down. Both of these are questions which he ought to answer for himself. So long as a pupil finds any difficulty whatever in recognising an exercise in a given rule, under any guise, however unfamiliar, be sure he does not understand that rule, and ought not to quit it for a higher. It is a very useful aid to this sort of versatility or readiness, not only to practise yourselves as teachers in the manufacture of new exercises, but also to encourage your pupils to invent new questions on each rule before you pass from it to the next. You will find a pupil's grasp of the real meaning and relations of an arithmetical rule much strengthened by the habit of framing new questions. Moreover you will find it a very popular and interesting exercise, which will kindle a good deal of spirit and animation in your class. Never permit any reference to be made to the answer Answers while the work is in progress. It would be a good thing to be kept out of if the printed answers to arithmetical questions could be sight. concealed from pupils altogether. But I fear this is impossible. At any rate, teachers should be on their guard against the tendency of children before they get to the end of the sum to glance furtively at the answer, and to work towards it. Perhaps if the right answer is evidently not coming the pupil alters a figure, or introduces a new multiplier in order to bring it right. But a sum so wrought is a very unsatisfactory and delusive performance. It is well at first rather to give a good number of Numerous short exercises irregularly formed, than to use those short exerlarge symmetrical masses of figures, which the school ferable to books are apt to give us, and which are so much more a few long convenient to the teacher, inasmuch as they take a good deal of time, and leave him a little more breathing space. ones. Recapituiation. A large square addition sum, in which all the lines are of the same length, and all extend to hundreds of millions, is far less likely to be useful than “Add seventeen to a hundred and twenty, that to three thousand and ninety-six, that to twenty-seven, and that to five." Many children in fact who can do the first will be unable to do the second. Now and then, however, it is a good thing to give a very long exercise, to test sustained attention and continuity of thought, and to ensure accuracy. It is good also to take care that before proceeding to any new rule, you give a few exercises, which call out not alone the previous rule, but all the preceding rules. There is no true progress if any one of the elementary rules is allowed to drop out of sight. I am often struck with the want of skill shewn in making sure at each step that all previous steps are understood. This arises no doubt from the way in which exercises are arranged in books, grouped under the heads of the various rules. A child gets a rule, works a number of sums all alike, and then leaves to go on to another. Whereas exercises ought to be so graduated, and sums so carefully framed as to bring into play all that has previously been learned, and to fix and fasten the memory of former rules. There is hardly any one text-book which I know that does this sufficiently. You should be supplied always therefore with a number of miscellaneous exercises, which you give the scholars from a book or manuscript of your own, and which they do not know to be illustrative of any special rule. Making out a fair copy of a sum in a book, garnished with ruled red ink lines and flourishes, is a favourite employment in some schools, and consumes a good deal Writing out sums in books Oral or Mental Arithmetic. 299 of time. It has its utility, of course, as an exercise not of much in neatness and arrangement, and in the mere writing of value. figures. Moreover, it is liked by some teachers because it pleases parents, and is the only visible evidence of arithmetical progress, which can be appreciated at home. Yet as a device for increasing or strengthening a child's arithmetical knowledge, it is very useless. I venture to warn you, therefore, against the inordinate use of what are called “ciphering books;" believing as I do, that in just the proportion in which you teach Arithmetic intelligently, you will learn to rely less on such mechanical devices. It will be well for us to consider, too, what use it is Oral or mental which a pupil makes of a slate or a paper when he is Arith. working a sum. The object of all rules is, of course, to metic. shew how a long or complex problem, which cannot be worked by a single effort of the mind, may be resolved into a number of separate problems each simple enough to be so wrought. As each separate result in multiplication, division, or addition is thus attained, we set it down as a help to the memory, and are thus at liberty to go on to the next. Now it is evident that the worth and accuracy of the general result depend upon the correctness with which we work out each of these single items. It is a good plan, therefore, to give a pupil some oral practice in the manipulation of single numbers, before setting him down to work a sum. This Oral or Mental Arithmetic has long been a favourite exercise in elementary schools, but it has not been very generally adopted in schools of a higher class. One reason for this is to be found in the very restricted and technical use made of the exercise. In manuals of Mental Arithmetic, advantage is taken of little accidental facilities or resemblances afforded by particular numbers, and rules are founded upon them : e.g. (1) To find the price of a dozen articles ; call the pence shillings, and call every odd farthing three pence. (2) To find the price of an ounce, when the price of 1 lb. is known; call the shillings farthings and multiply by three. (3) To find the price of a score, call the shillings pounds. (4) To find the interest on a sum of money at 5 per cent., for a year; call the pounds shillings, and for every additional month call the pound a penny. (5) To square a number; add the lower unit to the upper, multiply by the tens, and add the square of the unit. Its cbuses. Each of these rules happens to offer special facilities in computation. But the occasions on which a question actually occurs in one of these forms are rare; and the student who has his memory filled with these rules, is not helped, but rather hindered by them when for example he wants to know what fourteen articles will cost, or what is the interest at 3 per cent., or how to multiply 75 by 23. All such rules are apt to seem more useful than they are, and when children, who have learned the knack of solving a few such problems, are publicly questioned by those who are in the secret, the result is often deceptive. I attended an exhibition or oral examination of a middle school of some pretensions a short time ago; and the teacher of Arithmetic undertook to put the scholars through a little testing drill. All his questions fell within the narrow limits of some of these special rules. He also gave one or two exercises in rapid addition which were answered with what seemed astonishing rapidity and correctness: e.g. 73 + 27 + 65 = Answer 165. Not till six or seven such sums had been given did I notice that the first two numbers in each group amounted to 100, and the next two also; and that all the questions were framed on the same pattern. Many of the audience did not detect this, but of course the children were in the secret, and were, in fact, confederates with the teacher, in an imposture. It is because so much of what is called mental arithmetic consists of mere tricks of this kind, that the subject has been somewhat justly discredited by good teachers. But the mental Arithmetic which is of real service Iis uses. does not consist in exercise in a few special rules, but in rapid, varied, and irregular problems in all the forms which computation may take. It differs mainly from written Arithmetic, in that it uses small numbers instead of large ones. Before attempting to work exercises in writing in any rule, a good oral exercise should be given to familiarize the pupils with the nature of the operation.. I will give a few examples to illustrate my meaning: (1) Addition and Subtraction. Take the number 3, add it to i Examples and successively to the sums, up to 50. of oral exercise. 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, &c. &c. So with sevens : 1, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71. Then take 50 or 100 and go rapidly backwards taking away 3 every time, or seven, or eleven. You will observe as you do this that there are certain combinations less easy than others. He whose turn it is to say 21 after 18; or to take away 3 from 32 will halt a moment longer than the rest. You observe this, and make up a series of questions in which these two particular numbers shall be brought into relation : 28 and 3, 48 and 3, 19 and 3, 3 from 42, 3 from 21, &c. There are but nine digits, and if in succession you give nine short brisk lessons,—one on each, -requiring the number to be added and subtracted rapidly, you will come in succession upon every possible combination of these digits. You will bear in mind that when you yourself make an error in adding up a line of figures, |