Work and Play in Girls' Schools / By Three Head Mistresses

Then, as in teaching science or language we first awaken the powers of observation, and lead the child to reflect, so here, in the case of the self-conscious subject, we help the child to interpret the facts of the inner life as well as the outer.

Whilst recognising the danger of forcing the subjective in children, we ought in this, as in other things, to follow the guidance of nature, and surely our own experience, and that of most children, will show how much they are occupied with their own feelings, with the struggles of the higher to subdue the animal nature, and how through contest they are developing the will-power, which is the only safeguard of later life.

It is especially important early to correlate the subjective with the objective in early teaching. Surely much irreligion results in later life from the divorce of the two. As we guide the observing powers in the outward life, so as the power of reflection develops we should do with the inward life: the child is conscious of the pang which comes to all of us, when we act against conscience: that pang which makes our blood run cold, as we feel we have done wrong, is as much a fact of experience, as real, as the sensation of heat, when we touch hot iron. Would people grow up to deny the existence of the spiritual consciousness, if they had been led to question their own experience? A beautiful story is told by Parker and quoted by Armstrong.[22]

Conscience.“When a little boy in my fourth year, my father took me to the farm and sent me home alone. I had to pass a pond. A rhodora attracted my attention. I saw a spotted tortoise sunning himself in the shallow water at the root of the flaming shrub. I lifted the stick I had, to strike; though I had never killed any creature, I had seen boys destroy birds, squirrels and the like, and felt a disposition to follow their wicked example. All at once something checked my little arm, and a voice within said clear and loud, ‘It is wrong’. I held my uplifted stick in wonder at the new emotion, the consciousness of an involuntary but inward check upon my action, till the tortoise and rhodora both vanished from my sight. I hastened home to mother and asked what it was that told me it was wrong. Taking me in her arms, she said: ‘Some men call it conscience, but I prefer to call it the voice of God. If you listen and obey it, it will speak clearer and clearer, but if you turn a deaf ear, it will at last leave you in the dark without a guide: your life depends on your obedience to its voice.’ No event in my life has made so deep and lasting an impression.”

A witness for the spiritual, the universal.The fact that we cannot get rid of the consciousness of wrong, shows that there is a higher Self condemning the self, one other than ourselves; we must not force answers on the child, but we can bring into his consciousness the presence of the holy and righteous God. We may help to make clear and permanent in his consciousness the facts, which he will only later interpret—the conflict of the merely individual, the selfish life, with the larger, the all-embracing life of unselfish love.

We may appeal too to the experience of each child, who suffers punishment rather than disobey conscience. Such victories establish faith, convince us that we are more than creatures of time, that we are sons of God. Every true and self-denying act that a child is able to do is a ground of confidence; “I write unto you, young men, because ye have overcome the wicked one”. Each time that the mere animal desires are subdued by the love of truth and righteousness, we prove that we transcend the things of time and space. These are the eternal things, which eye sees not and thought cannot conceive, and yet for the sake of these unseen and eternal things men live and die, and count all earthly things as nought. Do not the hearts of all children “burn within them” as we expound to them the Scriptures which tell of heroes who have done battle, who laid down their lives for righteousness’ sake, of Him who triumphed from the Cross? We can appeal too to the inward experience of those who are naughty; they do not in their inmost heart wish to be so, but they try and fail; nothing is more touching than the penitence of children, when they find that we have seen the good which is hidden, and not only the evil that comes forth—that we know, not only what is done, but what is resisted. We can, as in the old myths, show that their deliberate choice is not for selfish pleasure; they would if offered the things most delightful to the mere animal, refuse all, if they could have it only on condition of becoming wicked and cruel and deceitful. Hauff’s Cold Heart is a beautiful story on the subject. Thus should we base healthy religious experiences upon facts, and foster habits of attention and obedience to the inward voice.

Right ambitions too should be fostered, the desire to enter into the Divine purposes in thought and word and deed, to be a fellow-worker with God. This will take more definite form in the later idealising period of life; still there will be developed sometimes at an early age earnest desires to become wise and good and to do some special work.

Order of teaching.For objective formal teaching the little ones would begin with the stories of the world’s childhood. The lessons first given in a simple form will be expanded in the higher classes. The child who has learned to trust his father, will learn from Abraham’s sacrifice that we can trust God; the higher classes will see how by the frustration of his purpose Abraham learned the true meaning of sacrifice; the Psalms and Prophets will carry on the subjective teaching, and the words of the old prophets will become a fact of experience; “the word of the Lord came unto me”.

