INSTRUCTIONS IN ARITHMETIC. Chapter I.—Introductory. In all countries where the subject of education is understood, we find arithmetic considered as one of the most useful branches of knowledge, and treated of by distinguished scholars, in respect to the development of the intellectual, moral, and even physical powers of a child, as a very important part of popular instruction. And this indeed with good reason; for, by being acquainted with numbers and their several combinations, a man learns to manage his affairs, acquires habits of order, foresight, and observation, strengthens the memory, and matures the judgment. In accordance with this view of the question we perceive, what might indeed have been all along expected, that the different methods by which arithmetic has been taught from the earliest down to the present time, have been, as far as we can learn any thing respecting them, more or less influenced by the spirit of the age or place in which they have prevailed; having, however, this one thing in common, that they all, either in precept or in practice, recommend in the case of children, that the appeal to the understanding should be made through the senses, and that all abstract arguments should be deferred until the powers of observation and memory have been in some degree exercised. Now, while we have at all times the greatest respect for the wisdom of experience, we think it quite within our province seriously to inquire, whether the works on arithmetic which are usually placed in the hands of masters and pupils in this country, duly attain the object proposed; whether the numerous, and in some respects excellent treatises on this subject, might not be systematized, or abridged, or improved; whether, in short, a time is not come, when, looking down as from an eminence on the several principles brought forward and illustrated by antient and modern writers, we might not produce a book which should be, as it were, the work of all, embodying the elements of common sense and reason which are to be found in each; collecting the scattered suggestions of the most practical and approved authors, and summing up, perhaps in short sentences, which might be committed to memory, the learned but diffuse exposition of some rule or rules, that may have been hitherto, for this very reason, lying neglected or forgotten. It is, however, right to observe, as increasing the difficulty of the task, that any one who has acquired a well-founded fame by his writings will always to a certain extent, and for a certain time, hold his ground against all innovators; and we have little right to complain of this if it is so. It is not every change, nor indeed the greater number of those changes which are proposed to us, that is desirable; and does not the name of Pestalozzi, and of several of his contemporaries and successors in Germany, in France, and in England, excite so much admiration as to raise at once a prejudice against any method which might either directly or indirectly seem to find fault with the theories or practice of such great men? Nevertheless, we would have our readers bear in mind, that it is through the light thrown on the subject by their works, that we have been led to make the following reflections, which may therefore be considered as in truth a homage" paid to their merit. Whilst, then, we give every credit to the Committee of Council for their good intentions in publishing the late book on arithmetic, which bears their sanction, and the excellence of which in very many respects we freely acknowledge, we trust we shall be understood, even by our strictures upon it, to be paying these gentlemen a still higher compliment, in attributing to them the love of truth above all other things, and the earnest and disinterested desire for the improvement of education generally throughout this kingdom. To guard, then, against any mistake on this point, we would once more repeat, that we entertain the sincerest respect for the eminent writers who have preceded us; we confess, indeed, that to them we are indebted for the greater part of the ideas which we have endeavoured to select, arrange, and explain in these pages; and as to the principles which have guided us in the task, they may be thus briefly stated : The first notions of number, and of the various combinations of which it is susceptible, are to be attained, and ought to be attained, at a very early age. They arise originally, perhaps, in the child's mind, from the instinctive desire of action, and thus become a preparation for the future man, so that we should seek to develop, as soon as possible the spirit of observation,—the very basis of all instruction,—as well as that of combination, which calls into play the faculty of invention. This applies, not only to infant schools, but also to national schools, as well as to those of a higher grade and character, where the same process has only to be continued and extended. As to the particular method of instruction, however, it will vary in each case. In institutions of the first kind, that which is called on the continent the Intuitive System, appears the most natural one to adopt; in the others, the reasoning faculties should be those principally addressed; and yet we must not be exclusive as regards either, seeing that they mutually assist each other, provided only that " intuition" predominate in the instruction of very young children; and the reasoning process, when we have to teach those of a more advanced age. Chapter II.