Produced by Jonathan Ingram and the Online Distributed Proofreading Team at http://www.pgdp.net THE MACMILLAN COMPANY NEW YORK · BOSTON · CHICAGO · DALLAS ATLANTA · SAN FRANCISCO MACMILLAN & CO., LIMITED LONDON · BOMBAY · CALCUTTA MELBOURNE THE MACMILLAN COMPANY OF CANADA, LIMITED TORONTO THE PSYCHOLOGY OF ARITHMETIC BY EDWARD L. THORNDIKE TEACHERS COLLEGE, COLUMBIA UNIVERSITY New York THE MACMILLAN COMPANY 1929 _All rights reserved_ COPYRIGHT, 1922, BY THE MACMILLAN COMPANY. Set up and electrotyped. Published January, 1922. Reprinted October, 1924; May, 1926; August, 1927; October, 1929. · PRINTED IN THE UNITED STATES OF AMERICA · PREFACE Within recent years there have been three lines of advance in psychology which are of notable significance for teaching. The first is the new point of view concerning the general process of learning. We now understand that learning is essentially the formation of connections or bonds between situations and responses, that the satisfyingness of the result is the chief force that forms them, and that habit rules in the realm of thought as truly and as fully as in the realm of action. The second is the great increase in knowledge of the amount, rate, and conditions of improvement in those organized groups or hierarchies of habits which we call abilities, such as ability to add or ability to read. Practice and improvement are no longer vague generalities, but concern changes which are definable and measurable by standard tests and scales. The third is the better understanding of the so-called "higher processes" of analysis, abstraction, the formation of general notions, and reasoning. The older view of a mental chemistry whereby sensations were compounded into percepts, percepts were duplicated by images, percepts and images were amalgamated into abstractions and concepts, and these were manipulated by reasoning, has given way to the understanding of the laws of response to elements or aspects of situations and to many situations or elements thereof in combination. James' view of reasoning as "selection of essentials" and "thinking things together" in a revised and clarified form has important applications in the teaching of all the school subjects.