Committees and Reports that Have Influenced the Changing Mathematics Curriculum

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. 2-15-05

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. The Reorganization of Mathematics in Secondary Education National Committee on Mathematical Requirements Final Report • 1923 http://www.mathcurriculumcenter.org

3. The National Committee on Mathematical Requirements (NCMR) The Reorganization of Mathematics in Secondary Education Appointed by MAA1916 Preliminary Report 1920Summary Report 1922Final Report 1923

4. Prepared in response to . . . the conflict of opinions on the problems of mathematics in secondary education with a focus on the following questions: • What should be taught? • How much of it? • To whom? • How? • Why?

5. NCMR Members Mathematicians: • J. W. Young, chairman, Dartmouth College, Hanover, NH • A. R. Crathorne, University of Illinois • C. N. Moore1, University of Cincinnati • E. H. Moore, University of Chicago • David Eugene Smith,Teachers College, Columbia University • H. W. Tyler, Massachusetts Institute of Technology • Oswald Veblen1, Princeton University Representatives of secondary mathematics teachers’ associations: • J. A. Foberg, vice chairman, State Department of Public Instruction, Harrisburg, PA, • Vevia Blair, Horace Mann School, Association of Teachers of Mathematics in the Middle States and Maryland • W. F. Downey2, English High School, Boston • G. W. Evans2, Charlestown High School, Boston Added later: • A. C. Olney, Commissioner of Secondary Education for California • Raleigh Schorling, The Lincoln School, New York City • P. H. Underwood, Ball High School, Galveston • Eula A. Weeks, Cleveland High School, St. Louis E. L. Thorndike, Columbia University, advised the committee on matters related to psychology 1 C. N. Moore took the place vacated in 1918 by the resignation of Oswald Veblen. 2 W. F. Downey took the place vacated in 1919 by the resignation of G. W. Evans.

6. Organization of Report Two Major Parts: • General Principles and Recommendations • Investigations Conducted for the Committee

7. Principles and Recommendations Chapter 1 – A brief outline of the report Chapter 2 – Aims of mathematical instruction— general principles Chapter 3 – Mathematics for grades 7, 8, 9 Chapter 4 – Mathematics for grades 10, 11, 12 Chapter 5 – College entrance requirements Chapter 6 – Listing of propositions in plane and solid geometry Chapter 7 – The function concept in secondary mathematics Chapter 8 – Terms and symbols in elementary mathematics

8. Investigations Conducted for the Committee Chapter 9 – The present status of disciplinary values in education Chapter 10 – The theory of correlation applied to school grades Chapter 11 – Mathematical curricula in foreign countries Chapter 12 – Experimental courses in mathematics Chapter 13 – Standardized tests in mathematics for secondary schools Chapter 14 – The training of teachers of mathematics Chapter 15 – Certain questionnaire investigations Chapter 16 – Bibliography on the teaching of mathematics

9. Aims of Mathematical Instruction • Practical Aims • Disciplinary Aims • Cultural Aims “The primary purposes of the teaching of mathematics should be to develop those powers of understanding and of analyzing relations of quantities and of space which are necessary to an insight into and control over our environment and to an appreciation of the progress of civilization in its various aspects, and to develop those habits of thought and of action which will make these powers effective in the life of the individual.” (NCMR, 1923)

10. Practical Aims • Understand and apply the fundamental processes of arithmetic • Understand and use the language of algebra • Understand and use elementary algebraic methods to solve problems • Understand and interpret graphical representations • Be familiar with common geometric forms and their properties and relations; develop and utilize space perception and spatial imagination

11. Disciplinary Aims 1. Acquisition of mathematical ideas or concepts that promote quantitative thinking 2. Development of ability to think clearly in terms of such ideas and concepts 3. Acquisition of mental habits and attitudes which enable use of these ideas and concepts (1 and 2 above) in the life of the individual 4. Development of “functional thinking”—thinking in terms of and about relationships between variables

12. Cultural Aims • Appreciation of beauty in the geometrical forms found in nature, art, and industry • Appreciation of the importance of logical structure, precision of statement and of thought, logical reasoning, discrimination between the true and the false • Appreciation of the power of mathematics and the role that mathematics and abstract thinking have played in the development of civilization

13. Mathematics for Years 7, 8, 9 All junior high students in Grades 7, 8, and 9 should have the opportunity to study and attain mathematical knowledge and training likely needed by all citizens. • Mathematics content should be presented in a correlated/unified fashion. • Mathematics should focus on concrete and verbal problems instead of formal exercises. • Mathematics should be practical for everyday life.

14. Mathematics for Years 7, 8, 9 Recommended content: • Arithmetic • Intuitive geometry • Algebra • Trigonometry • Demonstrative geometry • (optional) • History • Biography Five models for junior high school course sequencing were proposed, each reflecting some variations of a basic model.

