6 PREFACE TO TEACHERS EDITION. In putting this work before the public the author disclaims any ambitious schemes or "great expectations,” but he wishes to have the book for the use of his assistant teachers both as to methods and examples. The author ha used some parts it for many years and feels confident that excellent results may be obtained by using it. The aim of the book is to give so much practice as to fix each method in the pupil's inind, rather than to deal with the philosophy of each operation, leaving any teacher who believes that no step should be taken unless the pupil understands the reasoning process by which that step may be reached, to give it in his own way. It is possible that a few who see this book may have found that 7 times 8 are 56 by actual addition, yet those who have never added it may know the fact just as well for all practical purposes. If no one were to eat until he understood how food nourishes the system there would be a deal of hunger in the world. 'This book deals only with the fundamental rules of arithmetic. The intention is that they shall be so thoroughly mastered that much less time will be required for the remainder of the subject of arithmetic than would otherwise be needed. The teacher is to use the Teachers' Edition for one to two years before the pupil has advanced enough to use the Pupils' Edition or in fact any book on arithmetic. It is recommended that teachers begin to teach numbers as given in the first part of this book after pupils who have the alphabet and words to learn have been in school four to six months. In each new operation the examples are very easy ; as more problems are given they gradually increase in difficulty. By teaching the four operations of addition, subtraction, multiplication, and division, from the first, the examples are of such a kind as to compel some thoughtfulness on the part of the pupil. Much pains has been taken to make examples of a sort to interest the youngest pupils. Those teachers who wish to teach only addition and subtraction at first can designate those examples involving multiplication or division by some mark, and omitting them may return to them afterwards and so secure the variety of examples so essential to a pupil's real progress. It has been found, however, by actual trial that pupils may learn the four operations from the first without serious The first pages are devoted to what is known as the “ Grube Method.” If the teacher prefers it, the schedules may be omitted, and, in passing through the first time, the multiplication and division also, as before stated. The author hopes that teachers into whose hands this work may come will give it a thorough examination. Special attention is called to the treatment of numeration and notation. The examples are not all given in one place, to be forgotten, but are so placed as to review the subject often. Attention is called to the examples for rapid solving and the illustration of the easy exam. ples given under each rule. Also to the method of teaching long division and to the definition of addition. The method of teaching the addition, subtraction, multiplication and division tables is believed to be entirely new, so far as being published is concerned. The author discovered and used the method about ten years ago, and in his school has found it invaluable. To hear a recitation of a large class in tables and make the questions to each pupil promiscuous, and yet fall enough to satisfy the teacher that the pupil has a thorough knowl. edge of the tables gone over, is not only very wearying to the teacher but is exceedingly difficult also. By the old method a pupil frequently acquires the habit of saying the |