GEORGE WILLIAM MYERS PROFESSOR OF THE TEACHING OF MATHEMATICS AND ASTRONOMY SCOTT, FORESMAN AND COMPANY CHICAGO NEW YORK PREFACE THE last ten years have seen many substantial gains in both theory and practice in elementary education. Arithmetic has gained largely from this general advance. In the teaching of the elements of mathematical science we have learned much of late as to practical ways of attaining the larger and the more significant educational aims-strengthening the judgment and the will, and fostering the power to think and to do. There is no school subject in which foreshortened views and distorted perspective work more harm than in elementary mathematics. Children, as well as adults, learn new ideas by meeting them first in simple forms, intermingled with familiar ideas and fairly wellunderstood uses of the new ideas. After a little, the new idea makes itself felt as something new. This is the time PLACE FOR to differentiate it for formal study, to learn what it really is. This is the stage for the study of process and for drill enough to impress it and to make its use easy and facile.' The learner then desires to experience the added power the mastery of the process has given him, and this calls for the application stage. The treatment of new ideas, processes, and topics in this book is accordingly arranged on this three-fold plan of (1) its informal use, (2) its formal study, and (3) its application. Examples of this plan may be seen in the teaching of the tables. ORDER OF The arrangement of number work for the grades must be in accordance with the natural unfolding of the child's mind. Too often this important fact is lost sight of in the logic of the subject itself. Strictly speaking, there can be no contradiction between the demands of the child's mental development and the logical requirements of the The ideas of number and of the numerical processes must be derived from the concrete. Form and number are the two main developments of quantity. The process of numbering in its varied aspects is very closely paralleled in the physical world by the process of measuring in its varied applications. This does not imply that numbering and measuring are one and What it does imply the same process, or set of processes. is that numbering is the mental side of the same problem. of adjustment of activity that has its physical expression in measurement. It means that measurement is the most direct and certain route to correct notions of number, for one who has not yet acquired them. Part I is for the third grade. It begins by impressing the pupil with the need for estimating and measuring, by giving him considerable work in indefinite comparison, leading to definite comparison, measurement, and numberwork while interesting in itself to children, is PREFACE done not so much for its own sake, as to supply a rich groundwork of number judgments for arithmetical number and process. The year's work includes many of the uses of number that gather about the common standards and processes of measurement, and makes a sure and sound beginning on the tabular machinery of arithmetic. USE OF Part II, for the fourth grade, completes the work on the tables, gives a wide range of applications to easy and useful matters of common experience, and considerable practice in choosing processes and in estimating what results must Estimated results are then checked by calculating, and drill on the fundamental processes is continually kept up. There are numerous lists of problems involving real measurement, and incidentally also counting, at its best. These lists are carefully graded, and the teacher is urgently recommended at all times to have pupils solve all they can orally. The pencil and paper should be used only when the difficulties of the problem make it too hard for the pupil to do orally. Different pupils show very different degrees of aptitude for rapid oral work. No plan of isolating the oral from the written work can suit the varying needs of different pupils, and every pupil has a right to the best sort of training of which he is capable. The problems of life are handled in this way, and the pupil should early form the habit of using his head as much as possible and his pencil only as an aid to his head. CHOOSING AND FORMING It is also recommended that teachers follow the practice of having pupils work rapidly through many of the lists of problems, indicating the processes called for and giving and recording estimates of about what the answers must be before any figuring is done. Then the problems should be worked through and the correct results compared with the estimates. This work is of high value as training of judg ESTIMATES. |