The inner meaning of the sacred myths which had once been told as a mere story will now be felt; the story of the flood as interpreted by St. Peter, and quoted in our baptismal service, the deliverance from the bondage in Egypt, typifying redemption from the slavery of sin, the New Testament teaching of the synoptical gospels, especially the parables, will have supreme educative power.

Written work.It is essential that in this, as in other subjects, written exercises which require thought be set, and corrected and criticised—this is often the only subject in which pupils are not required to formulate their thoughts—hence there exists a vast amount of current religious phraseology to which no definite meaning is assigned; the words may be true in themselves, but not true for the person using them. An American writer tells of one who for years was a regular attendant at church, and often encouraged him by her attentive and responsive expression; when he came to know her later, he found to his surprise that she was as ignorant of the fundamental truths as if she had been brought up in a heathen land.

Sceptical phases.The later period, that of ripening experience, of adolescence, will give the maximum of reflective, as the earliest childhood, the maximum of the sensitive power. As the mysteries of their own being are more and more unfolded, the problems of philosophy and metaphysics have an attraction which should not be disregarded: there is a desire to be alone; the young feel that they must work out the problems for themselves, and they resent the attempt to force on them other people’s solutions. They must question ere they can fully believe; we must never give utterance to the profane idea, that God is angry with those who make mistakes in seeking truth, only show that truth like light is a good, that we may not rest in an indolent agnosticism, for we cannot grow vigorous and strong out of the sunlight; we must encourage them, in this as in all studies, to be ever seeking a fuller knowledge of truth, to live by the truth they have attained, and then they will gain more and more, even through the mistakes. The function of the teacher now is as Socrates described it, to be ready to give help, when needed, to bring to the birth the great thoughts which oppress the soul.

Need of leisure.Later the deep spiritual experiences of St. John and the arguments wherewith St. Paul convinced himself, will come home to the religious experience at least in some degree, and the words in which he describes the vision of God as seen from the spiritual heights, which he had reached in his later epistles. But there must be for the ripening of the character time for quiet, and the incessant activities of to-day, the filling up of every hour, the deprivation of quiet even on Sunday, are much to be regretted, and all educators should see that those who need time for spiritual thought, for working out the great questions which come to every thoughtful person, should not be deprived of it, because some would misuse it. There are two excellent articles in the Pedagogical Review for July, 1891, on the “Psychology and Pedagogy of Adolescence,” by E. Lancaster, and another, a study in “Moral Education,” by J. Street, both Fellows of Clark University; the second article is especially emphatic on this subject.

Systematic reading.In the highest classes, some systematic reading regarding the history and foundations of philosophy in general and Christian philosophy in particular should not, I think, be omitted: one cannot do better than begin with Plato; taking the Apology, the Crito and parts of the Phædo, or the two volumes of selections by Professor Jowett, or some less expensive edition. The Memorabilia of Xenophon is obtainable for 3d. Selections might be made from Aristotle’s Ethics, and some good history of philosophy be made accessible, e.g., Schwegler’s, edited by Dr. Hutchinson Stirling, which is not too long; and some such inspiring book as Fichte’s Vocation of the Scholar may be recommended; other books I might mention, e.g., Henry Jones on Browning; Professor Frazer’s selection from Berkeley; Mackenzie’s Social Philosophy and the series of small hand-books edited by Professor Knight. There might be meetings for discussion and reading under the presidency of one versed in such matters; this would give definiteness to thought, and would at least lead to the kind of wisdom which made the oracle pronounce Socrates the wisest of men; such meetings would be specially useful for the staff. Some effort should be made to establish the primary convictions which alone make life worth living, enable one to possess one’s soul in patience, live in the faith that each is working out the will of the All-Wise and All-Good—if willingly, then with the fullest joy and reward.

Higher teaching.The subject is not ignored at the University Colleges of the States, and there is much of deep interest in the article to which I have referred, viz., “Psychology of Adolescence”. In an article by Caswell Ellis, the special training of teachers of religion is insisted on, and the establishment of professorships. “A department of pedagogy cannot be called complete that does not deal with this important part of its field. Religious training is as much a problem for the pedagogue, as is physical or mental training. Surely we cannot entirely separate them. We have already at our command in the Universities, many helps in the study of the Bible, of theology, of philosophy, of psychology, etc.; why cannot there be found some man of broad culture, wide sympathies, reverent spirit, to focalise these in a chair of religious pedagogy, or whatever it may be called? it would give the opportunity while in college to look at the larger phases of the problem of religious training. No subject is more vital, and our best men need not leave college ignorant of the problem or the possibility of its solution—and find in the decline of life that (as editors, preachers, etc.) they have been spending their energies on reformation, while the great work of formation was never considered.”

