THE FRANKLIN ELEMENTARY ALGEBRA BY EDWIN P. SEAVER, A.M. SUPERINTENDENT OF THE PUBLIC SCHOOLS, BOSTON; AND GEORGE A. WALTON, A.M. BOSTON WILLIAM WARE AND COMPANY 1882 PREFACE. THE method of teaching algebra set forth in this book assumes as a leading principle that algebraic language, like other language, is best acquired, not through definitions and formal rules, but rather through actual use of the language itself as an instrument of thought. The singular power of algebraic language in analysis and reasoning is understood only when one has learned to use it in those processes. The earlier a beginner can be taught to use even a little of this language in reasoning, the better; for so may his interest in it be earlier awakened through the conscious exercise of a new power. Accordingly, the first part of this book is devoted to the solution of easy problems, by the use of simple equations, with one unknown quantity. The few symbols needed are briefly explained in the first section. The second section is a carefully arranged collection of problems, interspersed with explanations, as new features in the process of solution call for them. Small numbers are used in a portion of the problems, that the reasoning may with ease be carried on mentally and expressed orally. This oral use of algebraic language is found to be a very efficient means of instruction. Regarding the processes used in the solution of equations, the remark should be made here, that all these processes are based directly on the axioms given in Article 24. The process called "transposition" is not used, and the word "transpose" does not occur. It is believed to be better for the beginner to fix his thought directly on the real nature of the step he is taking than it is to run the risk of forming misconceptions through the use of terms which do not rightly mark the real nature of that step; for experience has shown that the operation called "transposition," though perfectly legitimate when properly understood, is, nevertheless, likely to be misleading and confusing to the beginner. Throughout this book, accordingly, equations are reduced by direct applications of the axioms, that is, the same quantity is added to or subtracted from each member, or each member is multiplied or divided by the same quantity. The third section is devoted to operations on algebraic quantities. The matter given in this section is more commonly given at the beginning of text-books; but the arrangement here adopted recognizes the principle that the meaning of algebraic expressions must be clearly apprehended, before rules for operating upon or with them can be well understood. Attention is asked to the manner in which negative numbers are introduced, to the distinction made between algebraic and arithmetical numbers, and to the distinction between algebraic addition, subtraction, multiplication, and division, and the arithmetical operations of the same names. All the algebraic operations are based on the nature of algebraic numbers as distinguished from arithmetical numbers. The learner is not in full possession of the algebraic method of reasoning until he has learned how to solve problems in a general manner, and to interpret the results for particular cases. The sixth section contains some illustrations of this process of generalization applied to problems already familiar. A degree of familiarity with generalized statements and processes, on the part of the learner, is assumed throughout the rest of the book. He is supposed to be ready to use the general language of algebra in the investigation of such topics as the binomial theorem, roots of numbers, arithmetical and geometrical progression, and logarithms. He is taught to guide his processes, not by rules, but by formulas, as in the solution of binomial equations in the ninth section, of quadratic equations in the tenth, and in various other instances. All this is in pursuance of the principal aim of algebraic training, which is to enlarge and generalize our notions of number and our methods of reasoning on numerical questions. The course of study embraced in this book is sufficient to meet the requirements for admission to any of our colleges, and is such as is pursued in our best high schools and academies. The work is submitted to teachers in the hope that it may prove helpful in promoting improved methods of mathematical instruction. BOSTON, May, 1882. |