boys to follow strictly the intuitive exposition of the principles of numbers, after they have been taught the operations of technical arithmetic. Boys are too apt to disregard the interests of their higher faculties: they almost invariably shirk the exercise of their reasoning powers, when they can attain their object by an effort of the memory; they seldom undertake the labour of a demonstration when they can apply a rule.

After the intuitive system of arithmetic has been taught, the pupil may then pursue with advantage that higher kind of mental arithmetic which is based upon the notation and processes of technical arithmetic.

Mental arithmetic, therefore, may be divided into two kinds; viz., intuitive arithmetic, and the arithmetic of figures, or the arithmetic depending upon the ordinary notation of numbers.

INTUITIVE ARITHMETIC.

It has been observed, that the intuitive properties and operations of numbers should be taught from natural objects apart from their symbolical representations. The objects usually employed for this purpose are dots, marks, balls, counters, &c. ; each object being regarded as a unit, or one thing. For ordinary school instruction, I prefer the use of dots and marks thus four will be represented by .... or by III. These dots, or marks, as the case may be, should be written distinctly and boldly with chalk upon the black board, as they are wanted, or as they may be required to demonstrate any particular property of numbers. I prefer this mode of extemporizing the arrangement of the units, to the ordinary form of the Pestallozzian Board.

This method of teaching arithmetic is pre-eminently suggestive; for example, if the teacher wants to show that three and two make five, he leads the pupils to the result in the following manner :—

... and

After counting the first group of dots, they go on to put the units in the second group to them; thus three and one more make four, and one more make five; that is, three and two make five. It is scarcely necessary to say, that the master should never tell the pupils anything which they are able to discover for themselves.

The master should never leave a subject until his pupils are thoroughly conversant with it. The different branches of arithmetic are so connected with one another, that you cannot reach the higher branches until you have made good your footing upon the lower ones.

Problems suited to the minds of the children, bearing on the actual business of life, should be given after each exercise.

And after the principles involved in each exercise have been sufficiently elucidated, the application of these principles to slate arithmetic should be fully explained and illustrated. In this way the symbols of arithmetic will be stripped of their repulsive abstractions and associated with actual operations and practical principles. In all these exercises, the teacher should regard utility as the means for attaining an important end, and that end is the intellectual culture of his pupils. His immediate object may be utility; but his final purpose is to lay the sure basis of future progress and development.

one and one make two; one from two, and one remains.

two and one make three; one from three, and two remain. .. and · make four; one from four, and three remain. And so on.

Problems.

1. A boy had three marbles; I gave him one more; how many had

he then?

2. I had four pence, and out of it I paid one penny for a pencil; how many pence had I left?

And so on.

These exercises and problems should be continued until the pupils are thoroughly conversant with the addition and subtraction by ones.

one and two make three; two from three, and one remains.

two and two make four; two from four, and two remain.

... and .. make five,

three and two make five; two from five, and three remain. ....and

make six,

four and two make six; two from six, and four remain.

And so on.

Problems.

1. I paid the grocer two pence for eggs, and five pence how much did I give him altogether?

for butter;

2. I bought some oranges for seven pence, and some apples for two pence; how much should I pay altogether?

3. A boy had seven pence, and spent two pence of it; how much had he left?

4. Show, by counting your fingers, that five and two make seven. 5. I had to pay the draper two pence; what change should he return me out of six pence?

And so on.

These exercises, &c., should be continued until the pupils are thoroughly conversant with the addition and subtraction by twos. This observation will also apply to all the exercises which may be hereafter given.

Addition and Subtraction by Threes.

one and three make four; three from four, and one remains.

.. and ..
••• make five,

two and three make five; three from five, and two remain.

... and ... make six,

three and three make six; three from six, and three remain.

and ... make seven,

four and three make seven; three from seven, and four remain.

And so on.

Problems.

1. I have five pence in one pocket, and three pence in the other; how many pence have I altogether?

2. I had seven shillings, and gave away three of them; how many shillings have I left ?

3. A woman bought some eggs for three pence; what change should the shopman give her out of a shilling, or twelve pence?

4. I paid four pence for sugar, and three pence for some barley; how much did I pay altogether?

5. A butcher had eleven sheep, and killed three of them; how many sheep had he left ?

And so on to the addition and subtraction by fours, fives, sixes, sevens, &c.

Mixed Questions in Addition and Subtraction.

1. I paid three pence for treacle, five pence for sugar, and two pence for barley? how much did I pay altogether? Ans. Ten pence. Here actually putting the pence down,

and .. make ten.

2. Add five and seven together, and then take away four from the sum. Ans. Eight.

3. I had nine pence in my pocket this morning; but I paid, out of it, three pence for a book, and four pence for paper; how many pence have I left? Ans. Two pence.

And so on.

or minus ;

The Symbols of Addition, Subtraction, and Equality. The sign of addition is + or plus; that of subtraction and that of equality or equal. The following figures stand for the number of marks placed over them :

=

0 1 2 3 4 5 6 7

Exercises showing the use of Symbols.

MINUTES OF COUNCIL.

TO THE EDITOR OF THE ENGLISH JOURNAL OF EDUCATION.

