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GEORGE WILLIAM MYERS
PROFESSOR OF THE TEACHING OF MATHEMATICS AND ASTRONOMY
COLLEGE OF EDUCATION, THE UNIVERSITY OF CHICAGO
SCOTT, FORESMAN AND COMPANY
COPYRIGHT, 1898, 1905, 1908
P. F. PETTIBONE & Co.
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 forePLACE FOR shortened 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 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.
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
subject. It is only when logic is construed to mean the procedure of adult mind that the demands of logic become mischievous in the elementary school. Rightly understood, logic means the natural procedure of the learning mind in mastering a subject. Recent work of educational experts has proved that the best nurture for the child as a child remains always the best nurture he could have had. In the language of biological science, while the frog is a tadpole, whatever is best for the tadpole will eventuate in the most perfect frog.
This doctrine is now generally accepted by all students of education. It has done much toward the general accrediting of childhood at its true worth. It has given flat and final denial of the right to quarrel with the child because he is not something else, by attempting to force upon him the logic of the adult. This modern doctrine is accepted by this book. It is believed the book has unified the interests of logic with those of the child by making its logic the logic of the learner at the stage he has reached.
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 the same process, or set of processes. What it does imply 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 numbering. This work, while interesting in itself to children, is
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.
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 be. Estimated results are then checked by calculating, and drill on the fundamental processes is continually kept up.
of problems involving real measurement, and incidentally MATERIAL. 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.
It is also recommended that teachers CHOOSING 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 ESTIMATES. 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