PART II. MATHEMATICS.

ARITHMETIC.

By Dorothea Beale.

Multiplication is vexation,

Division is as bad,

The Rule of Three doth puzzle me,

And Practice drives me mad.

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:—

····························
1234567
·+··+···+····+·····+······+······
1+11+21+31+41+51+6
2+22+32+42+5
etc.

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:—

Score.Dozen.One.

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—

£ S. D.
2„14„6
1„17„9
16„9

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.

hundredstensunits

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.

th.hun.tensunits
5 93
29 9
8 2

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:—

th.h.t.u.
5093
2909
2184

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.

hun.tensunits tenthshundth.thousandths
132.79
25.897

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.

 891
 891
 891
 891
3564

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:—

7times 1= 7 7 7 7 7 7 7 7 7 7 7 7
 7 7 7 7 7 7 7 7 7 7 7
7„ 2= 14 7 7 7 7 7 7 7 7 7 7
7„ 3= 21 7 7 7 7 7 7 7 7 7
7„ 4= 28 7 7 7 7 7 7 7 7
7„ 5= 35 7 7 7 7 7 7 7
7„ 6= 42 7 7 7 7 7 7
7„ 7= 49 7 7 7 7 7
7„ 8= 56 7 7 7 7
7„ 9= 63 7 7 7
7„10= 70 7 7
7„11= 77 7
7„12= 84

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:—

73
25
1460=20times.
365=5„
1825=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:—

73
5
365=5times.
5
1825=5×5times.
47300=100 times.
1825=1⁄4 of 100, or 25 times.
73
30
2190=30times.
365=5„
1825=25„
2730=10times.
365=5„
3650=50„
1825=1⁄2 50 = 25 times.

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.

874
43496

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.

 1909
23)43921
23   
209  
207  
221
207
14

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:—

123456789101112
Ones············
Twos · · · · · ·
Threes · · · ·
Fours · · ·
Fives · ·
Sixes · ·
Sevens ·
Eights ·
Nines ·
Tens ·
Elevens ·
Twelves ·

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.

Here- 230=2 × 5 × 23 -We want therefore as the common denominator 2 × 5 × 23 × 3 × 3 × 3, which is 6210.
46=23 × 2
561=3 × 3 × 3 × 23

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.

10,000 No=3294·294, etc.
10 No=   3·294, etc.
9990 No=3291
∴ No=3291
9990

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:—

Men.Long.Days.
 5143
1228?

First confine attention to the length of trench.

Ft. Ft. Days. Days.
14:28∷3:6

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

Men. Days.
 5 6
12 ?
Men. Men. Days. Days.
12: 5∷6:21⁄2.

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:—

£
984 price at£1.
Price of 984 yds. at £2 „ 15 „ 6.1968„£2.
492„10s.
246„5s.
24„ 12„6d.
2730„ 12at £2 „ 15 „ 6

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:

3·45 × ·00059692 = 345 × ·059692
·059692·059692
      345    543
17907600   2984
2387680  23876
   298460 179076
20·59374020·5936

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.

454523)145367·9(·31982
1363569
901100
454523
446577
409068
37509
36360
1149
908
241

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.

“Among the branches of this subject which it is proposed to omit are some which have survived from an epoch when more advanced mathematics was scarcely known in our schools, e.g., cube root, duodecimals; so far as any useful principles are embodied in them, they belong to algebra, and can be taught by algebraic methods with such facility, that there is no longer any sound reason for retaining them in the arithmetical course.”

I do not insist on algebra for all; it gives the same sort of mental discipline that arithmetic does, and so, educationally, is not of special value. Geometry, on the other hand, gives a different kind of training—opens a different set of ideas. Many girls, therefore, do not learn algebra, especially those who come late with no clear ideas about arithmetic. Those who have been taught arithmetic well from the beginning can be led on to use algebraic symbols and letters very early.

As soon as a pupil has gone through the course I have recommended, she is ready to take up algebra in a systematic way—I shall suppose she has already been familiarised with the use of letters as general symbols.

MATHEMATICS.

By Dorothea Beale.

How and when can we best introduce mathematical teaching? We have to do at present in girls’ schools with many who have come to the age, say of fifteen or sixteen, with no mathematical teaching except a very slight knowledge of arithmetical processes. For these it seems to me more important to give the mental training afforded by some initiation into geometrical ideas and methods, than to teach algebra.

Beginnings in the kindergarten.For the children (and they are happily a rapidly increasing number) who have had good teaching in the kindergarten, one may frame a course more approaching the ideal. Children can be quite early familiarised with geometrical forms and figures, and learn some of their simpler properties in connection with the drawing and modelling lessons.

