✓ HARVARD COLLEGE JUN 4 1923 LIBRARY J. Culver Pierce Entered. according to Act of Congress, in the year 1364, by In the Clerk's Office of the District Court of the District of Massachusetts PREFACE. Ta Pestalozzian or Inductive Method of teaching the science of numbers is now universally approved by intelligent teachers. The first attempt in this country to apply this method to Mental Arithmetic resulted in the publication of Colburn's First Lessons, a work whose success has not exceeded its merit. It was, however, a useful experiment rather than a perfect realization of the inductive system of instruction. That the subsequent books of the same class and purpose have failed to correct its defects, and thus meet the demand it created, is due evidently to their departure from the true theory as developed and exemplified by Pestalozzi. The Author of this work has endeavored to improve upon all his predecessors, by adhering more closely than even Colburn did to the original method of the great Swiss educator, and by presenting at the same time, in a practical and attractive form, such improvements in the application of his principles as have stood the test of enlightened experience. In accordance with this design, the subjects are so arranged that each step of the learner prepares him for that which follows. By this suggestive and natural order of arrangement, together with copious illustrations of principles and applications by means of small concrete numbers, the pupil is led to a clear apprehension of the properties and relations of numbers, and is enabled to understand everything as he advances, till he acquires a thorough knowledge of the nature and use of the essential numerical operations. While the general arrangement of the subjects and examples is strictly progressive and logical, the difficulty of the problems is occasionally varied, in order to prevent the weariness of a long, unbroken ascent, and to afford a grateful alternation of effort and relaxation, like that experienced by the traveler in crossing a country diversified by hill, valley, and plain. The analytical process which this method requires at every step is calculated to develop and strengthen the mental powers, and to form the habit of rapid and accurate thought. Some illustrations of modes of analyzing questions have been presented merely as suggestions to the pupil; but the plan of the work does not embrace set forms of analysis for the various classes of examples, a contrivance little likely to stimulate invention or promote self-reliance. On the contrary, its distinctive feature is its special adaptation to the mode of teaching which leads the learner to ascertain for himself each step to be taken, to think and reason independently, and to rely upon his own powers and resources, thus securing a vigorous and healthful discipline of his intellectual faculties. Though this work is intended as a connecting link between the Primary and Written Arithmetics of the Author, thus completing the Series on which he has been so long engaged, it is also complete in itself. It presents a mental analysis of Arithmetic adapted to the younger pupils by its easy gradations, and to advanced pupils by its scientific arrangement and its logical development of the art of computation; and yet it has been limited to the true province of Intellectual Arithmetic, which is to serve as an introduction to Written Arithmetic, and not as a substitute for it, as some authors seem to imagine. In the spirit of the inductiye method, concrete numbersnumbers applied to physical objects - have been largely employed in treating of each topic, as the only fit preparation for the exercises upon abstract numbers, which are far more difficult for the youthful mind to grasp. A few pages of Written Arithmetic have been appended, embracing examples in the ground rules and compound numbers, which may be profitably studied in connection with the mental lessons illustrating the same principles. Fully aware of the difficulty of the task he has undertaken, the Author has spared no pains in its execution, and he gratefully acknowledges his obligations for the numerous valuable suggestions with which he has been favored by several eminent practical teachers. The favorable reception of the other books of his Series, encourages him to hope that this attempt to perfect and modernize the original Inductive System of Mental Arithmetic, and adapt it to the wants of schools of the present day, will meet with the general approbation of teachers and educators. THE teacher who would attain high success must study methods, and never take it for granted that he is perfect in his art. Why does one teacher accomplish twice as much as another, with no greater expenditure of time and strength? Because he has twice as much skill. Skill is acquired. It is gained by experimenting; that is, by experience guided by good judgment, and enlightened by the study of methods and expedients. The following Suggestions, derived from long experience and much study of the subject of teaching Mental Arithmetic, are submitted for your consideration, and not as rules which you are to blindly follow without the exercise of independent thought. 1. Take great pains in assigning the lesson, adapting its length to the capacity of the class, stating explicitly how it is to be learned and in what manner it is to be recited, and giving sufficient time for its thorough preparation. 2. See that the lesson is faithfully studied. Many teachers waste time over lessons which have not been properly prepared. Sometimes study a lesson with the pupils, to show them how, 3. Do not require the pupils to commit the questions to memory. This is a waste of time. Nor should they commit the answers, excepting the answers to that class of examples which involve a single operation upon abstract numbers; that is, such questions as are usually comprised in the tables of addition, subtraction, multiplication, and division. 4. Never require a pupil to analyze questions according to a set form of analysis, but encourage originality in methods of solution. The fewer words in the solution the better, if it is correct and intelligible. By all means avoid long and complicated formulas. 5. Do not demand reasons for answers which require no process of analysis. If the child knows that 4 from 6 leaves 2, what is gained by requiring him to say, Because 4 and 2 are 6? The thing is no better understood, and time is consumed. 6. The teacher will read the questions himself, the class dispensing with the book, or he will allow the pupils to have the book and read the examples, as he may prefer. In questions requiring analysis the pupils should not be called in turn, but promiscuously or by cards, and, if the example is read by the teacher, time should be given, after the reading, for the class to think, before any pupil is designated to answer. Examples like those in Lesson II, page 11, may be recited by the members of the class in rotation, the questions being read rapidly. 7. The answer to a question requiring a process of solution should not be given before the solution, but it should be given at the conclusion of the solution. Nor should pupils be required, as a practice, to give what may be called an abstract or general answer before the solution, like the following: of 36 is of |