be directed to put alternately A and B, or more letters, on their slates. After which, for each letter a separate set of sums will be given out. During the working the teacher, as before, will carefully keep watch and at once warn by name any whose eye wanders or whose lips he may see moving. He should do his utmost to keep all employed at once, either in taking down, or in working, or in proving sums. When he has been teaching new matter, and is testing their knowledge of what he has just taught, he should adopt similar tactics, if he would not be misled by the lazy taking their cue from such as have really grasped the lesson.

From the first a teacher should not merely state but prove by numerous examples and illustrations of every kind, that multiplication is only a short way of adding. Short Division is usually taught before Long, but whether this be really well seems questionable. Certainly children so taught, being from sheer force of habit constantly troubled by a desire to ascertain the remainder, feel the utmost difficulty in tackling sums involving the division of large numbers.

When children have completely mastered the first four simple rules then, and not till then, have they laid a sure and solid foundation for all that has to follow. The next subject usually taught is Compound Addition and Subtraction; but if these are to be taught really intelligently, the children should at the same time receive instruction also in Vulgar Fractions. In the early teaching of this subject, frequent appeals should be made to matters of common observation, and the lessons should be copiously illustrated by such examples as actual sticks equally subdivided, lines drawn on the board, &c. &c. The meaning of every rule should be made thoroughly clear by continual reference to the actual things signified. Children once thoroughly grounded in Fractions will learn all else in Arithmetic, not only with ease, but with pleasure and intelligence. In short, though the subject be not required for examination purposes, it will be found to pay even for them. Unless well grounded in Fractions, children have no solid ground to tread upon when they come to the higher branches of Arithmetic, but can only learn to work by rule of thumb Practice, Proportion, Rule of Three, and all

sums involving fractional tables, or farthings, square measure, &c. As a knowledge of Fractions is not tested by actual examination in elementary schools below the Sixth Standard, and as at present a very small number of children have the advantage of remaining in attendance at school till they reach that Standard, many teachers who have not had the benefit of a thorough grounding in Arithmetic do not take the trouble to master the subject, and are therefore not unnaturally loth to teach it.

Mental Arithmetic has been too much neglected since it has formed no part of the subjects set down for examination, but its omission is a serious loss. A few minutes a week devoted to it would not be lost time, even for the Standard work. Where Mental Arithmetic is practised the children acquire a quick insight into the relations of figures which lessens their paper work, and facilitates their adoption of short and easy methods. These should be always encouraged. It may be as well here to warn young teachers against setting sums which involve long working while girls are at needlework, as is sometimes done merely to fill up time, or to keep boys employed. A number of short problems gives them far better training, though it will of course give the teacher more trouble to set the questions and look over the working.

Of Decimals some teachers seem to entertain still greater dread than of Vulgar Fractions. Yet children who have been taught before the age of seven that figures added to the left increase by powers of ten can surely be taught three years later that figures added to the right decrease by powers of ten. If a teacher's mind be once cleared of the idea that there is any real difficulty about Decimals, and if, whenever it is feasible, he will only refer everything to what children can see, he will be surprised to find what an interest boys at any rate can be made to take in Decimals, as also in squaring and cubing. A child requires at first not merely to be told that ·2 = , but he should be shown that two-tenths of a line are of the same length as one-fifth. Many a child will wonder in his mind why 42 is 16 when he thinks it ought to be 8, until he is shown by actual drawing that a square of which each side is divided into 4 equal lengths

contains 16 squares; why 23 should be 8, not 6, till he is shown by wooden cubes that there are actually eight one-inch cubes in any cube of which each side is two inches long.

If the teacher thus habitually refer figures to things, base his instruction on objects familiar to the children, and set them little problems such as they may any day wish to solve for their own or their friends' use, he may make Arithmetic one of the most popular, as it is one of the most useful, subjects in his school. Otherwise treated, it is necessarily one of the most irksome and wearisome both to teachers and taught. In this, as in all other subjects, it cannot be too often repeated that a teacher will do little good as long as he is content to tell children facts to be believed on his assertion, and to take no trouble to make them teach themselves by reasoning out every step. The following humorous speech of Bartle Massey the schoolmaster in Adam Bede aptly illustrates the above remarks :—

