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BOOK ONE

BY

ERNST R. BRESLICH

Assistant Professor of the Teaching of Mathematics,

The College of Education

and

Head of the Department of Mathematics,

The University High School

The University of Chicago

New York
THE MACMILLAN COMPANY

1925

All rights reserved

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AUTHOR'S PREFACE

The study of certain phases of mathematics deserves an important place in a liberal education. Modern thought and enterprise are steadily increasing in mathematical precision, as is apparent in the statistical aspects of the biological and human-nature sciences. To understand modern civilization in many of its aspects one must be able to appreciate its precise quantitative character. One must be able to read intelligently quantitative accounts of modern enterprises. Familiarity with modern methods of measurement and skill in doing quantitative thinking are essential. This text offers a general course intended for those who continue studies in the senior high school as well as for those who do not continue. It aims to contribute to the pupil's liberal education by preparing him to understand the quantitative aspects of contemporary civilization.

Since this is a first course in junior high school mathematics, it presupposes knowledge of the fundamental operations with whole numbers, with common fractions, and with decimals. As soon as proficiency in these operations has been attained, the pupil is prepared to take up the study of this course.

The following aims have been set up in the selection and organization of material:

1. The material used must fill a real need in the life of the pupil. It must be useful to him in his present

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school studies as well as in preparing him to perform activities in later life. It must have social value.

In this textbook the social realities of the material are emphasized throughout. Free use is made of discussions, pictures, and diagrams to make the social worth of the material appeal to the pupil.

2. The material must be adapted to the abilities of pupils of the early adolescent period and lie within their experience.

Since the success of a course must depend largely on the extent to which this adaptation is accomplished, all the material has been tried out with junior high school classes. On the basis of results of carefully made tests, subject matter not adapted to the mental ability of these pupils has been rejected or changed in treatment. Thus, business applications have been limited to such matters as pupils may be expected to appreciate and understand. Ideas of percentage, interest, etc., which everybody should know, are presented concretely. This is considered sufficient for the majority of pupils. If in a school there are enough pupils intending to enter commercial work, they may be given an additional vocational course in commercial arithmetic. It is advised that vocational courses be given near the end of the junior high school course, or in the senior high school, rather than in the first year.

3. It is not sufficient to teach mathematics merely as a body of principles. There ought also to result training in mathematical methods of thought, effective habits of study as applied to mathematical situations, a conviction of the universal applicability of powers of concentration, and insight into the method of sound

generalization in any field. Such larger values cannot be depended upon to come of themselves. Their achievement has been a matter of constant attention throughout the course.

4. Quantitative relations are to be studied in three ways: geometrically, as in length, area, and graphs; algebraically, as in formulas, equations, and functions; and arithmetically, as in tables and evaluation.

Geometry in its simplest form, because of its usefulness and concreteness, has been made the core of the course. It is experience getting in space relations. It is intuitional, experimental, constructional, not demonstrative. The principles established are those that appeal to the pupil as valuable information. This experience in geometry makes it possible to make the beginning of topics in algebra concrete and then to pass from the concrete to the abstract.

Algebra is not taught in this text as an organized science, but as a helpful tool in the study of other topics. Formulas and equations are the outcome of concrete problems and relate to real things. Since the geometry is concerned only with the conception of plane figures, such as line segments, angles, and areas, no algebraic functions of degree higher than the second are introduced. These functions are to be considered in the second and third course where three-dimensional figures are introduced and where algebra is studied as a science.

An abundance of work in arithmetic has been provided. The arithmetic aims to secure proficiency in the fundamental manipulative processes needed in the course of ordinary life and in the acquisition of further

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