Foundations of faith.The means of giving a thorough and systematic teaching regarding the strong foundations of faith, is one that should be considered by all educators. It is true that the emotions and affections are, as in the case of all personal relations, the appropriate means of intercommunion; but the religious life, if it is not to become weak and sentimental, needs the bracing power of intellectual study, and the Scriptures, especially the writings of St. John and St. Paul, afford such exercise.

I may perhaps summarise the lines on which the grounds of a rational faith seem to be established, and which should surely be formulated, as we formulate the principles on which we base our faith in matters of science. They may be arranged under two heads—objective and subjective:—

1. Sense compels us to recognise the existence of a universe, to which we can set no bounds of space or time. We find everywhere at work forces adapted to produce results immeasurably greater, yet similar in character, to those produced by our own exercise of thought and will; we are unable to conceive of either except as ultimately proceeding from a personal mind and will.

Since our mind interprets the phenomena of sense, which is the language of Nature; since the intelligent mind is related to an intelligible universe, the finite mind must be related to the infinite, man must be the child of God.

The facts of history show us man in all ages renouncing all that the animal craves for, for the sake of the ideal, the transcendent.

2. Man is self-conscious, he can become an object to himself; that he can do this proves him to have a dual nature. The higher sits in judgment on the lower, or animal nature (identified with the individual), seeks to bring it into obedience to the universal. Since we can identify conscience with the universal mind and will, we infer that we are on the one side in communion with God, as on the other with the universe.

Man has the power of sympathy. As we cannot conceive of light without postulating an all-comprehending æther, through which all things are related, so the fact that we are affected, actually feel physically and mentally with others, is inconceivable without postulating one all-embracing Personality.

The faith that good must ultimately triumph is an axiom of the moral life; we find it impossible to believe the reverse.

These are some of the broad bases on which rest the Christian dogmas of the relation of man to God the All-Father, which tell of a perfect Son, and of the power given to all to rise through grace into the spiritual life.

I have dwelt on the subject at some length, because it seems to me that the intellectual relation to God has been too much ignored; we should love with the mind as well as with the heart; with the developing of the physical and psychical life, the soul craves to root itself more firmly on the consciousness of the universal, it desires to be at one with the All-Wise and the All-Good Father of spirits to work out the purpose of its own existence. It seeks to be in harmony with all who are living by the highest ideal; hence the impulse to work in associations, specially in the spiritual life, for life must overflow into action! It seeks evermore to be at one in its being, and to bring the individual self into harmony with the all-embracing Spirit in whom we are one.

I may recommend to teachers the recently published volume on Religious Teaching in Schools, by Dr. Bell of Marlborough.

PART II. MATHEMATICS.

ARITHMETIC.

By Dorothea Beale.

Never will such lines express the feelings of properly taught children.

It may be convenient to work out the process of teaching arithmetic on strictly psychological principles.

Concrete teaching first.(1) From the concrete to the abstract. Let the children learn to count with the actual things.

Once the teacher would have set the child down to a slate, taught it to count, and write down the figures, and work sums in addition and subtraction, and then to learn the multiplication table. Now the child has actual things—stones, coloured beads, sticks, bricks—anything but marbles (which one of H.M. Inspectors recommends) or things which run about freely. A box of china buttons, which cost only a few pence the gross, is perhaps best.

(2) Associate doing and knowing. Let the child add actual things: Mary has 3 buttons, Anna gives her 2, she now has 5.

(3) Put thoughts into words. Get the child to say exactly what addition is—“giving to”—and let her find out from words she already knows or may know, as donation, donor, etc., the meaning. The sign + for addition may also be given.

Similarly, subtraction ought to be actually performed by drawing away, and the word explained—its connection with drag, traction, tray, dray, etc. Thus the common fault of writing “substraction” may be avoided. It should be thought of as undoing addition. The signs - and = may now be given.

Analysis of numbers.(4) We learn by analysis and synthesis, i.e., to see the parts in the whole, and the whole as made up of parts. It is very useful at this stage to get children to group numbers, to think of 2, e.g., as 1 + 1, of 3 as 1 + 1 + 1 and 1 + 2, of 8 as 1 + 7, 2 + 6, 3 + 5, 4 + 4, 2 + 2 + 2 + 2. This is much insisted on in Germany and America. In kindergartens there are many pictures which are used for grouping numbers, thus, e.g., a seven-branched candlestick.

