subserves the general method of education; for the formation of a good mental habit is manifestly promoted by substituting for a method and aim purely mechanical a method characterized by intelligence, promotive of discipline, and instinct with a solid purpose.

OBJECTS AND METHOD OF TEACHING ARITHMETIC.

Intellectual discipline of Arithmetic-School Arithmetic should be practical and economic-Method of teaching: the concrete method-Moral uses of School Arithmetic.

Were the teacher free from all limitations in the formation of his plans for attaining the ultimate end of primary education, he might possibly choose to eject reading and writing from the schoolroom, in so far as these arts are merely technical, and to substitute the intellectual and moral discipline which can be best effected through conversation and personal influence, assisted by the objects of external nature, and the lessons that might be drawn from the hourly occurrences of life. It is true that, in the initiatory stages of reading and writing, there is a right and a wrong method, contributing more or less to the discipline of the pupil, as well as to his facility and certainty of acquisition; but beyond those stages the educative purpose is really attained, we have seen, through the identification, as much as possible, of the arts taught with the objects for which they are taught, viz., instruction, discipline, and moral

and æsthetic cultivation. It is otherwise with Arithmetic. The art of manipulating numbers with dexterity, and the rationale of the expedients whereby the processes are abbreviated and guaranteed, are merely the evolving and strengthening, in the most direct way, of a special intellectual power which exerts itself spontaneously in all men. Irrespectively, therefore, of the future necessities of the child, this power would, in its relation to the general and theoretic object of education alone, demand and amply reward cultivation. The combination of parts into wholes, the dissolution of wholes into parts, and of these parts themselves into lower unities, are exercises in the relation of particulars to generals, and of generals to particulars, of great value to the intellect in other applications of its powers. The visible, we may even say the palpable, effects of error, which renders nugatory the most strenuous efforts, if vitiated by the most trifling flaw, must exercise a wonderful influence in giving the habit of accuracy and caution in all exercises of comparison and inference. Indeed, so universally diffused is the discipline given by means of the science and art of numbers, that we are perhaps scarcely able to estimate fully the extent to which it contributes to the intelligence of the people, and above all, to a rapid and easy movement of the human understanding in the conduct of ordinary affairs.

Nor are the above the most important of the disciplinary effects of adding, multiplying, and dividing wholes and parts; for it is impossible, notwithstanding

the numerous contrivances for saving excessive tension of mind which enter into the rules in obedience to which the pupils work, to elaborate a correct answer to the questions which a good arithmetical manual supplies, without a certain amount of conscious intellectual concentration. A habit of mind is thereby strengthened which more than any other constitutes the intellectual superiority of one man over another, and of man himself over the lower animals. In acquiring other subjects, the pupil may give or withdraw his attention almost at his will, and yet make sensible progress in the acquisition of knowledge. In arithmetic there is a certain amount of deliberate and sustained attention essential to even the most elementary processes. This discipline cannot be evaded without leaving the work undone. The conscious exertion of the will to keep certain powers of the understanding in operation on a special question until a certain result be reached, is not only valuable in relation to the acquisition of the subject which for the moment may engage the mind; but, in addition to this, it increases the force available for the study of every other subject. This kind of discipline belongs peculiarly to arithmetic, even when taught merely as an exercise in abstract figures.

All this is true of arithmetic, apart from its practical relations to life, on which alone ultimately rest its claims to enter into the curriculum of the primary school. A consideration of these practical relations yields us at once the final purpose towards the realiza

tion of which the teacher must direct all his efforts, the methods which he ought to follow, and a further insight into the educative nature of the art.

Arithmetic is the science and art of numbers: school-arithmetic must always be, more or less, the adaptation of the art to the future uses of the pupil. Those uses tell us that the purpose of teaching arithmetic in elementary schools, apart from its influence as a discipline, is attained when such a command has been given over numbers as enables a young man or woman to calculate with facility all those questions which arise in the ordinary course of life. This may be called Economic Arithmetic. It embraces the addition, subtraction, and division of money, proportion, and vulgar fractions. Beyond these subjects no elementary teacher ought to attempt to go if he desires to be impartial in his instructions and do justice to other subjects much more important than advance in arithmetic. His aim should be thoroughness rather than extent of acquirement.

Economic or school arithmetic embraces the domestic, but also extends to the general out-of-door, relations of the head of the family. The relation of his wages to the size of his family, to the several heads of legitimate expenditure, such as food, clothing, insurance, sick-clubs, saving, gives full occupation for the application of his knowledge, and ought to be constantly present to his thoughts. "Tell me how a man

1 Except in those schools in which pupils stay beyond the age of eleven or twelve.

spends his money, and I will tell you the character of the man," was a remark in a special sense true of the labouring man. Almost the whole range of the duties of benevolence and justice fall under the head of income and expenditure, and resolve themselves into questions of arithmetic, which cannot be encountered, much less solved, by a man unfamiliar with figures. People of the middle class are themselves so much accustomed to economic calculation that it does not occur to them how serious an obstacle a deficiency of arithmetical training is to a labouring man, still more to a labouring man's wife.

Schoolmasters are frequently to blame for the meagre practical issue of their arithmetical teaching in the operative class; and the cause of their failure is to be found in this, that having omitted to define to themselves clearly the ultimate object of their labours, they necessarily fail to find a true method, and thus expend much well-meant labour in vain. The quantity of instruction given is generally ample, but much of it is irrelevant.

By the word Economic, the purpose of arithmetical teaching in schools has been defined: the method follows from the purpose, and is called the Concrete Method. And here I come on ground so much beaten by theoretical educationists that though it is yet untrodden by the great majority of practical teachers, I shall omit those details of ways and means, which have been so frequently reiterated. It will be suffi