1. 1 This resource was developed by CSMC faculty and doctoral students with support from the National Science Foundation under Grant No. ESI-0333879. The opinions and information provided do not necessarily reflect the views of the National Science Foundation. 12-6-04

2. 2 Committees and Reports that Have Influenced the Changing Mathematics Curriculum This set of PowerPoint slides is one of a series of resources produced by the Center for the Study of Mathematics Curriculum. These materials are provided to facilitate greater understanding of mathematics curriculum change and permission is granted for their educational use. Report of the National Committee of Fifteen on the Geometry Syllabus Report 1912 http://www.mathcurriculumcenter.org

3. 3 Final Report of the National Committee of Fifteen on the Geometry Syllabus National Education Association American Federation of Teachers of the Mathematical and Natural Sciences Mathematics Teacher, December 1912

4. 4 Committee of Fifteen on the Geometry Syllabus Appointed 1909 Final Report 1912

5. 5 Forces at Work Increasing enrollments placed pressure on high schools to provide education to a broader student population Geometry courses varied widely between two extremes –Formal proof approach –Utilitarian approach There were rising failure rates in algebra and geometry Psychology of “Mental Discipline” was losing favor and both mathematicians and teachers were calling for reform A common syllabus and direction for geometry was needed

6. 6 Committee of Fifteen 7 members from higher education 8 members from secondary schools Charles Bouton, Harvard University Florian Cajori, Colorado College Herbert Hawkes, Columbia University Earle Hedrick, University of Missouri Henry Rietz, University of Illinois David Eugene Smith, Teachers College, Columbia University Herbert Slaught, University of Chicago, Chair William Betz, East High School, Rochester, NY Edward Brown, North High School, Denver, CO William Fuller, Mechanic Arts High School, Boston, MA Walter Hart, Shortridge High School, Indianapolis, IN Fredrick Newton, Andover Academy, Andover, MA Eugene Smith, Park School, Baltimore, MD Robert Short, Technical High School, Cleveland, OH Mabel Sykes, Bowen High School, Chicago, IL

7. 7 Final report had 5 general sections Historical overview of school geometry Logical considerations Special courses Exercises and problems Syllabus of geometry

8. 8 Historical Overview of School Geometry Reviewed the nature of school geometry in several European countries. Provided a brief review of popular geometry textbooks being used in the United States. Identified books and resources that illustrated attempts to reform the teaching of geometry.

9. 9 Logical Considerations Recommended axioms, postulates, and definitions that should be included in the syllabus for a geometry course. Addressed the strengths and dangers associated with informal proofs, and argued that nearly 100 of the propositions (theorems) “must receive formal proof in any well-regulated course in geometry.” Recommended algebra as a ninth-grade course, geometry in the tenth grade, and algebra and geometry in the eleventh. Recommended that algebraic approaches and notations be utilized in geometry. Recommended incorporation of some solid geometry and trigonometry in geometry.

10. 10 Special Courses Rather than special courses, a single syllabus for high school geometry was recommended. The syllabus could be altered by omission of some theorems and by increased emphasis on theorems supporting practical applications. Guidelines were provided for the study of geometry across the grades, including informal geometry in the elementary grades. Definitions should be “stated formally only after the concept is clearly forms in the student’s mind.” Theorems need not all be proved; for some, heuristic arguments to convince students of their truth would be sufficient. Making scale drawings and models that apply important theorems on measurement was recommended.

11. 11 Exercises and Problems Expressed concern about the lengthy sets of exercises and their lack of application as found in prevailing textbooks. Recommended distributing exercises by difficulty so that each theorem was accompanied by immediate concrete exercises and applications and more difficult exercises were delayed to a later part of the course. Urged a balance of formal reasoning exercises and application problems to make geometry more appealing to the average student.

12. 12 Geometry Syllabus Proposed a list of theorems, often stated algebraically, that would be a basic set for a geometry course. For plane geometry, 106 theorems were listed; and for solid geometry, 75 theorems were listed. Used different fonts to convey to teachers and examiners the relative importance of listed theorems. Identified some theorems that could be developed with informal arguments.

13. 13 Significance of the Report of the Committee of 15 Called for a balance of informal and formal work in geometry and more concrete examples. Emphasized the need for good pedagogy to support the learning of geometry. Recommended a full-year of plane geometry in 10 th grade to be followed by more geometry and algebra in 11 th grade. Geometry textbooks published after 1910 acknowledged the Committee of Fifteen and noted their alignment with the Committee’s report. Theorems listed by the Committee of Fifteen were included in forthcoming high school geometry textbooks. College Entrance Examination Board limited testing of propositions to the Committee’s list of basic theorems.