We may give 7, as 3 + 1 + 3, as 1 + 2 + 2 + 2, as 1 + 6. This makes numbers, so to speak, easily fall into their constituents, which will be shown to be of use later. I knew a child who habitually thought of the written figures as picturing the number. Children might arrange the 9 digits in various ways, thus, giving also the written figures:—

At this stage the question would naturally arise why there are only 9 figures, and an historical digression could be conveniently made. I give a sketch of such a lesson before coming to more difficult and abstract things.

Historical methods.Dogs are very clever. A collie will go with the shepherd and take care that none stray. Suppose one has disappeared over a cliff when he was not looking, would he know one was gone, would he count like the shepherd? No, he will track out a lost sheep, by scent, as we cannot, but I never heard of a shepherd setting the dog to count. If puss has 3 kittens and you take 1, she seems not to know. Some savage races can count only a few numbers, but man carries a ready-reckoner in his fingers, and most can easily count up to 5 or 10, or, if taking in the toes, up to 20; all the higher races are marked out by their greater power of doing long and difficult sums.

Now, suppose some great owner of sheep, as Abraham or Jesse, sent out a shepherd with many sheep, how would he know each day whether they were all right? Well, the simplest way would be to have two stones for each—the master could have one bag and the man another, and then they could calculate each night; calculus is the Latin for a stone. The shepherd would need a long bag for his stones. Was that how David happened to have the one which he used as a sling to kill Goliath?

Suppose, however, the flock was very large, a bag of stones would be heavy. Has a shepherd something else, which, instead of his exactly carrying, seems to help to carry him? The shepherd’s staff. Could he not put notches on this for his sheep? It would hold a good many; but in days when people had to use stones for knives, it was not so easy to cut a great many notches, and besides it would get used up with a large flock. Could he not make a sign like a hand, V, for every 5 sheep? That was what the Romans did, and next they said, why not have a sign for two hands, X, and let that stand for 10? So, if they wanted to write sixteen sheep, they would put XVI instead of sixteen strokes. You see in the Bible the Roman numbers. The Greeks used letters, too, as the Romans did, for numbers.

Money.When people began to trade they wanted something more than tally sticks and stones—something the value of which all knew. Amongst pastoral people the most ready things to calculate by were sheep or cattle. A piece of land would be sold for so many sheep, but it would be very inconvenient to have to drive your money about, and so people seem very early to have had pieces of metal which were reckoned to be equal in value to sheep or cattle, and to save weighing, each piece had, perhaps, a sheep scratched on it; and this was called in Latin (from pecus, cattle) pecunia, i.e., the piece of metal representing the value of cattle. This would be carried about and exchanged. Lawyers now put in our wills “goods and chattels”; by the first they mean houses and lands, which cannot be moved; by the latter, things which, like cattle, can be moved. Then people could have larger and smaller pieces of money, representing half or a quarter of a sheep, or many sheep.

Account-keeping.You wonder, perhaps, that people did not have books to keep their accounts in, as we do; but in early days people’s books were made of clay, and were more like our slates, and they scratched on them with a sharp instrument called a stylus, which looks something like our stylograph, but had no ink inside, and they could not put these in their pockets.

Modern arithmetic.It was not till the beginning of the third century before Christ, that the Greek Archimedes proposed a plan not altogether unlike ours, because he was a very clever scientific man, and he wanted to do difficult sums, which he could not with the old Greek system. And something of the kind was used in India. But it was not introduced into Europe until about 1000 years after Christ by the Arabs, who had made many conquests. The first English book about it seems to have been written in the reign of Edward III. Chaucer, who died in 1400, talks of the “figures newe,” i.e., the figures we use now, instead of those difficult Roman characters which we find in the Bible.

But I think that before that, people had begun to use some such plan as ours. Have you ever heard of public-houses being called “The Chequers,” and seen a painted board hung up covered with squares of different colours? This was once a sign for a house of public entertainment, where people could make reckonings, and the place where they reckoned the money they paid was called a “counter,” and the court belonging to the king where the people paid their taxes was called the Court of Exchequer.

Suppose a man came into an inn, he would find the counter marked with lines thus:—

and he could have say 3 glasses of beer; the landlord would put a chalk mark for each, but when he had had 12, one mark would be put instead in the next row, or in the third row if he had had a score, i.e., 20, and these marks would correspond with pieces of money. Thus we have pence and shillings and pounds, and we put dots between instead of lines to mark them off.

