Weights and Measures. 279 in writing. The dimensions of the school-room and of the principal furniture should be known, and a foot or a yard, or a graduated line of five or ten feet should be marked conspicuously on the wall, as a standard of reference, to be used when lengths are being talked about. The area of the playground; the length and width of the street or road in which the school stands; its distance from the church or some other familiar object, the height of the church spire, should all be distinctly ascertained by the teacher, and frequently referred to in lessons wherein distances have to be estimated. Children should be taught to observe that the halfpenny has a diameter of exactly one inch, and should be made to measure with it the width of a desk or the dimensions of a copy-book. It constantly happens, that if I ask elder children, who have “gone through" as it is called a long course of computation in "long measure," to hold up their two hands a yard apart, or to draw a line three inches long on their slates, or to tell me how far I have walked from the railway station, or to take a book in their hands and tell me how much it weighs, their wild and speculative answers show me that elementary notions of the units of length and weight have not been, as they ought to be, conveyed before mere ciphering" was begun. 66 As to weights and measures, they are, as we all know, a great stumbling-block. The books give us a formidable Weights and list of tables, and children are supposed to learn measures. them by heart. But a little discrimination is wanted here. It is needful to learn by heart the tables of those weights and measures which are in constant use, e.g. avoirdupois weight, long measure, and the number of square yards in an acre; but it is not worth while to learn apothecaries' weight, cloth measure, or ale and beer measure, because in fact these measures are not in actual or legal use; and because the sums which the books contain are only survivals from an earlier age when the technical terms in these tables, puncheons, kilderkins, scruples, and Flemish ells, had a real meaning, and were in frequent use. Keep these tables in the books by all means, and work some sums by reference to them: they are of course all good exercises in computation; but here, as elsewhere, abstain from giving to the verbal memory that which has no real value, and is not likely to come into use. in sums. It seems hardly necessary to refer to the efforts some teachers Moral lessons have made to use Arithmetic as a vehicle for the inculcation of Scriptural or other truths. Such efforts have been commoner in other countries than our own. "How admirably," says an enthusiastic French writer on Arithmetic, does this science lend itself to moral and religious training." Père Girard composed a manual of Arithmetic in which, for the most part, the problems given had a distinctly hortatory character, and were meant to embody economic and moral instruction. Here is an example. "Un père de famille avait l'habitude d'aller tous les soirs au cabaret et laissait souvent sa famille sans pain à la maison. Pendant quatre ans qu'il a mené cette vie il a depensé la première année 197 fr., la seconde 204 fr., la troisième 212 fr., et la quatrième 129 fr. Combien de francs aurait epargné ce malheureux père s'il n'eut pas eu le gout de la boisson?" And another French writer seeks on this wise to give a moral tone to his arithmetical lessons. He supposes the Curé to visit the school, and the teacher to say, What does the number 7 remind you of? The 7 deadly sins, the 7 sacraments, and the 7 golden candlesticks. The number of the Apostles, the number of the minor prophets, and of the gates of the Apocalyptic Jerusalem." And then he turns to the children, "Mes enfants," he says, we have thus shown to our worthy pastor that we establish true relations between the art of computing and the principles of virtue and religion. Who will say after this that Arithmetic is not a moral and edifying study?"-Who indeed? Of course sums founded on Bible facts, on the age of Methusaleh, or the length of Goliath's spear, are innocent enough. But I suspect Rapid Computation. 281 that all attempts of this kind to kill two birds with one stone, so to speak, are very unsatisfactory. Moreover it does not seem quite reverent to use books or names with which some of us have very sacred associations for the sake of manufacturing arithmetical puzzles for school-boys. After all, in just the proportion in which children pay attention to the sum and do it well as a question in arithmetic they will disregard the moral or religious lessons which have been thus artificially forced into the exercise of counting. Arithmetic has indeed its own moral teaching. Rightly learned, it becomes a discipline in obedience, in fixed attention, in truthfulness and in honor. These are its appropriate lessons, and they are well worth learning. But if you want to deal with drunkenness and extravagance, or to teach Bible History, it is better to adopt some other machinery than that of an arithmetic lesson. And touching one of these habits, that of fixed and concentrated intellectual attention, it may be well to bear Rapid comin mind how greatly it is helped by exercises in putation. rapid counting. Now and then