ON ARITHMETIC.
The man who is ignorant that two and two make four, is stigmatized
with the character of hopeless stupidity; except, as Swift has
remarked, in the arithmetic of the customs, where two and two do not
always make the same sum.
We must not judge of the understanding of a child by this test, for
many children of quick abilities do not immediately assent to this
proposition when it is first laid before them. "Two and two make
four," says the tutor. "Well, child, why do you stare so?"
The child stares because the word _make_ is in this sentence used in a
sense which is quite new to him; he knows what it is to make a bow,
and to make a noise, but how this active verb is applicable in the
present case, where there is no agent to perform the action, he cannot
clearly comprehend. "Two and two _are_ four," is more intelligible;
but even this assertion, the child, for want of a distinct notion of
the sense in which the word _are_ is used, does not understand. "Two
and two _are called_ four," is, perhaps, the most accurate phrase a
tutor can use; but even these words will convey no meaning until they
have been associated with the pupil's perceptions. When he has once
perceived the combination of the numbers with real objects, it will
then be easy to teach him that the words _are called_, _are_, and
_make_, in the foregoing proposition, are synonymous terms.
We have chosen the first simple instance we could recollect, to show
how difficult the words we generally use in teaching arithmetic, must
be to our young pupils. It would be an unprofitable task to enumerate
all the puzzling technical terms which, in their earliest lessons,
children are obliged to hear, without being able to understand.
It is not from want of capacity that so many children are deficient in
arithmetical skill; and it is absurd to say, "such a child has no
genius for arithmetic. Such a child cannot be made to comprehend any
thing about numbers." These assertions prove nothing, but that the
persons who make them, are ignorant of the art of teaching. A child's
seeming stupidity in learning arithmetic, may, perhaps, be a proof of
intelligence and good sense. It is easy to make a boy, who does not
reason, repeat by rote any technical rules which a common
writing-master, with magisterial solemnity, may lay down for him; but
a child who reasons, will not be thus easily managed; he stops,
frowns, hesitates, questions his master, is wretched and refractory,
until he can discover why he is to proceed in such and such a manner;
he is not content with seeing his preceptor make figures and lines
upon a slate, and perform wondrous operations with the self-complacent
dexterity of a conjurer. A sensible boy is not satisfied with merely
seeing the total of a given sum, or the answer to a given question,
_come out right_; he insists upon knowing why it is right. He is not
content to be led to the treasures of science blindfold; he would tear
the bandage from his eyes, that he might know the way to them again.
That many children, who have been thought to be slow in learning
arithmetic, have, after their escape from the hands of pedagogues,
become remarkable for their quickness, is a fact sufficiently proved
by experience. We shall only mention one instance, which we happened
to meet with whilst we were writing this chapter. John Ludwig, a Saxon
peasant, was dismissed from school when he was a child, after four
years ineffectual struggle to learn the common rules of arithmetic. He
had been, during this time, beaten and scolded in vain. He spent
several subsequent years in common country labour, but at length some
accidental circumstances excited his ambition, and he became expert in
all the common rules, and mastered the rule of three and fractions, by
the help of an old school book, in the course of one year. He
afterwards taught himself geometry, and raised himself, by the force
of his abilities and perseverance, from obscurity to fame.
We should like to see the book which helped Mr. Ludwig to conquer his
difficulties. Introductions to Arithmetic are, often, calculated
rather for adepts in science, than for the ignorant. We do not pretend
to have discovered any shorter method than what is common, of teaching
these sciences; but, in conformity with the principles which are laid
down in the former part of this work, we have endeavoured to teach
their rudiments without disgusting our pupils, and without habituating
them to be contented with merely technical operations.
In arithmetic, as in every other branch of education, the principal
object should be, to preserve the understanding from implicit belief;
to invigorate its powers; to associate pleasure with literature, and
to induce the laudable ambition of progressive improvement.
