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_,