Here we will take real pieces of money. Suppose £1 „ 14 „ 6 has to be added to S.7 „ D.9. I say 9 and 6 make 15 pence. I change the 12 pence into one silver shilling, add that to the 14 shillings and the 7, and I get 22 shillings. 20 shillings is one pound, so I change that and leave the 2 shillings. Thus I get altogether £2 „ 2 „ 3. We can now write that in figures and add, as before. Suppose I had to pay to A £1 „ 17 „ 9, and I had £2 „ 14 „ 6. We can first do the sum with real money. I find I have not enough pence to give 9, so I have to change one of the shillings, then I shall have 18 pence, out of which I give 9, and write down 9 left. Now, I have only 13 shillings, and I want to pay 17, so I change one pound, then I have 33 shillings, out of which I take 17 and have 16 left. When I have given the pound, I have none left, and there remains in my purse S.16 „ D.9. We can then also write it down thus—

putting the money we have to take away below, pounds under the pounds, shillings under shillings, etc.

Decimals.After a while people all agreed to have for general arithmetic what we call the decimal notation, or reckoning by tens, and so lines were drawn, and figures in the first row were worth one, in the second ten, in the third ten tens, i.e., 100; after that would come figures representing ten hundreds or a thousand, and then ten thousands, and then a hundred thousands; and so we could go on to any length. Ten seemed such a natural number to use, because we all have our ready-reckoner in our ten fingers.

Addition.We can have bags containing 10 buttons, 100 buttons, and then we can get change. Sonnenschein’s box makes carrying very clear. Suppose I want to put down 5 thousands, 9 tens and 3 units or ones. I should write it thus, and if I wanted to add to this 2 thousands, 9 hundreds and 9, I should write that below.

Then I should say 9 units and 3 units make 12 units. But this is equal to 1 ten and 2 units, so I should carry on 10 to the second row, and write down 2 in the unit row. Then I add the 1 to the 9, that makes 10, but 10 in the second row is the same as 1 in the third, so I carry that on; 9 and 1 make 10, but 10 in the third row makes 1 in the fourth, so I carry again, and get 5 + 2 + 1 = 8 thousands, and we should read it 8 thousands and 2.

Subtraction.Then after a while people said, “Why need we have all the chequers? suppose we put a nought when there is no number, just to mark that there is a row, and all will come right;” so they wrote thus:—

And a little later they left off writing anything at the top of the line, because every one knew. Here is a subtraction sum. We cannot take 9 units from 3 units, so we get change from the next row, that gives 13 units, from which we take 9, and have 4 left. We have nothing to take from our 8 remaining tens, so we write 8. We have no hundreds, so we cannot take away 9, but we change one of our thousands into 10 hundreds, and take away 9, leaving 1; lastly we take away 2 from our 4 thousands, and get 2—altogether 2184.

Decimal fractions.Now would come in naturally the extension of this system of notation to decimal fractions, marking the unit by a full stop. If numbers decrease as we go from left to right, they might get smaller than one; the next row to the right would be one-tenth of a penny or of an inch, and the next one-hundredth, and so on. Sums in addition and subtraction might be worked at this stage with decimal fractions. Then it should be pointed out that to push the number a row farther from the point which marks the unit row increases it tenfold, and pushing to the right diminishes tenfold.

It is good practice and interests young children to work in different scales of notation—one may suggest that Goliath would prefer the 6 or 12 scale.

It would be well now to give children some practice in counting backwards, and in rapid viva voce addition, which the exercises in analysis of numbers will have made easy. E.g., 15 + 7, the number naturally falls apart into 5 + 2, and we get 22; 29 + 7, it falls into 6 + 1, at the next step into 3 + 4.

Multiplication.We should next proceed to continued addition or multiplication. Many children come to school not knowing that multiplication is continued addition, and still fewer have any idea that division is continued subtraction. In entrance papers I have had sheets covered in reply to such questions as “How often can 19 be subtracted from 584?”

A few multiplications should be worked with real things. Thus, we have to give to 5 people 3 buttons each. We arrange them in parcels of 3 and add 3 to our pile five times. Now, if we have 15 and want to know how many times we can take away threes, we find we can do it five times over; this is subtraction or undoing the addition. It is the same as making little parcels of 3 each, and so continued subtraction is called division. Some continued addition sums should be given, thus: Find 4 times 891.

It will be easily seen that such sums are done much more quickly if we know by heart how much 4 nines come to, and how much 4 eights; and so people learn their addition tables by heart, and children make them out for themselves thus, generally up to 12 times, some learn up to 20 times. Here is part of 7 times worked out:—

The signs × and ÷ may now be given. All tables should be written out and learned, and it is well to say both ways, 6 × 7 = 42, and 7 × 6 = 42. There are certain numbers that are easily remembered, others in which children habitually make mistakes: it is a waste of time to hear the tables therefore all through after a time, but these difficult ones, 7 × 8, 6 × 9, 11 × 11, etc., should be insisted on; then, finally, the whole heard through, and any about which there is the slightest hesitation asked for daily. If children can learn up to 20 times without much trouble, it is an advantage.

