PART IV. Chapter X. RECENT TENDENCIES AND DEVELOPMENTS. The development of arithmetic since 1892 has been more intimately connected with the general educational development than it was during the period beginning with 1821. An attack upon the importance of the disciplinary function of arithmetic grew out of two more general movements. The Herbartian movement.-Beginning about 1890, American educators were greatly interested in the educational principles enunciated by Herbart, a German educator who lived from 1776-1841. Some of the important events in the rise of this movement were: The publication of educational books on Herbartian principles; Essentials of Method, by Charles De Garmo, 1889; General Method, by Charles McMurry, 1892; The Method of the Recitation, by Frank McMurry and Charles McMurry, 1897; and the formation of a national Herbartian society in 1892. The principle of apperception, which is one of the most important accredited to Herbart, was emphasized by his followers in America. Briefly the principle is this: New experiences are given meaning and interpreted by means of the ideas which one has obtained from his past experience and which are present in his consciousness at the time. This principle, coupled with Herbart's concept of the immediate end of education as the development of a "many-sided interest," means that education is to give the child (1) a "many-side" acquaintance with the external world, and (2) to give this acquaintance in such a way that it will be accompanied by an active "interest" in each "side" of this experience. The child will then be equipped to meet new situations as they arise. This theory, which places the emphasis upon the content of a subject, is fundamentally opposed to the disciplinary concept of education, and the wave of enthusiastic interest in the work of Herbart which swept over the United States did much to counteract the great emphasis upon the disciplinary function of instruction in arithmetic. The Herbartians emphasized history and literature as subjects in the elementary school, and by so doing were a factor in reducing the amount of time given to arithmetic. 1 For Herbart's own account, see Outlines of Educational Doctrines. The psychological movement.-In his "Principles of Psychology," 1890, William James stated that one's native ability to retain can not be changed,1 which means that a general capacity to remember can not be trained by specific exercises. This assertion, which was "supported by some plausible experimental evidence" was extended by other educators to a complete refutation of the theory of formal discipline as then interpreted." The reaction against the disciplinary value of arithmetic. Coupled with these two movements, partly as a result of them, both educators and the public became more actively critical of the work of the public schools. There were reports of investigations and more general utterances based upon general observations. The following are typical of this latter type: In almost all of the arithmetics, first come the definitions, then the rules, then a problem with full explanation, then the problems for the children to work according to rule and like the sample given. And this is called discipline! God save the mark!5 From one-sixth to one-fourth, or even one-third, of the whole school time of American children is given to the subject of arithmetic—a subject which does not train a single one of the four faculties that it should be the fundamental object of education to develop. It has nothing to do with observing correctly, or with recording accurately the results of observation, or with collecting facts and drawing just inferences therefrom, or with expressing clearly and forcibly logical thought." In 1892 the Committee of Ten, a committee appointed by the National Education Association, recognized the existence of a formal disciplinary value of arithmetic,' but insisted that it was "greatly inferior to what may be obtained by a different class of exercises." Essentially this is a refutation of the doctrine of formal discipline as it had been applied to arithmetic. Simon Newcomb, who was chairman of the subcommittee on mathematics, stated in another place that "the main end of mathematical teaching—we might say of teaching generally-is to store the mind with clear conceptions of things and their relations."8 Charles A. McMurry stated that the "chief aim of arithmetic is the mastery of the world on the quantitive side through number concepts. 1 Vol. I, p. 667. * See W. C. Bagley: Educational Values, p. 188 ff., for a historical account of the reaction against formal discipline. See the proceedings of the National Educational Association following 1880 for criticisms by the leading educators of the country. These have particular reference to arithmetic, 1880, p. 114; 1881, pp. 24, 85, 106; 1892, pp. 617, 620. See pp. 126-27 for the report of investigations of the teaching of arithmetic in Connecticut. Statement by Charles W. Eliot, 1892. Quoted by O. F. Bright, "Changes in Schools," Proc. of Nat. Educ. Assoc., 1895, p. 264. See also, Charles W. Eliot: Educational Reform, pp. 186-187, Address, 1890. 7 Report of Committee of Ten, p. 108. Simon Newcomb: "Methods of Teaching Arithmetic," Educ. Rev., vol. 31, pp. 339-340. This point of view was originally expressed in 1892. • Special Method in Arithmetic, 1895, p. 16. These last two statements are representative of the pure Herbartian point of view and of the extreme reaction against the importance of the disciplinary function of arithmetic.1 Arithmetic a psychical and social demand. The most important constructive contribution of this period was made by Prof. John Dewey, whose fundamental thesis was that the psychical and social environment in which we live presents problems which the human mind solves by measurement, i. e., by number and number relations.3 Thus number is not a property of objects, but rather it is "the product of the way in which the mind deals with objects in the operation of making a vague whole definite." The necessity for making these vague wholes definite grows out of the fact that (1) material things are "limited," (2) that energy must be economized, and (3) that remote ends must be attained. Dewey illustrates these reasons as follows: (1) If every human being could use at his pleasure all the land he wanted, it is probable that no one would ever measure land with mathematical exactness. There might be, of course-Crusoe like a crude estimate of the quantity required for a given purpose; but there would be no definite numerical valuation in acres, rods, yards, feet. There would be no need for such accuracy. If food could be had without trouble or care, and in sufficiency for everybody, we should never put our berries in quart measures, count off eggs and oranges by the dozen, and weigh out flour by the pound. (2) Because there is a limit to human energy, when we employ this energy for the attainment of a purpose, the most fruitful results are attained when there is the most accurate balancing of the energy over against