Educative Value of Arithmetic.—Of all his early studies, perhaps none is more important to the child
as a means of education than that of arithmetic. That he should do sums is of comparatively small importance; but the use of those functions which ‘summing’ calls into play is a great part of education; so much so, that the advocates of mathematics and of language as instruments of education have, until recently, divided the field pretty equally between them.
The practical value of arithmetic to persons in every class or life goes without remark. But the use of the study in practical life is the least of its uses. The chief value of arithmetic, like that of the higher mathematics, lies in the training it affords to the reasoning powers, and in the habits of insight, readiness, accuracy, intellectual truthfulness it engenders. There is no one subject in which good teaching effects more, as there is none in which slovenly teaching has more mischievous results. Multiplication does not produce the ‘right answer,’ so the boy tries division; that again fails, but subtraction may get him out of the bog. There is no must be to him; he does not see that one process, and one process only, can give the required result. Now, a child who does not know what rule to apply to a simple problem within his grasp, has been ill taught from the first, although he may produce slatefuls of right sums in multiplication or long division.
Problems within the Child’s Grasp.—How is this insight, this excuse of the reasoning powers, to be secured? Engage the child upon little problems within his comprehension from the first, rather than upon set sums. The young governess delights to set a noble ‘long division sum,’—953,783,465÷873—which shall fill the child’s slate, and keep him occupied
for a good half-hour; and when it is finished, and the child is finished too, done up with the unprofitable labour, the sum is not right after all: the two last figures in the quotient are wrong, and the remainder is false. But he cannot do it again—he must not be discouraged by being told it is wrong; so, ‘nearly right’ is the verdict, a judgment inadmissible in arithmetic. Instead of this laborious task, which gives no scope for mental effort, and in which he goes to sea at last from sheer want of attention, say to him—
“Mr Jones sent six hundred and seven, and Mr Stevens eight hundred and nineteen, apples to be divided amongst the twenty-seven boys at school on Monday. How many apples apiece did they get?”
Here he must ask himself certain questions. ‘How many apples altogether? How shall I find out? Then I must divide the apples into twenty-seven heaps to find out each boy’s share.’ That is to say, the child perceives what rules he must apply to get the required information. He is interested; the work goes on briskly: the sum is done in no time, and is probably right, because the attention of the child is concentrated on his work. Care must be taken to give the child such problems as he can work, but yet which are difficult enough to cause him some little mental effort.
Demonstrate.—The next point is to demonstrate everything demonstrable. The child may learn the multiplication-table and do a subtraction sum without any insight into the rationale of either. He may even become a good arithmetician, applying rules aptly, without seeing the reason of them; but arithmetic becomes an elementary mathematical training only
in so far as the reason why of every process is clear to the child. 2+2=4, is a self-evident fact, admitting of little demonstration; but 4×7=28 may be proved.
He has a bag of beans; places four rows with seven beans in a row; adds the rows, thus: 7 and 7 are 14, and 7 and 21, and 7 are 28; how many sevens in 28? 4. Therefore it is right to say 4×7=28; and the child sees that multiplication is only a short way of doing addition.
A bag of beans, counters, or buttons should be used in all the early arithmetic lessons, and the child should be able to work with these freely, and even to add, subtract, multiply, and divide mentally, without the aid of buttons or beans, before he is set to ‘do sums’ on his slate.
He may arrange an addition table with his beans, thus—
0 0 0 =3 beans
0 0 0 0 =4 ”
0 0 0 0 0 =5 ”
and be exercised upon it until he can tell, first without counting, and then without looking at the beans, that 2+7=9,etc.
Thus with 3,4,5,—each of the digits: as he learns each line of his addition table, he is exercised upon imaginary objects, ‘4 apples and 9 apples,’ ‘4 nuts and 6 nuts,’ etc.; and lastly, with abstract numbers—6+5, 6+8.
A subtraction table is worked out simultaneously with the addition table. As he works out each line of additions, he goes over the same ground, only taking away one bean, or two beans, instead of adding, until he is able to answer quite readily, 2 from 7? 2 from 5? After working out each line of
addition or subtraction, he may put it on his slate with the proper signs, that is, if he have learned to make figures. It will be found that it requires a much greater mental effort on the child’s part to grasp the idea of subtraction than that of addition, and the teacher must be content to go slowly—one finger from four fingers, one nut from three nuts, and so forth, until he knows what he is about.
When the child can add and subtract numbers pretty freely up to twenty, the multiplication and division tables may be worked out with beans, as far as 6×12; that is, ‘twice 6 are 12’ will be ascertained by means of two rows of beans, six beans in a row.
