steps in mastering algebraic tasks. The concepts of lines, rectilinear figures, and solids are so much space material, always and everywhere available for concreting, visualizing, and vivifying number laws and relations, at no great cost in money or effort. The high school youth has lived long enough in this world of space to have become familiar with it, and his spatial experiences need only to be drawn upon to enable him to lay firm hold on the highly abstract fundamentals of beginning algebra. Really to see that algebra merely generalizes mensuration laws, that algebraic numbers, laws, and problems picture into vivid forms, and to learn the secret of laying before his eyes diagrammatically the conditions of algebraic problems as an aid in formulating these conditions into algebraic language and technique, are of the highest interest and value to the beginner. The professional duty, of employing the concreting agencies of pictures, diagrams, geometrical figures, and graphs to vivify and vitalize algebra will be readily accepted by the teacher who strives to realize in practice the educational merits of well-taught algebra. No clumsy laboratory equipment of extensive and expensive apparatus is required to enable the algebra teacher through space-materials to supply genetic backgrounds for algebraic problems, truths, and laws. III. To show the pupil that algebra will enable him to do much more than he can do with either arithmetic or geometry, or both. The first and second professional duties are really preliminary, through which motivating and clearing the way for effective attack are accomplished. This third duty is peculiarly due to algebra. It is in fact due to both pupil and subject that the particular gains to be secured by a mastery of the subject-matter shall appear in the learning acts. For example, the pupil should see such things as, that by arithmetic he cannot subtract if the subtrahend happens to be greater than the minuend; that he cannot solve so simple an equation as x+9=3; but that if he include the negative numbers among his number notions he can do both easily. He should see that he can square and cube numbers geometrically, but that he can go no further with involution than this. If, however, he will learn the symbolism of algebra he may easily express and work with 4th, 5th, 6th, even with nth powers. He should be shown that while he can solve equations in one, two, and perhaps in three unknowns with graphical pictures, i.e., geometrically, the great power he gains by mastering the algebraic way enables him to go right on easily to the solution of simultaneous equations in 4, 5, 6, and even n unknowns. He should be made to feel that while arithmetic would enable him, by a slow process of feeling about, to find one solution of many problems, algebra, if he will learn its language and method, will lead him directly not to one, but to all possible solutions. It will thus enable him to know when he has solved his problem completely. These and similar gains of power over quantitative problems are the real reasons why the educated man of today cannot afford not to know algebra. Let teachers perform this professional duty well and the foes of algebra as a school subject will be confined to those who are ignorant of it. The one who has learned the subject will then regard it as the emancipator of quantitative thinking. It is desired to call particular attention to the introductory pages on Reasons for Studying Algebra, and to Suggestions on Problem-solving on page 113, and to the careful treatment of factoring. The treatment of the function notion, on pages 50-56, will appeal to many teachers. It will be noted also that this elementary course is divided into half-year units. The problem and exercise lists are full, varied, and carefully chosen. Teachers who employ supplementary lists of exercises with the regular text should not require pupils to try to solve all the problems and exercises given here. These lists are made full and varied to afford choice and range of material. Great care has been exercised to cover all the standard difficulties of first-year algebra, for this book makes its primal task to teach good algebra. This text is to be followed presently by a second course on Intermediate Algebra. The two together will cover the standard requirements of secondary algebra. The pleasant task now remains to acknowledge the assistance the authors have received from Mr. John DeQ. Briggs of St. Paul Academy, St. Paul, Minn.; from the Misses Ellen Golden and Estelle Fenno of Central High School, Washington, D. C.; and from Professor H. C. Cobb of Lewis Institute, Chicago, all of whom read and criticized the proofs of the book. Their criticisms and suggestions have resulted in numerous improvements. May this book find friends amongst teachers and pupils, and a deserving place amongst the influences now making for the improvement of the educational results of high school algebra. THE AUTHORS. Chicago, September, 1916. : |