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2. But he must, never be allowed to carry his desire for abbreviation so far as to write (which he will probably do if not checked) "5 apples 24d." or "5 apples cost 2 (omitting the d which denotes pence).'

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3. Before beginning any sum of this kind, the pupil should be asked to give a rough answer to the problem by common sense. Thus, in the question about the price of 21 apples, he should be asked, "Will the price demanded be more or less than 24d.?" More. "How much more?" As much more as 21 apples are more than 5 apples. "And how many times 5 is 21, roughly?" Four times. "Then roughly the new price will be how many times more than 24d.?" About four times. "And 4 times 24d. amount to -?" 10d. This will enable the pupil at once to detect any gross inaccuracy in the answer.

4. In spite of this and other precautions, most children, after "doing" the Method of Unity for a few days, will probably-from natural aversion to thinking-fall into a mechanical way of writing their sums.

When the teacher sees signs of this, he should set the pupil a sum on the same principle, but in a different shape, thus:

"If 5 men do a piece of work in 2 days, how long will it take 23 men?"

The average pupil will do this sum very rapidly, thus:

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It would be a most valuable antidote to thoughtlessness and to the slight conceit that is sometimes bred by a confidence in mechanical methods, to point out the extreme absurdity of this answer, and to convince the boy thereby of the utility (1) of the preliminary question, "Will the time demanded be more or less than the time given?" (2) of the necessity of reasoning, as well as writing figures.

If the pupil had reasoned before he began to write, he would have seen at once that 23 men will take less, not more, time than 5 men to

1 The Author has been for many years in the habit of setting almost every week a sum of this kind in an entrance examination: If 216 lbs. of soap cost £5 88., what will 800 lbs. of soap cost? And a very large number (a fourth or fifth, at least, of those who have attempted it) have stated it thus, omitting not only the sign lbs., but also the signs £. s.:

216: 800 :: 58: 2.

Then, having taken for granted that 5,8 means 58. 8d., they proceed to show that 800 lbs. of soap cost about £1. 58, less than a quarter of the price of 216 lbs.

do the same piece of work, and might further have seen that the answer would be about the fifth part of 24 days.

41. DECIMAL FRACTIONS.

Only one or two hints on this subject will be given. It presents very little difficulty, if ordinary fractions have been thoroughly mastered, and if at first the pupil is constantly reminded of the unexpressed Denominator.

(1) The Rule for the multiplication of Decimals may be illustrated experimentally by small numbers; thus: To multiply .2 by .2. This is the same as, or 1, which is expressed by .04. Similarly, .12 multiplied by .12 12 × 120 == .0144. In these two instances we see the general Rule.

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144

Multiply as in whole numbers, and point off decimal places in the product equal to the sum of the decimal places in the multiplier and multiplicand, adding naughts to the left if necessary to complete the number.

(2) For the Division of Decimals it is a good rule, at all events for beginners, to multiply the Divisor and Dividend by such a power of 10 as to convert both into whole numbers.

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(3) The process of expressing a circulating Decimal as an ordinary Fraction is commonly taught by a mechanical rule which in no way exercises the reasoning powers. But no rule at all should be given. The logical process itself can be easily understood, and is very little longer than the ordinary mechanical one. It should therefore be not only understood, but regularly employed by the pupil, as follows: Express 13.34567 as an ordinary fraction.

The fraction may be represented by F.

Then F=

13.34567567, etc., (1).

Multiply both sides of (1) by 100.

Fx 100 = 1334.567567567, etc., (2).
Multiply both sides of (2) by 1,000.

Fx 100 x 1000 1334567.567567, etc., (3).

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Whence in time the pupil may discover for himself the general rule that:

The Numerator of the new Fraction is formed by subtracting the nonrepeating part of the decimal from the whole decimal, including the whole number, and the Denominator by writing down as many nines as there are repeaters, and as many naughts as there are non-repeaters.

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1. Children should not be allowed (as they are in many schools) to have the answers to their sums.

The possession of the answers encourages them to scamper over the working of a sum without thought about the reasonableness of the method, knowing that "the answer will tell them" whether their method has been correct.

2. As to the correction of errors, it is well that error of miscalculation (as well as method) should be corrected, as far as possible, by the pupil himself. A child should not be allowed to say, or fall into the way of thinking, that a sum "won't come out right." On the contrary, he ought to be made to believe that a sum "must come out right"; and if a sum, right in principle, has resulted in an erroneous answer, owing to some miscalculation, the sum should be returned to him, that he may work it over again and detect his error. "But may not a child weary himself thus endlessly, by repeating some error into which he may have fallen by a temporary lapse of memory, fancying, for example, that 8 × 9 73, or that 28 cwt. = 1 ton? A mistake of this kind cannot be detected by the child himself, though he may labor for twenty-four hours."