the thing to be done. If the arrow of the savage is too heavy for his bow, or if it is too light to pierce the skin of the deer, there is in both cases a waste of energy. If the bow is so thick and clumsy that all his strength is required to bend it, or so slight or uneven that too little momentum is given to the arrow, there is but a barren show of action, and the savage has his labor for his pains. Bow and arrow must be accurately adjusted to each other in size, form, and weight; and both have to be equated (as the mathematician would say) or balanced to the end in view-the killing of the game. (3) In working out a certain purpose, for example, one of a series of means is a journey to be undertaken; it is of a certain length; it is to be completed in a given time and within a certain maximum of expense, etc.; and this involves careful calculation, measurement, and numerical ideas. To this principie Dewey added his more general educational principle that the process of education is most efficiently carried on when the child is placed in the physical and social environment which demand psychical activity. Applied to arithmetic, this means that to teach it efficiently the school must produce situations which call for measurement and the relating of quantities. According to this thesis the immediate purpose of the author of a text and of the teacher 1 For the other extreme see p. 91. James A. McLellan and John Dewey: The Psychology of Number, 1895. Op. cit., p. 32. Op. cit., p. 23. would be to provide these situations. When taught from this point of view, arithmetic affords "an unrivaled means of mental discipline," but Dewey does not use "mental discipline" in the sense in which it was used by the authors of arithmetics during the previous period. Both the disciplinary and the utilitarian functions of arithmetic recognized at present.-Through the publication of The Psychology of Number, and through the exemplification of his principles in the University Elementary School, Dewey combined with the reaction against the doctrine of formal discipline in influencing the concept of the aim of instruction in arithmetic. Soon after the publication of The Psychology of Number two series of arithmetics were published which the authors claimed embodied the principles enunciated by Dewey. Other authors builded upon them in part. These texts have been a factor in increasing the emphasis upon problems taken from practical situations, and hence upon the utilitarian value of arithmetic. 2 The reaction against formal discipline was followed by a counter action in which educators have recognized the disciplinary function of arithmetic, but in general they accorded the utilitarian value equal rank, and this appears to be the present status. A recent questionnaire was sent to 185 State normal schools and to 8 city training schools. Replies were received from 65 State normal schools and 3 city training schools. In training teachers of mathematics 51 per cent of these schools claim to pay equal attention to mathematics as a science (the so-called culture value) and to mathematics as an art (the so-called utilitarian value). About 28 per cent claim to emphasize more the cultural aspect (except in arithmetic), and 21 per cent put greater stress upon the utilities.3 In all of this agitation there seems to have been the underlying purpose to adapt arithmetic to the nature of the child and to the social demands which will be made upon him when he leaves school. Definition of the aim of instruction. Recently scientific investigation has revealed that the product of instruction in arithmetic is not a single ability, but consists of many abilities. The ability "to add columns three figures long is not the same ability as to add columns five figures long." Each type of example calls for a different ability, and thus the product of instruction in arithmetic includes as many different abilities as there are different types of examples. Courtis has identified 15 different addition abilities,5 8 for subtraction, 11 for multiplication, and 14 for division. 1 One series was by McLellan and Ames, 1898; the other by Belfield and Brooks, 1898. 2 See J. W. A. Young: The Teaching of Mathematics, 1906, p. 204; and D. E. Smith: The Teaching of Arithmetic, 1909, p. 11ff. 3 Training of Teachers of Elementary and Secondary Mathematics, U. S. Bu. of Ed. Bul., 1911, No. 12, p. 15. C. W. Stone: Arithmetical Abilities and Some Factors Determining Them, 1908, pp. 42-43. S. A. Courtis: Teachers Manual, 1914, p. 2. Op. cit., p. 2. 1 This analysis of the product of instruction in arithmetic has made possible more exact and objective definitions of the aim of instruction. At present this has been done by Courtis for the fundamental operations with integers. For any particular grade the teacher and pupils have for their aim, in so far as it involves the fundamental operations with integers, to attain the ability to solve examples of certain types with a specified speed and accuracy. These standards are based upon extensive experimental data gathered from both schools and the commercial world. These detailed and objective statements of aims of the instruction in arithmetic are not opposed to, but will supplement, the more general statement of aim. Less time given to arithmetic in the schools.-During the preceding period a large per cent of the total school time was given to arithmetic. Estimates ranged as high as 50 per cent or more. The Committee of Ten, 1892, stated that a "radical change in the teaching of arithmetic was necessary," and recommended that the course be both "abridged and enriched." In 1895 the Committee of Fifteen reported as follows: Your committee believes that, with the right methods and a wise use of time in preparing the arithmetic lesson in and out of school, five years are sufficient for the study of mere arithmetic-the five years beginning with the second school year and ending with the close of the sixth year; and that the seventh and eighth years should be given to the algebraic method of dealing with those problems that involve difficulties in the transformation of quantitative indirect functions into numerical or direct quantitative data. Your committee is of the opinion that the so-called mental arithmetic should be made to alternate with written arithmetic for two years, and that there should not be two lessons daily in this subject (arithmetic).3 In another place the committee reports "that the practice of teaching two lessons daily in arithmetic, one styled 'mental' or 'intellectual' and the other 'written' arithmetic, is still continued in many schools." Although there was a marked tendency even before 1892 to combine "mental" and "written" arithmetic in the texts, this practice persisted in some places until very recently. Separate classes in mental arithmetic were discontinued in Kansas City, Mo., in 1913. The following data show the change in the relative amount of time given to arithmetic in several American cities. 1 Op. cit., p. 3. 2 See page 126. Proc. Nat. Ed. Assoc., 1895, pp. 300-301. |