-Of Intuitive Arithmetic. This is in some countries called Verbal or Oral Arithmetic, calculations in the head, &c., according to the ideas prevalent in particular times and places; but for our own part we prefer the first appellation, as expressive of the way in which the notion itself is acquired. As to the advantages it offers, however, we believe all parties are agreed; whilst there is nothing which tends more to improve the habit of attention, or to strengthen the memory in respect of numbers themselves, particularly when applied in the study of history, and of other similar branches of knowledge. But is it desirable, we would ask, to dwell so much upon it according to the practice of most of our schools and educational`institutions? We think not: but rather that an undue importance has been given to it as compared with something better; for though it is highly useful to exercise those faculties which this particular method calls into play, why should not the judgment be addressed in a greater, or at least in an equal degree? Why should it not, we ask again, be much more directly, positively, and, in consequence, more effectually appealed to than has been the custom hitherto ? seen A time very soon arrives in the instruction of children, when the intuitive method ceases to benefit the pupil, and something more intellectual is required. The boy or girl is no longer content with only seeing or touching what is placed before them; they wish to understand it as well; a blind belief of the truth or existence of that which they have or heard, no longer satisfies them; they desire a more intimate persuasion; and consequently, expressions, which address themselves exclusively to the senses, no longer interest as heretofore; but in their own minds at least, are superseded by words and arguments which speak to the intellect and reason. But it may be said, at what age does this change take place? It is the master's province to decide this point. It is evidently at that time, come when it may, when the scholar comprehends the relation of the several ideas which enter into the resolution of the question proposed to him: so that it will not be possible to fix upon 8, or 10, or 12 years of age, or to say how much or how little time we should devote to one or to the other method. This must depend partly on the pupil, partly on the master, and partly perhaps on the system of instruction which has been adopted. Or, again, it may be asked, by what means we may ascertain which is the best method in any particular instance or instances? It would certainly be the gradual and relative progress of the pupils which would determine this point, were it not that a good or bad master does, to a great degree, practically supersede or nullify any method; so that in one case more success would be obtained with a very indifferent system, or even no system at all, than with the best, when the directing influence is below mediocrity. But as we know that something, nay much may be in the long run effected by well devised rules and regulations, and that there are certain principles founded on truth, and to which a good method can never be in contradiction, we will now briefly state those which occur to us as being the most important, and which, whilst philosophy would suggest them à priori, the experience of past and present times has illustrated and confirmed. 1st. We should exercise our pupils in those combinations of numbers, chiefly, which they are hereafter to apply in practice, and which seem best suited as a basis upon which to build our instruction in the principles of arithmetic (arithmetique raisonnée). 2ndly. We should explain the nature of numbers, independently of numeration properly so called, and it is very desirable to impress upon the child's mind as soon as possible the meaning and object of numeration itself, as being the foundation of all subsequent calculations, so that in this case, and at this stage, the intuitive method will immediately suggest itself as the most suitable. 3rdly. Every science has a language peculiar to itself, and it is of great importance that this language be rightly constructed. Now children naturally express themselves correctly, and in accordance with their own views so far as these reach, when we do not attempt to imitate them, or to distort the words and phrases they make use of; and as the language of true philosophy is after all the best constructed, we must from the very beginning address it to our pupils, nor should there be any difference in this respect between the students of tender or mature age, except in the extent of the knowledge imparted to them. 4thly. That which the pupil does himself is much more likely to profit him, than that which he sees others do, and still more than that which he finds already done. 5thly. When we use the intuitive method, we should take care to select such examples and illustrations as are likely to strike the child's mind, and that he should not have to look for them in any table or list that is not of the simplest character, seeing that hitherto all combination, and consequently all connection, of ideas is left out of the question. 