Practical geometry.The Conference on Mathematics, called by the Committee of Ten, U.S.A., recommends that children from the age of ten should have some systematic instruction in concrete or experimental geometry. “The mere facts of plane and solid geometry should be taught, not as an exercise in logical deduction and exact demonstration, but in as concrete and objective a form as possible; the simple properties of similar plane figures and solids should not be proved, but illustrated and confirmed by cutting up and rearranging drawings and models. The course should include the careful construction of plane figures by the eye and by the help of instruments, the indirect measurements of heights and distances by the aid of figures drawn to scale, and elementary mensuration plane and solid.”

A small book by Paul Bert, First Elements of Experimental Geometry (Cassell), is very suggestive, and would throw much interest into the subject. Spencer’s Constructive Geometry may be referred to, but it is not altogether satisfactory. A useful and practical book is Geometry for Kindergarten Students, by Pullar (Sonnenschein).

Geometry before algebra.I consider that geometry should be preferred to algebra in order of time, because, as I have said, arithmetic gives the same kind of mental training as algebra, whereas from geometry the learner gains a unique mental discipline.

Its educational value.Thus the learner is taught to frame a definition; he has to put before the imagination the abstract generalised idea, and then describe, in words clear and precise, what is in the mind. Each proposition begins with a general statement regarding what is to be proved, or to be done, and compels us to have a clear idea of what we are going to talk about before we begin. The sub-enunciation makes us bring the general into the region of the particular, and infer the general from it. We must for the demonstration select certain relations relevant to the subject and omit all others, and we must be ready to give a reason for every assertion. Thus geometrical teaching trains the judgment and forms a most useful and logical habit of mind. One finds the tendency is greatly checked to use words without any clear idea of their meaning, to plunge into a subject without having set in order in the mind, what is the matter to be discussed, or the problem to be solved, and order is introduced into the general work in all other subjects of study.

Leads up to the region of ideas.But geometry has still higher uses in the process of mental development. It is, so to speak, the link between the real and the ideal; as Professor Cayley has said, “imaginary objects are the only realities, the οντως οντα, in regard to which the corresponding physical objects are as the shadows in the cave”;[23] if, on the one hand, it opens the gates of science, on the other it leads us to philosophy, and so Plato is said to have placed over the door of the Academy, “Let none enter here ignorant of geometry”.

[23] Presidential Address, Brit. Assoc.

To study geometry is to enter a new path, and we do not see at first to what heights it leads, upwards to the universe of ideas; ideas are nothing for sense, and yet they are the most necessary things for the everyday life we lead. Thus, a point, though it exists not, yet as a thought-dynamic is—it moves and traces out lines which do not exist, and yet give us direction, and are of most practical use; by them we calculate the height of real things, we guide our ships, we find paths in the heavens. Again, moving lines give us planes, and these, which exist only in thought, as they move, form what we call solid figures, i.e., something which occupies space.

Forming definitions.Of course, no one who is grounded in the principles of real education, would think of letting children begin by learning definitions; they must be made to put their vague notions into words; and it will be well for them to see how difficult this is, e.g., in the case of a straight line, an angle, though the notion is quite clear to the mind’s eye. It is surprising to those who have not taught the subject how long it takes girls, who have not been trained to exactness, to bring out, e.g., the definition of a circle. They will say, all lines drawn from the centre are equal; or all lines drawn from the centre to the circumference are equal.

No child should be allowed for a long time to see a Euclid. Each proposition must be treated as a rider, and a copious supply of riders provided in addition; the child helped to discover the solution or the proof, then set to write it; if wrong it must be gone over again and again; it will take a long time to get through a very few propositions thus, but later all is easy.

Methods of teaching.It appears from the report of the Oxford Local Examinations, August, 1897, that the methods of the dark ages still prevail in too many schools; we read: “In many cases candidates who wrote out correctly all propositions for the first six books sent up attempts at problems that can only be described as grotesque, and showed their complete failure to understand the subject, giving the unpleasing impression that all they knew was learned by heart”.

