easier to work. It is that most generally adopted in tions.

schools. But before beginning to do a sum by it, it is
worth while to explain to your class the very simple
principle that “the difference between unequal quantities
is not altered, if we add equal quantities to both.” If I
have five shillings in one pocket and seven in another,
the difference is two (7 - 5 = 2); but if I afterwards put
three shillings into each pocket, the difference is still
two (10 - 8= 2). By very simple illustration of this
kind you may easily bring children to the conclusion,
that if, for any reason, we think it convenient to add
equal sums to two numbers whose difference we want to
find, we are at liberty to do so without affecting the
accuracy
of the answer.

When this has been explained, the sum may be thus worked :

853 + 100 + 10
479 + 100 + 10

8 hundreds, 15 tens 13
5 hundreds, 8 tens ୨
3 hundreds + 7 tens + 4

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“9 from three cannot be taken. Add 10 to the upper

line. 9 from 13 leaves 4. Set down 4. “Having added 10 to the upper line, I add ten to the lower. 8 tens from 5 tens cannot be taken. Add 10 tens to the upper line, 8 tens from 15 tens leaves 7 tens. Set down 7.

Having added 10 tens, or i hundred to the upper line, we must add 1 hundred to the lower; 5 hundreds from 8 hundreds leave 3 hundreds. Set down 3."

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But here it is observable that you have not performed the problem proposed. You have not taken 479 from 853; but you have added first 10, and afterwards i hundred to each, and the real problem performed has been to take 479 +110, or 500 + 80 +9 from 853 + 110, or from 800 + 150 + 13. But this, according to the principle first explained gives the same result as to take the first number from the second without the addition of the hundred and ten.

Yet the common phraseology employed about borrowing and carrying is equally inappropriate, and therefore equally bewildering, in both these processes. For by the first method there may be borrowing, but there is no carrying; and by the second, there is neither borrowing nor carrying, but equal addition.

Another device to which a good teacher resorts early Learners is the making of the Multiplication Table in the presence may make of the class, and with its help. Generally the whole of

own tables. that formulary is placed before the scholars, and they are required to learn it by heart, without knowing how it is formed or why. Now if the teacher says he is going to make up the table of multiplication by two's, and then writes 2 on the board, and requires the scholars to repeat the number, so that he writes down each result and records at the side the number of twos which have been added, he makes it clear to the scholars that multiplication is only a series of equal additions, and that the rule is only a device for shortening a particular form of addition sum.

He will then deal in like manner with each of the 9 digits in succession, and afterwards efface what he has written and require the scholars to manufacture their own table before learning it.

One very effective way of making the theory of a Arithmetiprocess clear, is to adopt the method to which I may cal parsing give the name of arithmetical parsing.' It consists in drawing out before the class the whole of a given process without abridgment, and then analysing it in such a way that a separate account shall be given of every figure in succession, shewing clearly how and why it plays a part in obtaining the final result. I take an example from Simple Division although almost every other rule would do as well. I will suppose that by simple examples you have shewn what Division is, that you have deduced from the division of the parts—say of a shilling, and from some such example as this :

Because 27 = 12 + 9 +6 Therefore the third of 27 or 1 = 1 + + for 4 +3+2, the general truth that we divide one number by another when we divide each of the parts of the first successively by the second, and add the quotients together.' It is then seen that when the dividend is a large number, it has to be resolved into such parts as can be dealt with one by one, in order that all the several results as they are obtained shall be added together to make the whole. An example may be worked thus : Divide 34624 by seven :

7) 34624

4000 = 28000 = 7
900 = 6300 - 7
40=

280 = 7
6 =

42 -- 7 움 2 +7 4946 = 34624 = 7

=

This method of analysis is especially effective in what is called Long Multiplication, in Division, and also in Practice; for in these rules the answer comes out piece

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meal, and it is both easy and interesting to challenge pupils for the separate significance and value of each line of figures as it is arrived at.

In the exercise just given it is well to call attention to the fact that the whole problem has not in fact been solved, for that all the dividend except 2 has been divided; but the seventh part of two remains undiscovered, and must for the present remain in the form ; or the seventh part of two.

Here then is the proper place to begin the explana- Fractions tion of fractions. They ought not to be postponed later,

should be

begun certainly not placed as they often are most improperly, early. after proportion. The remainder of a division sum suggests the necessity of dealing with the parts of unity. Here an appeal may be made to the eye :

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and it may be demonstrated that one seventh of two inches is the same as two sevenths of one inch. I need

that in your early lessons on fractions, the method of visible illustration is especially helpful, and that by drawing squares or other figures and dividing them first into fourths and eighths, then into thirds, sixths and ninths, or by the use of a cube divided into parts, you may make the nature of a fractional expression very evident even to young children, and may deduce several of the fundamental rules for reduction to a common denominator, and for addition and subtraction.

Fractions afford excellent discipline in reasoning and reflection. No one of the rules should be given on authority, every one of them admits of being thought out and arrived at by the scholars themselves, with very

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