Copyright, 1917, by EDWARD LEE THORNDIKE In the compilation of this book certain matter from The Thorndike 3d Ed. 1922. 25 M THE 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, and the 592056 minimum of interference with, others. Other things being equal, no habit is formed that must be later 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 problemsolving 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 TWO Parts One and Two are intended for Grades V and VI respectively. Part One provides for mastery of the four operations with such common fractions as the pupils will meet in life, and for ability in the four operations with decimal numbers, in simple cases. Part Two completes the training with decimals, gives mastery of computations with percents and with such denominate numbers as the pupil will meet, and provides experience in simple accounting. The training of Part One in observing and using the relations of numbers is extended and systematized. The applications of arithmetic include simple problems in computing areas, volumes, wages, commissions, discounts, and advances. The traditional so-called logical arrangement of topics is abandoned in favor of an order that fits the learner's needs, the book being an instrument by which children acquire a rounded, organized, working knowledge of arithmetic, not a display of such knowledge as an adult finally possesses it. The traditional methods of securing ability with fractions, decimals, and percents are also replaced in cases where educational science has found a better way. The resulting selection of topics and methods by expert teachers of arithmetic needs no explanation except perhaps in five particulars. Concerning each of these a brief note is in place. The pupil learns to add and subtract fractions without any formal treatment of least common multiples, being taught specifically to use fourths, sixths, eighths, twelfths, and sixteenths where each is appropriate, and for other cases to reduce to any denominator which is satisfactory. This will be found to save time, prevent ponderous treatment of simple tasks, and in the end be the best introduction to learning what a least common multiple is, if that information is desired. Division by a fraction is made the occasion (pages 51 to 54) of solid general instruction concerning the reciprocal rule. If any rule is worth teaching in arithmetic it is the rule "To divide by a number is the same as to multiply by the reciprocal of that number." This rule helps to make rational a number of procedures and is often the means of reducing labor greatly in technical and commercial computations. The equation with a missing number to be supplied is often used in place of verbal forms, such as "24 is what part of 30?" "How much is two thirds of 18?" "$75 less 10% of itself is how much?" and "What percent of 40 is 32?" These exercises in equation form with missing numbers are harder than routine drills with question and answer, but are more productive of ability, and of ability of a higher type. They also penalize mere memoriter acquisition and serve as an ideally clear, brief, and unrestricted form for mental imagery of arithmetical facts and relations. Their value as preparation for the use of formulae in shop arithmetic and for algebra is obvious. The meanings of decimal numbers are taught directly from an extension of the "thousands, hundreds, tens, ones" series to tenths, hundredths, and so on, as well as from the comparison with 6, 10, and T. The latter is used chiefly to emphasize the smallness of the magnitudes and the commensurability of the two sorts of expressions, and to clarify the general concept of a fraction by experience with fractions with very large denominators. Experience shows that place value, United States money, and railroad distance tables are more useful in explaining decimal numbers than the unfamiliars, Ts, and obs. |