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A CHALLENGE
OF NUMBERS
People in the Mathematical Sciences
Prepared by
Bernard L. Madison
and
Therese A. Hart
for the
Committee on the Mathematical Sciences
in the Year 2000
Mathematical Sciences Education Board
Board on Mathematical Sciences
Commission on Physical Sciences, Mathematics, and Resources
National Research Council
National Academy Press
Washington, D.C. 1990

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NOTICE: The project that is the subject of this report was
approved by the Governing Board of the National Research
Council, whose members are drawn from the councils of the
National Academy of Sciences, the National Academy of
Engineering, and the Institute of Medicine. The members
of the committee responsible for the report were chosen for
their special competences and with regard for appropriate
balance.
This report has been reviewed by a group other than the
authors according to procedures approved by a Report
Review Committee consisting of members of the National
Academy of Sciences, the National Academy of Engineer-
ing, and the Institute of Medicine.
The National Academy of Sciences is a private, nonprofit,
self-perpetuating society of distinguished scholars engaged
in scientific and engineering research, dedicated to the fur-
therance of science and technology and to their use for the
general welfare. Upon the authority of the charter granted
to it by the Congress in 1863, the Academy has a mandate
that requires it to advise the federal government on scien-
tific and technical matters. Dr. Frank Press is president of
the National Academy of Sciences.
The National Academy of Engineering was established
in 1964, under the charter of the National Academy of Sci-
ences, as a parallel organization of outstanding engineers.
It is autonomous in its administration and in the selection of
its members, sharing with the National Academy of Sciences
the responsibility for advising the federal government. The
National Academy of Engineering also sponsors engineer-
ing programs aimed at meeting national needs, encourages
education and research, and recognizes the superior achieve-
ments of engineers. Dr. Robert M. White is president ofthe
National Academy of Engineering.
The Institute of Medicine was established in 1970 by the
National Academy of Sciences to secure the services of
eminent members of appropriate professions in the exami-
nation of policy matters pertaining to the health of the pub-
lic. The Institute acts under the responsibility given to the
National Academy of Sciences by its congressional charter
to be an adviser to the federal government and, upon its own
initiative, to identify issues of medical care, research, and
education. Dr. Samuel 0. Thier is president of the Institute
of Medicine.
The National Research Council was organized by the
National Academy of Sciences in 1916 to associate the broad
community of science and technology with the Academy's
purposes of furthering knowledge and advising the federal
government. Functioning in accordance with general poli-
cies determined by the Academy, the Council has become
the principal operating agency of both the National Acad-
emy of Sciences and the National Academy of Engineering
in providing services to the government, the public, and the
scientific and engineering communities. The Council is
administered jointly by both Academies and the Institute of
Medicine. Dr. Frank Press and Dr. Robert M. White are
chairman and vice chairman, respectively, of the National
Research Council.
The Committee on the Mathematical Sciences in the Year
200O, which was appointed at the beginning of 1988, is a
three-yearproject ofthe Mathematical Sciences Education
Board and the Board on Mathematical Sciences. Its purpose
is to provide a national agenda for revitalizing mathemati-
cal sciences education in U.S. colleges and universities.
Support for this project and for the publication and dis-
semination of this report was provided by grants from the
National Science Foundation and the National Security
Agency.
Cover photograph reprinted courtesy of the University of
Maryland and with permission from John Consoli, pho-
tographer. Copyright (I) 1989 by John Consoli.
Library of Congress Catalogue Card Number 89-64078
International Standard Book Number 0-309-04190-2
Available from:
National Academy Press
2101 Constitution Avenue, N.W.
Washington, D.C. 20418
First Pnniing - April 1990
Second Printing - November 1990
Third Printing - January 1992
Summaries of this report may be obtained from MS 2000,
2101 Constitution Avenue, N.W., Washington, D.C. 20418
Printed in the United States of America