Euclid.As a formal introduction to Euclid for young pupils, I know nothing better for the teacher to study and use than Bradshaw’s First Step. Many others might be named. The Harpur Euclid is good (Longmans), and Books I. and II., by Smith and Bryant, may be specially recommended. Still I regret that the text-book in England is Euclid; its inconsistencies are manifest; we stand alone in keeping it. Yet a good workman will make the best of his tools, and there are editions which remedy many of the defects. One would, however, hope that some day Societies for the Improvement of Geometrical Teaching and Reformed Spelling will rejoice together. It does seem an anachronism not to have an angle as large as 180°; to use the circle, and think of a circumference, yet refer to no other loci, and work out in a cumbrous manner the propositions of Books III. and IV.—to talk of lines touching and not make use of limits. The more a teacher knows of the higher mathematics, and looks forward for the pupil, the better will he teach the rudiments. The treatment of the subject by Professor Henrici (London Science Class-books, Longmans) seems excellent, but I do not know how far it would answer for young beginners. I should be glad to have the experience of some who have tried it. The professor derives the notion of a point from a solid, particular figures from infinite planes, and proceeds generally in an inverse direction from that of Euclid; the nomenclature is admirably compact, and must result in a large economy of thinking power—the notion of a locus is introduced early, and the methods employed lead up to the modern or projective geometry.

I once spent some time at Zurich, a town especially remarkable for its intellectual activity, and chiefly for its mathematical school. Through the kindness of Professor Kinkel and other friends, I easily obtained permission to be present at various lessons in the Polytechnic and Canton School. I found the method there similar to that which we follow. The pupils used as a text-book Wolff’s Taschen-buch, a duodecimo of less than 300 pages, which contains the principal results in pure mathematics and the applied sciences, but no demonstrations. I heard a lesson given in the Canton School. Professor Weileman first read the proposition; it was the same as Euclid, XI. 2: to draw a perpendicular to a plane from a given point without it. About a dozen held out their hands to show they were ready to demonstrate. The professor selected one, who took his place at the board, and, subject to correction, worked the problem. The professor gave as little direct instruction as possible, appealing rather to the class. I was much struck with the eager interest that the class (I think it was Class II. B) took in the work. The next proposition (in Wolff) afforded much amusement. The demonstrator jumped to the conclusion that the lines required to complete the construction would meet, and could not be made to see he had assumed what required proof. Other members of the class offered to take the matter up; he was accordingly superseded by No. 2, who having surmounted this difficulty, also broke down before he reached the end. No. 3 therefore took his place at the board. Thus were the reasoning and inventive powers of the boys developed, and a keen interest awakened; there was no weariness, no apathy.

I make a few remarks on what may seem to some trivial matters, yet which are of importance to beginners.

In giving the proof at the board, there is no need to use three letters, and drag children by their help round every angle; we can write a Greek letter or a number, as we constantly do in trigonometry, or we could colour the angles; say the red is equal to the blue, and let the children write out the propositions in an abbreviated form first; or we might adopt the convenient and concise plan of Professor Henrici: let capitals stand for points, small letters for lines, and let angles be represented by the small letters with ∠ prefixed. Thus we have line PQ or a; PR or b; and ∠ QPR or ∠ ab; anything to avoid tediousness is good; children are so bored by verbosity.

Riders need not be always mere lines without any human or scientific interest. Suppose instead of saying—From two points to draw lines to a given line, which shall make equal angles with the given line, we say—Let CD be a mirror or a wall, a ray or a ball strikes it at P, draw the direction it will take after—or, There is a big house A, and a little house B, near a river—the man in B has to fetch water for A daily, where should he draw the water so as to go the shortest possible distance?

The method of determining the distance of the moon can be made clear long before a child is able to conceive the trigonometrical ratios, and if we are able to arouse an interest in astronomy, we may excite ardour in some which will make hard thought and work delightful. The distant prospect of the mountain top has a wonderful power of leading us on. The writer can never forget the joyful enthusiasm with which she threw herself into the study of mathematics in consequence of hearing courses of lectures on astronomy from Mr. Pullen of Cambridge, Gresham Professor of Astronomy, and the late Vice-Chancellor of Cambridge has described to her the power which the first realisation of the wonders of the boundless universe had over him when a boy of fourteen.

Mr. Glazebrook has suggested that some insight may be given to those who have no high mathematical ability into what seems so marvellous to the uninitiated, the development of curves from equations.

Algebra.The close relation between algebra and geometry becomes apparent in Euclid, Book II., but this might be shown somewhat earlier by methods such as those recommended by Mr. Wormell in the first pages of Plotting or Graphic Mathematics. We can see by a figure that 1 + 2 + 3 + 2 + 1 = 32, and lead the pupil on to the general proposition which is in constant use, when treating of falling bodies.

Or we can show similarly that the sum of an arithmetical series equals a + l2.

As regards the formal introduction of generalised arithmetic or algebra, one cannot lay down any limit of age, owing to the very untrained state in which girls come to secondary schools, but with children who have been taught thoroughly the principles of arithmetic up to fractions, it is easy to introduce literal symbols and so prepare the way: this should be done much earlier than is usual.