We could next point out that this continued addition is called multiplication, and all the numbers made up by continually adding threes would be called multiples of 3, i.e., many times 3. So 12 would be a multiple of 2 or 3 or 4.

Then examples should be worked, but here let me say that at the early stages concrete examples should abound. Many good books there are containing miscellaneous examples of concrete quantities, such as, There are 319 fruit trees planted in each field for making jam, and there are 12 fields; how many fruit trees? Or, 7 labourers have to be paid on Saturday £17 each; how much will they get in 12 weeks?

When children know the effect of pushing numbers to the left, multiplication by two figures will be easy, but the child should be accustomed to write at the end of each row the real sum, thus: 73 × 25:—

and to work the same sum in a variety of ways, e.g., multiply by 5 × 5; by 100, and divide by 4; by 30, and take off 5; by 10, halve and by 10 again and halve:—

It is well to accustom children to begin to multiply with the left-hand figure, as we shall see later. Thus we get the most important part first.

Division.It should be insisted on that division is undoing multiplication—that if we divide 63 by 9, we are finding a number 7 which when multiplied by 9 gives 63. In working division sums it is better to put the quotient over the dividend, and the children should be ready to explain each step thus: Divide 3496 nuts amongst four schools equally. None will get as many as 1 thousand. They will get, out of 34 hundreds, 8 hundreds each; of 29 tens, 7 tens each; of 16 units, 4 each.

Long division should be fully explained thus: Divide 43921 amongst 23 people. We see that no one will have as much as 1 ten-thousand. Out of 43 thousands, each can have 1 thousand, and there will be 20 thousands left, that is, 200 hundreds; adding 9 we get 209 hundreds. We give 9 to each and 2 hundreds or 20 tens are left. 22 tens do not give one each; they equal 220 units. Of the 221 units we give 9 to each. Some dispense with the written multiplication. This seems to me to strain too much young children’s attention, and to lead to loss of time.

Factors, measures, multiples.Here, while continuing to work many miscellaneous examples, it may be well to interpose some useful exercises on matters interesting and yet puzzling to children, on factors and measures of numbers, and primes and squares. If they get quite familiar with factors, they will not have such difficulty as they do when they come upon the whole set at once: factors, common factors, measure, common measure, G.C.M., multiple, common multiple, L.C.M.

Let us bring out the box of buttons once more and arrange the numbers, finding the factors. 1, 2, 3 have only the number itself, and so these are called primes, because they have no other factor than 1, the first number.

But 4 is not only 4 × 1, it is 2 × 2, and we may notice that the dots form a square—it is a compound number. 5 is again a prime; 6 can be arranged in three ways—in a row of ones, in three rows of 2 or two rows of 3, but these are the same if we look at them a different way round, i.e., 2 × 3 is 3 × 2. 7 is a prime, but for 8 we can have 2 × 4 and 4 × 2, which are the same. 9 is again a square number; it has no factors except 3. Here we might give the expressions 2² for 2 × 2, 3² for 3 × 3 and 3³ for 3 × 3 × 3.

We might go on to pick out all the primes by what is called the sieve of Eratosthenes, and to give all squares and cubes, say up to 100. Sometimes we speak of measure of numbers; 4 can be measured into rows of twos, 6 into rows of twos or threes, so 2 is said to be a common measure of 4 and 6.

After working some examples in factors and measures, it will be well to leave the matter, returning to the subject later. I should pass over for girls the wearisome exercises in weights and measures, bills of parcels, etc., very slightly. These things belong to the shop rather than the school, and waste the time that should be given to learning principles.

Vulgar fractions.We may proceed at once to fractions. In nothing is the advice Festina lente more valuable than now. Once give the children a clear idea of what a fraction is, how the two numbers represent respectively the size of the pieces and the number taken, and all will be easy. They are already familiar with ¹⁄₂d. and ³⁄₄d., so we can get from them that the lower figure stands for the number of pieces into which the penny is divided, and that the figure above shows the number of pieces taken. Many fractions should be drawn by the children—⁵⁄₆ of a line, a circle, a square, etc. The fraction may be written thus: 56 numberernamer,

5 gives the number of pieces taken; is numberer or numerator; 6 gives the number of pieces into which the whole is cut,
the size, the name, the denominator.

