(2) Problems should appeal to the interests and understanding of the children in their re spective school years. Arithmetic was formerly taught only to boys who could read and write and who were preparing for business. When the subject found its way into the earlier school years it carried many difficult problems of busi ness down to immature minds. The modern tendency is to replace such problems by others that relate to children's interests. Thus in the primary grades there should be the study of home purchases, of the application of number to the large interests of the country, especially such as appeal to a child's love of nature and of the heroic, and such as relate to the sources of food and clothing. Later, the problems should refer to the more detailed features of the national and world life, to the great industries, trades and transportation facilities. Finally they should relate to the details of the indus trial and commercial life, thus preparing both the boy and the girl for earning a livelihood. In all this there should be an effort to make arithmetic interesting, since when the interest of the pupil is secured the work is prosecuted with more zeal and is attended with better and more permanent results.
(3) In the effort to modernize the problems care must be taken to avoid the extreme of withdrawing from arithmetic all topics involv ing effort, thus making the subject insipid from its very lack of fibre.
IX. Sequence of Topics.— Formerly arith metic was taught from a single book, each im portant topic being met but once. Then came the two-book series, the second book covering the ground of the first, but with more difficult ex amples, thus forming a spiral of two revolu tions. In this way there arose the so-called Spiral Method of treatment, which certain devotees have carried to the extreme of return ing to each topic every 'few days. Between the topical method and the radical spiral method there has been much strife. Advocates of the latter said that the former encouraged forget fulness through lack of review, while advocates of the former said that the latter gave the pupil no feeling of mastery of any subject. The result has been a compromise, seen in all modern American courses. Such important topics as percentage are treated several times, with progressive difficulty, applications like simple interest offering new features on each succeeding occasion. On the other hand, such relatively unimportant chapters as that on longitude and time (semi-geographical) are met but once. In the same spirit, the fundamental operations with integers, decimal fractions, and those common fractions often met in business, are frequently reviewed, while compound num bers and fractions involving unusual numer ators and denominators are less emphasized. The technicalities of business, including the study of investments, insurance, banking and exchange, are reserved until the last years of the grammar school, when a child beginning to look forward to being self-supporting is pre pared to understand them.
X. Methods.—Various methods have been suggested for presenting arithmetic to children, especially in the primary grades. The serious consideration of this phase of the subject be gan toward the close of the 18th century, particularly in Germany and Switzerland. With it are connected such names as Trapp, von Busse, Kranckes, Pestalozzi, Tillich, Grube, Tandc, Knilling and Kaselitz. Each of these
writers stood for some principle which he car ried to such an extreme as to render the method generally unusable. Pestalozzi, for example, did great good in his judicious use of objective illustration, but he went to an unwarranted ex treme in his emphasis of the unit and in his devotion to abstract work. Tillich suggested a valuable set of number blocks, but his follow ers went to the extreme of eliminating all other material. Grube wrote a condensed manual for teachers, and systematically treated num bers in concentric circles of progressive dif ficulty, but he went to several extremes that made the system so absurd that it is now nearly forgotten. On the other hand, every promi nent writer of this class has usually suggested some slight improvement which has gradually worked its way into the schools. It has been the universal experience that no advocate of a single -method has been able to impress this method on any considerable number of fol lowers. The best teacher has been the one who, being interested in the subject, has im parted that interest to the pupils, who has not been limited to any one set of objects or to any peculiar device, who has made arithmetic modern in its applications, and who has fol lowed the best curricula of the day.
XI. Time Required for the Subject in the Schools.— There has been a gradual. diminution in the time allowed to arithmetic in American schools for a number of years past, on account of the demands of more modern studies for a place in the curriculum. As a result there has been decreased attention to the subject, there is less ability on the part of pupils to grapple with problems, and the question has arisen as to the amount of time necessary to secure a reason able facility in the arithmetical processes. Al though the textbooks and the teaching have both improved, the curtailment of time and the scattering of the pupils' attention over more subjects have left the results far from satisfac tory. It has even been urged that arithmetic be not taught before the third or after the seventh school year, thus allowing five instead of eight years to the subject. But although it is true that the necessary parts of arithmetic can be covered in five school years, it is equally true that the child has as much delight in his work with numbers in his first school year as he has in the other subjects studied, and quite as much need for this work. It is also true that the number facts are more easily impressed on the memory if the work is begun, as Pes-, talozzi advised, when a child first enters school. It is therefore better to allow arithmetic to ex tend throughout the elementary grades, com bining with it, if the class is well advanced, some constructive geometry and the first steps in algebra in the eighth school year.
Bibliography.-- Smith, 'The Teaching of Elementary Mathematics' (New York 1900) ; 'The Outlook for Arithmetic in America' (Boston 1904) ; Brooks, 'The Philosophy of Arithmetic' (Philadelphia, 2d ed., 1901) ; Unger, 'Die Methodik der praktischen Arith metik' (Leipzig 1880) ; McMurray, 'Special Method in Arithmetic' (New York 1911); Sussallo, The Teaching of Primary Arithmetic' (Boston 1911).