MYERS ARITHMETIC BOOK III BY GEORGE WILLIAM MYERS PROFESSOR OF THE TEACHING OF MATHEMATICS AND ASTRONOMY COLLEGE OF EDUCATION, THE UNIVERSITY OF CHICAGO CHICAGO PREFACE The present book is an outgrowth of the notion that arithmetic as a science of pure number, and arithmetic as a school science, must be treated from essentially different standpoints. Viewed as a finished mental product, arithmetic is an abstract science, taking its bearings solely from the needs of the subject; but viewed as a school subject, arithmetic should be an abstracted science, taking its bearings mainly from the needs of the learner. The former calls only for logical treatment, while the latter calls for psychological treatment as well. In other words, to be of high educational value the school science of arithmetic must take into full account the particular stage of the pupil's development. The abstract stage must be approached by steps which begin with the learner, rise with his unfolding powers, and end leaving him in possession of the outlines of the science of arithmetic. To break vital contact with the learner at. any stage of the unfolding process is fatal. A controlling principle in the development of the various topics of this book is that any phase of arithmetical work, to be of value, must make an appeal to the life of the pupil. But the social and industrial factors in American communities enter largely into the pupil's life. This gives to material drawn from industrial sources and from everyday affairs, high pedagogical value for arithmetic. The recent infusion of new life into the curricula of elementary schools through the wide introduction into them of nature study, manual training, and geometrical drawing furnishes a basis for a closer unifying of the pupil's work in arithmetic with his work in the other school subjects. Wide use has been made of all these sources of arithmetical material. jji A rational presentation of the processes and principles of arithmetic can be secured as well through material representing real conditions as through material representing artificial conditions. Not only have most of the problems been drawn from real sources but an earnest effort has also been made to have all data of problems correct and consistent, to the end that inferences from them may be relied upon. With so rich a store as the book contains, however, it is perhaps too much to hope that no errors remain. The authors will deem it a favor to be notified of any errors that may be detected. The majority of the pupils of the elementary school never reach the high school. Even these pupils, whose circumstances cut them off from advanced mathematical study, have a right to claim some useful knowledge of the more powerful instruments of algebra and geometry. For those who will continue their studies into the high school it is important that the roots of the later mathematical subjects be well covered in the soil of the earlier. The present book meets the needs of both classes of pupils through the organic correlation of the elements of geometry and of general number with the arithmetic proper. Treated thus, the geometry serves to illustrate the work of arithmetic, and the algebra emerges from the arithmetic as generalized number. In particular, this book aims to accomplish four main purposes, viz.: (1) To present a pedagogical development of elementary mathematics, both as a tool for use and as an elementary science; (2) To base this development on subject-matter representing real conditions; (3) To open to the pupil a wide range and variety of uses for elementary mathematics in common affairs—to aid him to get a working hold of his number sense; and (4) To give the pupil some training in ways of attacking common problems arithmetically and some power to analyze quantitative problems. The work makes continual call for estimating magnitudes and for actual measurement by the pupils. Supply the children with measures, foot-rules, yardsticks, metersticks, etc., and encourage continual use of them. Make so regular a feature of this work that pupils form the habit of estimating distances, areas, volumes, weights, etc., always correcting their estimates by actual measurement. All models, scales, or standards of measure made by pupils should be carefully kept and used in the later work. As in Books I and II of this Arithmetic, so here there are groups of problems that bear on the development of some important idea or law, having an interest on its own account. In these groups each problem is a step in a connected line of thought culminating in an important truth. This plan furnishes numerous problems, miscellaneous as to process, thereby requiring original mathematical thought, and still organically related to a central idea, thereby calling for the constant exercise of judgment. For the stage of maturity of pupils of the later grades this is believed to be an important feature. It avoids the danger, always present with lists of promiscuous problems when classified under the arithmetical processes to be exemplified, of reducing to the mechanical what should never be allowed to become mechanical, viz.: the analysis of relations. But due regard has been had to the necessity of sufficient drill in pure number to enable the pupil to obtain both a conscious recognition of processes and considerable facility in their automatic use. This is done through the conviction that the fundamental arithmetical operations should be reduced to the automatic stage as early as possible consistently with a clear understanding of them. The attention of teachers is called to the section on Short Methods and Checking at the close of the book. After pupils have clearly grasped the meaning of the arithmetical |