Excerpt from First-Year Mathematics for Secondary Schools

This volume aims to present in teachable form an inter weaving Of the more concrete and the easier portions of the first courses in both algebra and geometry. It is intended as a text for students of the first year of secondary schools. The type of work begun here will be continued by a second year text, to appear soon and to include not less than the rest of the subject-matters that make up the customary two years of required secondary work in algebra and plane geometry.

The present book places chief emphasis upon algebra, but weaves in a considerable body of related fundamental notions and principles Of geometry, even advancing somewhat in one direction or another upon the most concrete, graphic, and practical aspects of elementary geometry. Geometrical treat ments are at first intuitive, inductive, and experimental, fol lowed in a few instances by the quasi-experimental methods of superposition.

A like informality as to method characterizes the first half of the algebra. The algebraic treatments feature distinctly: (i) the liberal use of inductions from arithmetic; (2) the early and persistent use of the equation, both as an instrument for problem - solving, and as a means of suggesting and of treating topics; (3) the early, varied, and systematic use Of pictured and graphic modes of rendering algebraic truths Vivid and appealing to beginners.

The algebra of the last half of the book is more formally deductive in character. Even as early as p. 26 the four pro cess-axioms are given as simple statements of the four ways in which equations have been changed in solving arithmetical and mensurational problems. They are stated as laws forusing the equation, and are given only as much generality as the antecedent work justifies. Transformations of equations are then made on the basis of these laws and of others brought out from time to time as the work proceeds, until p. 216 is reached, where a fifth general axiom is added, and the five are here given final statement. These five axioms are hence forth made the reasons for all changes in equations. The root-axiom is added later when an ocrasion to use it arises. Thus the transition from the informal procedure of the earlier part of the algebra to the formal procedure of the later part is gradual, while throughout the latter half the reasoning becomes more and more highly deductive. This is in accord with what both Classroom practice and a-priori reasoning Show to be the natural order for secondary mathematical education.

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