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.
[Illustration]
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:--
· ·· ··· ···· ····· ······ ·······
1 2 3 4 5 6 7
·+· ·+·· ·+··· ·+···· ·+····· ·+······
1+1 1+2 1+3 1+4 1+5 1+6
2+2 2+3 2+4 2+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. D.
7 „ 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. D.
16 „ 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.
£ S. D.
2 „ 14 „ 6
1 „ 17 „ 9
-----------
16 „ 9
[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.
hundreds tens units
+----------+----------+----------+
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
+----------+----------+----------+
[Addition.]
th. hun. tens units
+-----+------+------+------+
| 5 | | 9 | 3 |
| | | | |
| 2 | 9 | | 9 |
+-----+------+------+------+
| 8 | | | 2 |
+-----+------+------+------+
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:--
th. h. t. u.
5 0 9 3
2 9 0 9
------------
2 1 8 4
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. tens units tenths hundth. thousandths
1 3 2 . 7 9
2 5 . 8 9 7
--------------------------------------------
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?”
891
891
891
891
----
3564
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:--
+-------------------------------------+
| 7 times 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 = 20 times.
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 = 5 times.
5
----
1825 = 5 × 5 times.
4 | 7300 = 100 times.
+-----
1825 = ¹⁄₄ of 100, or 25 times.
73
30
----
2190 = 30 times.
365 = 5 „
----
1825 = 25 „
2 | 730 = 10 times.
+----
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.
874
+----
4 |3496
+----
[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.
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
5 numberer
written thus: - --------,
6 namer
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 _a_⁄_b_. 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:
16 2 × 8
-- = ------;
24 2 × 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
16 8 × 2
-- = -----;
24 8 × 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 ⁸⁰⁰⁸⁄₉₀₀₉, 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 × 1001
--------,
9 × 1001
and the greatest common measure comes to light. We see that the
numerator of
1176
----
2205
is commensurable by 4 and 3, _i.e._, by 12, the denominator by 9:--
1176 3 × 4 × 98 3 × 4 × 2 × 49
---- = ----------- = ---------------;
2205 3 × 3 × 735 3 × 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:--
1 2 3 4 5 6 7 8 9 10 11 12
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:
7 3 11
--- + -- + ---.
230 46 621
{ 230 = 2 × 5 × 23 } We want therefore
Here { 46 = 23 × 2 } as the common
{ 561 = 3 × 3 × 3 × 23 } denominator
2 × 5 × 23 × 3 × 3 × 3, which is 6210.
[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:
7 3 11
--- + -- + ---.
230 46 621
They would factorise and put down
7 3 11
---------- + ------ + --------------.
2 × 5 × 23 2 × 23 3 × 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._:--
·35
·000035 ÷ 5·9623 = -----.
59623
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
3291
∴ No = ----
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.
5 14 3
12 28 ?
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:--
14 28
-- : -- ∷ 3 : x
12 5
3 x 28 x 5 3⃥ × 2⃥ × 1⃥4⃥ × 5 5
---------- = -------------- = - = 2¹⁄₂.
14 x 12 1⃥4⃥ 3⃥ × 4⃥ 2
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 „ 12 at £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
-------- -------
20·593740 20·5936
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 _a/b_.
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.
[Illustration]
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?
[Illustration]
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_ + _l_
---------.
2
| | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
14| * | * | * | * | * | * | * | * | * | * | * | * | * | * | | |2
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
11| * | * | * | * | * | * | * | * | * | * | * | | | | | |5
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
8| * | * | * | * | * | * | * | * | | | | | | | | |8
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
5| * | * | * | * | * | | | | | | | | | | | |11
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
2| * | * | | | | | | | | | | | | | | |14
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | |
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 _a_^{_m_} × _a_^{_n_} = _a_^{_m_ × _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.”