21. Monroe, Walter S. Measuring the Results of Teaching. New York: Houghton Mifflin Company (1918). 22. Myers, G. W. "A Plan for Testing Methods of Secondary Mathematics." The School Review, xxii: 91-97 (February, 1914). 23. Pintner, Rudolph and Marshall, Helen. "A Combined Mental-Educational Survey." Journal of Educational Psychology, xii: 32-43 (January, 1921). 24. Pintner, Rudolph and Marshall, Helen. "Results of the Combined Mental-Educational Survey Tests." Journal of Educational Psychology, xii: 82-91 (February, 1921). 25. Rugg, H. O. Statistical Methods Applied in Education. New York: Houghton Mifflin Company (1917). 26. Schorling, Raleigh and Clark, John R. "A Program of Investigation and Coöperative Experimentation in the Mathematics of the Seventh, Eighth, and Ninth School Years." The Mathematics Teacher, xiv: 264-275 (May, 1921). 27. Starch, Daniel. Educational Measurements. (1916). New York: Macmilla 28. Symonds, Percival M. "Mathematics as Found in Society: With Curriculum Proposals." The Mathematics Teacher, xiv: 444-450 (December, 1921). 29. Thorndike, E. L. An Introduction to the Theory of Mental and Social Measurements. New York: Bureau of Publications, Teachers College (1919). 30. Thorndike, E. L. "Measurement in Education." Teachers College Record, xxii: 371-379 (November, 1921). 31. Thorndike, E. L. "Changes in the Quality of the Pupils Entering High School." The School Review, xxx: 355-359 (May, 1922). 32. Thorndike, E. L., and Symonds, P. M. "The Occupations of HighSchool Graduates and Non-Graduates." The School Review, xxx: 443-451 (June, 1922). 33. Thorndike, E. L. "Instruments for Measuring the Disciplinary Values of Studies." Journal of Educational Research, v: 269-279 (April, 1922). 34. Touton, Frank C. "The Frequency of Certain Problem Solving in the High-School Curriculum and a Suggested General Method of Solution." School Science and Mathematics, xxii: 330-343 (April, 1922). 35. Wilson, G. and Hoke, K. J. How To Measure. New York: Macmillan (1920). 36. The Seventeenth Yearbook of the National Society for the Study of Education, Part II. Bloomington, Illinois: Public School Publishing Company (1918). I. The Training of Teachers of Mathematics for Secondary Schools in 430 II. The Training of Teachers of Mathematics for Secondary Schools in the United States 447 A. North Atlantic Division-Pennsylvania; Cambridge, Mass.; B. North Central Division-Illinois; Cleveland, O.; Detroit, C. South Atlantic Division-District of Columbia (Washington); 448 460 480 D. South Central Division-Kentucky; Birmingham, Ala.; Dallas, 490 E. Western Division-California; Portland, Ore. 494 III. Accrediting Agencies 502 IV. A Tentative Standard for the Training of Teachers of Mathematics, and Courses Primarily Intended for such Teachers 506 Foreword The general plan of this selection of material from a very much longer incomplete report appears on inspection of the table of contents; the method of selection is indicated in the introductory paragraph of Part II, page 447. In the longer report is a detailed statement of: (a) The actual standards for certification of teachers of mathematics in secondary schools for every state in the Union, and for each of its cities of not less than one hundred thousand inhabitants; (b) the courses offered in every college in the country for giving the prospective teacher of mathematics in the secondary school his professional and scientific training. Only a mere fraction of this material is here selected. In order adequately to picture the quality of recently appointed teachers in the secondary schools of a state, it would be necessary not only to indicate the various methods by which such teachers may have been appointed, but also the percentage of teachers added by each of these methods. Too often do we find reports referring to the manner of appointment involving the highest standards only-as if such were characteristic of the state, whereas not five percent of the teachers may have been so appointed. These percentages are naturally very difficult to obtain, and since they could not be procured for all states to which we refer, they are not included in this report. A complete record of the methods of certification in the ground covered has, however, been attempted. Discussion of the very extensive literature of the subject is entirely omitted; the original section on Accrediting Agencies (Part III, page 502) is greatly abbreviated; and argument and discussion on the Summaries and on the tentative standard formulated at the end are deleted. The population figures are those for 1920 and are taken from advance sheets supplied by the Bureau of Census. I. The Training of Teachers of Mathematics for Secondary Schools in Foreign Countries In 1912 about 150 reports' regarding the teaching of mathematics in seventeen foreign countries were presented to the International 1 The International Commission on the Teaching of Mathematics stated in July, 1921 (Enseignement