thod, with notation and numeration also, having been put into the manufacturer's hands before it was pointed out by this letter:

To the Editor of the Educational Magazine.

MR. EDITOR,

I HAVE read the two numbers of your magazine with a considerable degree of satisfaction, but not unmixed with regret on observing that so small a portion of the matter bears on the actual business of teaching. As a schoolmaster, I gladly hailed the announcement of your forthcoming periodical, in the hope that it would occupy the field-a very extensive one--between that of the Sunday School Teacher's Magazine, which is almost restricted to religious information, and that of the Quarterly Journal of Education, which, with all its talent, which I would not gainsay,- in devoting its pages to criticism on Latin, Greek, and Hebrew, and the mathematics, or to reviews of works on political economy, or on the increase and state of education, and benevolent institutions in their corporate state,—has taken a higher, and to me, a teacher of a public school, a comparatively useless position. Such a work is valuable to the legislators on education, to the members of superintending committees, and to the general reader, rather than to the working schoolmaster. Your Magazine appears to be tending to the same course. To us, as men, loving our kind, and anxious for the spread of Christian principle, and the decrease of mental and physical suffering among our countrymen, or our species generally, such articles as those which occupy your two numbers are highly interesting; but as Schoolmasters, we wish to find in your pages information which may be brought immediately to bear on children;-practical exercises, actual examinations, lessons, which may, on the instant, be given in the school, with the magazine in hand. What are the great schools doing? and how are they doing it? are enquiries of far more importance to schoolmasters, and through them to the rising generation, than to be informed on the state of education in a general or national point of view, or even of the lives of its friends. One good practical lesson, taken verbatim as taught in some excellent school (which from the circumstance of all day-schools being open at the same hour, other masters are precluded from hearing), or a detailed account of any new apparatus of a decidedly improved kind, is a real boon; yet such as they are justified in expecting from you. All the articles I have yet seen, may be written by men who have only read on the subject of education. I wish to see articles written by men who draw their information rather from experience than from books; and as one of them, I now venture to appear in your columns, hoping that the example will be followed by other masters more able than I am, and of whom I also may learn something. The subject I have chosen is Arithmetic, one to which I was early attached, and one which, I am inclined to think, I have had some success in teaching.

The two methods which divide the schools at the present day, are the Tangible and the Abstract methods. The Abstract method which has been, hitherto, the almost exclusive one, is now seen to possess great defects. The Tangible, or Pestalozzi's, is more talked about than understood. It is certainly absurd to employ boys, hour after hour, with abstract calculations, without an attempt to explain the principles on which they are acting: it is equally unprofitable to keep them handling balls, tracing fractional parts, and demonstrating the truth of principles among series of traversing lines; so that should elapse before they are able to perform any mercantile calculation.

years

The communication of a correct idea of numbers should be first aimed atThis idea is to be given by reference to feasible-to tangible objects;-lines, balls, marbles, pebbles, &c. The child, being led to see that this applies to all things around him, has then the abstract idea of number. The names may

then be used without a direct application to things, Thence we advance to the combination of numbers,-numeration and the four primary rules, with simple numbers; and thus approach those general principles, which form the basis of all future arithmetical reasoning. This is the natural order, and in this way the pupil may be led without difficulty, from the most simple to the solution of the most complicated arithmetical or algebraic questions. The understanding keeps pace with the practice; the process of reasoning strengthens and matures the mind; and the peculiar pleasure which mathematical operations alone furnish, the proof or certainty of the result arrived at, act as a constantly recurring stimulus to the succeeding effort. Indeed I am convinced, both by long practice and by the testimonies of the best teachers, that arithmetic should be taught in schools universally, though, as a science, it were never required in life, from the peculiar power it gives the teacher of developing the intellect of the child, of invigorating his judgment, and of habituating him to reason with accuracy and precision. How forcibly to my mind flashes the conviction of the incompetency of the teacher, when I hear it said by him 'my boys are too dull to understand mental arithmetic." If they are dull, as it is called, they need this sharpner of the intellect the more; as the duller the edge, the more the whetstone is required. In such a view of the case, if any be kept from the study, it should be the sharp and clever boys. I cannot conceive of any child, above the degree of an idiot, who cannot learn it. If, however, the teacher proceeds out of the natural course, as above stated;-if he commences with numbers too large to be comprehended, so that the child cannot reason on them;—if he should reverse the natural process, and attempt to teach his pupils general principles before they have attained the simple ideas of number;-if, in a word, he attempts to commence where he ought to leave his pupil, his failure will be no more wondered at, by the better informed teacher, than a gardener would be surprised at a plant not growing, which some novice had planted with its branches in the earth, and the root in the air.

