PREFACE

UNTIL recently upper elementary and high school work in mathematics was planned for the pupil who was expected to continue it in the university. Although logical, its arrangement was neither psychological nor pedagogical, but some progress has been made recently in adapting the study to the needs and abilities of pupils.

In the junior high or intermediate school, work in mathematics in the seventh, eighth, and ninth grades should be complete in itself and at the same time preparatory to senior high school work. No effort should be made to "finish" arithmetic in the eighth grade and algebra in the ninth, while denying the child the interest and beauty that lie in geometry and trigonometry until his taste for mathematics has been destroyed. Nor will alternate bits of formal algebra, geometry, and trigonometry solve the problem. The result is a mastery of none and a confusion of all.

Experience has proved that the necessary elements of arithmetic can be taught and certain definite skill developed in the first six grades. In the seventh grade business applications of arithmetic with the simplest elements of bookkeeping should be given. In the eighth grade, mensuration should be taught experimentally or through observational geometry, and through that, in a natural and meaningful way, the

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very beginnings of algebra. Optional courses should be offered in the ninth grade.

Experience in junior high schools has shown that much of the content and the whole of the organization of subject matter must be changed to make the course of study fit the needs of the pupils.

The definite aims of this study are:

1. To extend the pupil's knowledge of arithmetic through its practical applications in mensuration.

2. To train the hand to use the simple drawing instruments.

3. To familiarize the pupil with common geometric forms.

4. To train him to see geometric forms in nature and in the various buildings and other structures in his surroundings, and to appreciate their use in design.

5. Through experiment and observation to develop the formulas of mensuration.

6. Through a continued study of formulas to introduce general number in a natural way that will give algebraic expressions such a meaning to the pupil that he will use them as convenient and practical tools.

7. To permit a pupil to live so continuously in the atmosphere of geometry that he may be enabled to think naturally and without confusion, in its terms, about its relations.

This book is planned for a year's work in the eighth grade, with the idea that the pupil should advance slowly by doing and thinking for himself. If necessary it can be condensed into one semester's work. It

should always precede and form an introduction to the more formal algebra and geometry.

The nucleus of this book grew out of the author's experience as an instructor in mathematics in a large city high school, in teaching the first book of plane geometry without a text. Four years ago an outline of it was given by the author to the teachers under her supervision because no suitable text in experimental geometry could be found. By way of further suggestion, one or two topics were expanded in more detail and discussed with the teachers. This book is the result of their urgent request for more. Since then, over a score of teachers have used the outline, and their unanimous opinion is that the pupils take an increased delight in mathematics of this kind. The author was in a position to observe the effects of teaching it in various types of schools and found it most gratifying.

The author is very much indebted to Dr. John H. Francis, Superintendent of Schools, Columbus, Ohio, for reading the manuscript and for giving helpful suggestions. She hereby acknowledges her indebtedness also to Miss Meta Philbrick of Mt. Vernon Intermediate School and to Miss Amy Preston of Roosevelt Intermediate School, Columbus, for their coöperation and assistance in gathering problems and program material.

MARIE GUGLE