CONTENTS. Page. The National Committee on Mathematical Requirements... Chapter I. A brief outline of the report... Chapter II. Aims of mathematical instruction-general principles.. Chapter III. Mathematics for years seven, eight, and nine.. Chapter IV. Mathematics for years ten, eleven, and twelve... Chapter V. College entrance requirements... Chapter VI. List of propositions in plane and solid geometry.. Chapter VII. The function concept in secondary school mathematics.. Chapter VIII. Terms and symbols in elementary mathematics. Synopsis of the remaining chapters of the complete report... 421320 THE NATIONAL COMMITTEE ON MATHEMATICAL REQUIREMENTS. (Under the auspices of The Mathematical Association of America.) OFFICERS. J. W. Young, chairman, Dartmouth College, Hanover, N. H. MEMBERS. 1 A. R. Crathorne, University of Illinois. Representing the Association of Teachers of Mathematics in New England.” Vevia Blair, Horace Mann School, New York City. Representing the Association of Teachers of Mathematics in the Middle States and Maryland. J. A. Foberg, director of mathematical instruction, State Department, Harrisburg, Pa.3 Representing the Central Association of Science and Mathematics Teachers. A. C. Olney, commissioner of secondary education, Sacramento, Calif. Raleigh Schorling, The Lincoln School, New York City. P. H. Underwood, Ball High School, Galveston, Tex. Eula A. Weeks, Cleveland High School, St. Louis, Mo. 1 Prof. Moore took the place vacated in 1918 by the resignation of Oswald Veblen, Princeton University. : Mr. Downey took the place vacated in 1919 by the resignation of G. W. Evans, Charlestown High School, Boston, Mass. * Until July, 1921, of the Crane Technical High School, Chicago, Ill. IV INTRODUCTION. The National Committee on Mathematical Requirements was organized in the late summer of 1916 under the auspices of the Mathematical Association of America for the purpose of giving national expression to the movement for reform in the teaching of mathematics, which had gained considerable headway in various parts of the country, but which lacked the power that coordination and united effort alone could give. The original nucleus of the committee, appointed by Prof. E. R. Ledrick, then president of the association, consisted of the following: A. R. Crathorne, University of Illinois; E. H. Moore, University of Chicago; D. E. Smith, Columbia University; H. W. Tyler, Massachusetts Institute of Technology; Oswald Veblen, Princeton University; and J. W. Young, Dartmouth College, chairman. This committee was instructed to add to its membership so as to secure adequate representation of secondary school interests, and then to undertake a comprehensive study of the whole problem concerned with the improvement of mathematical education and to cover the field of secondary and collegiate mathematics. This group held its first meeting in September, 1916, at Cambridge, Mass. At that meeting it was decided to ask each of the three large associations of secondary school teachers of mathematics (The Association of Teachers of Mathematics in New England, The Association of Teachers of Mathematics in the Middle States and Maryland, and the Central Association of Science and Mathematics Teachers) to appoint an official representative on the committee. At this time also a general plan for the work of the committee was outlined and agreed upon. In response to the request above referred to the following were appointed by the respective associations: Miss Vevia Blair, Horace Mann School, New York, N. Y., representing the Middle States and Maryland association; G. W. Evans, Charlestown High School, Boston, Mass., representing the New England association;' and J. A. Foberg, Crane Technical High School, Chicago, Ill., representing the central association. At later dates the following members were appointed: A. C. Olney, commissioner of secondary education, Sacramento, Calif.; Raleigh Schorling. The Lincoln School, New York City; P. H. Underwood, Ball High School, Galveston, Tex.; and Miss Eula A. Weeks, Cleveland High School, St. Louis, Mo. From the very beginning of its deliberations the committee felt that the work assigned to it could not be done effectively without adequate financial support. The wide geographical distribution of its membership made a full attendance at meetings of the committee difficult if not impossible without financial resources sufficient to defray the traveling expenses of members, the expenses of clerical assistance, etc. Above all, it was felt that, in order to give to the ultimate recommendations of the committee the authority and effectiveness which they should have, it was necessary to arouse the interest and secure the active cooperation of teachers, administrators, and organizations throughout the country--that the work of the committee should represent a cooperative effort on a truly national scale. For over two years, owing in large part to the World War, attempts to secure adequate financial support proved unsuccessful. Inevitably also the war interfered with the committee's work. Several members were engaged in war work 2 and the others were carrying extra burdens on account of such work carried on by their colleagues. 