6thly. Repose in any one place or posture is naturally distasteful to very young children. An apparatus or instrument, therefore, which changes its shape or form, as may be required at the moment, will affect them much more than any picture or table where the combinations are already made and set down. 7thly. The only tables of calculations, which it is desirable should be learnt by heart, are those of addition, of multiplication, and of the squares and cubes of simple numbers. To carry on this exercise of the memory to greater extent is burdensome to the pupil, and detracts from the respect which is due to the intellect. 8thly. We must take care to lay down with great strictness and precision those principles, which have been termed from their intrinsic importance, fundamental; and also, that these should, both literally and figuratively, hold the first places, if such an expression be allowable; that is, they should not be lost sight of in a multitude of details, or meet thereby with any obstacle or impediment to their being duly distinguished and comprehended. Having then premised thus much, we would further remark, that whilst a treatise on mental or intuitive arithmetic may save some trouble to the master, it can only be as one plan among many. Any intelligent teacher might form one for himself, and such a work can only serve as a guide, in order that by its means and suggestions there may be a certain method in what is done. The outline itself must be filled up as occasion requires; so that to give many minute details, would be to take away the advantage always attending extemporaneous illustrations and applications of the kind, and would rather embarrass than assist the teacher, who would most probably lay them aside, or not attend to them. In a word, whatever is really superfluous is an objection in any system of instruction. The child, as well as the man, prefers that something should be left to the intellect; and every one of a certain age must have remarked, that even in his own case the mind has not stood still, but has undergone a modification in its ideas and views within the last ten or twenty years. The same observation will apply to every system of instruction, past, present, and future; and we believe that the forms we are now laying down, however suitable to present wants and occasions, will have to undergo modification and improvement no less than those existing methods which we have taken upon ourselves to criticise; and which, we doubt not, their own authors would correct and amend much better than we can do, if they would only give themselves the necessary trouble. Thus, till the last moment of his useful career, Pestalozzi was continually occupied in endeavouring to bring his plans to greater perfection, and, in the words of Mr. Chavannes, to give to popular instruction that direction which should be most certainly conformable to the wants of the age in which he lived. We cannot do better than follow his noble example.-Would that it may be with like devotion and success! F. CURIE. (To be continued.) ON THE ESTABLISHMENT OF A SCHOOLMASTERS' MUTUAL BENEFIT ASSOCIATION. DEAR SIR,-The subject of this letter will, I trust, appear deserving of the attention of your readers. If properly carried into effect, it would remove many of the evils which now beset the schoolmaster's profession, and which deter many from entering upon duties so valuable to society and honourable in themselves. Generally speaking, the office of a schoolmaster is so badly remunerated, that few can save out of their incomes when young and healthy, the means of maintaining themselves in sickness and old age; much less of leaving at their death a provision for those who depend upon them. The most ready remedy for these evils seems to be a properly constituted Mutual Benefit Association. Such societies are too many and well known to need that much should be said on this occasion to prove the great advantages they have conferred on this country. The exact period when they were introduced in their modern form, is difficult to ascertain. From the imperfection of their regulations, few have had a long season of prosperity. One still exists in London, said to have been established in 1715; but even this society has no documents which can aid us in tracing its own history, much less in tracing the history of others of a corresponding date. But the principle of friendly societies is much older than the ordinary reader is aware of; it is essentially connected with the earliest steps in civilization by our Saxon forefathers. Mr. Hallam, in his history of the Middle Ages, traces the resemblance between the guilds of the Saxon period and our friendly societies.* "The guilds or social corporations of the Anglo-Saxons," says another modern writer,† seem on the whole to have been friendly associations for mutual aid and contribution, to meet the pecuniary exigencies which were perpetually arising from burials, legal exactions, penal mulcts and other payments.' Thus the principle of our mutual benefit associations, must needs be consisidered as essentially connected with the state of society in this coun * Vol. III, p. 34. The word gild comes from the Saxon gildan, to contribute. + S. Turner. Vide also Blomfield's History of Norfolk, vol. III, p. 494. |