As soon as a child can read, he should be accustomed to count, and to
have the names of numbers early connected in his mind with the
combinations which they represent. For this purpose, he should be
taught to add first by things, and afterwards by signs or figures. He
should be taught to form combinations of things by adding them
together one after another. At the same time that he acquires the
names that have been given to these combinations, he should be taught
the figures or symbols that represent them. For example, when it is
familiar to the child, that one almond, and one almond, are called two
almonds; that one almond, and two almonds, are called three almonds,
and so on, he should be taught to distinguish the figures that
represent these assemblages; that 3 means one and two, &c. Each
operation of arithmetic should proceed in this manner, from
individuals to the abstract notation of signs.
One of the earliest operations of the reasoning faculty, is
abstraction; that is to say, the power of classing a number of
individuals under one name. Young children call strangers either men
or women; even the most ignorant savages[15] have a propensity to
generalize.
We may err either by accustoming our pupils too much to the
consideration of tangible substances when we teach them arithmetic, or
by turning their attention too much to signs. The art of forming a
sound and active understanding, consists in the due mixture of facts
and reflection. Dr. Reid has, in his "Essay on the Intellectual Powers
of Man," page 297, pointed out, with great ingenuity, the admirable
economy of nature in limiting the powers of reasoning during the first
years of infancy. This is the season for cultivating the senses, and
whoever, at this early age, endeavours to force the tender shoots of
reason, will repent his rashness.
In the chapter "on Toys," we have recommended the use of plain,
regular solids, cubes, globes, &c. made of wood, as playthings for
children, instead of uncouth figures of men, women and animals. For
teaching arithmetic, half inch cubes, which can be easily grasped by
infant fingers, may be employed with great advantage; they can be
easily arranged in various combinations; the eye can easily take in a
sufficient number of them at once, and the mind is insensibly led to
consider the assemblages in which they may be grouped, not only as
they relate to number, but as they relate to quantity or shape;
besides, the terms which are borrowed from some of these shapes, as
squares, cubes, &c. will become familiar. As these children advance in
arithmetic to square or cube, a number will be more intelligible to
them than to a person who has been taught these words merely as the
formula of certain rules. In arithmetic, the first lessons should be
short and simple; two cubes placed _above_ each other, will soon be
called two; if placed in any other situations near each other, they
will still be called two; but it is advantageous to accustom our
little pupils to place the cubes with which they are taught in
succession, either by placing them upon one another, or laying in
columns upon a table, beginning to count from the cube next to them,
as we cast up in addition. For this purpose, a board about six inches
long, and five broad, divided into columns perpendicularly by slips of
wood three eighths of an inch wide, and one eighth of an inch thick,
will be found useful; and if a few cubes of colours _different from
those already mentioned_, with numbers on their six sides, are
procured, they may be of great service. Our cubes should be placed,
from time to time, in a different order, or promiscuously; but when
any arithmetical operations are to be performed with them, it is best
to preserve the established arrangement.
One cube and one other, are called two.
Two what?
Two cubes.
One glass, and one glass, are called two glasses. One raisin, and one
raisin, are called two raisins, &c. One cube, and one glass, are
called what? _Two things_ or two.
By a process of this sort, the meaning of the abstract term _two_ may
be taught. A child will perceive the word _two_, means the same as the
words _one and one_; and when we say one and one are called two,
unless he is prejudiced by something else that is said to him, he will
understand nothing more than that there are two names for the same
thing.
"One, and one, and one, are called three," is the same as saying "that
three is the name for one, and one, and one." "Two and one are three,"
is also the same as saying "that three is the name of _two and one_."
Three is also the name of one and two; the word three has, therefore,
three meanings; it means one, and one, and one; _also_, two and one;
also, one and two. He will see that any two of the cubes may be put
together, as it were, in one parcel, and that this parcel may be
called _two_; and he will also see that this parcel, when joined to
another single cube, will _make_ three, and that the sum will be the
same, whether the single cube, or the two cubes, be named first.
In a similar manner, the combinations which form _four_, may be
considered. One, and one, and one, and one, are four.
One and three are four.
Two and two are four.
Three and one are four.