Mathematique), that it had published 187 volumes or fascicules which it listed, containing 310 reports of 13565 pages. Congress of Mathematicians at Cambridge, England. With these as a basis the writer prepared an extensive report, published by the Bureau of Education, Washington, D. C., in 1918, on The Training of Teachers of Mathematics for the Secondary Schools of the Countries represented in the International Commission on the Teaching of Mathematics. For each country considered, rather detailed information and full bibliographies were given with reference to the training of teachers to serve in better secondary schools for boys. A summary of the material thus set forth is here adapted for an introductory chapter which may be suggestive to those who thoughtfully consider in what way desirable reforms in connection with our secondary schools may be brought about. That the contrast in standards and ideals may be more marked, this summary is placed in immediate contiguity to some indications of corresponding conditions in the United States. In instituting such a comparison it should be borne in mind that there is no concensus of opinion throughout the world as to the periods to be devoted to primary, or to secondary, education. Most foreign countries agree that twelve or thirteen years normally lead to a university. But while the United States and Australia hold that only four years should be devoted to secondary education, seven countries (Austria, Finland, Hungary, Italy, Japan, Roumania, and Russia) set aside eight years, and three countries (France, with her two extra years for the mathematical or military or naval or engineering specialist, Germany, and Sweden) nine years. In contrast to eight years of elementary education2 in the United States, France and Germany consider that the best results are obtained when even the three or four years allotted to primary instruction are given in connection with secondary schools. It is then not surprising to find that there is great difference between the scholastic equipment of students coming from these two general types of school. The graduates of the classe de mathématiques spéciales in France, or of the German Gymnasium are about on a par with the youth who has finished his junior year in one of the better American colleges. And in other countries also, like Denmark, Japan, and Sweden, the graduate of a higher secondary school has covered considerable of the work of which the equivalent is offered in colleges in the United States. It 2 The recent movement towards the establishment of junior high schools tends to reduce the elementary period to six years and to extend the secondary period to six years. is only in the light of such considerations that the full force of, say, Sweden's requirement of ten or eleven years of preparation before a graduate of a Gymnasium may return as a regularly appointed professor can be adequately appreciated. But we are anticipating; let us consider our inquiry in connection with each country in turn. Australia. (Primary education, about 7 years; secondary education, 4 years.)—The standards for the training of secondary school teachers in Victoria and New South Wales are those to which all states of the Commonwealth are approaching. In the Teachers' College and University at Melbourne, Victoria, the usual course of training lasts four years three years to obtain the B. A. or B. Sc. degree and one year for special professional training. The professional training includes: (a) Attendance at lectures in the university on the theory of education with special reference to the method of teaching the various subjects; and (b) 120 hours of teaching under supervision in primary and secondary schools. In New South Wales similar requirements of at least four years of training after leaving the high school are maintained. Professional work is taken up after graduation from the university. This consists in part: (a) Of study in philosophy, education, principles of teaching, and school hygiene; (b) of continuous practice teaching (8-10 hours a week). In neither Victoria nor New South Wales is it necessary that the teacher shall have had special courses in mathematics in the university. On the other hand the better schools prefer graduates with "honors" in mathematics. Austria. (Primary education, 4 years; secondary education, 8 years.) The training required for candidates as teachers in the leading Austrian secondary schools, namely the Gymnasium and the Realschule, is practically the same. There must be not less than three and one-half years of scientific and other preparation (not always good) at a university. After five semesters the preliminary examination in philosophy and pedagogy may be taken. When this is passed, application may be made, not earlier than the end of the seventh semester, for the Lehramtsprüfung (teachers' examination), with its Hausarbeiten (theses) and oral examinations. After this, in theory only one year of |