SIR,-At a recent meeting of the "Ayrshire Certificated Masters' Association," I was appointed to draw your attention, and through your Journal; that of the teachers of the country, to a recent minute of council, by which we are informed future volumes of reports will not be sent to teachers. This we consider a loss, and I doubt not many others will think so too. It is proposed to send a memorial to London regarding it; but with the advice of an inspector we are anxious to carry along with us the teachers of the country. The plea of expense set forth in the minute is an exceedingly paltry one. But something like the following plan might perhaps do. Let only the administration part be sent to committees, and the Report, as heretofore, to certificated teachers. The hints and suggestions of inspectors are almost of no use, unless they are in the possession of the teacher. Their very variety helps to sustain an independent teacher against the vague and impracticable theories of particular inspectors. In the assurance that you will continue to plead on behalf of the teacher, I have taken the liberty of addressing these lines. I am yours, &c., J. G.

Kilmarnock.

VALUE OF INSPECTION.—I have inspected numerous schools in various parts of the country, and connected with different religious denominations-for, in an educational point of view, I feel an interest in all,-and although a majority of these have been connected with the National Society, yet I have come to the conclusion that Government aid and inspection, in the way in which it is given-I do not say it is impossible to improve it—is invariably a benefit, and that, generally speaking, schools not inspected, and not receiving aid, are inferior in teaching power and efficiency. I say this not merely on my own authority, but also on the authority of others who have ample means of judging; and I see many schools suffering from the crotchets of individual managers, who would, I am confident, if they had time and opportunity to look into the matter in all its bearings, see good reason for altering their opinions.-Dean Dawes' able Lecture at Huddersfield.

PROGRESS IN KNOWLEDGE.-He that would make a real progress in knowledge must dedicate his age as well as youth, the latter growth as well as the first fruits, at the altar of truth.—Berkeley.

Entelligence.

NEW BOARD OF EDUCATION.— One of the measures contemplated by the Cabinet for early adoption this session, is, we understand, the re-organization of the Committee of Council on Education, so as to vest the administration of its present, and probably of enlarged functions, in a new government department, similar in organization to the Board of Trade, with a "minister of education" at its head.

This important post will, we have reason to believe, be offered to Lord John Russell, whose signal and efficient services to the cause of public instruction have earned for him the gratitude of its friends in all ranks, classes, and religious bodies, and whose return to the Cabinet, and to official usefulness in that dignified sphere, will meet with general approval, should our expectations be verified.

UNITED ASSOCIATION OF SCHOOLMASTERS. THE following notice of the Second Anniversary of this Association appeared in the Times of Friday, the 28th December :

"A body which, under this title, has entered the second year of its existence, opened its annual meeting yesterday at the rooms of the Society of Arts, in the Adelphi, under the presidency of Mr. Tate, a gentleman well known for his educational works. The proceedings of this meeting will not be concluded until to-day, but are of a character so professional in the main, as to be unsuited for any lengthened notice. From the report of the acting committee we gather that the association now numbers nearly 250 members, of whom about 100 reside in or near the metropolis. As there are said to be some 40,000 teachers in the United Kingdom, the association is obviously still in its infancy, and it seems to be struggling at the outset with financial difficulties; but there seems little reason to doubt, from the earnestness of the men engaged in the union, from the usefulness of the course which they have chalked out for themselves, and from the temperate and able manner in which their discussions are conducted, that every year will find them increasing in importance and influence. The association is founded on a strictly professional basis, religious distinctions, except in the case of Roman Catholics and Jews, being entirely ignored. Among the subjects brought forward for consideration at this meeting are included the following: The best Method of Teaching Mechanics in Schools,' 'The Teaching of Reading,'' Rational Gymnastics as a Branch of Education,''The Teaching of History and Geography,' and 'Infant Gardens.' Besides, however, topics of this description, it came out in the course of the proceedings yesterday, that the association aspires hereafter to overlook and protect the interests of schoolmasters as professional men, especially in their relations with the clergy and the Committee of Council on Education: not that they propose to do so in any intemperate or hostile spirit, for the services rendered by the Committee of Council to the cause of education were most warmly acknowledged by the speakers; but some of the minutes are objected to, and on other points it is felt that the schoolmasters' experience may be turned to account in giving an increased impulse to to the spread of elementary instruction. Thus, it was made the subject of complaint that the Privy Council, in its anxiety for superior scholarship in the master, did not pay sufficient attention to his qualifications as a teacher, and that the remuneration given to him for the time spent on his pupil-teachers was made dependent on their progress instead of his services. It was also suggested that, the parents of the children whom it was most important to get into schools, being, in a large number of cases, recipients of out-door relief, an education test should be attached to such relief. There seems, on the whole, good ground for believing that an association of this kind may do good service to a great public cause, and we hope to see it gradually increase in strength and usefulness."

A full report of the meeting may be had on application to the Corresponding Secretary, Mr. J. Tilleard, 17, Scarsdale Terrace, Kensington.

The following is the programme of the meetings to be held during the remaining portion of the present session :

MEETINGS OF THE UNITED ASSOCIATION OF SCHOOLMASTERS. ST. MARTIN'S HALL, LONG ACRE, LONDON.

Session 1855-56.

SATURDAY, FEBRUARY 2, 1856.

Discussion on the Method of Teaching Geography, to be opened by Mr. TILLEARD.