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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
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The Reorganization of
Mathematics in Secondary
The National Committee on
Mathematical Requirements (NCMR)
Appointed by MAA 1916
Preliminary Report 1920
Summary Report 1922
Final Report 1923
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?
NCMR Members
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.
W. F. Downey took the place vacated in 1919 by the resignation of G. W. Evans.
Organization of Report
Two Major Parts:
General Principles and Recommendations
Investigations Conducted for the Committee
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
Chapter 7 – The function concept in secondary
Chapter 8 – Terms and symbols in elementary
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
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)
Practical Aims
1. Understand and apply the fundamental processes
of arithmetic
2. Understand and use the language of algebra
3. Understand and use elementary algebraic
methods to solve problems
4. Understand and interpret graphical
5. Be familiar with common geometric forms and
their properties and relations; develop and utilize
space perception and spatial imagination
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
Cultural Aims
1. Appreciation of beauty in the geometrical forms
found in nature, art, and industry
2. Appreciation of the importance of logical
structure, precision of statement and of
thought, logical reasoning, discrimination
between the true and the false
3. Appreciation of the power of mathematics and the
role that mathematics and abstract thinking have
played in the development of civilization
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.
Mathematics for Years 7, 8, 9
Recommended content:
Intuitive geometry
• Demonstrative geometry
• History
• Biography
Five models for junior high school course sequencing were
proposed, each reflecting some variations of a basic model.
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
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.
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.
Mathematics for Years 10, 11, 12
Recommended course offerings, in various
configurations, included:
Plane geometry
Solid geometry
Elementary statistics
Elementary calculus
Additional electives
Four plans for high school course sequencing with slight
variation to the above were proposed by the Committee.
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
• Focus on elementary algebra and plane geometry.
College admissions should be based on more than just
test scores.
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
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.
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
• Proposed simplification of terms in elementary
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.
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.
But ultimately. . . .
• No major change in practice was seen due to traditional
inertia in educational practice and the depression of the
• 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).
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
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.