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and early education; other departments of pure science being reserved for what is called higher or university instruction. But all the arguments in favour of teaching algebra and trigonometry to advanced students, apply equally to the teaching of the principles or theory of arithmetic to school boys. It is calculated to do for them exactly the same kind of service, to educate one side of their minds, to bring into play one set of faculties which cannot be so severely or properly exercised in any other department of learning. In short, relatively to the needs of a beginner, Arithmetic, as a science, is just as valuable—it is certainly quite as intelligible—as the

higher mathematics to a university student. Arithme- It is probably because the purely utilitarian or practical tic has the view of school Arithmetic has so generally prevailed same rela. tion to a that it has never been a favourite study in girls' schools. girl's as to Mistresses, as a rule, do not take a strong interest in it, a boy's edu. cation. or seek to kindle their pupils' enthusiasm in it. Girls at

school are, if not actually encouraged to dislike arithmetic, apt to take for granted that it is rather an unfeminine pursuit, that it is certainly unnecessary, and probably vulgar. And, no doubt, if the conventional notion about the purpose of Arithmetic is well founded, they are right. If cyphering means a collection of artifices for doing sums; if the great object of learning the art is to be fitted for the counting-house or the shop; then the instinct which makes governesses and their pupils shrink from Arithmetic is a true one. But if Arithmetic is a study capable of yielding intellectual fruit, if it helps to quicken and concentrate the attention, to bring under control the reasoning faculty, to shew by what method we can proceed from the known to the unknown, to enable us to perceive the nature of a fallacy, and to discriminate the two sides of the fine line by which the true is often separated

Two main purposes to be served.

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from the false; if, in short, the study of Arithmetic is
mainly helpful in shewing what truth is, and by what
methods it is attained, then surely it bears just as close a
relation to the needs of a woman's life as to those of a ·
man. For she, too, has intellectual problems to solve,
books to read, and opinions to form; and she will do all
this to good purpose in just the proportion in which she
brings to her work a trained and disciplined under-
standing, accustomed to analyse the grounds of belief,
and to proceed by slow and careful steps from premisses
to inference.

So much will suffice for the present as to the greater The pracpurposes to be served in the teaching of Arithmetic. tical sig

of ArithBut the lesser purpose is not insignificant, and must not metic. be overlooked. It is no slight thing to be a good computer, and to know how to apply arithmetical rules deftly and accurately to the management of an income, to the conduct of business, to statistics, to averages, to scientific and political data, and to the manifold problems which life presents. And even though the higher aims of Arithmetic are altogether overlooked, it cannot be said that time is wasted in achieving the lower aim. So much of arithmetical knowledge as is fairly tested by setting sums to be worked, and as is required in order to work them promptly and accurately, is well worth attaining. Its relative importance to genuine mathematical training may be, and often has been, exaggerated, but of its absolute importance there can be little question.

Thus then the two distinct uses of Arithmetic, (1) Its direct or practical use as an instrument for the solution of problems, and (2) Its indirect or scientific use as a means of calling out the reasoning faculty require to be separately apprehended, and I am intending

to ask you to-day to look at the first, and in my next lecture at the second, and to enquire how each of the two objects thus to be kept in view can be best fulfilled. Of course two objects may be logically separable; and for purposes of discussion here may be treated apart; while as a matter of fact, they are pursued together. In attaining either object you cannot help doing something towards the attainment of the other. For you cannot teach practical arithmetic, even by mere rule of thumb, without giving some useful intellectual discipline; and you cannot make the theory and laws of Arithmetic clear to a boy's understanding without also giving him some serviceable rules for practical use. Still we may with advantage treat the two purposes of Arithmetic separately, and at present ask ourselves

only how to teach Arithmetic as an Art. Computa- A really good computer is characterized by three tion.

qualities—promptitude, perfect accuracy, and that skill or flexibility of mind which enables him at once to seize upon the real meaning of a question, and to apply the best method to its solution. How are these qualities

best to be attained ? Early Now the first thing necessary to be borne in mind is exercises

the familiar truth, that a child's earliest notions of

number are concrete, not abstract. He knows what stract.

three roses, or three chairs mean before he can make abstraction of the number 3 as a separate entity. Hence it will be seen that the earliest exercises in counting should take the form of counting actual objects. For this purpose the ball-frame or abacus is generally employed, and with great advantage. He should count also the objects in the room, the panes of glass in the window, a handful of pebbles, the pictures on the wall, and the number of scholars in the class. It must

concrete not ab.

The discipline of an Arithmetic class.

295

not be set down as a fault if at first he counts with his fingers. Let him do so by all means if he likes. The faculty of abstracting numbers, and of learning to do without visible and tangible illustrations comes more slowly to some children than others. So long as they get the answer right, let them have what help they want till this power comes. It is sure to come ere long. At first, too, the little questions and problems which are given to children may fitly refer to marbles or apples, or to things which are familiar to them. But the mistake made by many teachers is to continue using these artifices too long; to go on shewing an abacus, or talking about nuts and oranges after the children have fully grasped the meaning of 6+5 in the abstract, and are well able to do without visible help. It is a sure test of a good teacher that he knows when and how far to employ such artifices, and when to dispense with them. The moment that concrete illustrations have served their purpose, they should be discarded.

Remember also that Arithmetic is one of the lessons Strict in which discipline is more important than in any other."

needec. The amount of order and drill which may suffice for a good lesson in reading or geography will not suffice for arithmetic. Undetected prompting and copying are easier in this subject than in any other, and they are more fatal to real progress. It is important that in computing a scholar should learn to rely on the accuracy of his own work. If he has any access to the answer, and works consciously towards it; if he can get a whispered word or a surreptitious figure to guide him, the work is not his own, and he is learning little or nothing. It is therefore essential that your discipline should be such, that copying or friendly suggestion during the working of a sum shall be impos

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sible. It is idle, in this connexion, to talk of honour.

The sense that it is dishonourable to avail oneself of any such chance help as comes in one's way in solving a problem, is, after all, only a late product of moral training. You do not presuppose its existence in grown men at the Universities, who are undergoing examinations for degrees, or even for Holy Orders. You have no right to assume its presence in the minds of little children. They will at first copy from one another without the smallest consciousness that there is any harm in it. After all there is nothing immoral in copying until we have shewn it to be so. It is inconvenient to us, of course, and it happens to be inconsistent with genuine progress in Arithmetic, and it is for these reasons that it becomes necessary to stop it. The truth is that if you want to train children in the habit of doing their own work well, and depending on its accuracy, you must do habitually that which is done at all public examinations—make copying impossible. And this may be done by divers expedients, e.g. by giving different exercises to scholars as they sit alternately, so that no two who are together shall have the same sum, or by placing them in proper attitudes, and at needful distances, and under vigilant

supervision.. Exercises Again, I suggest that a good many sums should be given out given out in words, not in figures. Remember that the in words, 8 not figures. actual questions of life are not presented to us in the

shape of sums, but in another form which we have to translate into sums; and that this business of translating the question out of the ordinary form into the form adopted in the arithmetic books is often harder than the working of the sum itself, e.g. Take 3018 from 10,000. In an ill-taught school a child is puzzled by this; he first asks what rule it is in. He next asks how to set it

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