All these assertions mean the same thing, and the term _four_ is
equally applicable to each of them; when, therefore, we say that two
and two are four, the child may be easily led to perceive, and indeed
to _see_, that it means the same thing as saying one _two_, and one
_two_, which is the same thing as saying two _two's_, or saying the
word _two_ two times. Our pupil should be suffered to rest here, and
we should not, at present, attempt to lead him further towards that
compendious method of addition which we call multiplication; but the
foundation is laid by giving him this view of the relation between two
and two in forming four.
There is an enumeration in the note[16] of the different combinations
which compose the rest of the Arabic notation, which consists only of
nine characters.
Before we proceed to the number ten, or to the new series of
numeration which succeeds to it, we should make our pupils perfectly
masters of the combinations which we have mentioned, both in the
direct order in which they are arranged, and in various modes of
succession; by these means, not only the addition, but the
subtraction, of numbers as far as nine, will be perfectly familiar to
them.
It has been observed before, that counting by realities, and by
signs, should be taught at the same time, so that the ear, the eye,
and the mind, should keep pace with one another; and that technical
habits should be acquired without injury to the understanding. If a
child begins between four and five years of age, he may be allowed
half a year for this essential, preliminary step in arithmetic; four
or five minutes application every day, will be sufficient to teach him
not only the relations of the first decade in numeration, but also how
to write figures with accuracy and expedition.
The next step, is, by far the most difficult in the science of
arithmetic; in treatises upon the subject, it is concisely passed over
under the title of Numeration; but it requires no small degree of care
to make it intelligible to children, and we therefore recommend, that,
besides direct instruction upon the subject, the child should be led,
by degrees, to understand the nature of classification in general.
Botany and natural history, though they are not pursued as sciences,
are, notwithstanding, the daily occupation and amusement of children,
and they supply constant examples of classification. In conversation,
these may be familiarly pointed out; a grove, a flock, &c. are
constantly before the eyes of our pupil, and he comprehends as well as
we do what is meant by two groves, two flocks, &c. The trees that form
the grove are each of them individuals; but let their numbers be what
they may when they are considered as a grove, the grove is but one,
and may be thought of and spoken of distinctly, without any relation
to the number of single trees which it contains. From these, and
similar observations, a child may be led to consider _ten_ as the name
for a _whole_, an _integer_; a _one_, which may be represented by the
figure (1): this same figure may also stand for a hundred, or a
thousand, as he will readily perceive hereafter. Indeed, the term one
hundred will become familiar to him in conversation long before he
comprehends that the word _ten_ is used as an aggregate term, like a
dozen, or a thousand. We do not use the word ten as the French do _une
dizaine_; ten does not, therefore, present the idea of an integer till
we learn arithmetic. This is a defect in our language, which has
arisen from the use of duodecimal numeration; the analogies existing
between the names of other numbers in progression, is broken by the
terms eleven and twelve. _Thirteen_, _fourteen_, &c. are so obviously
compounded of three and ten, and four and ten, as to strike the ears
of children immediately, and when they advance as far as twenty, they
readily perceive that a new series of units begins, and proceeds to
thirty, and that thirty, forty, &c. mean three tens, four tens, &c. In
pointing out these analogies to children, they become interested and
attentive, they show that species of pleasure which arises from the
perception of _aptitude_, or of truth. It can scarcely be denied that
such a pleasure exists independently of every view of utility and
fame; and when we can once excite this feeling in the minds of our
young pupils at any period of their education, we may be certain of
success.
As soon as distinct notions have been acquired of the manner in which
a collection of ten units becomes a new unit of a higher order, our
pupil may be led to observe the utility of this invention by various
examples, before he applies it to the rules of arithmetic. Let him
count as far as ten with black pebbles,[17] for instance; let him lay
aside a white pebble to represent the collection of ten; he may count
another series of ten black pebbles, and lay aside another white one;
and so on, till he has collected ten white pebbles: as _each_ of the
ten white pebbles represents ten black pebbles, he will have counted
one hundred; and the ten white pebbles may now be represented by a
single red one, which will stand for one hundred. This large number,
which it takes up so much time to count, and which could not be
comprehended at one view, is represented by a single sign. Here the
difference of colour forms the distinction: difference in shape, or
size, would answer the same purpose, as in the Roman notation X for
ten, L for fifty, C for one hundred, &c. All this is fully within the
comprehension of a child of six years old, and will lead him to the
value of written figures by the _place_ which they hold when compared
with one another. Indeed he may be led to invent this arrangement, a
circumstance which would encourage him in every part of his education.