When the child can say readily, without even a glance at his beans, 2×8=16, 2×7=14, etc., he will take 4,6,8,10,12 beans, and divide them into groups of two: then, how many twos in 10, in 12, in 20? And so on, with each line of the multiplication table that he works out.
Problems.—Now he is ready for more ambitious problems: thus, ‘A boy had twice ten apples; how many heaps of 4 could he make?’ He will be able to work with promiscuous numbers, as 7+5-3. If he must use beans to get his answer, let him; but encourage him to work with imaginary beans, as a step towards working with abstract numbers. Carefully graduated teaching and daily mental effort on the child’s part at this early stage may be the means of developing real mathematical power, and will certainly promote the habits of concentration and effort of mind.
Notation.—When the child is able to work pretty freely with small numbers, a serious difficulty must be faced, upon his thorough mastery of which will
depend his apprehension of arithmetic as a science; in other words, will depend the educational value of all the sums he may henceforth do. He must be made to understand our system of notation. Here, as before, it is best to begin with the concrete: let the child get the idea of ten units in one ten after he has mastered the more easily demonstrable idea of twelve pence in one shilling.
Let him have a heap of pennies, say fifty: point out the inconvenience of carrying such weighty money to shops. Lighter money is used—shillings. How many pennies is a shilling worth? How many shillings, then, might he have for his fifty pennies? He divides them into heaps of twelve, and finds that he has four such heaps, and two pennies over; that is to say fifty pence are (or are worth) four shillings and twopence. I buy ten pounds of biscuits at fivepence a pound; they cost fifty pence, but the shopman gives me a bill for 4s. 2d.; show the child how put down: the pennies, which are worth least, to the right; the shillings, which are worth more, to the left.
When the child is able to work freely with shillings and pence, and to understand that 2 in the right-hand column of figures is pence, 2 in the left-hand column, shillings, introduce him to the notion of tens and units, being content to work very gradually. Tell him of uncivilised peoples who can only count so far as five—who say,’ five-five beasts in the forest,’ ‘five-five fish in the river,’ when they wish to express an immense number. We can count so far that we might count all day long for years without coming to the end of the numbers we might name; but after all, we have very few numbers to count with, and
very few figures to express them by. We have but nine figures and a nought: we take the first figure and the nought to express another number, ten; but after that we must begin again until we get two tens, then, again, till we reach three tens, and so on. We call two tens, twenty, three tens, thirty, because ‘ty’ (tig) means ten.
But if I see figure 4, how am I to know whether it means four tens or four ones? By a very simple plan. The tens have a place of their own; if you see figure 6 in the ten-place, you know it means sixty. The tens are always put behind the units: when you see two figures standing side by side, thus, ’55,’ the left-hand figure stands for so many tens; that is, the second 5 stands for ten times as many as the first.
Let the child work with tens and units only until he has mastered the idea of the tenfold value of the second figure to the left, and would laugh at the folly of writing 7 in the second column of figures, knowing that thereby it becomes seventy. Then he is ready for the same sort of drill in hundreds, and picks up the new idea readily if the principle have been made clear to him, that each remove to the left means a tenfold increase in the value of a number. Meantime, ‘set’ him no sums. Let him never work with figures the notation of which is beyond him, and when he comes to ‘carry’ in an addition or multiplication sum, let him not say he carries ‘two,’ or ‘three,’ but ‘two tens,’ or three hundreds,’ as the case may be.
Weighing and Measuring.—If the child do not get the ground under his feet at this stage, he works arithmetic ever after by rule of thumb. On the same
principle, let him learn ‘weights and measures’ by measuring and weighing; let him have scales and weights, sand or rice, paper and twine, and weigh, and do up, in perfectly made parcels, ounces, pounds, etc. The parcels, though they are not arithmetic, are educative, and afford considerable exercise of judgment as well as of neatness, deftness, and quickness. In like manner, let him work with foot-rule and yard measure, and draw up his tables for himself. Let him not only measure and weigh everything about him that admits of such treatment, but let him use his judgment on questions of measure and weight. How many yards long is the tablecloth? how many feet long and broad a map, or picture? What does he suppose a book weighs that is to go by parcel post? The sort of readiness to be gained thus is valuable in the affairs of life, and, if only for that reason, should be cultivated in the child. While engaged in measuring and weighing concrete quantities, the scholar is prepared to take in his first idea of a ‘fraction,’ half a pound, a quarter of a yard, etc.
