learning the tables of twos and threes. According to this plan the teacher says to the pupils: "If we are to make large scores, what table must we learn next? How many think they can learn half of the table of fours to-day? If you learn it, we will play our game ten minutes." In this case the "association" of the game with half the table of fours consists of holding up the game and its attendant pleasure as a bribe for memorizing three multiplication facts. The subject matter bears no intrinsic relation to the value to be controlled.

In contrast to this emphasis upon making arithmetic interesting, there is an increasing tendency to recognize that parts of arithmetic, perhaps most of arithmetic, are in themselves immediately interesting to children. This is particularly true of the work of the primary and intermediate grades. D. E. Smith says: "Such statistical information as we have shows that arithmetic has always been looked upon by children as one of the most interesting subjects of the course.”

Since the publication of The Psychology of Number there has been an increasing tendency to secure motive for work upon arithmetic by having the child feel the need for number and number relations before he is asked to study them. D. E. Smith says: "This ideal is not always easy to realize, but we are approaching it in our education of children, and the tendency is a healthy one." The teacher is to cause the pupil to feel a need by taking advantage of the quantitative situations the pupil meets in his life outside of school and by so setting the stage that he will meet others. The latter plan is illustrated when the pupil undertakes a project in the manual-training shop or in the domestic-science laboratory and discovers that he needs arithmetic, or when arithmetic is taught incidentally. Need is also felt when a pupil experiences difficulty in controlling a situation efficiently. In such a case he needs a better method of control or drill upon his present method. The attempt to make arithmetical problems "real" and "concrete" has been prompted, in part, by the desire to secure motive.

Objective methods. Prior to this period the use of objective materials had become rather indiscriminate, and often was looked upon as an end in itself. The significant feature of the objective teaching of this period has been a tendency toward a more refined correlation of the pupil's "experience with the social problem or subject involved." The objective materials have become more varied, and there is an increasing tendency to look upon them simply as a means to an end. These changes have resulted in a wider distribution of objective methods, but at the same time a clearer understanding of the function of objective materials has resulted in the total objective teaching being reduced.

1 Strayer: A Brief Course in the Teaching Process, p. 180. 2 The Teaching of Arithmetic, p. 87.

Correlation. When attention was focused upon the child the unitary nature of his life outside of school was revealed. In contrast, the course of study portioned out the child's time to the several school subjects, and each subject jealously guarded its apportioned period, resenting any encroachment. Each school subject was taught isolated from the other subjects. The topical arrangement of texts, as in the case of arithmetic, tended to isolate topics within a subject. And to a very considerable extent the work of successive days was isolated. Within the recent period plans for relieving this isolation have been proposed. Because of their bearing upon the teaching of arithmetic, some of them are worthy of our notice. The subcommittee of the Committee of Fifteen appointed by the National Education Association, 1893, reporting on the correlation of studies recognized five "staple branches of the elementary course of study." These were grammar, literature, arithmetic, geography, and history. They contended that "there should be rigid isolation of the elements of each branch."

In opposition to this, plans of concentration were proposed. A subject, or a closely related group of subjects, was taken as a center and all other school subjects were made subsidiary. A well-known attempt at concentration was made by Col. F. W. Parker at the normal school of Cook County, Ill. He concentrated the curriculum around the scientific subjects, elementary science, geography, myth, and history. Arithmetic was simply a means for controlling arithmetical situations within these subjects. By using it as a tool, arithmetic would be sufficiently learned. In fact Col. Parker believed that geography alone is sufficient.

If the child had no other study than that of geography, and the exercise of the numbering faculty met the necessities of the child's increasing knowledge, both of observation and imagination, the opportunities for the acquisition of the knowledge of arithmetic, as it is now understood, would be fully adequate.1

Charles A. McMurry, in The Elements of General Method, 1903, advocates correlation. This he defines as "such a connection between the parts of each study and such a spinning of relations and connecting links between different sciences that unity may spring out of the variety of knowledge." This is opposed to a plan of concentration such as proposed by Col. Parker. Each important study is to be isolated for purposes of instruction. But correlation also means that arithmetic is to be taught so that every important topic will be seen "in its natural relations to topics in other studies, thus binding the studies together in a multitude of close interrelations." In this way arithmetic, though taught as a separate subject, is to be correlated with geography, elementary science, history, etc. This is to be done by taking problems from

1 Talks on Pedagogics, p. 71.

these subjects for part of the work in the arithmetic class and by using the knowledge learned in the arithmetic class as a tool for the better understanding on these other subjects.

By some, correlation was given additional meaning. Connection was to be made between topics within a subject, and even between the lessons of successive days. This was to be accomplished by a proper ordering of the topics and by reviews. To review the previous lesson to secure the connection became with many a necessary mark of good teaching.

Drill. If one may draw conclusions from the texts, the emphasis upon drill as a factor of the teaching process has, in general, increased in this period. All of the more popular texts give much space to exercises for rapid drill. Some are in the form of special devices whose function is to assist the teacher in calling for combinations rapidly and in a variable order. A device which seems to be standard, but which appears in several variations, consists of a number surrounded by other numbers placed along some contour. By choosing appropriate numbers this device may be used for drill upon any of the fundamental operations. The device may be used directly from the text, or it may be transferred to the blackboard. In either case the teacher designates a number on the contour, and the pupils are to perform the required operation upon t. For example, if the process is division, the number in the center s 8, and the number designated on the contour is 72, the pupils are to give the quotient of 72 divided by 8.

