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the mathematics, already sufficiently looked down upon in our public schools, be exhibited to youths who are likely enough to despise the study, in a shape which will furnish some excuse for their contempt.
LESSONS ON NUMBER.
Lessons on Number, as given in a Pestalozzian School, Cheam, Surrey. John Taylor, London, 1831.
THIS work forms a part of the same series of publications as the Lessons on Objects noticed in the first number of this Journal. They deserve general attention, as professing to be the result of actual experience; and the treatise now under consideration has accidentally also a peculiar claim on our notice, from the circumstance that, in the article already referred to, the science of number was one especially pointed out as requiring elucidation on the principles adopted in the Lessons on Objects. We are, therefore, desirous of examining with what success this has been furnished by the publication before us.
The plan on which it is intended that the first processes of arithmetical instruction should be conducted, is stated by Dr. Mayo, in a preface which he has contributed to the work, the body of which is written by a foreigner; and a few short extracts will best show the scope of the treatise, and the principles on which it is meant to depend. After a wellmerited panegyric on Professor De Morgan's Treatise on Arithmetic, Dr. Mayo observes
The aim of the little work now offered to the public is different; it does not propose to explain processes, but to unfold principles. The pupil is not taught to comprehend a rule, but to dispense with it, or form it for himself. The path along which he may be led may be longer than the usual route; but then it is in broad daylight, he is more independent of his guide, and derives more health and vigour from the exercise.' . . . When the true end of intellectual education shall be admitted to be, first, the attainment of mental power, and then the application of it to practical and scientific purposes, that plan of early instruction, which dwells long on first principles, and does not make haste to make learned, will be acknowledged as the most economical, because the most effectual, Experience will show, as it has indeed already shown, that while superficial teaching may prepare for the mere routine of daily business, whensoever a question, not anticipated in the manual, occurs, none but the pupil whose faculties have been exercised in the investigation of truth, who is the master-not the slave of rules, will
solve the unexpected difficulty, by a novel application of the principles of the science.'-Preface, pp. viii. x.
In this preparatory course' (a course of instruction previously described as rather intended to train the mind for the study of the science, than to communicate the knowledge of it) the order is determined by a consideration of the mind of the pupil; it commences with what is already known to him, and proceeds to the proximate truth; the more easy precedes the more difficult, the individual prepares for the general truth, the example for the rule.'
It is strongly recommended that it be used as a course of mental arithmetic; that is, the questions should be solved in the head, without any figures being written on the slate by the pupil. In this manner it retains more of its character as an intellectual instead of a mechanical exercise. The vigour, freedom, activity, clearness, and retentiveness of mind, which a persevering adherence to the plan imparts, will prove an ample compensation for any additional trouble which it may seem to occasion.'-Preface, pp. xi. xii.
With the latter part of this extract, that which enforces the importance of exercises in mental arithmetic,--we entirely agree. They furnish very much the best foundation of all arithmetical knowledge; and probably have only failed to be generally used for that purpose, because it is less trouble for a teacher, if idle, to look over a written sum, than to attend to all the steps as orally expressed; or because it is easier for him, if ignorant or unthinking, to examine results by applying a mere rule, than to attend to, and explain the difficulties which a child may meet with in the course of an operation. If this publication shall induce any parents or teachers to attach more importance to the exercises which it recommends, it will not fail to be of great and enduring value. And it ought to have this effect; for, whatever may be its imperfections in other respects, it at least furnishes a large collection of questions suited to the faculties of children in the early stages of mental arithmetic, carefully arranged in a progressive order *, from the very
*This commendation of the order observed must not be taken without some allowance. The general principle of the work is to observe it, but there are some few singular inversions. At page 78 a very remarkable confession of disorder occurs: The pupils should, therefore, be well acquainted with similar divisions, which, by means of the subdivisions of a line, may easily be obtained by them. The following exercises ought, therefore, to precede the above. If this were an after-thought, the hurry of publication may explain why the transposition thus recommended was not actually made; but it will not justify the failing to make it. There are other symptoms of a haste to publish, which is a good deal to be regretted. We had occasion, in our notice of Professor De Morgan's Treatise, to speak of the incorrectness with which it was printed, and the observation is too often applicable to works of science; but some parts of the present work, for it is very unequal in this respect, very far exceed any ordinary limits of inaccuracy. In page 81, there are no less than eight errors of the press. There is also a good
simplest possible, to the more complicated examples which it comprehends. The observation of this order is strongly recommended in the work itself; and undoubtedly, although it sometimes occasions a little tediousness, especially when combined with another cause, which we shall hereafter have to mention, it is, on the whole, very desirable to adhere to it. Another important merit of the work consists in the manner in which, in its earliest pages, the idea of number is extracted from the consideration of the objects by which it must in the first instance be exemplified. It is done without any parade of abstraction, but successfully and completely; and the author, without being in too great a hurry to get rid of the sensible objects which he at first employs, is very soon able to do so.