Arithmetic a Means of Training.—Arithmetic is valuable as a means of training children in habits of strict accuracy, but the ingenuity which makes this exact science tend to foster slipshod habits of mind, a disregard of truth and common honesty, is worthy of admiration! The copying, prompting, telling, helping over difficulties, working with an eye to the answer which he knows, that are allowed in the arithmetic lesson, under an inferior teacher, are enough to vitiate any child; and quite as bad as these is the habit of allowing that a sum is nearly right, two figures wrong, and so on, and letting the child work it over again. Pronounce a sum wrong,
or right—it cannot be something between the two. That which is wrong must remain wrong: the child must not be let run away with the notion that wrong can be mended into right. The future is before him: he may get the next sum right, and the wise teacher will make it her business to see that he does, and that he starts with new hope. But the wrong sum just just be let alone. Therefore his progress must be carefully graduated; but there is no subject in which the teacher has a more delightful consciousness of drawing out from day to day new power in the child. Do not offer him a crutch: it is in his own power he must go. Give him short sums, in words rather than in figures, and excite in him the enthusiasm which produces concentrated attention and rapid work. Let his arithmetic lesson be to the child a daily exercise in clear thinking and rapid, careful execution, and his mental growth will be as obvious as the sprouting of seedlings in the spring.
The A B C Arithmetic.—Instead of entering further into the subject of the teaching of elementary arithmetic, I should like to refer the reader to the A B C Arithmetic by Messrs Sonnenschein & Nesbit.
The authors found their method upon the following passage from Mill’s Logic:—“The fundamental truths of the science of Number all rest on the evidence of sense; they are proved by showing to our eyes and our fingers that any given number of objects, ten balls for example, may by separation and re-arrangement exhibit to our senses all the different sets of numbers the sum of which is equal to ten. All the improved methods of teaching arithmetic to children proceed on a knowledge of this fact. All who wish to carry
the child’s mind along with them in learning arithmetic, all who wish to teach numbers and not mere ciphers, now teach it through the evidence of the senses in the manner we have described.”
Here we may, I think, trace the solitary source of weakness in a surpassingly excellent manual. It is quite true that the fundamental truths of the science of number all rest on the evidence of sense; but, having used eyes and fingers upon ten balls or twenty balls, upon ten nuts, or leaves, or sheep, or what not, the child has formed the association of a given number with objects, and is able to conceive of the association of various other numbers with objects. In fact, he begins to think in numbers and not in objects, that is, be begins mathematics. Therefore I incline to think that an elaborate system of staves, cubes, etc., instead of tens, hundreds, thousands, errs by embarrassing the child’s mind with too much teaching, and by making the illustration occupy a more prominent place than the thing illustrated.
Dominoes, beans, graphic figures drawn on the blackboard, and the like, are, on the other hand, aids to the child when it is necessary for him to conceive of a great number with the material of a small one; but to see a symbol of the great numbers and to work with such a symbol are quite different matters.
With the above trifling exception, which does not interfere at all with the use of the books, nothing can be more delightful than the careful analysis of numbers and the beautiful graduation of the work, “only one difficulty at a time being presented to the mind.” The examples and the little problems could only have been invented by writers in sympathy with children. I advise the reader who is interested in the teaching
of arithmetic to turn to Mr Sonnenschein’s paper on ‘The Teaching of Arithmetic in Elementary Schools,’ in one of the volumes published by the Board of Education.
Preparation for Mathematics.—In the ‘forties’ and ‘fifties’ it was currently held that the continual sight of the outward and visible signs (geometrical forms and figures) should beget the inward and spiritual grace of mathematical genius, or, at any rate, of an inclination to mathematics. But the educationalists of those days forgot, when they gave children boxes of ‘form’ and stuck up cubes, hexagons, pentagons, and what not, in every available schoolroom space, the immense capacity for being bored which is common to us all, and is far more strongly developed in children than in grown-up people. The objects which bore us, or the persons who bore us, appear to wear a bald place in the mind, and thought turns from them with a sick aversion. Dickens showed us the pathos of it in the schoolroom of the little Gradgrinds, which was bountifully supplied with objects of uncompromising outline. Ruskin, more genially, exposes the fallacy. No doubt geometric forms abound,—the skeletons of which living beauty, in contour and gesture, in hill and plant, is the covering; and the skeleton is beautiful and wonderful to the mind which has already entered within the portals of geometry. But children should not be presented with the skeleton, but with the living forms which clothe it. Besides, is it not an inverse method to familiarise the child’s eye with patterns made by his compasses, or stitched upon his card, in the hope that the form will beget the idea? For
the novice, it is probably the rule that the idea must beget the form, and any suggestion of an idea from a form comes only to the initiated. I do not think that any direct preparation for mathematics is desirable. The child, who has been allowed to think and not compelled to cram, hails the new study with delight when the due time for it arrives. The reason why mathematics are a great study is because there exists in the normal mind an affinity and capacity for this study; and too great an elaboration, whether of teaching or of preparation, has, I think, a tendency to take the edge off this manner of intellectual interest.
 See Appendix A.