ON MENTAL ARITHMETIC, ་ IN THEORY AND PRACTICE. BY THE HECT RODLE REV. ISAIAH STEEN, HEAD MASTER IN THE MATHEMATICAL DEPARTMENT OF SECOND EDITION. GREATLY ENLARGED AND IMPROVED. LONDON: SIMMS AND M'INTYRE, 13, PATERNOSTER ROW; AND 26, DONEGALL STREET, BELFAST. PREFACE TO THE FIRST EDITION. MENTAL ARITHMETIC has, of late, been introduced into some of the best conducted schools in the empire. In teaching it, the author of the following treatise has, for several years, felt the want of a systematic work on the subject. He was obliged to prepare a treatise in manuscript, for the use of his own classes, which he has used for a considerable time, with good results. He has found, by experience, that it was not sufficient to possess a copy himself; but that the pupils required to have access to the rules, to study and commit them to memory. Before venturing to publish a work that is so elementary, he had numerous opportunities of conversing with the teachers of some of the most distinguished, and best conducted schools, in England, Scotland, and Ireland; and they having felt the same want, and approving of the author's plan, he was induced at last to publish it, to supply a desideratum, which others felt, in common with himself. What the writer found in previous publications, he has reduced to a system of rules, commencing with those of the most simple and easy kind, founding each additional rule, as much as possible, on some preceding one, already explained. It will be found that a considerable number of new, and, it is hoped, useful, rules have been added on the same plan; and, though the entire number may appear to some unnecessarily great, they are all founded on a few elementary principles, and are applicable in a great variety of cases. Several of the rules might be useful to persons buying and selling in markets and shops. But the principal use of Mental Arithmetic is to train the mind of pupils at school, and to accustom them not only to think, but to do so quickly. Though some may suppose that explanations have been given too much in detail, and that there is too much space occupied with worked examples, yet teachers, in general, will admit that this is the safe side to err on. The great aim throughout has been to render the book useful to every person into whose hands it may fall, and to make it explain itself; so that any one may use it, and even with a very little trouble become familiar with the practical application of its rules, without having had his attention previously turned to the subject. The following plan of teaching is recommended, after considerable experience, to those who may introduce this useful branch into their schools. The tables, including particularly the extended multiplication and pence tables, having been carefully prepared, the pupils, after having been sufficiently exercised in performing, mentally, exercises in Reduction, Simple and Compound Addition, Subtraction, Multiplication, and Division, should then be introduced to the SECOND and other PARTS, in succession, as they may be found capable. The teacher should write out the questions with chalk on a black board, so as to be perfectly legible to the whole class. This prevents the necessity of a boy asking aloud what the question is, and thereby disturbing his class-fellows, should he have the misfortune, after an unsuccessful attempt at solution, to have forgotten it. It is best to give each pupil, in succession, a separate question or two in each rule, and then to give some general questions, to be answered by any pupil, so soon as he is ready, or after allowing a reasonable time to the class to think, to give an opportunity to all, or to a section of the class, to answer simultaneously. The giving of a separate question to each pupil, and seeing that he solves it, and understands the principles of the rule, prevents any from escaping, through inattention; while the general questions, proposed alike to all, afford a stimulus to attention, on the principles of a well-regulated emulation. It will, no doubt, facilitate the progress of the pupils, to explain, at the termination of each lesson, the rule for the next, and to require them to be well prepared on it, on the next Mental Arithmetic day. It is proper also to remark, that, at the commencement of each lesson, the class should be examined briefly, by way of recapitulation, and for the sake of connexion, on the rules taught on the last Mental Arithmetic day. Monthly, quarterly, and half-yearly repetitions of all the rules learned in the previous period, are most useful. To prevent this branch from interfering with the ordinary routine of a school, the author has found it quite sufficient for every purpose to devote an hour each week to each class in Mental Arithmetic. In this, as well as in every other department of education, there will likely be pupils of different degrees of 'proficiency; these may be formed into different classes, and each receive one hour's instruction on this branch once a week. The different classes may either be taught at different hours on the same day, or at the same or different hours on different days, so as to save the teacher the necessity of giving more than one hour, on any day, to this subject. Should the exercises on each rule, though tolerably numerous, be found insufficient for the wants of any particular teacher, he will find it quite easy to prepare additional questions, which he may preserve, either in a separate book, or in an interleaved copy of this book. The miscellaneous questions, which have been added, on the suggestion of an experienced teacher, will, however, it is hoped, supply a sufficient number. These are given without any regard to order, to afford mental exercise, in finding out, as quickly as possible, by what particular rule the proposed question may be solved. In many cases, it will be found an excellent exercise, to require the pupil to solve a question by several methods, and to give his opinion, which is the shortest and easiest. All these things will prove most useful in stimulating thought, and in aiding the development of the powers of the mind. ROYAL ACADEMICAL INSTITUTION, Belfast, August, 1846. PREFACE TO THE SECOND EDITION. THE demand for a second edition of this treatise on Mental Arithmetic, shortly after a year from the publication of the first, is a convincing proof, that such a work was not altogether uncalled for, as well as a gratifying testimony to the manner in which it has been received. A considerable quantity of new matter has been added, and to render the volume uniform with works on common arithmetic, generally used in schools, the form has been changed. Great care has been taken, in revising the work, to render the rules and explanations as clear as possible, whilst the additions and alterations have been so introduced as not to interfere with the original arrangement of the rules. The importance of Mental Arithmetic, as a means of mental culture, not to speak of its practical advantages, cannot be overrated. It is much to be regretted that the number of schools in which it is systematically taught, is comparatively small. The Author takes the liberty of observing, for the benefit of such teachers as may not have introduced it into their schools, as a regular branch, that this treatise is self-explaining; so that any person having a tolerable knowledge of common arithmetic, may, with the greatest ease, use it as a text-book, and teach it to his pupils. Let any one, who is sceptical on this point, only make the trial, and he will be equally surprised at the simplicity of the principles and the success of his pupils. The defect of most books on this subject is, that they do not combine principles with practice. Some, for example, contain little else than rules; others are vast collections of questions, with scarcely an attempt to unfold the principles on which they are solved. In this treatise, both are kept equally in view. The following is the plan according to which the matter connected with each rule is arranged. 1st. There is the object proposed to be accomplished. 2nd. There is a rule given for effecting this. 3rd. The arithmetical principles, on which the rule is based, are clearly unfolded; or, in other words, the rule is demonstrated. 4th. Worked examples are given to illustrate the manner in which the rule is to be used. 5th. Numerous questions are given under each rule, to be proposed by the teacher to his class. Pupils should be required to come to school prepared to answer on all these points, in reference to as many rules as the teacher may have prescribed. The work is again offered to the public, in the hope that it may, in some degree, prove instrumental in promoting the cultivation of a too much neglected branch of education. ROYAL ACADEMICAL INSTITUTION, Belfast, July, 1848. |