Goals for School Mathematics

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. 3-10-05

2. Committees and Reports that Have Influenced the Changing Mathematics Curriculum Goals for School Mathematics 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 Cambridge Conference on School Mathematics • 1963 http://www.mathcurriculumcenter.org

3. Cambridge Conference onSchool MathematicsPurposes • Look to the future to establish goals for mathematics education that would extend current reform efforts. • Involve mathematicians in brainstorming about the content of mathematics curriculum, free of the existing realities of schools and teacher education. • Stimulate discussion of radical change in the school mathematics curriculum that might be appropriate in the 1990s.

4. Maurice Auslander Edward G. Begle* Jerome S. Bruner* R. Creighton Buck George Francis Carrier Julian D. Cole Robert Davis Robert P. Dilworth Bernard Friedman H. L. Frisch Andrew M. Gleason (co-director)* Peter J. Hilton J. L. Hodges, Jr. Mark Kac* Seymour H. Koenig C. C. Lin Earle L. Lomon William Ted Martin (co-director)* Edwin E. Moise* Frederick Mosteller Henry O. Pollak Mina S. Rees* Max M. Schiffer George Springer Patrick Suppes* A. H. Taub Stephen White* Samuel S. Wilks* Jerrold R. Zacharias* *indicates members of the steering committee Conference Participants

5. Contents of the Conference Report Foreword by Francis Keppel—U.S. Commissioner of Education Section 1: Introduction Section 2: Broad Goals of the School Mathematics Curriculum Section 3: Pedagogical Principles and Techniques Section 4: Some Overall Observations Section 5: Curriculum for Elementary School (K-6) Section 6: Curriculum for Grades 7-12 Appendices

6. Basis of the Report • Focus was on developing an advanced mathematical foundation. • Report was to serve as a “discussion document” not a “prescription.” • Proposals were not limited by the mathematical knowledge of the current teacher population. • Expectations for student learning were not limited by existing research on developmentally appropriate ages.

7. Broad Goal of the School Mathematics Curriculum “The subject matter which we are proposing can be roughly described by saying that a student who has worked through the full thirteen years of mathematics in grades K to 12 should have a level of training comparable to three years of top-level college training today.” (Goals for School Mathematics, 1963, p. 7)

8. Basic Principles • With the realization that fewer and fewer students would elect to take mathematics in the later years, the most important concepts should be pushed down— especially probability and statistics. • The inclusion of additional content in each year could be accomplished by decreasing drill for its own sake, by reorganizing subject matter, and by providing practice in the context of learning new concepts. • There should be a balance between pure and applied mathematics. • There should be a balance between rigor and intuition.

9. Pedagogical Principles/Techniques • Use spiral curriculum to increase level of rigor throughout schooling. • Use guided discovery methods in instruction and in the design of exercises. • Emphasize understanding of algorithms justifying manipulations. • Foster independent and creative thinking. • Develop language and notation after concepts have been formed. • Use historical discussions, applications, and a mathematics laboratory to motivate learning.

10. Some Overall Observations • Children should be able to perform at levels necessary to succeed in learning the proposed ontent. • Different inquiry methods should focus on mathematical ideas and be used to promote greater depth of understanding. • Future teachers need greater content knowledge to teach the proposed curricular ideas. • Discovery problems should be encountered before the results are explicitly stated in the text. • Focus of assessments should be placed on student understanding instead of memorized facts.

11. Curriculum Topics for Elementary School • Grades K-2 Real numbers, geometry, logic and set theory, function, applications • Grades 3-6 Real numbers, geometry, logic and foundations, theory of real functions, applications, longer projects

12. Curriculum for Grades 7-12 Proposal I Grades 7/8: Algebra and Probability Grade 9: Geometry Grade 10: Geometry, Topology, Algebra, and Linear Algebra Grades 11/12: Analysis

13. Curriculum for Grades 7-12 Proposal II Grades 7/8: Algebra, Geometry, and Probability Grade 9: Introductory Calculus, Algebra, and Geometry Grade 10: Analysis, Probability, and Algebra Grades 11/12: Analysis

14. Differences Between ProposalsAlgebra Proposal I: polynomial forms over fields, applications of polynomial functions, difference operators, complex functions, tangent of the graph of a polynomial. Proposal II: emphasis on polynomial functions, trigonometric, functions studied earlier, different presentation of complex numbers.

15. Differences Between ProposalsCalculus • Proposal I Calculus concepts and methods treated in grades 11 and 12 to provide a thorough treatment of calculus • Proposal II Introductory calculus concepts included beginning in grade 9

16. Appendices • Probability and Statistics • Logarithms in Elementary School • The Introduction to Formal Geometry • Exploration • Elementary Modern Mathematics from the Advanced Standpoint • Opportunities for Proof-Making in Elementary School • The Use of Units • Remarks on Significant Figures

17. Goals for School MathematicsSignificance • Stimulated the formation of the Comprehensive School Mathematics Project (CSMP) to develop the recommended curriculum. • Influenced the work of the Secondary School Mathematics Curriculum Improvement Study. • Provoked much discussion among academics and educators. • Provided a basis for numerous hypotheses to be tested about curriculum, teaching, and learning.

18. Goals for School MathematicsSignificance and Criticism • Inspired a closer look at teaching methodology such as the discovery method. • Increased awareness of the need for better teacher preparation programs. • Criticized for its elitist feel and bias toward the most capable student.

19. References Adler, I. (1966). The Cambridge Conference report: Blueprint or fantasy? Mathematics Teacher, 59(3), 210-217. Education Development Center, Inc. (1967). Goals for mathematical education of elementary school teachers: A report of the Cambridge Conference on Teacher Training. Boston: Houghton Mifflin. Educational Services, Inc. (1963). Goals for school mathematics: The report of the Cambridge Conference on School Mathematics. Boston: Houghton Mifflin. Ferrini-Mundy, J., & Graham, K. (2003). The education of mathematics teachers in the United States after World War II: Goals, programs, and practices. In G. M. Stanic & J. Kilpatrick (Eds.), The history of school mathematics (pp. 1193-1308). Reston, VA: NCTM. Hilton, P. (1966). The continuing work of the Cambridge Conference on School Mathematics. Arithmetic Teacher, 13(2),145-149. National Council of Teachers of Mathematics. (1970). A history of mathematics education in the United States and Canada. Reston, VA: NCTM. Stone, M. H. (1965). Review of Goals for School Mathematics: The Report of the Cambridge Conference on School Mathematics. Mathematics Teacher, 58(4), 353-360.