In proceeding to the examination of other parts of the work before us, we enter upon more questionable ground. We shall not have occasion for much of particular criticism on the execution of different portions of the design: the point mainly to be considered is the principle adopted; for it is in this respect that the work puts forth the strongest claims to attention. If we are entitled to take the earlier portions of the extracts already made from Dr. Mayo's preface, in their most obvious sense, as furnishing the exposition of this principle, it contains nothing from which we should dissent, and very little which we should wish to qualify. But there is some ambiguity in the expressions used, and when we take the 'Lessons on Number' themselves, as a commentary on the text of the preface, we fear that there may perhaps be more difference between our notions than we should otherwise have suspected.
Dr. Mayo, after explaining that upon his system no technical rules are given antecedently to examples, informs us that the pupil is not taught to comprehend a rule, but to dispense with it, or to form it for himself;' that the individual prepares for the general truth, the example for the rule.'
deal of inconsistency in the manner in which fractions are represented; they are generally reduced to their lowest terms, but by no means uniformly so, and the exceptions do not appear to depend on any fixed principle, or to have any particular object. In another respect, also, the employment of a very little time might have produced much benefit. Dr. Mayo, in his preface, claims indulgence for any inaccuracies of style or inelegancies of expression in the body of the work, on the score of its author being a foreigner. Surely it would have been more to the purpose to have had the proof sheets revised by an Englishman. The alterations required would have been very trivial, and might have been made with the utmost ease; but they would not have been unimportant. The inaccuracies which now occur are not frequent, but they sometimes occasion difficulty in understanding the passages where they are found,
The author of the work, to which these remarks are prefixed as an introduction, says
Thus, without assistance of rules, generally little understood, by a chain of simple reasonings, easily ascertained by facts equally simple, we arrive at results hitherto inaccessible to the understanding of a child.' p. 97. 'Such were the answers, nor early such, that have actually been given by children of the age of nine or ten years. We give no rules but those found by the pupils themselves; they are fetters which enchain the powers of the mind, and deprive it from ever attaining strength and vigorous health.' pp. 69, 70.
The reader must not understand from the last passage, that the child is generally, in this treatise, led up to the construction of rules for himself, or that such rules are the results' spoken of in the preceding extract. There are, indeed, some instances of the attainment of such rules, but they are few; and the results generally consist merely in the solution of particular questions. Are we, therefore, to understand that when the pupil is taught either to dispense with a rule, or to form it for himself,' it is immaterial which he does? and that when the example prepares for the rule,' it is the object of this treatise to furnish such a preparation only? The former of these is a question of general importance; the latter chiefly affects the value of the particular work, and perhaps calls for little remark, except that, if the treatise be meant to furnish only an introduction to rules, it furnishes one of extraordinary and, we think, unnecessary length.
On the main question, we entertain no doubt of the thorough soundness and great importance of the principle, that it is desirable, as far as possible, to conduct a child gradually, by his own observations and induction, to the rules which he is hereafter to apply. All knowledge acquired by reasoning and observation is more valuable, and is better known, than that which is received on the authority of others. It is better known, because, in the process of its acquisition, it has been seen in various bearings and connexions, and because the principles on which it depends have been fairly worked into the mind, and remain there, capable of the same and further applications, even if the results originally deduced from them are confused or forgotten. It is, for the same reason, more valuable; and yet more so, because the process of acquisition has exercised the most important faculties of the mind, instead of being confined to the exercise of others of inferior dignity-namely, memory, and a certain degree of attention and distinctness in comprehending the application of a set form of words, and performing the operations which they direct. Whenever, therefore, a child can be led JULY, 1831,
to form, to think out as it were, a rule for himself, it is most desirable that he should; but it does not, therefore, follow, that in cases where he is unable to do so, the rule should be suppressed or omitted; nor does the mere fact that he may be taught, without the rule, to perform the same operations, prove that the rule is superfluous, or ought not to be communicated, when the operations without it are much more laborious and circuitous than when it is applied. Where, indeed, the principle of the rule is unintelligible, even when communicated, it may generally be desirable to suppress it: there may be more harm in accustoming the mind to take things upon trust, than in leaving it without the practical assistance to be derived from the rule itself. But the more common case will be that of a rule not within the compass of the learner to discover, but admitting of full explanation and proof, such as he can comprehend, when it is once announced to him. And these rules it appears to us desirable to communicate; not in the first instance, indeed, before the want of them has been found, and their value consequently appreciated, by examples of the same operations performed without them; but as soon as these preliminary steps have been gone through, and without waiting till the same cautious process has been carried into other departments of the subject.
Perhaps the importance of the alteration thus suggested will most fully appear by one or two instances of the inconvenience and imperfection occasioned by the plan adopted in the work before us; and we will take one where it would be difficult, or perhaps impossible, to conduct a child to the discovery of the rule required, yet the rule is perfectly intelligible when explained; another, where the rule itself, by a series of questions artificially combined, might become the result of the pupil's own speculation. The first of these is the principle of numeration.
It may appear that, as the great value of a system of numeration is the facility which it gives to the operations of written arithmetic, it would be superfluous to take any notice of it in a collection of examples designed for exercises in mental arithmetic only. This, however, is hardly the case, when we look to the execution of the work before us; for the pupils, though forbidden the use of pencil and slate in performing the operations required for them, are occasionally allowed the assistance of seeing the question itself written before them (p. 70), so as to keep the data on which they are to proceed before their eyes, during the course of the operation founded upon them; and in page 45 it is said to be desirable that the pupil should write the answer to