Math. 9-29-23 EDITOR'S PREFACE This text derives its origin and character from the view that geometry is a valuable study for high school pupils in the degree to which they understand and appreciate it at the time they are studying it. Appreciation of the worth of what is being done and insight into the meaning of its tasks go very far toward a real motivation of geometry. The system of thought it embodies is of course the chief claim of geometry to a place in the curriculum, but this system of thought is of no great value to one who neither appreciates nor understands its spirit. Furthermore, this system of thought is a derived product. It arises from the habits formed from doing things geometrically and reasoning about what is done and why it is so done. It must accordingly wait upon many an act of intelligent judging and discriminating. The acts of systematic judging and discerning find a rich genetic background in measurement, in constructive exercises, in comparison of figures, and in geometrical experimentation by the student. This text so organizes the material as to give a concrete, experimental, and somewhat informal approach to the rather highly wrought scheme of demonstrative geometry. This informality pervades the first half of the first chapter, thereby laying a firm conceptual basis for the more systematic geometry. Throughout the text, however, subjects of special difficulty to high school students are approached experimentally. Mental possibilities are nowhere sacrificed to logical refinements. Logical accuracy that is beyond the reach of the learner is held to be only apparent and largely specious. Besides the closer conformity to pedagogical standards of this inductive and informal approach to systematic geometry, there is also an important scientific gain from it. It is commonly regarded as bad scientific practice to allow unlimited use of hypothetical constructions. With beginners in geometry it is always a question how far it is desirable to permit them to give proofs that involve constructions which they have not yet learned. For example, before the student has learned how to bisect an angle he may be allowed, openly or tacitly, to assume that a bisector can be drawn, while he is proving the theorem about the base angles of an isosceles triangle. The real question is how far shall this sort of practice be allowed. The prevailing view is that it should be practiced no longer than necessary. Too long continued, it blunts the learner's perception of the nature of real geometrical proof. By including in the preliminary work such constructions as are needed early in proofs, the need for hypothetical constructions is reduced to a minimum. The present organization thus scores a real gain in soundness of early geometrical thinking in the very act of reaching an important pedagogical end. To the beginner no motives are so wholesomely appealing as the all-around usefulness of the study. "What benefit can be derived from the subject?" is the question most on his mind and lips. A good pedagogical organization must so marshal its ideas for presentation that the most attractive ones are in the foreground. This important principle has been kept prominent in the authors' minds in the organization of the subject as a whole, and also of its important parts. The exercise lists are exceptionally rich and varied in really practical problems, without at any point failing to include an ample supply of the standard geometric type as well. Many of the abstract exercises are followed at once by practical problems that give a life-setting to the geometric principles involved. An earnest attempt has been made to give a great variety to the types of practical exercise. This feature will assist in reaching a wide range of taste and aptitude among pupils, and also in keeping up the spirit of study through a many-sided motivation. This cannot fail to appeal to those who hold the view that it is of maximum importance to make the learner's work seem worth while to him while he is yet in school. The text lays no claim to logical precision beyond that which good students can appreciate. It recognizes that boys and girls can prove things long before they can demonstrate them. That is to say, they can bring to bear enough relevant evidence to heighten materially the feeling of certainty of the truth of a theorem before they can intelligently master the technique of formal deductive proof. This treatment recognizes the learner's right to acquire the ability to demonstrate through the exercise of his ability to prove. To do otherwise is to endanger needlessly the very spirit of geometric reasoning in a vain show of mechanical steps and logical technique. In gauging this treatment, pedagogical rather than logical standards should be applied. Logic has been intentionally sacrificed to insight whenever it was believed that the general geometric interests of the student would be thereby materially subserved. Finally, this book has been prepared by actual teachers of long and successful high school experience. They know high school boys and girls. Visionary theories about what ought to be, but cannot be, have been ruthlessly set aside. They have kept an eye single to what is feasible, practicable, and remunerative in the class-room. Much of the manuscript was used for years in mimeograph form and the roughnesses have been smoothed off through class-room findings. The plan has also worked successfully in the hands of other teachers than the authors and in much less practicable form than that in which it is here given to the public. The editor feels that the text has an important rôle to play in the attempt now being made to bring school geometry into closer conformity with the needs, standards, and possibilities of those who are to study it. Chicago, August, 1915. G. W. MYERS. |