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Committee on the Mathematical Sciences in the Year 2000
William E. Kirwan (Chairman), President, University of Maryland
Ramesh A. Gangolli (Vice Chairman), Professor of Mathematics, University of Washington
Lida K. Barrett, Dean, College of Arts & Sciences, Mississippi State University
Maria A. Berriozabal, Councilwoman, City of San Antonio, Texas
Ernest L. Boyer, President, Carnegie Foundation for the Advancement of Teaching
William Browder, Professor of Mathematics, Princeton University
Rita R. Colwell, Director of the Maryland Biotechnology Institute and
Professor of Microbiology, University of Maryland
John M. Deutch, Provost, Massachusetts Institute of Technology
Ronald G. Douglas, Dean, Division of Physical Sciences and Mathematics,
State University of New York, Stony Brook
Patricia A. Dyer, Vice President for Academic Affairs, Palm Beach Community College
Lloyd C. Elam, Distinguished Service Professor of Psychiatry, Meharry Medical College
Sheldon L. Glashow, Higgins Professor of Physics, Harvard University
Nancy ]. Kopell, Professor of Mathematics, Boston University
Donald W. Marquardt, Consultant Manager, E. I. du Pont de Nemours & Co.
David S. Moore, Professor of Statistics, Purdue University
Jaime Oaxaca, Vice Chairman, Coronado Communications
Moshe F. Rubinstein, Professor of Engineering and Applied Science, University of California,
Los Angeles
Ivar StakgoicI, Chairman, Department of Mathematical Sciences, University of Delaware
S. Frederick Starr, President, Oberlin College
Lynn Arthur Steen, Professor of Mathematics, St. Olaf College
Staff
'lames A. Voyluk, Project Director
Bernard L. Madison, Project Director (through August 1988)
Therese A. Hart, Research Associate (through November 1989)
Craig E. Hicks, Project Assistant
. . .

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Mathematical Sciences Education Board
Alvin W. Trivelpiece (Chairman), Director, Oak Ridge National Laboratory
Shirley A. Hill (Past Chairman), Curators' Professor of Mathematics and Education,
University of Missouri- Kansas City
Iris M. Car! (Vice Chairman), Elementary Mathematics Instructional Supervisor,
Houston Independent School Distnct, Texas
Lillian C. Barna, Supenntendent of Schools, Tacoma Public Schools, Washington
L`ida K. Barrett, Dean, College of Arts and Sciences, Mississippi State University
C. Diane Bishop, Supenntendent of Public Instruction, State of Arizona
Constance Clayton, Superintendent of Schools, School District of Philadelphia, Pennsylvania
Paula B. Duckett, Elementary Mathematics Teacher, River Terrace Community School,
Washington, D.C.
Joan Duea, Elementary School Teacher, Price Laboratory School and Professor of Education,
University of Northern Iowa
Joseph W. Duncan, Corporate Vice President and Chief Economist,
The Dun & Bradstreet Corporation
Wade Ellis, fir., Mathematics Instructor, West Valley College, California
Shirley M. Frye, Director of Curriculum and Instruction, Scottsdale School District, Arizona
Ramanathan Gnanaclesikan, Head, Information Science Research Division,
Bell Communications Research
Donald L. Kreider, Vice Chairman, Mathematics and Computer Science Department,
Dartmouth College
Martin D. Kruskal, Professor of Mathematics, Rutgers University
Katherine P. Layton, Mathematics Teacher, Beverly Hills High School, California
Steven J. Leinwand, Mathematics Consultant, Connecticut State Department of Education
Richard S. Lindzien, Sloan Professor of Meteorology, Massachusetts Institute of Technology
Gail Lowe, Principal, Acacia Elementary School, Thousand Oaks, California
Steven P. Meiring, Mathematics Specialist, Ohio State Department of Education
Jose P. Mestre, Associate Professor of Physics, University of Massachusetts
1V

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Calvin C. Moore, Associate Vice President, Academic Affairs, University of California,
Berkeley
Jo Ann Mosier, Mathematics Teacher, Fairdale High School, Louisville, Kentucky
Leslie Hiles Paoletti, Chairman, Department of Mathematics and Computer Science,
Choate Rosemary Hall, Connecticut
Lauren B. Resnick, Director, Learning Research and Development Center,
University of Pittsburgh; liaison with the Commission on Behavioral and Social Sciences
and Education, National Research Council
Yolanda Rodriguez, Middle School Mathematics Teacher, Martin Luther King School,
Cambridge, Massachusetts
Thomas A. Romberg, Professor of Curriculum and Instruction, University of Wisconsin,
Madison
Isadore M. Singer, Institute Professor, Department of Mathematics,
Massachusetts Institute of Technology
Lynn Arthur Steen, Professor of Mathematics, St. Olaf College
William P. Thurston, Professor of Mathematics, Princeton University
Manya S. Ungar, Past President, The National Congress of Parents and Teachers
Zalman Usiskin, Professor of Education, The University of Chicago
John B. Walsh, Vice President/Chief Scientist, Boeing Military Airplanes
Nellie C. Weil, Past President, National School Boards Association
Guido L. Weiss, Elinor Anheuser Professor of Mathematics, Washington University;
liaison with the Commission on Physical Sciences, Mathematics, and Resources,
National Research Council
Staff
Kenneth M. Hoffman, Executive Director
Marcia P. Sward, Executive Director (through August ~ 989)
v