Let there be plenty of such questions as these: What is the effect of increasing the numerator or the denominator? Of doubling each? Of halving each? Notice that most things grow larger the larger the number, but with a fraction the larger the denominator the smaller the pieces. Children should not have books giving explanations. They must discover these by the dialectic process, and then in their own words answer questions, and sometimes explain every step in the sum they are working. All we require in books are well-chosen examples. Those who have not taught, have no idea how hard children find it to get really hold of the nature of a fraction. Homely illustrations should not be spared. For instance, there are two ways of getting much cake. To take many pieces, that is have a large numerator,—or to look out the biggest piece, that is have a small denominator.

Multiplication and division by integers.We are now ready for multiplication and division by integers. Take ⁵⁄₁₂. There are two ways of making the fraction twice as large, that is by taking twice as many pieces, that is ¹⁰⁄₁₂, or twice as large pieces, ⁵⁄₆. The shortest way must always be insisted on. Similarly, ⁴⁄₅ may be divided by 2 in two ways. Many examples should be worked out in detail thus:—

³⁄₁₄ × 7 ÷ 3 ÷ 4 × 5 ÷ 8.

³⁄₁₄ × 7 = ³⁄₂; ³⁄₂ ÷ 3 = ¹⁄₂; ¹⁄₂ ÷ 4 = ¹⁄₈; ¹⁄₈ × 5 = ⁵⁄₈; ⁵⁄₈ ÷ 8 = ⁵⁄₆₄.

Nearly all children will write thus: ³⁄₄ × 7 = ³⁄₂ ÷ 3, etc., and leave the whole unreadable.

Next should come the proposition 7 is 8 times as large as ⁷⁄₈. (Some pupils might be ready to use letters by this time, a is b times as large as ab. The teacher must be on the watch for such.) It is very difficult for young children to see this, and also that ⁷⁄₈ is the same as 7 ÷ 8. This should be illustrated by drawings in a variety of ways.

By fractions.On that would follow multiplication of fractions by fractions, which is explained as making a mistake and correcting. Thus if we have to multiply ⁵⁄₇ × ²⁄₃, we know how to multiply by 2, so we do that first: ⁵⁄₇ × 2 = ¹⁰⁄₇. But we have multiplied by a number three times too large; to correct the mistake, we must divide by 3; ¹⁰⁄₇ ÷ 3 = ¹⁰⁄₂₁. Similarly, we explain division. Not until some sums have been worked in detail should pupils be allowed to get hold of the rules. They should work with factors only, whenever possible.

Reduction.Now we might return to the subject of multiples and measures. We have ¹⁶⁄₂₄. We want to have it in its simplest form. We divide it into factors: 1624 = 2 × 82 × 12; 2 is a common measure of both; the 2 above makes the fraction twice as large, the 2 below twice as small, so both may be taken out. But we might have said 1624 = 8 × 28 × 3; 8 is the largest number that will measure both, so it is called the greatest common measure. I think it better not to give the ordinary rule for finding G.C.M. until its proof can be given algebraically. It is very seldom that children will fail in the attempt to analyse numbers, and so find out all their common measures.

G.C.M. and L.C.M.The common rules should now be given for finding at sight when a number is commensurable by each digit, though the reason of these rules will not perhaps appear yet. These children know at a glance whether a number can be measured by 2, 4, 8, 3 or 9, and remove the common factor.

Suppose we have 80089009, we cannot see a common factor, but we can proceed to break it up, one being commensurable by 8 and the other by 9. Then we get 8 × 10019 × 1001, and the greatest common measure comes to light. We see that the numerator of 11762205 is commensurable by 4 and 3, i.e., by 12, the denominator by 9:—

11762205 = 3 × 4 × 983 × 3 × 735 = 3 × 4 × 2 × 493 × 3 × 5 × 147;

so the G.C.M. is 49 × 3, or 147.

I may here notice there is an ingenious table by Mr. Ellis, published by Philip at 6d., showing graphically the common measures and multiples of numbers up to 36, which makes this matter clear. I give a section of it:—

We find at a glance the primes.

Looking down the line we see the multiples thus, 12 is a multiple of 1, 2, 3, 4, 6. Looking horizontally and moving down, we come to all the measures of each number.

It is also useful for teaching fractions.

Common denominators.We should next proceed to bring fractions to a common denominator preparatory to addition and subtraction. It is not always easy to find a number that will do for all the denominators. We want a common multiple, and of course the smallest we can have is the best. For this we have only to break up the denominators into factors and make up a number which shall contain all these. I would not let the pupils work at first by the mechanical methods sometimes given: 7230 + 346 + 11621.

