as are given in this chapter is not to give practice in operations, but to test the pupils' power of reasoning. If the pupils could give instantly each result in the several operations involved without performing the indicated operations, they could proceed with the necessary reasoning without delay, and could analyze several problems in the time usually taken to perform the operations. This saving of time is made. possible by merely indicating the operations and using letters as symbols to represent the various results. The pupils thus get, in a given time, very much more practice in arithmetical reasoning, and in the statement of such reasoning, than is possible if they are required actually to perform all the indicated operations before they begin the analysis. A half hour devoted to work of this kind will enable the pupils to review rapidly the problems of several days preceding, or to analyze in advance many problems to be worked out completely, if thought desirable, on succeeding days.

While the number of problems in the body of the book is not so great as in some arithmetics, it is believed that it is sufficient for ample practice in the important operations. The thorough drills for accuracy and rapidity in the fundamental processes given both in Book One and in the first part of this book make it possible, in the subsequent applications of these processes, to reduce the number of problems, many of which are included in other books for no other purpose than to afford practice in such operations. The aim has been to provide material adequate for the purposes for which arithmetic is taught, and at the same time to reduce the amount of time usually given to this subject in the schools.

For the mastery of any topic in arithmetic there are certain things that the pupils must know and do. As a rule, very few of these things are entirely new, yet the

knowledge and skill previously acquired by pupils are not always available by them at the time when needed. It is evident, therefore, that at every stage the choice and arrangement of matter should be made with reference to the proper preparation of pupils for the next stage of work. While the things that must be known and done for the mastery of each new topic are not stated in the book, they have been carefully considered in its preparation. The subject-matter, accordingly, is so arranged as to insure the pupils' power to attack what is new with the greatest success.

The introduction to a new topic or to a difficult phase of a topic by means of the Study Recitation, in which the pupils work under the guidance and inspiration of the teacher in their preparation for the mastery of the new, is a distinctive feature of this book as well as of Book One. The Study Recitation shows the teacher the present state of the pupils' knowledge of what is necessary for the next subject and the degree of their skill in using that knowledge. It also enables the teacher to determine what, if any, review work is necessary and what special difficulties are likely to be encountered. Then, by careful guidance of the children's activities in the proper direction, the instructor may become a real teacher instead of a mere testing machine.

The Notes to the Teacher have been prepared with care, and are introduced at those points where experience has shown that they will be helpful for the pupils' advancement.

The Review Questions at the close of each important topic are designed not only to test the pupils' knowledge and power of clear statement, but also to test their power of reasoning and their ability to put that reasoning into proper form. Many of the questions are framed for the purpose of leading the pupils to arrive at and formulate the reasons for arith

metical operations not previously explained. Thus they require the pupils to reason, instead of confining their work to an attempt to understand, state, and apply the reasoning given in the book. Both lines of work are regarded as valuable and both are provided for.

The books of this series are the result of many years of practical experience in teaching arithmetic and in supervising the teaching of others in all classes of public schools and in professional schools for the training of teachers. It has not been the aim to make a series of books whose chief merit is striking novelty in the character or arrangement of matter, but rather to make a series of workable arithmetics adapted to the needs of pupils and teachers in the schools of to-day. Acknowledgment is made to M. A. Bussewitz of the Milwaukee State Normal School for valuable assistance rendered in the preparation of material for this book; and to George A. Works of the Menomonie High School for assistance in reading the proofs.

L. D. HARVEY.

SUGGESTIONS TO TEACHERS

PART I of this book should be used in sixth grade classes, Part II in seventh grade classes, and Part III in eighth grade classes. Where the work in Book One and in Part I of Book Two has been properly done, Parts II and III can be satisfactorily completed in about one half the time usually given to arithmetic in the seventh and eighth grades. It is probably better to distribute this time over the two years than to attempt to complete the work in arithmetic in the seventh grade.

Upon the first introduction of this book, it will be well for the teachers of seventh and eighth grade classes to familiarize themselves with the drill exercises in Book One and in the first part of Book Two, and to give such of these drill exercises to their classes as may be found necessary to secure accuracy and rapidity in the handling of numbers.

While the Notes to the Teacher are not intended to interfere with the teacher's use of those methods which his experience has justified, it is believed that if careful consideration is given to them they will prove valuable aids. They are the outcome of wide experience not only in teaching arithmetic but in the training of teachers.

Under the heading "Study Recitation" matter is presented which should be worked out by pupils in the class under the direction and guidance of the teacher. In order to do this properly it will be necessary for teachers to make themselves thoroughly familiar with the order and plan of work in the study recitation. This work must in no sense be considered

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a waste of time. If properly done, it will prepare the class to enter with intelligence and interest upon the work of preparation of the lesson that is assigned, and will make possible a surer and more rapid advance.

In the assignment of lessons the teacher should be sure that pupils understand the meaning of terms used and that they know exactly what is expected of them, both in the way of solution of examples and problems and in the requirements for oral and written recitations. It will be well to devote a considerable portion of the time of a number of recitation periods to such study of definitions and rules as may be necessary to enable the pupils of the class to recite readily and accurately. This exercise gives good training in concentration of attention upon the subject-matter of a lesson, and presents proper standards for a recitation. With such training pupils will understand better what is required in the assignment of a lesson.

At the close of the most important subjects, a set of review questions is given. As the pupils proceed in their work upon a topic it is an excellent plan for the teacher to assign as a part of the lesson such of these questions as have been covered by the work already done. In this way pupils will be required to prepare upon not more than one or two questions a day, and will need but little time, when the subject is finished, to prepare for a complete summary

of it.

New and unusual words are underscored as in Book I. The teacher should make sure that pupils understand the meaning of the underscored words.