In this respect the frames invented by Mr. Martin are defective, though in other respects, the most complete and efficient mechanical contrivance ever introduced into a public school; in fact, so complete, that they have left nothing to be wished in this department for monitorial teaching. But the defect consists in their not having an introductory frame to show the connection between arithmetical figures and things—the first step,—lessons on tangible arithmetic. They pre-suppose a knowledge of this, and of numeration and notation, which they ought to give, instead of expect to find in the children. If they were preceded by a frame, similar to the others in size, but made somewhat after the model of the Ball Frame, or Arithmeticon used in Infant Schools, by which lessons on tangible and palpable arithmetic might be made intoductory to a knowledge of figures, and by which the principles of addition, subtraction, multiplication, and division, might be rendered familiar, before the pupils enter on the abstract study of those rules, appending also some exercises both in numeration and notation, a valuable service would be rendered to every master, and they could scarcely fail, within a short time, of becoming the standard medium of teaching the four first rules throughout the country. If to these frames be added the little work called "The Intellectual Calculator," to which Mr. Martin's name is also attached, and which embraces, besides the usual arithmetical rules, “a complete course of mental arithmetic," a knowledge of all the useful rules, and a power and quickness of mental arithmetic will, in a comparatively short time, be obtained, such as not even the best conducted boarding school will exceed. In proof of this, I will refer to the Borough-Road School, in which both the Frames and the Intellectual Calculator (which was written by the master of that school in conjunction with Mr. Martin) are used, and are the only means used. I refer to this school in preference to others, because it is a large public institution, in which any person

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can test the truth of what I now assert, and because, as there is only one master, and he almost constantly engaged with visitors, the teaching must be done exclusively by the monitors."

In unison with a remark I made above, on the importance of a practical exercise, I shall give the questions proposed at that school, on a number given by a visitor. The number was 7859. The following questions were put by the master, as the visitors declined to question :-78 times 78? 59 times 59? Square of 85? 7859 pence? In £7 8s. 5d. how many pence? In £8 5s. 9d. In £9 5s. 8d., &c. 7859 yards of cloth, at 7d., 8d., 5d., 9d.? £7859, at 5 per cent., for 7 months? £78, at 5 per cent., for 9 days? 14 per cent. on £785. Cost of 7 cwt. at 8d. per lb.? Of 5 tons, at 9d. per lb.? 5 ounces at 8s. per lb.? 5 lbs., at 9d. per ounce? The value of the decimal £.7859? .7859 of cwt.? 7d. per day for a year? 8d., 5d., 9d., with and without Sundays? 17 yards, at 18d.? 18 yards, at 15d.? 15 yards, at 19d.? The square root of 7859? The cube root of 7859? If 7 be the first term in proportion, 8 the third, and 5 the fourth, what is the second term? In like manner, the other terms were asked, and a variety of other questions, and nearly the whole were answered instantly as proposed, but in every case a few, very few seconds, produced the answer. This is as mental arithmetic is now studied, astonishing to a stranger, seeing that the time these boys remain in the school is very limited, and that the teaching must be by the monitors. But the examination did not end with this; for a great number of other questions, of a difficult kind, were put and answered as readily; it was evidently of no moment to the pupils what numbers were taken. They were then examined on the means and principles of working, and in several cases different means were described, and in every case a full and clear explanation of the principle, and a ready reply to the why did you so? Neither slate, nor paper, or any help of the kind, was near them.

This is the test. The true end of intellectual teaching is the attainment of mental power, and then the application of it to practical and scientific purposes; and all this is accomplished by analytic induction, that is, by learning the principles by the examples performed. Much is said, but I think, not wisely said, about lengthy explanations in books of arithmetic, development of the principles of the rules, &c. My experience has found but little advantage, from even the very best printed explanations. The illustration of principles must be made by the teacher or monitor, viva voce, and be accompanied by much questioning. This I know to be the course pursued in that school, and it doubtless will be increasingly so. Good examples or questions are the grand object for a master, and I rather expect to see books on arithmetic with questions only, than with few sums and long explanations. The explanations must accompany the example, and the only effectual method of explaining is by questioning, that is, by applying the interrogative principle to this as to every other intellectual branch of study. Each difficulty, as it presents itself, is then brought fairly out-is put in a right light; and the mind of the pupil being brought to bear fully on it, the difficulty is so completely conquered that it can never present itself to the same mind again.

But by this digression I am departing from the intention which I proposed to myself, and stating what is done, rather than answering that question, so interesting to every one ambitious of putting time to best purposes,-how it is done. In reply to this question, I must briefly describe the frames, of which there are, in a complete set, eight frames, one for each of the simple, and one for each of the compound rules. They are about 20 inches long and 14 inches broad. Each frame contains one or more cubical rollers, ingeniously and strongly placed, so as to turn round and exhibit, in succession, four lines of figures. Each frame contains also, as part of it, a slate. The pupils are ar