Addition of fractions.Suppose we want to add ²⁄₃ + ³⁄₄ - ⁷⁄₈ + ¹¹⁄₂₄. I should write what we may call skeleton fractions below; I mean simply the line; next enter the denominator 24. This is 8 times as large as 3, i.e., we have made the pieces in the first 8 times as small, so we take 8 times as many. Only after working a fair number of sums should children write all in a single fraction thus:—

16 + 18 - 21 + 22 24.

If we have larger numbers, the pupils must never be allowed to make a number of long-division sums, but work thus: 7230 + 346 + 11621. They would factorise and put down 72 × 5 × 23 + 32 × 23 + 113 × 3 × 3 × 23. To get the common denominator we see we must multiply the first by 3 × 3 × 3; the second denominator by 5 × 3 × 3 × 3, the third by 5 × 2:—

7 × 3 × 3 × 3 + 3 × 5 × 9 + 11 × 5 2 × 5 × 23 × 3 × 3 × 3.

I have not given a complete exposition, but touched on what seems essential as regards the method and the order of teaching, derived from my experience of children’s difficulties, some will think, I fear, at unnecessary length.

In regard to the later rules for decimals, I need only make two remarks: that the points should be always removed from the divisor, e.g.:—

·000035 ÷ 5·9623 = ·3559623.

and the point put in as soon as we reach the decimal fraction. In working circulators it is well for a time to express the equations thus: ·32̇94̇ = No.

Proportion.As regards proportion, I need add little. But there is one vexed question: Shall we let children work by the unitary method? I think not, at least not those who are likely to go on to mathematics. We cannot get the thought of proportion too ingrained, and the unitary method evades it.

In compound proportion I would make pupils work out the double process in detail, and then with factors only, e.g.:—

If 5 men dig a trench 14 ft. long in 3 days, how long ought 12 men to take to dig one 28 ft. long? Put in tabular form thus:—

First confine attention to the length of trench.

Now we have to consider the consequences of altering the men:—

But we could have arranged it thus and worked it out fractionally at once:—

1412 : 285 ∷ 3 : x

3 x 28 x 514 x 12 = 3⃥ × 2⃥ × 1⃥4⃥ × 51⃥4⃥ × 3⃥ × 4⃥ 2 = 52 = 2¹⁄₂.

If practice sums are done, the meaning of each line should be marked at the end thus:—

Approximations.Approximate methods should be practised, and for this reason it is well to get the habit of multiplying by the larger number first.

Suppose we want a sum accurate, say to 3 decimal places. We remove the point from one of the factors, pushing it, of course, an equal distance in the other. We make the whole number reversed the multiplier, and begin with the fourth decimal figure (one beyond the one we need). This will give the fourth place as the first number, since we are multiplying by units. In the next row we must take in the fifth decimal, since we are multiplying by 10, and so on. Here is a sum worked out at length and an abbreviated one:—

Find correct to 3 places of decimals 3·45 × ·00059692:

In division we approximate by cutting off a figure each time from the divisor as soon as we have come to the number which is one less than the number of digits still to be found. Get correct to five places.

Summary.I might summarise the order of teaching fractions thus:—

What a fraction is—mixed numbers, improper fractions.

Effect of increasing or diminishing numerator or denominator.

Multiplication and division by integers.

Proposition a is b times as large as ab.

Multiplication and division by fractions.

Meaning of ²⁄₃ of ⁷⁄₈.

Measures, common measures, factors, common factors.

Reduction by inspection.

Meaning of common multiple, common measure, L.C.M. and G.C.M.

Bringing to common denominator.

Addition and subtraction.

Exclusion of some subjects.There are interesting papers by Potts of Cambridge, 2d., published by the National Society, giving the history of arithmetic. I have found it throws much interest into the subject to teach it historically. It seems to me that various things at present included in arithmetic books should be deferred; e.g., present values, annuities, etc., which no one would be likely to attempt who is unacquainted with algebra.

The Mathematical Conference called by the Committee of Ten, U.S.A., writes as follows, and I quite agree with its view: “The conference recommends that the courses in arithmetic be abridged and enriched—abridged by omitting entirely those subjects which perplex and exhaust without affording any really valuable mental discipline, and enriched by a greater number of exercises in simple calculation and in the solution of concrete problems. Among the subjects which should be curtailed or omitted are compound proportion, cube root, abstract mensuration and the greater part of commercial arithmetic. Percentage should be reduced, and the needs of practical life—profit and loss, bank discount, compound interest, with such complications as result from fractional periods of time—are useless and undesirable. The metric system should be taught in application to actual measurements, and the weights and measures handled.