Other plans for securing rapid drill upon the fundamental number facts are counting by twos, by threes, by fours, etc., adding numbers as the teacher writes them on the board or dictates them, and using drill cards which have exercises upon them. The pupils may be divided into groups for number contests, the group winning who does the work the most rapidly or the most accurately; or, instead of dividing the class into groups, all may work all the exercises, and scores be kept. At the end the total scores for the set of exercises are computed, the pupil making the highest score being the winner. It has been urged that some time each day be devoted to rapid drill. One authority states "about five minutes a day devoted to rapid oral work are sufficient to keep grammar-school pupils in practice." Besides this "rapid oral work," he contends that there should be a definite amount of rapid written work every day. The median per cent of time given to strictly drill work in arithmetic in 564 cities is as follows for the several grades: First grade 43 per cent, second grade 50 per cent, third grade 53 per cent, fourth grade 47 per cent, fifth grade 39 per cent, sixth grade 31 per cent, seventh grade 22 per cent, and eighth grade 17 per cent. But notwithstand

1 W. A. Jessup, "Economy of Time in Arithmetic," Elementary School Teacher, Vol. XIV, p. 474.

ing this increased emphasis upon drill, skill is regarded less as a primary aim than heretofore. The function of drill is being better understood.

Scientific investigation and experimentation.-The pioneer in this field was J. M. Rice,' 1902, who attempted to evaluate the excellence of instruction in arithmetic by measuring the results of that teaching and to determine what factors contribute to superior results. He gave a test to 6,000 pupils in the fourth to eighth grades, inclusive. On the basis of the data obtained he eliminates as controlling factors home environment, size of classes, time of day which a class recites, age of pupils, time devoted to arithmetic, amount of home work required, method of teaching, and general qualifications of teachers, and concludes that the quality of the supervision is the controlling factor in determining the achievement of pupils in arithmetic.

The procedure of Rice's investigation is open to criticism as might be expected of a pioneer study, but it stimulated and inspired other scientific investigations and experimentation. The major problems attacked have been: (1) What is the nature of the product of instruction in arithmetic? (2) What factors are most effective in producing arithmetical abilities? (3) How to measure these abilities and to set standards of attainment in these abilities. (4) The determination of superior methods of instruction and courses of study by scientific experimentation. (5) A scientific analysis and study of the learning process as it occurs in the case of arithmetic. The most extensive work on these problems has been by S. A. Courtis, who received his inspiration and stimulus from an investigation by C. W. Stone. In addition to identifying elementary arithmetical abilities, which we have mentioned on p. 131, Courtis has devised tests for measuring these abilities, and has set tentative standards of attainment in them. His standard practice tests and the manual which accompanies them represent the product of his study of methods of instruction and the learning process. The most significant feature is a plan for giving individual instruction to pupils when formed in classes.

At present there is much scientific investigation and experimentation which is resulting in an accumulating body of data which can be used as a basis for directing the development of arithmetic as a school subject in the decades to come. This is the most conspicuous tendency at present, and the indications are that future development will be made in this way.

1 "Educational Research, A Test in Arithmetic," The Forum, XXXIV: 281–297. "Causes of Success and Failure in Arithmetic," The Forum, XXXIV: pp. 437-452.

2 See bibliography for a list of his published material.

Arithmetical Abilities and Some Factors Determining Them.

4 Manual of Instructions for Giving the Courtis Standard Tests, Department of Cooperative Research, Detroit. Also the Courtis Standard Practice Tests, World Book Co.

Chapter XI.
SUMMARY.

The place of arithmetic in education.-During the ciphering book period arithmetic was a part of the school curriculum in those towns where it was demanded as a tool of commerce. In communities whose interests were not commercial and in the rural districts it was frequently not given a place in the plan of education and was conceded to possess little or no educational value. When arithmetic was taught under these conditions, it was simply as a concession to its practical utility. This early attitude was modified somewhat before the close of the ciphering book period. The commercial need for arithmetic had become more widespread and more universally recognized. When the Colonies became a free and independent Nation and a Federal currency was established, interest in arithmetic was greatly augmented. In 1729 the publication of the first arithmetic by an American author had passed unnoticed, but the appearance of Nicolas Pike's text in 1788 marked the beginning of interest in improving the subject matter of arithmetic which was manifested by the publication of many texts. By the beginning of the nineteenth century arithmetic had been given a place in the schools, though not one of first importance. There is some indication of the recognition of an educational value in addition to the practical value. But to Warren Colburn is due the credit for initiating in this country the movement which gave to arithmetic the place of first importance in the curriculum of the elementary school and which caused some to exalt it as a newly discovered "royal road" to learning.

Recently there has been a reaction from this extreme disciplinary conception of arithmetic and a return to arithmetic as a practical subject. But the meaning of practical is not that of the eighteenth century. Arithmetic now represents tools which the child needs to control his present and potential quantitative situations. These tools are to be organized in accord with the nature of the child and as the child works out methods for controlling these quantitative situations and organizes the arithmetical tools which he has acquired, arithmetic fulfills its disciplinary function in his education.

The content of arithmetic and its organization. Two complementary tendencies are revealed in the modifications of the content of arithmetic. Practical demands and the desire of the arithmetician for a