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True; and, therefore, after a child has made one attempt to correct an error, the teacher may come to his assistance in one of two ways: (1) either he may tell the pupil to refresh his memory as to "8 times" in the Multiplication Table, or to study the Table of Weights and Measures, and then to try again; or else, (2) the pupil may work the sum aloud before the teacher, and have his error or errors pointed out; but, in either case, the teacher should not be satisfied till the pupil has worked through the whole sum by himself correctly.

(3) Children should be taught to be slow, and to think, in the reasoning part of arithmetic, but to be quick, and not to think, in the mechanical part.

If a child is slow in calculation, he is likely to be inaccurate; for his slowness increases the chance that he will not be able to keep up the strain of attention for the necessary time to finish the process of calculation.

Working in competition with others, working against time, and constant repetition of Tables, whenever an error in any Table has been made—these are the best means for securing rapidity in the mechanical processes of Arithmetic.

ENGLISH COMPOSITION AND GRAMMAR.

43. A NATURAL STYLE.

In teaching children English Composition, the teacher must be on his guard against destroying the naturalness of their style. A child must not be expected to use the ample vocabulary or flexible phraseology of his elders; and if his rudimentary attempts at composition are corrected by the standard of a mature composer, he is likely to be discouraged by the multiplicity of the corrections, and also to fall into a premature and affected employment of language that has for him but little meaning. Of all dangers, artificiality in composition is the most to be avoided. It is difficult, and, indeed, hardly possible, to recover the power of writing naturally when once lost; and an unnatural style is an obstacle to thinking clearly, as well as to writing forcibly. Therefore:

Let children write, as they speak and as they think, after the manner of children.

But of course we are not to leave children at a stand, to be children always in thought and language. We must endeavor both to improve their style and to develop their faculty of thought, taking care that the former may keep pace with, but not outrun, the latter. Even a child can understand, besides grammatical errors, (1) the mischief of ambiguity. (2) the utility of brevity, and (3) to some extent, the superiority of pointed, forcible, and picturesque expressions over those which are flat, dull, and colorless. Later on, he may also be made to understand (4) the advantages of order. On the whole, we may say, as a general rule:

Let the teacher insert no correction of which the pupil cannot see the advantage.

44. THE USE OF CONVERSATION FOR THE PURPOSE OF COMPOSI

TION.

It is obvious that if we are to improve a child's power of writing, and yet to encourage him to write as he speaks, we must not allow him to speak in a slovenly way.

Care will be required here, on the one hand, not to pass over so many inexact or uncouth expressions as to confirm the child in bad habits of expression, and, on the other hand, not to correct him so constantly, especially before strangers, as to make the very act of speaking a burden to him. It must be remembered, also, that in conversation, as in everything else, the child will imitate those around him, and will be fluent or hesitating, exact or inexact, weak or forcible, very much after the pattern of those with whom he has to do.

45. THE USE OF LETTERS IN COMPOSITION.

The best exercises for young children are letters; because in letters they may most easily acquire the art of writing naturally, that is, the art of writing as one would speak.

If possible, the letters should be bona fide, i.e., letters written to some one who is, or may be supposed to be, interested in reading them. They may be corrected by pointing out, (1) how nouns or pronouns have been unnecessarily repeated; (2) how facts have been inexactly or ambiguously expressed; (3) how incidents, or features in incidents likely to be interesting to the intended reader, have been omitted by the child. Corrections of the kinds (1) and (2) will increase ease, neatness, and exactness; corrections of the kind (3) will increase picturesqueness of style.

Without any direct praise of the style, the teacher may sometimes apply a useful stimulus by finding occasion to say of a better letter than usual, "I think will be interested in reading this letter."

"In correcting letters," says Preceptor, "the teacher must carefully distinguish between differences of thought and differences of expression, and must very seldom correct the former. For example, a brother and sister having seen a hare in the field, may describe the sight in two totally different manners. I remember that a boy of a statistical and matter-of-fact turn of mind, actually described such a sight thus: 'Yesterday, while we were driving along the road to a hare started in a field about twenty yards to the right of us, and ran some sixty yards in a northwest direction, after which, it turned into a wood on the left, and disappeared.' But his sister, seeing precisely the same thing, might describe it thus: 'As we were out on a drive, we saw such a pretty brown hare, quite close to us; and as soon as it saw us, it rushed away over the grass, and hid itself in a thick wood.'"

Here our teacher will probably agree with Preceptor that it would be equally unwise to try to make the boy's style more picturesque, and the girl's more statistical. Each must write what is in his own mind. All that the teacher can do with advantage will be to make an occasional comment on the boy's style, to the effect that "will not be much interested in this letter;" or, on the girl's, that will not be able to understand from this letter when, or where, this or that took place, or how it happened."

46. THE USE OF TALES IN COMPOSITION.

Letters cannot be regularly used as exercises; for it cannot regularly happen that a child will have matter for a letter; and more harm than good will be done by compelling him to write letters when he really has nothing to say.

Another method of teaching English Composition is to tell children

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