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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 honour. 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 Rapid concentrated intellectual attention, it may be well to bear computain mind how greatly it is helped by exercises in 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 a 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 Exactness. other exact sciences, has the advantage of dealing with results which are absolutely certain, as far as we can claim certainty for anything 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 mathe

matics no detail is insignificant. Exercises You will occasionally get answers not only wrong, in forecasting ap

but preposterously and absurdly wrong; e.g. you ask proximate what percent. 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. Ingenuity'. In teaching the art of computation it is legitimate to

devise special exercises in order to cultivate ingenuity. Such exercises may often be found in connexion with different methods of proving or verifying the answers to

answers.

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sums.

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 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.

It is perhaps hardly necessary to remind any one Comhere that it is a mistake to measure the practical utility

mercial

rules. of 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 artizans. Simple examples of receipts, and of the use of a ledger and a balance-sheet, should also be given in connexion 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, are modified a good deal in the counting-house and bank, and are in their immediate application to business often far less serviceable than they seem. An eminent London Banker once said to me “The chief qualifications I want in a clerk are, next to good character and associations, that he should write a good hand, that he should have been taught intelligently, especially in Arithmetic, and that he should not have learned bookkeeping. We have our own method of keeping accounts, and a pretentious system of school book-keeping has a number of technical terms which we do not use, and which hinder a lad from learning that method. But let him only have a good general knowledge of the principles of arithmetic and counting, and we will undertake to teach him all that is peculiar to the books of our house in less than a week.” Perhaps this is an extreme case, but I am convinced that attempts to anticipate the actual application of arithmetic to the particular business in which a pupil may be hereafter

engaged are generally mistakes. Other The application of arithmetic to the solution of forms of problems is often limlted in the books to what is called practical applica- business. But commerce is after all only one, though the Decimalizing English !oney.

most prominent, of the uses to which arithmetic has to be put in life. There are many interesting and varied

tion.

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applications to other purposes, which might be used with advantage: e.g.

The computation of the time of falling bodies.

The conversion of our weights and measures into French.

Finding the length of circumference and radii, and the area of circles and squares.

Actual measurement of the play ground or a neighbouring field, and elementary land surveying.

The right use of annuity and insurance tables, e.g. the tables at the end of the Post Office Guide will suggest many interesting forms of sums.

The use of logarithmic tables, and the solution of triangles by means of them : their application to the determination of the heights of mountains or spires or the breadth of rivers.

The difference of time between various places whose longitude is given.

The measurement of distances on a map which has a scale of miles attached to it.

The readings of the thermometer and the conversion of Fahrenheit to centigrade.

The statistics of attendance in the school itself, and the method of computing its average attendance.

One great help to the easy solution of money questions Reduction is the habit of using decimal equivalents, or reducing sums of English of money at sight to decimals of £1. We are at present decimals. far from the adoption of a decimal coinage in England; but we can by anticipation enjoy, in our accounts at least, many of the advantages of a decimal system of money, by the adoption of a simple rule. Let it be observed that two shillings = £'l, that one shilling = £

£.05, that sixpence = £.025, and that a farthing differs only from £.001 by a very small fraction; and it then becomes very

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