2d Ed. Copyright, 1917, by EDWARD LEE THORNDIKE Copyright, 1920, by THE PEOPLE OF THE STATE OF CALIFORNIA In the compilation of this book certain matter from The Thorndike 1921. 25M. A EDUC. DEPT. PREFACE HESE books apply the principles discovered by the psychology of learning, by experimental education, and by the observation of successful school practice, to the teaching of arithmetic. Consequently they differ from past practice in the following respects: Nothing is included merely for mental gymnastics. Training is obtained through content that is of intrinsic value. The preparation given is not for the verbally described problems of examination papers, but for the actual problems of life. In particular, problems whose answers must be known to frame the problems or whose conditions are fantastic are rigorously excluded. Reasoning is treated, not as a mythical faculty which may be called on to override or veto habits, but as the coöperation, organization, and management. of habits; and the logic of proof is kept distinct from the psychology of thinking. Interest is secured, not in pictures, athletic records, and the like, but in arithmetic itself and its desirable applications. Interest is not added as a decoration or antidote, but is interfused with the learning itself. Nothing that is desirable for the education of children in quantitative thinking is omitted merely because it is hard; but the irrelevant linguistic difficulties, the unrealizable pretenses at deductive reasoning, and the unorganized computation which have burdened courses in arithmetic are omitted. The demand here is that pupils shall approximate 100 percent efficiency with thinking of which they are capable. The formation and persistence of useful habits is not left to be a chance result of indiscriminate drill and review. Every habit is formed so as to give the maximum of aid to, 575855 and the minimum of interference with, others. Other things being equal, no habit is formed that must later be broken; two or three habits are not formed where one will do as well; each is formed as nearly as possible in the way in which it is required to function; each is kept alive and healthy by being made to coöperate in the formation of other and higher habits in the arithmetical hierarchy. If a pupil carries through the projects in computing and problem-solving of these three books under competent supervision, he will have abundant practice for the arithmetical insight, knowledge, and skill that the elementary school is expected to provide. E. L. T. NOTES ON BOOK ONE Part One of this book is for use as a supplement to the informal work of Grade II, and as a basic text in Grade III. Part Two, or so much of it as the course of study makes advisable, is for use in Grade IV. Experienced teachers will, by examining and using this book, understand the reasons for the choice of the exercises and problems, for the order in which they appear, and for the methods used, with three possible exceptions. These are: (1) the early, varied, and extended use of the equation form with a missing number or quantity to be supplied, (2) the introduction of two-place and three-place multiplicands before the products of 6, 7, 8, and 9 are learned, (3) the rationalizing of procedures by verifying the fact that they are right rather than by arguments to show that they must be right. Such uses of the equation form as the book contains will be found admirable as preventives of rote memorizing without understanding, as stimuli to mathematical thinking, and as means to an economical organization of arithmetical knowledge. The time spent on them will be saved twice over in later work. The introduction of two-place and three-place multiplicands provides a genuine use for the multiplication facts learned, organizes the knowledge of the products of 5, 2, 3, and 4, gives a needed review, relieves the monotony of learning the tables, and enables the pupil to utilize rather than memorize the products of 6, 7, 8, and 9, as fast as these are learned. The rationalization of procedures by the pupil's own experience in verifying the results obtained is superior to the use of formal proofs of the validity of the procedures before they are learned and used. With all save the most gifted, |