When once he clearly comprehends that the third place, counting from
the right, contains only figures which represent hundreds, &c. he will
have conquered one of the greatest difficulties of arithmetic. If a
paper ruled with several perpendicular lines, a quarter of an inch
asunder, be shown to him, he will see that the spaces or columns
between these lines would distinguish the value of figures written in
them, without the use of the sign (0) and he will see that (0) or
zero, serves only to mark the place or situation of the neighbouring
figures.
An idea of decimal arithmetic, but without detail, may now be given to
him, as it will not appear extraordinary to _him_ that a unit should
represent ten by having its place, or column changed; and nothing more
is necessary in decimal arithmetic, than to consider that figure which
represented, at one time, an integer, or whole, as representing at
another time the number of _tenth parts_ into which that whole may
have been broken.
Our pupil may next be taught what is called numeration, which he
cannot fail to understand, and in which he should be frequently
exercised. Common addition will be easily understood by a child who
distinctly perceives that the perpendicular columns, or places in
which figures are written, may distinguish their value under various
different denominations, as gallons, furlongs, shillings, &c. We
should not tease children with long sums in avoirdupois weight, or
load their frail memories with tables of long-measure, and
dry-measure, and ale-measure in the country, and ale-measure in
London; only let them cast up a few sums in different denominations,
with the tables before them, and let the practice of addition be
preserved in their minds by short sums every day, and when they are
between six and seven years old, they will be sufficiently masters of
the first and most useful rule of arithmetic.
To children who have been trained in this manner, subtraction will be
quite easy; care, however, should be taken to give them a clear notion
of the mystery of _borrowing_ and _paying_, which is inculcated in
teaching subtraction.
From 94
Subtract 46
"Six from four I can't, but six from ten, and four remains; four and
four _is_ eight."
And then, "One that I borrowed and four are five, five from nine, and
four remains."
This is the formula; but is it ever explained--or can it be? Certainly
not without some alteration. A child sees that six cannot be
subtracted (taken) from four: more especially a child who is
familiarly acquainted with the component parts of the names six and
four: he sees that the sum 46 is less than the sum 94, and he knows
that the lesser sum may be subtracted from the greater; but he does
not perceive the means of separating them figure by figure. Tell him,
that though six cannot be deducted from four, yet it can from
fourteen, and that if one of the tens which are contained in the (9)
ninety in the uppermost row of the second column, be supposed to be
taken away, or borrowed, from the ninety, and added to the four, the
nine will be reduced to 8 (eighty), and the four will become fourteen.
_Our_ pupil will comprehend this most readily; he will see that 6,
which could not be subtracted from 4, may be subtracted from fourteen,
and he will remember that the 9 in the next column is to be considered
as only (8). To avoid confusion, he may draw a stroke across the (9)
and write 8 over[18] it [8 over (9)] and proceed to the remainder of
the operation. This method for beginners is certainly very distinct,
and may for some time, be employed with advantage; and after its
rationale has become familiar, we may explain the common method which
depends upon this consideration.
"If one number is to be deducted from another, the remainder will be
the same, whether we add any given number to the smaller number, or
take away the same given number from the larger." For instance:
Let the larger number be 9
And the smaller 4
If you deduct 3 from the larger it will be 6
From this subtract the smaller 4
--
The remainder will be 2
--
Or if you add 3 to the smaller number, it
will be 7
--
Subtract this from the larger number 9
7
--
The remainder will be 2
Now in the common method of subtraction, the _one_ which is borrowed
is taken from the uppermost figure in the adjoining column, and
instead of altering that figure to _one_ less, we add one to the
lowest figure, which, as we have just shown, will have the same
effect. The terms, however, that are commonly used in performing this
operation, are improper. To say "one that I borrowed, and four"
(meaning the lowest figure in the adjoining column) implies the idea
that what was borrowed is now to be repaid to that lowest figure,
which is not the fact. As to multiplication, we have little to say.