15. Basic Curriculum Model For Years 7, 8, 9 First year: Applications of arithmetic, particularly as they relate to home, thrift, and to the various school subjects such as intuitive geometry. Second year: Algebra and applied arithmetic, particularly as they relate to commercial, industrial, and social needs. Third year: Algebra, trigonometry, demonstrative geometry. In this model, arithmetic is practically completed in the second year and demonstrative geometry is introduced in the third year.

16. Mathematics for Years 10, 11, 12 All high schools should offer mathematics courses for years 10, 11, 12 and encourage a large proportion of students to take them. • Courses should prepare students for possible vocations and life in the real world. • Content should include ideas and processes important to contemporary applications. • Material should be logically organized to facilitate the development of effective habits of mind.

17. Mathematics for Years 10, 11, 12 Recommended course offerings, in various configurations, included: • Elementary calculus • History • Biography • Additional electives • Algebra • Plane geometry • Solid geometry • Trigonometry • Elementary statistics Four plans for high school course sequencing with slight variation to the above were proposed by the Committee.

18. College Entrance Requirements Entrance requirements in mathematics should reflect the special mathematical knowledge and training required for the successful study of courses in the physical sciences and in the social sciences which the student will take in college. Entrance exams should: • Assess candidate’s ability to benefit from college instruction. • Focus on elementary algebra and plane geometry. College admissions should be based on more than just test scores.

19. Propositions in Geometry Identified a minimum set of propositions to be included in any standard geometry course (reduced list from the Committee of Fifteen) Selection based on: • Usefulness in other proofs and exercises • Value in completing important pieces of theory

20. The Function Concept Proposed use of function as a unifying concept in the secondary curriculum • At the junior high school level, function was seen embedded in and relevant to work with formulas, graphing, and interpretation of data. • At the high school level, function provided a way of unifying the study of dependency relationships in algebra, geometry, trigonometry, and everyday life.

21. Terms and Symbols in Elementary Mathematics • Recommended words and symbols to be used, and not to be used (e.g., trapezium) • Proposed standardization of mathematical exposition in texts and mathematical journals (e.g. try to avoid vulgar mathematical slang such as “tan,” “cos,” and “math”) • Proposed simplification of terms in elementary instruction

22. Significance of 1923 NCMR Major areas of impact: • The purpose of mathematics in secondary education was defined and defended. • The theory of mental discipline was rejected in favor of ideas of transfer. • The function concept was suggested as a unifier of algebra and geometry. • College entrance requirements were amended to include general tests to predict collegiate success in addition to examining achievement for specific mathematics courses. • Model curricula were offered based on the work of the committee as well as descriptions of experimental work both nationally and internationally.

23. Significance of 1923 NCMR Other impacts: • The junior and senior high school curriculums (6-3-3) were solidified. • An integrated course called “general mathematics” FOR ALL, less dominated by arithmetic, was created for junior high. • New texts and methods were designed, nontraditional material was placed at the end of the book, implementation was minimal. • Calculus was recommended for study in high school. • Geometry texts with emphasis on thinking through “original” exercises, as opposed to memorization of “book” theorems evolved. • Many Mathematics Teacher articles and yearbook chapters concerning the aims of mathematics were written in the two decades that followed. • Teacher training programs began to include general knowledge, professional knowledge, and specialized knowledge. • Mathematicians involvement with school mathematics increased.

24. But ultimately. . . . • No major change in practice was seen due to traditional inertia in educational practice and the depression of the 1930s. • Over the next two decades, the views expressed in the Kilpatrick report, The Problem of Mathematics in Secondary Education, exerted more influence than the 1923 Report (Klein, 2003).

25. References Bidwell, J. K., & Clason R. G. (1970). Readings in the history of mathematics education. Washington, DC: National Council of Teachers of Mathematics. Kilpatrick, W. H. (1920). The problem of mathematics in secondary education. A report of the Commission on the Reorganization of Secondary Education, appointed by the National Education Association. Bureau of Education Bulletin 1920, 1, 1-24. Klein, D. (2003). A brief history of K-12 mathematics education in the 20th century. In J. Royer (Ed.), Mathematical cognition. Information Age Publishing. National Committee on Mathematical Requirements (NCMR). (1923). The reorganization of mathematics in secondary education. The Mathematical Association of America. National Committee on Mathematical Requirements (NCMR). (1927). The reorganization of mathematics in secondary education (Part I).Boston: Houghton Mifflin. National Council of Teachers of Mathematics. (1970). A history of mathematics education in the United States and Canada. Reston, VA: National Council of Teachers of Mathematics.