"Now, you see, you don't do this thing a bit better than you did a fortnight ago; and I'll tell you what's the reason. You want to learn accounts; that's well and good. But you think all you need do to learn accounts is to come to me and do sums, for an hour or so, two or three times a week; and no sooner do you get your caps on and turn out of doors again, than you sweep the whole thing clean out of your mind. You go whistling about, and take no more care what you're thinking of than if your heads were gutters for any rubbish to swill through that happened to be in the way; and if you get a good notion in 'em, it's pretty soon washed out again. You think knowledge is to be got cheap; you'll come and pay Bartle Massey sixpence a week, and he'll make you clever at figures without your taking any trouble. But knowledge is not to be got with paying sixpence let me tell you: if you're to know figures, you must turn 'em over in your own heads, and keep your thoughts fixed on 'em. There's nothing you can't turn into a sum, for there's nothing but what's got number in it,— -even a fool. You may say to yourselves, I'm one fool, and Jack's another; if my fool's head weighed four pound, and Jack's three pound three ounces and three quarters, how many pennyweights heavier would my head be than Jack's? A man that had got his heart in learning figures would make sums for himself, and work 'en in his head when he sat at his shoe-making, he'd count his stitches by

1 The one rule in which some who are otherwise fair arithmeticians seem liable to flounder is division of decimals. The difficulties usually experienced disappear if both divisor and dividend be first reduced to whole numbers by multiplication. Thus 13.5 15 135015; 10005 ÷ 106.3 = 100051063000; 1080 008 1080000 ÷ 8.

fives, and then put a price on his stitches, say half a farthing, and then see how much money he could get in an hour; and then ask himself how much money he'd get in a day at that rate; and then how much ten workmen would get working three, or twenty, or a hundred years at that rate —and all the while his needle would be going just as fast as if he left his head empty for the devil to dance in. But the long and short of it is— I'll have nobody in my night school that doesn't strive to learn what he comes to learn as hard as if he was striving to get out of a dark hole into broad daylight. I'll send no man away because he's stupid: if Billy Taft, the idiot, wanted to learn anything, I'd not refuse to teach him. But I'll not throw away good knowledge on people who think they can get it by the sixpenn'orth, and carry it away with them as they would an ounce of snuff. So never come to me again if you can't show that you've been working with your own heads, instead of thinking you can pay for mine to work for you. That's the last word I've got to say to you."

CHAPTER III.

1. Grammar.-2. Composition.-3. Learning by Heart.

1. Grammar.

IN teaching grammar it is a common mistake of young teachers to overload children's minds with definitions and to take parrotlike repetition of phrases for a real knowledge of the things spoken of. A child over eight years of age must have been illtaught or ill-disciplined or both if on being told to underline all the nouns in a passage which he has written down from dictation out of a reading book used in his class, he either omits to do so in the case of many nouns, or, still worse, underlines adjectives and verbs.

After a few simple lessons the child's knowledge should be easily kept up by occasional practice from reading and dictation lessons, which for this purpose may be lengthened from the thirty minutes usually allotted to each to forty or forty-five minutes.

The main difficulties arise from words which are both nouns and verbs-as blow, stroke, love, look, box, cuff, sleep, sow, name, leaves, face, fish, thought, play, rock, walk, tears, &c. These

very difficulties give a teacher opportunities he might otherwise overlook of developing the intelligence of his class in reading. Let it not be forgotten how much in this as in all subjects may be learnt from children's mistakes. The timely and judicious correction of one blunder may enable them to avoid a hundred similar mistakes into which they would otherwise fall. As per contra the leaving one uncorrected may lead to the commission of a hundred more. When sufficient practice has been given in picking out nouns, and distinguishing the three kinds of nouns, the teacher should go on to verbs rather than adjectives, because he will then be able to point out and make clear to the minds of his scholars the simplest form and framework of any sentence. Indeed, a clear-headed teacher may make such good use of his lessons on the noun and verb, and their invariable connection and necessity in every sentence that it will be afterwards comparatively easy to teach them to analyse a simple sentence. As in teaching arithmetic it was suggested as on the whole better that children should be taught fractions immediately after mastering the first four rules, so also it may be urged that analysis of sentences should be taught before syntactical parsing. Quite apart from any question of examination and grants, were it only to improve reading, it seems desirable that analysis of sentences should be taught much earlier than is customary.

This point is ably argued in the following passage from Mr. Fearon's work on School Inspection:

"In the case of English it is absurd to waste time over learning the cases of nouns which have lost all their case-endings, and have substituted for those case-endings structural position or logical relation in the sentence. What is wanted is to get as quickly as possible a notion of the structure of the sentence and of the logical relation of its parts. And for this purpose the teaching of English grammar should be begun, and based throughout its course, on the analysis of sentences. The teacher should, immediately after imparting the first elementary notions and general definitions, proceed to the subject and predicate, beginning with the noun and pronoun as the subject, and with intransitive verbs, as verbs of complete predication. He should then pass on to the direct objective relations of nouns and pronouns with verbs of incomplete predication, introducing no more study of case-endings than is absolutely necessary for the purposes of the pronouns. Number, gender, person, tense, mood, and voice, should

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