Children well taught in arithmetic might perhaps begin the subject formally about thirteen, and I think it well for the first term to drop arithmetic altogether, so as to get as much time as possible for overcoming the initial difficulties, and making use of the zeal which a new study gives; but of course every good teacher of arithmetic will train his pupils to use letters for numbers very much earlier. There is a good deal put into arithmetic books, which would be much better dealt with by algebraical methods, and should be postponed, e.g., involution and evolution, and much time should be saved by omitting long sets of examples on weights and measures, etc., and giving sums to be worked out mechanically with large numbers. As in arithmetic, it is extremely important to give an insight into the composition of quantities, so that de-composition may be easy, subsequent mechanical work in multiplication, division, involution, etc., minimised, and the pupil reach sooner the more attractive branches of the subject, and feel the power it gives.

Mixed mathematics.If children have acquired early a fair knowledge of geometry and algebra, they may, say at sixteen, be ready to pass on to those branches in which the alliance of the two is most intimate, and which are so closely correlated with all the teaching in mechanics and physics. It takes most girls some time to assimilate the ideas of the trigonometrical ratios, and it is fatal to hurry them.[24] Those who are able to proceed further, and enter upon the study of co-ordinate geometry, usually take great delight in it; and it is well, too, to lead them gradually on by some such books as Proctor’s Easy Lessons in the Differential Calculus, to form some idea of what a powerful instrument the Calculus is, before they actually make use of it or formally study it; it takes time for a new method to infiltrate the mind of an ordinary student.

[24] I may add that there is an interesting chapter in Herbart’s A B C of Sense-Perception, in which he works out trigonometrical ratios on the basis of his philosophical system: this chapter would interest those teaching mathematics.

Historical method.Finally, I would once more recommend that, whenever it is possible, pupils should be led along the path of discovery pursued by original investigators, both in physics and applied mathematics; I have found the interest of logarithms greatly increased by this method.[25]

[25] Professor Salford (Monographs on Education and Health) insists on the importance of teaching logarithms as a part of scientific arithmetic. “Often logarithms are first taught in connection with trigonometry, and the average pupil does not learn the difference between a logarithmic and a natural sine; there is no cure for this confusion but to teach logarithms where they belong and to apply them to purely arithmetical problems.” He advises the introduction of logarithms “as soon as the pupil has reached in algebra the proposition am × an = am × n, and he should be shown that the practical method of dealing with powers and roots is the logarithmic. Teachers will then abstain from annoying young pupils with difficult and needless problems solved in the antiquated manner; they will learn how to calculate a compound interest table, an excellent exercise in itself, as well as a labour-saving contrivance in arithmetic. The reason why logarithms are so little appreciated, is that teachers of arithmetic have not as a rule really learned their use; they go on wasting time in arbitrary exercises in evolution, interest, etc., done by tedious methods, and do not appreciate how instinctively the best calculators employ logarithms.”

Professor Lodge’s popular book, Pioneers of Science, is very much appreciated by the young, and I may quote à propos evidence given by Dr. Bryce of Glasgow before the Royal Commission of 1864:—

“Pure mathematics cultivates the power of deductive reasoning, and as soon as boys are capable of forming abstract ideas, and grasping general principles, as soon as they have got correct notions of numbers, and an accurate knowledge of the essential parts of arithmetic, and have made some progress in geometry, then natural philosophy may be advantageously taught. I speak on this matter from experience. My relative and colleague, who had charge of the mathematical department in the Belfast Academy, introduced natural philosophy as part of the work of all the mathematical classes. After these classes had gone a certain length in geography and algebra, he took up the elements of natural philosophy two days in the week, as part of the work of every mathematical class. He began with simple experiments, and according as the progress of the boys in Euclid and algebra admitted of it, more mathematical views of natural philosophy were introduced. The great advantage of the study of physical science is that, when properly taught, it interests boys in intellectual pursuits generally. For instance, Newton’s great discovery, the identity of the power which retains the moon in her orbit with terrestrial gravity, was being explained to a class of from twelve to eighteen boys. The teacher did not tell them the result; he enumerated the phenomena by which Newton arrived at it, taking care to present them in the order most likely to suggest it. As fact after fact was marshalled before them, they became eager and excited more and more, for they saw that something new and great was coming; and when at last the array of phenomena was complete, and the magnificent conclusion burst upon their sight, the whole class started from their seats with a scream of delight. They were conscious that they had gone through the very same mental operation, as that great man had gone through. The consciousness of fellowship with so great a mind was an elevating thing, and gave them a delight in intellectual pursuits. An unusual proportion of those boys who passed through the Belfast Academy during the twenty years that I was able to have natural and physical science taught on those principles, have, as men, been distinguished and successful; and they owe it, I am convinced, in a large degree to the taste for intellectual pursuits thus formed.”