it is a useful exercise to have a match, and to let the scholars work a given number of sums against time, say so many within half an hour. One great advantage of this is that it keeps the scholar's whole power and faculty alive, and keenly bent on the one object. No irrelevant or foreign thought can for the time intrude into the mind. And quick work is not in arithmetic, as in so many other subjects, another name for hasty and superficial work. In this one department of school life slowness and deliberation are rather ensnaring than otherwise. Intervals are here of little or no value for reflection. They merely give an opportunity for the thoughts to wander. The quickest calculators are those who for the time during which they are engaged on a sum shut everything else but the sum out of their thoughts; and they are for that very reason the best calculators. It must not be forgotten that arithmetic, like all the other exact sciences, has the advantage of dealing with results which are absolutely certain, as far as we can claim certainty for any Exactness. thing we know. In mathematical and purely logical deduction we always know when we get at a result that it is either correct or incorrect. There are no degrees of accuracy. One answer is right, and every other possible answer is wrong. Hence if we want to get out of arithmetic the training in precision and conscientious exactness which it is calculated to give, we must never be content with an answer which is approximately right; right for all practical purposes, or right in the quotient, but a little wrong in the remainder. The perfect correctness of the answer is essential, and I counsel you to attach as great importance to the minute accuracy of the remainder and what seems the insignificant part of the answer, as to the larger and more important parts of it. In mathematics no detail is insignificant. Exercises in forecasting approximate answers. You will occasionally get answers not only wrong, but preposterously and absurdly wrong; e.g. you ask what per cent. of profit is gained, and receive some thousands of pounds for the answer; or you ask a question the answer of which has to be time, and the pupil brings it you in pence. It is well to check this by often asking a scholar to tell approximately, and before he does his sum, what he expects the answer to be,-about how much; why e.g. it cannot be so great as a million, or so small as twenty, or in what denomination the answer is sure to come. And if he has not expected anything, nor exercised himself in any prevision as to what sort of answer should emerge, you are in a position at once to discern that he is not making the best sort of progress, and when you see this to apply a remedy at once. In teaching the art of computation it is legitimate to devise special exercises in order to cultivate ingenuity. Ingenuity. Such exercises may often be found in connection with different methods of proving or verifying the answers to When the answer has been found, the data and the quæsita should be made to exchange places, and the scholars may be asked to construct new questions, so that each of the sums. Commercial Rules. 283 factors in the original problem shall be made in turn to come out as the answer. Another method is to work out before the class in full a solution to a long and complex sum, and then invite the scholars to tell how the process might have been abridged; which of the figures set down was not essential as a means of obtaining the answer, or might have been dispensed with. Indeed the invention of contracted methods of working, whether by cancelling or otherwise, ought always to be at the suggestion of the scholar, and grow fairly out of his own experience in working by a needlessly long process. It should seldom or never be enunciated as a rule by the teacher. rules. It is perhaps hardly necessary to remind any one here that it is a mistake to measure the practical utility of Commercial the arithmetical exercises you adopt by their visible relation to commerce, and to the affairs of life. Of course it is important that many of the problems you set should be as like the actual problems of business as possible. Mere conundrums, obviously invented by the bookmakers, are apt to seem very unreal to boys and girls; and they prefer to confront the sort of difficulties which they are likely to meet with out of school. So I think it desirable that you should make sums out of the bills you pay, and bearing on what you know to be the rents of the houses, the income and expenditure of families of the class of life to which your pupils belong. You should keep your eyes open, and invent or take from the newspapers of the day little problems on the changing prices of goods, the weekly returns of births and deaths, the returns of the railway companies, or the fluctuations in the weekly wages of artisans. Simple examples of receipts, and of the use of a ledger and a balance-sheet, should also be given in connection with the smaller transactions, with which the scholars are most familiar. But do not suppose that exercises which have no ostensible relation to real business are of inferior value even for practical purposes. What are often called commercial rules, such as discount, and tare and tret, arc modified a good deal in the |