The central purpose of this book is to provide maximum help for the student and maximum service for the teacher.
This book has been designed to improve the learning of geometry far beyond that of the typical and traditional book in the subject. Students will find this text useful for these reasons:
(1) Learning Each Rule, Formula, and Principle
Each rule, formula, and principle is stated in simple language, is made to stand out in distinctive type, is kept together with those related to it, and is clearly illustrated by examples.
(2) Learning Each Set of Solved Problems
Each set of solved problems is used to clarify and apply the more important rules and principles. The character of each set is indicated by a title.
(3) Learning Each Set of Supplementary Problems
Each set of supplementary problems provides further application of rules and principles. A guide number for each set refers the student to the set of related solved problems. There are more than 2000 additional related supplementary problems. Answers for the supplementary problems have been placed in the back of the book.
(4) Integrating the Learning of Plane Geometry
The book integrates plane geometry with arithmetic, algebra, numerical trigonometry, analytic geometry, and simple logic. To carry out this integration:
(a) A separate chapter is devoted to analytic geometry.
(b) A separate chapter includes the complete proofs of the most important theorems together with the plan for each.
(c) A separate chapter fully explains 23 basic geometric constructions. Underlying geometric principles are provided for the constructions, as needed.
(d) Two separate chapters on methods of proof and improvement of reasoning present the simple and basic ideas of formal logic suitable for students at this stage.
(e) Throughout the book, algebra is emphasized as the major means of solving geometric problems through algebraic symbolism, algebraic equations, and algebraic proof.
(5) Learning Geometry Through Self-study
The method of presentation in the book makes it ideal as a means of self-study. For able students, this book will enable then to accomplish the work of the standard course of study in much less time. For the less able, the presentation of numerous illustrations and solutions provides the help needed to remedy weaknesses and overcome difficulties, and in this way keep up with the class and at the same time gain a measure of confidence and security.
(6) Extending Plane Geometry into Solid Geometry
A separate chapter is devoted to the extension of two-dimensional plane geometry into three-dimensional solid geometry. It is especially important in this day and age that the student understand how the basic ideas of space are outgrowths of principles learned in plane geometry.
Teachers of geometry will find this text useful for these reasons:
(1) Teaching Each Chapter
Each chapter has a central unifying theme. Each chapter is divided into two to ten major subdivisions which support its central theme. In turn, these chapter subdivisions are arranged in graded sequence for greater teaching effectiveness.
(2) Teaching Each Chapter Subdivision
Each of the chapter subdivisions contains the problems and materials needed for a complete lesson developing the related principles.
(3) Making Teaching More Effective Through Solved Problems
Through proper use of the solved problems, students gain greater understanding of the way in which principles are applied in varied situations. By solving problems, mathematics is learned as it should be learned–by doing mathematics. To ensure effective learning, solutions should be reproduced on paper. Students should seek the why as well as the how of each step. Once students sees how a principle is applied to a solved problem, they are then ready to extend the principle to a related supplementary problem. Geometry is not learned through the reading of a textbook and the memorizing of a set of formulas. Until an adequate variety of suitable problems has been solved, a student will gain little more than a vague impression of plane geometry.
(4) Making Teaching More Effective Through Problem Assignment
The preparation of homework assignments and class assignments of problems is facilitated because the supplementary problems in this book are related to the sets of solved problems. Greatest attention should be given to the underlying principle and the major steps in the solution of the solved problems. After this, the student can reproduce the solved problems and then proceed to do those supplementary problems which are related to the solved ones.
This book can be used profitably by others besides students and teachers. In this group we include: (1) the parents of geometry students who wish to help their children through the use of the book’s self-study materials, or who may wish to refresh their own memory of geometry in order to properly help their children; (2) the supervisor who wishes to provide enrichment materials in geometry, or who seeks to improve teaching effectiveness in geometry; (3) the person who seeks to review geometry or to learn it through independent self-study.
BARNETT RICH
Brooklyn Technical High School
April, 1963