This book presents the applications of this newer dynamic psychology to the teaching of arithmetic. Its contents are substantially what have been included in a course of lectures on the psychology of the elementary school subjects given by the author for some years to students of elementary education at Teachers College. Many of these former students, now in supervisory charge of elementary schools, have urged that these lectures be made available to teachers in general. So they are now published in spite of the author's desire to clarify and reinforce certain matters by further researches. A word of explanation is necessary concerning the exercises and problems cited to illustrate various matters, especially erroneous pedagogy. These are all genuine, having their source in actual textbooks, courses of study, state examinations, and the like. To avoid any possibility of invidious comparisons they are not quotations, but equivalent problems such as represent accurately the spirit and intent of the originals. I take pleasure in acknowledging the courtesy of Mr. S. A. Courtis, Ginn and Company, D. C. Heath and Company, The Macmillan Company, The Oxford University Press, Rand, McNally and Company, Dr. C. W. Stone, The Teachers College Bureau of Publications, and The World Book Company, in permitting various quotations. EDWARD L. THORNDIKE. TEACHERS COLLEGE COLUMBIA UNIVERSITY April 1, 1920 CONTENTS CHAPTER PAGE INTRODUCTION: THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL SUBJECTS xi I. THE NATURE OF ARITHMETICAL ABILITIES 1 Knowledge of the Meanings of Numbers Arithmetical Language Problem Solving Arithmetical Reasoning Summary The Sociology of Arithmetic II. THE MEASUREMENT OF ARITHMETICAL ABILITIES 27 A Sample Measurement of an Arithmetical Ability Ability to Add Integers Measurements of Ability in Computation Measurements of Ability in Applied Arithmetic: the Solution of Problems III. THE CONSTITUTION OF ARITHMETICAL ABILITIES 51 The Elementary Functions of Arithmetical Learning Knowledge of the Meaning of a Fraction Learning the Processes of Computation IV. THE CONSTITUTION OF ARITHMETICAL ABILITIES (_continued_) 70 The Selection of the Bonds to Be Formed The Importance of Habit Formation Desirable Bonds Now Often Neglected Wasteful and Harmful Bonds Guiding Principles V. THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE STRENGTH OF BONDS 102