Our pupil should be furnished, in the first instance, with a table
containing the addition of the different units, which form the
different products of the multiplication table: these he should, from
time to time, add up as an exercise in addition; and it should be
frequently pointed out to him, that adding these figures so many times
over, is the same as multiplying them by the number of times that they
are added; as three times 3 means 3 added three times. Here one of the
figures represents a quantity, the other does not represent a
quantity, it denotes nothing but the times, or frequency of
repetition. Young people, as they advance, are apt to confound these
signs, and to imagine, for instance, in the rule of three, &c. that
the sums which they multiply together, mean quantities; that 40 yards
of linen may be multiplied by three and six-pence, &c.--an idea from
which the misstatements in sums that are intricate, frequently arise.
We have heard that the multiplication table has been set, like the
Chapter of Kings, to a cheerful tune. This is a species of technical
memory which we have long practised, and which can do no harm to the
understanding; it prevents the mind from no beneficial exertion, and
may save much irksome labour. It is certainly to be wished, that our
pupil should be expert in the multiplication table; if the cubes which
we have formerly mentioned, be employed for this purpose, the notion
of _squaring_ figures will be introduced at the same time that the
multiplication table is committed to memory.
In division, what is called the Italian method of arranging the
divisor and quotient, appears to be preferable to the common one, as
it places them in such a manner as to be easily multiplied by each
other, and as it agrees with algebraic notation.
The usual method is this:
Divisor
71)83467(1175
Italian method:
Dividend
83467| 71
| ----
| 1175
The rule of three is commonly taught in a manner merely technical:
that it may be learned in this manner, so as to answer the common
purposes of life, there can be no doubt; and nothing is further from
our design, than to depreciate any mode of instruction which has been
sanctioned by experience: but our purpose is to point out methods of
conveying instruction that shall improve the reasoning faculty, and
habituate our pupil to think upon every subject. We wish, therefore,
to point out the course which the mind would follow to solve problems
relative to proportion without the rule, and to turn our pupil's
attention to the circumstances in which the rule assists us.
The calculation of the price of any commodity, or the measure of any
quantity, where the first term is one, may be always stated as a sum
in the rule of three; but as this statement retards, instead of
expediting the operation, it is never practised.
If one yard costs a shilling, how much will three yards cost?
The mind immediately perceives, that the price added three times
together, or multiplied by three, gives the answer. If a certain
number of apples are to be equally distributed amongst a certain
number of boys, if the share of one is one apple, the share of ten or
twenty is plainly equal to ten or twenty. But if we state that the
share of three boys is twelve apples, and ask what number will be
sufficient for nine boys, the answer is not obvious; it requires
consideration. Ask our pupil what made it so easy to answer the last
question, he will readily say, "Because I knew what was the share of
one."
Then you could answer this new question if you knew the share of one
boy?
Yes.
Cannot you find out what the share of one boy is when the share of
three boys is twelve?
Four.
What number of apples then will be enough, at the same rate, for nine
boys?
Nine times four, that is thirty-six.
In this process he does nothing more than divide the second number by
the first, and multiply the quotient by the third; 12 divided by 3 is
4, which multiplied by 9 is 36. And this is, in truth, the foundation
of the rule; for though the golden rule facilitates calculation, and
contributes admirably to our convenience, it is not absolutely
necessary to the solution of questions relating to proportion.
Again, "If the share of three boys is five apples, how many will be
sufficient for nine?"
