The guiding rule in the preparation of the present work has been to present the pupil with but one new idea at a time, and, by examples and exercises, to insure its assimilation be fore passing to another. This system is carried out by a minute subdivision of all the algebraic processes, and the end kept in view is that in no case where it can be avoided shall the pupil have to go through a process of which he has not previously learned all the separate steps. As a part of the same plan, "definitions are so far as practicable brought in only as they are wanted, the first exercises in indicated operations are performed with numbers, and the pupil is set to work on exercises from the start.
Correlative with the system of subdivision is that of ex tending the scope of the exercises so as to include not only the elementary operations of algebra, but their combinations and applications. It is hoped that the wider range of thought and expression in which the pupil is thus practised will be found to tell in his subsequent studies.
As another part of the same general plan the subject has been divided into three separate courses. The First Course, which extends to Simple Equations, is intended to drill the student in all the fundamental processes by exercises which are, for the most part, of the simplest character. The varied exercises in algebraic expression which are scattered through this course form a feature to which the attention of instructors is especially solicited.
In the Second Course the processes are combined and the whole subject is treated on a higher plane. The general arrangement of this course is the same as that of the Elementary Course in the author's College Algebra. But the exercises are all different, and greater simplicity of treatment is aimed at. This course terminates with Quadratic Equations.
The Third Course consists of three supplementary chapters which, however, should be mastered before entering college.
An attempt has been made to treat quadratic equations with such fullness as to avoid the usual necessity of reviewing that subject after entering college.
In the preparation and use of such a work no question is more difficult than that of the extent to which rigorous de monstrations of the rules and processes shall be introduced. At one extreme we have the old method, in which the teaching is purely mechanical; at the other, the modern demand that nothing be taught of which the reason is not fully explained to and understood by the pupil. The latter method should of course be preferred, but we meet the insuperable difficulty that we are dealing with a subject of which the reasoning cannot be understood until the pupil is familiar with it. The rule adopted in the present case has been to present a proof, reason or explanation wherever it was thought it could be clearly mastered. Most teachers will, however, admit that long explanations of any kind weary the pupil more than they instruct him, and that the best course is to present the examples and exercises in such a form that their logical correctness shall gradually become evident without much further help.
Were the author to make a suggestion respecting the system of teaching to be adopted, it would be to commence the study of algebra at least one year earlier than usual, and to devote the whole of that year to the first course, taking two or perhaps three short lessons a week. The habit of using algebraic symbols in working and thinking would thus become established before taking up the subject in its more difficult forms.