Bulletin

logically rounded-out science have caused subject matter to be added, and tradition has tended to keep subject matter which has once been added. New subject matter has been much more rapidly incorporated than obselete subject matter has been discarded. The most conspicuous change of emphasis has been in reference to the rule of three. Formerly it was the great topic, the "golden rule” of arithmetic. Now it has been reduced to the inconspicuous topic of proportion. Evolution has been reduced from an array of specific rules for roots up to the "squared square-cube root" until now only square root is frequently given. Some topics, such as permutations and combinations, position, and infinite series, have been transferred to more advanced courses in mathematics. Other topics, such as fellowship, certain tables of denominate numbers, much of exchange, tare and trett, alligation, duodecimals, annuities, etc., have been dropped as topics because the need which they satisfied no longer exists. On the other hand decimal fractions now occupy a much larger place. This has been due to the introduction of a decimal currency. In this way the relative importance of common fractions has been lessened, but they now occupy more space than formerly. More significant than the increased space given to fractions is the fact that they have been moved forward in the course.

The first great change in the subject matter of arithmetic came with the work of Warren Colburn. He introduced primary arithmetic and intellectual or mental arithmetic, gave a place of prominence to common fractions, and omitted the rule of three and other topics as such. Many of the omissions for which he took a stand have since been made, and others are at present being urged.

Arithmetic being a practical subject, the problems, for the most part, have been practical when they were introduced. As conditions changed, some problems were no longer practical. Tradition tended to keep these in the texts, the result being that our texts have contained a number of problems from obsolete or obsolescent situations. A few arithmetical puzzles have always found a place in our texts. When the disciplinary function of arithmetic was emphasized, the number of such puzzles was much increased, particularly in the mental arithmetics. Recently the force of tradition has been very much weakened, and there has been a tendency to reduce the number of arithmetical puzzles and to insist that practical problems be practical. These practical problems are to be drawn from a wide range of human activities and from the child's own life.

In the larger features of organization we have had many variations and combinations of the original topical plan and the more recent spiral plan. From a strictly topical organization we have come to a moderate spiral for grades one to four, followed by a transition to a topical organization in grades seven and eight. In the details of

organization the logical, deductive order of the past has been replaced by an attempted psychological order. Here again credit is due Warren Colburn for making the break with the past by organizing his texts upon the inductive plan. Following Colburn there was a partial relapse to the old logical deductive order, but recently there has been a movement toward the form and spirit of Colburn's organization.

Methods of teaching arithmetic.-Before 1821 the teacher's function was to set "sums," tell rules, and pass upon the correctness of the pupil's work. This instruction was given to the pupils individually. After 1821 pupils were usually instructed in classes, and in practice the technique of dealing with pupils in classes became almost synonymous with methods of teaching. However, the concept of the function of the teacher was enlarged to include explaining the process and problems. Colburn and some others believed that the teacher should guide the pupil in developing his own rules. Some emphasis was placed upon drill, and much emphasis upon exact forms of analyses. Colburn's ideas concerning the teaching of arithmetic were as progressive as were his texts, but he failed to exert much direct influence upon the mode of teaching. Recently, starting with an analysis of the nature of the child, a clearer conception of the subject matter involved and the goal to be attained, more rational methods of teaching arithmetic are being worked out. In these rational methods direct instruction and drill have a place. Motive by appeal to artificial incentives has been supplemented by motive secured by interest and by need. The spirit of present-day methods is to assist the child by making the instruction coincide with the natural working and development of the child's mind. Although Warren Colburn has influenced the present methods of teaching arithmetic scarcely at all, yet we are distinctly returning to the spirit and form of his methods. We are now, like him, studying the child for the basis of our methods.

The men who have made our arithmetic.-Warren Colburn without a doubt occupies first place, because of his influence in stimulating and directing the development. Much of our present arithmetic we owe to him directly or indirectly. He himself was much greater than his influence has been, and his writings are still sources of information as well as inspiration. To Joseph Ray we should give second place. He was not a great constructive writer and thinker as was Colburn, but his greatness consists rather of his ability to write clearly, to organize, and to adapt. Because he could do these things well, his texts have been given a wide and long-continued use in our schools. Following these two men, there are many others who have materially contributed to the molding of our present arithmetic and the methods of teaching it.

Some inferences.-The story of human activity, human progress, is always interesting, and it may be of value to the present generation in their attempts for improvement. The story of the development of arithmetic which we have traced repeatedly suggests that permanent improvement of content, organization, or methods of teaching must be based upon a clear conception of the child. In this was Colburn's greatness, and here also is the foundation of our recent progress.

In their enthusiasm to improve arithmetic and its teaching, the teachers have not maintained a critical attitude toward proposed reforms. Using methods which have since been shown to be fundamentally wrong, they have secured results which they interpreted as an improvement over previous results. The judgment of results has often been based upon biased opinions and has seldom been the result of a clear comprehension of the aim of arithmetic teaching and a comprehensive survey of the effect of the teaching upon the pupils. For this reason the judgments have at times been defective. But the fact remains that the belief of a teacher in a method has been a large factor in the determination of the measure of its success.