ranged in order before it, and the monitor commences by working out a question on the frame, as it hangs against the wall. The boys' eyes are directed to each figure of the question as the monitor points to it, and to each figure of the answer as the monitor puts it down on the slate. The monitor, during the whole operation, is questioning on what he is doing. The pupils then work a question by themselves, aloud, on which again the monitor questions as they proceed; each answer is proved by the pupils as it is wrought. At the third exercise the monitor takes the frame down, dictates a question to be written by each boy on his slate, and requires the operation to be performed aloud, each one taking his portion. A great number of sums are proposed and wrought in this way, the monitor having nothing to do but to ask questions, explain the principle of the rule, and glance occasionally at the previously wrought answers, which are now so many keys. There are several modes of proceeding with them, at the choice of the master, which I need not describe; but there are two excellencies, of the first importance, which I must notice:-the one is, that each rule calls for considerable exercise in each of the preceding rules-multiplication, for instance, exercises them in subtraction and addition; and the other excellence is, that to each rule there is a master-key, so ingeniously contrived, that whatever sums may be wrought, (and the inventor speaks of millions of sums being presented by the revolution of the rollers) the teacher can detect an error at a glance in a whole morning's work, although even both monitor and boys should combine to deceive him. So ingeniously are the keys involved that no monitor, however long he might work at a frame, would suspect, from using it, that it had such a key, and if suspected in consequence of the master's ready detection of error, he could never find a clue to it. This last has ever been a desideratum with masters, and it is venturing nothing to say, that fancy's self could not have pictured one more complete. The frames are succeeded by the Intellectual Calculator,' which has been advertised in your two numbers. This work, if we may judge by its results, is the best of its kind, but as its price is so low that every teacher will have it who has any anxiety for the increased success of his school, it is not necessary to describe it: its course of mental arithmetic, with its rules for shortening calculations, appear to me to be its chief recommendation. This branch of study, which is now daily gaining advocates, the authors appear to have been the first properly to appreciate and to carry out as a distinct study. Although I have dwelt so long on this, I will yet venture to make an extract from the introduction to this part of the work, as it may awaken some to a consciousness of its importance. "Mental arithmetic possesses so great a superiority in common calculations, and presents so many advantages to persons engaged in business, especially in ready-money trades, that instead of forming NO PART of common education, it ought to form its most prominent feature. Its importance is second only to scriptural instruction, not only from its usefulness as an acquirement, but from its tendency to draw forth the faculties of the mind, and to fix and concentrate them on particular objects; it is the mind's power exercised in this way, that makes the acute thinker and solid reasoner.”

This has extended to greater length than I proposed, by my being led to give a detailed account of the apparatus, and as I advocate the use of it from a conviction of its superiority to any other means, and as it is not generally known, I was forced to this lengthened explanation,-in actual teaching detail is nearly all. I shall rejoice to see, in any subsequent number, the experience of other masters, who may, by pursuing another course, have arrived at other conclusions, and made discoveries which have escaped the notice of Your obliged correspondent,

London, Feb. 15, 1835.

A SCHOOLMASTER.

REVIEW OF BOOKS.

The Principles of Physiology, applied to the Preservation of Health, and to the Improvement of Physical and Mental Education. By Andrew Combe, M.D. Third edition; Maclachlan and Stewart, Edinburgh; Simpkin and Marshall, London.

POPULAR works are generally thermometric indications of the changes of the public mind; and the tone of society, to a certain extent, may be ascertained by their circulation. It is one of the most pleasing signs of a correct estimation of the principles of human nature, both physical and moral, being forming by society, when we observe an anxious avidity manifested for the perusal of works in which these principles are scientifically explained. During the last seven years a spirit of desultory reading, of pamphlet and magazine studying, has trenched most deeply upon works of a higher and more important character; and philosophy has rather come by bits and shreds, than in a manner calculated to afford a physic, as well as a food, for the soul. The first principle in the ephemeral works of the day has been to amuse; instruction has followed perchance, but not with any determinate certainty, or with any particular aim. The result has been that with all the extensive and extended means of information, drawn from the abundant resources of nature, of science, and of art, but few works of a purely philosophical character, having for the chief object the elaborate illustration of one particular theory or science, have found purchasers. The most accomplished and the most talented men have been obliged to communicate their ideas to the public through inferior channels, and a standard work has but rarely become the popular book. The work before us is an exception to the general rule, and this arises partly from the existence of a better feeling in the public, and partly from the subject being one in which every organized human being must feel a great and paramount interest. Such a work might, however, have been written in a very different spirit; it might have taken up the subject in a manner calculated to shackle still further than to emancipate the mind. It might have been narrow, contracted, exclusive, and illiberal, instead of breathing a fine spirit of freedom, and advocating principles of the first importance to man, with the first and highest sentiments of which man is capable. are, therefore, as also every liberal friend of his, deeply indebted to Mr. Combe for the excellent work before us. With the views entertained by the author, in almost every important point, we most cordially agree. Perhaps there is no subject of which men talk so much, and of which they know so little, as health: every one has his specific for this or that disorder when it makes its appearance; but few understand or take the trouble to make enquiry regarding the principles that constitute the blessing of health: or to secure that blessing to them. selves by rational cure. It was a saying of Dr. Franklin, that his three physicians were Dr. Air, Dr. Diet, and Dr. Merryman. Mr. Combe's work will do much to establish the truth of the principle contained in the facetious remark of the “ Household Philosopher.”

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We

In a very sensible preface, Mr. Combe remarks that, notwithstand