Our pupil will attempt to proceed as in the former question, and will
begin by endeavouring to find out the share of one of the three boys;
but this is not quite so easy; he will see that each is to have one
apple, and part of another; but it will cost him some pains to
determine exactly how much. When at length he finds that one and
two-thirds is the share of one boy, before he can answer the question,
he must multiply one and two-thirds by nine, which is an operation _in
fractions_, a rule of which he at present knows nothing. But if he
begins by multiplying the second, instead of dividing it previously by
the first number, he will avoid the embarrassment occasioned by
fractional parts, and will easily solve the question.
3 : 5 : 9 : 15
Multiply 5
by 9
--
it makes 45
which product 45, divided by 3, gives 15.
Here our pupil perceives, that if a given number, 12, for instance, is
to be divided by one number, and multiplied by another, _it will come
to the same thing_, whether he begins by dividing the given number, or
by multiplying it.
12 divided by 4 is 3, which
multiplied by 6 is 18;
And
12 multiplied by 6 is 72, which
divided by 4 is 18.
We recommend it to preceptors not to fatigue the memories of their
young pupils with sums which are difficult only from the number of
figures which they require, but rather to give examples _in practice_,
where aliquot parts are to be considered, and where their ingenuity
may be employed without exhausting their patience. A variety of
arithmetical questions occur in common conversation, and from common
incidents; these should be made a subject of inquiry, and our pupils,
amongst others, should try their skill: in short, whatever can be
taught in conversation, is clear gain in instruction.
We should observe, that every explanation upon these subjects should
be recurred to from time to time, perhaps every two or three months;
as there are no circumstances in the business of every day, which
recall abstract speculations to the minds of children; and the pupil
who understands them to-day, may, without any deficiency of memory,
forget them entirely in a few weeks. Indeed, the perception of the
chain of reasoning, which connects demonstration, is what makes it
truly advantageous in education. Whoever has occasion, in the business
of life, to make use of the rule of three, may learn it effectually in
a month as well as in ten years; but the habit of reasoning cannot be
acquired late in life without _unusual_ labour, and uncommon
fortitude.
FOOTNOTES:
[15] V. A strange instance quoted by Mr. Stewart, "On the Human Mind,"
page 152.
[16]
NOTE.
1 Two is 1 the - name for 2 =
1 1 1 1 2 - - 3 3 = =
1 1 1 1 1 1 2 1 2 3 2 - - - - 4 4 4 4 = = = =
1 1 1 1 1 1 1 1 1 1 1 2 3 1 2 3 4 2 2 - - - - - - 5 5 5 5 5 5 = = = =
= =
1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 2 3 4 2 3 1 2 3 4 5 2 2 2 2
3 3 - - - - - - - - - - - 6 6 6 6 6 6 6 6 6 6 6 = = = = = = = = = = =
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 1 1 1 1 1 2 2 3 4
5 2 3 4 1 2 3 4 5 6 2 2 2 2 2 3 3 3 - - - - - - - - - - - - - - 7 7 7
7 7 7 7 7 7 7 7 7 7 7 = = = = = = = = = = = = = =
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 2 1
2 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 4 1 2 3 4 5 6 7 2 2 2 3 3 3 3 3 -
- - - - - - - - - - - - - - 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 = = = = = =
= = = = = = = = =
1 1 2 1 1 5 2 2 3 4 2 2 3 4 4 4 4 5 6 - - - - - - - 8 8 8 8 8 8 8 = =
= = = = =
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1
1 2 1 1 1 1 1 1 1 1 1 2 2 1 2 2 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 1 2 3
4 5 6 7 8 2 2 2 3 3 3 3 - - - - - - - - - - - - - - - 9 9 9 9 9 9 9 9
9 9 9 9 9 9 9 = = = = = = = = = = = = = = =
1 1 1 1 1 1 2 3 1 2 1 1 2 1 1 2 1 3 3 4 4 5 6 2 2 4 5 2 2 2 2 3 3 3 3
3 3 4 4 4 4 5 5 6 7 - - - - - - - - - - - - - - 9 9 9 9 9 9 9 9 9 9 9
9 9 9 = = = = = = = = = = = = = =
[17] The word calculate is derived from the Latin calculus, a pebble.
[18] This method is recommended in the Cours de Math, par Camus, p.
38.