1 Mr. Evans resigned in the summer of 1919, owing to an extended trip abroad; his place was taken by W. F. Downey, English IIigh School, Boston, Mass. 2 Prof. Veblen resigned in 1917 on account of the pressure of his war duties. His place was taken on the committee by Prof. C. N. Moore, University of Cincinnati. In the spring of 1919, however, and again in 1920, the committee was fortunate in securing generous appropriations from the General Education Board of New York City for the prosecution of its work.2 This made it possible greatly to extend the committee's activities. The work was planned on a large scale for the purpose of organizing a truly nation-wide discussion of the problems facing the committee, and J. W. Young and J. A. Foberg were selected to devote their whole time to the work of the committee. Suitable office space was secured and adequate stenographic and clerical help was employed. The results of the committee's work and deliberations are presented in the following report. A word as to the methods employed may, however, be of interest at this point. The committee attempted to establish working contact with all organizations of teachers and others interested in its problems and to secure their active assistance. Nearly 100 such organizations have taken part in this work. A list of these organizations will be found in the complete report of the committee. Provisional reports on various phases of the problem were submitted to these cooperating organizations in advance of publication, and criticisms, comments, and suggestions for improvement were invited from individuals and special cooperating committees. The reports previously published for the committee by the United States Bureau of Education 3 and in The Mathematics Teacher 4 and designated as “preliminary” are the result of this kind of cooperation. The value of such assistance can hardly be overestimated and the committee desires to express to all individuals, organizations, and educational journals that have taken part its hearty appreciation and thanks. The committee believes it is safe to say, in view of the methods used in formulating them, that the recommendations of this final report have the approval of the great majority of progressive teachers throughout the country. No attempt has been made in this report to trace the origin and history of the various proposals and movements for reform nor to give credit either to individuals or organizations for initiating them. A convenient starting point for the history of the modern movement in this country may be found in E. H. Moore's presidential address before the American Mathematical Society in 1902.5 But the movement here is only one manifestation of a movement that is world-wide and in which very many individuals and organizations have played a prominent part. The student interested in this phase of the subject is referred to the extensive publications of the International Commission on the Teaching of Mathematics, to the Bibliography of the Teaching of Mathematics, 1900–1912, by D. E. Smith and C. Goldziher (U.S. Bur. of Educ., Bull., 1912, No. 29) and to the bibliography (since 1912) to be found in the complete report of the national committee (Ch. XVI). The national committee expects to maintain its office, with a certain amount of clerical help, during the year 1921–22 and perhaps for a longer period. It is hoped that in this way it may continue to serve as a clearing house for all activities looking to the improvement of the teaching of mathematics in this country, and to assist in bringing about the effective adoption in practice of the recommendations made in the following report, with such modifications of them as continued study and experimentation may show to be desirable. 2 Again in Nov., 1921, the General Education Board made appropriations to cover the expense of publishing and distributing the complete report of the committee and to enable the committee to carry on certain phases of its work during the year 1922. 8 The Reorganization of the First Courses in Secondary School Mathematics, U.S. Bureau of Education, Secondary School Circular, No. 5, February, 1920. 11 pp. Junior High School Mathematics, U.S. Bureau of Education, Secondary School Circular, No. 6, July, 1920. 10 pp. The Function Concept in Secondary School Mathematics, Secondary School Circular No. 8, June, 1921. 10 pp. 4 Terms and Symbols in Elementary Mathematics, The Mathematics Teacher, 14: 107–118, March, 1921. Elective Courses in Mathematics for Secondary Schools, The Mathematics Teacher, 14: 161-170, April, 1921 College Entrance Requirements in Mathematics, The Mathematics Teacher, 14: 224–245, May, 1921. 6E. H. Moore: On the Foundations of Mathematics, Bulletin of the American Mathematical Society, vol. 9 (1902–3), p. 402; Science, 17: 401. THE REORGANIZATION OF MATHEMATICS IN SEC ONDARY EDUCATION. Chapter I. A BRIEF OUTLINE OF THE REPORT. The present chapter gives a brief general outline of the contents of this pamphlet for the purpose of orienting the reader and making it possible for him to gain quickly an understanding of its scope and the problems which it considers. The valid aims and purposes of instruction in mathematics are considered in Chapter II. A formulation of such aims and a statement of general principles governing the committee's work is necessary as a basis for the later specific recommendations. Here will be found the reasons for including mathematics in the course of study for all secondary school pupils. To the end that all pupils in the period of secondary education shall gain early a broad view of the whole field of elementary mathematics, and, in particular, in order to insure contact with this important element in secondary education on the part of the very large number of pupils who, for one reason or another, drop out of school by the end of the ninth year, the national committee recommends emphatically that the course of study in mathematics during the seventh, eighth, and ninth years contain the fundamental notions of arithmetic, of algebra, of intuitive geometry, of numerical trigonometry and at least an introduction to demonstrative geometry, and that this body of material be required of all secondary school pupils. A detailed account of this material is given in Chapter III. Careful study of the later years of our elementary schools, and comparison with European schools, have shown the vital need of reorganization of mathematical instruction, especially in the seventh and eighth years. The very strong tendency now evident to consider elementary education as ceasing at the end of the sixth school year, and to consider the years from the seventh to the twelfth inclusive as comprising years of secondary education, gives impetus to the movement for reform of the teaching of mathematics at this stage. The necessity for devising courses of study for the new junior high school, comprising the years seven, eight, and nine, enables us to |