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Board on Mathematical Sciences
Phillip A. GrifO~ths (Chairman), Provost and James B. Duke Professor of Mathematics, Duke
University
Lawrence D. Brown, Professor of Mathematics, Cornell University
Ronald G. Douglas, Dean, College of Physical Sciences and Mathematics, State University of
New York, Stony Brook
David Eddy, J. Alexander McMahon Professor of Health Policy and Management, Duke
University
Frederick W. Gehring, Professor of Mathematics, University of Michigan
James G. Glimm, Chairman, Department of Applied Mathematics and Statistics, State
University of New York, Stony Brook
William H. Taco, Executive Director, American Mathematical Society
Joseph Kadane, Professor of Statistics, Carnegie-Mellon University
Gerald J. Lieberman, Professor of Operations Research and Statistics, Stanford University
Alan Newell, Head, Department of Mathematics, University of Arizona
Jerome Sacks, Head, Department of Statistics, University of Illinois
Guido L. Weiss, Elinor Anheuser Professor of Mathematics, Washington University
Shmue! Winograd, Director, Mathematical Sciences Department, IBM Thomas J. Watson
Research Center
Staff
Lawrence H. Cox, Director
V1

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Commission on Physical Sciences, Mathematics, and Resources
Norman Hackerman (Chairman), Chairman, Scientific Advisory Board, Robert A.Welch
Foundation
Robert C. Bearclsley, Senior Scientist and Chairman, Department of Physical Oceanography,
Woods Hole Oceanographic Institution
B. Clark Burchf.'el, Schlumberger Professor of Geology, Massachusetts Institute of Technolo~v
George F. Carrier, Professor Emeritus, Harvard University
Ralph J. Cicerone, Chair, Geosciences Department, University of California at Irvine
Herbert D. Dean, The Dow Chemical Company (retired)
~, ~. ~. . ., ~_ . . . ~. .
my,
refer a. ~ag~eson, Immune K. ~ umer Professor of c~v~1 Engineering, Massachusetts Institute of
Technology
Dean E. Eastman, Vice President, IBM T. I. Watson Research Center
Marye Anne Fox, Roland Pettit Centennial Professor of Chemistry, University of Texas
Gerhart FriedIander, Consultant, Brookhaven National Laboratory
Lawrence W. Funkhouser, Chevron Corporation (retired)
Phillip A. Grimths, Provost and James B. Duke Professor of Mathematics, Duke University
Neal F. Lane, Provost, Rice University
Christopher F. McKee, Professor of Physics and of Astronomy, University of California at
Berkeley
Richard S. Nicholson, Executive Director, American Association for the Advancement of
Science
Jack E. Oliver, Director of INSTOC, Cornell University
Jeremiah P. Ostriker, Chairman, Department of Astrophysical Sciences, Director, Princeton
University Observatory, Princeton University
Philip A. Palmer, Principal Consultant, E. I. du Font de Nemours & Company
Frank L. Parker, Professor of Civil and Environmental Engineering, Vanderbilt University
Denis J. Prager, Deputy Director, Health Program, MacArthur Foundation
David M. Raup, Professor, Department of Geophysical Sciences, University of Chicago
Roy F. Schwitters, Director, Superconducting Super Collider Laboratory
Larry L. Smarr, Director of National Center for Supercomputing Applications, Professor of
Astronomy and Physics, University of Illinois at Urbana-Champaign
Karl K. Turekian, Silliman Professor of Geology and Geophysics, Yale University
Staff
Myron F. Uman, Acting Executive Director
. .
V11

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Preface
During the last decade of this century, the U.S. educational system will face many challenges, but
few more important than the renewal and revitalization of the mathematical sciences. Problems in
mathematical sciences education exist at all levels elementary, high school, and postsecondary.
These problems are caused by many factors, including a static curriculum, insufficient resources
devoted to instruction, inadequate numbers of well-trained teachers, declining student interest, and
more generally, the public's failure to understand fully the importance of education in science for the
well-being of our society. The problems manifest themselves in the product of the educational
enterprise-the scientifically trained work force that must serve the needs of our society. The
demands of the next decade and the twenty-first century will require that large numbers of America's
work force be equipped with higher levels of mathematical sophistication. Yet, the National Science
Foundation and other organizations project that by the year 2000, unless there is a dramatic change,
our nation will face a significant shortfall of people with the necessary skills in the mathematical
sciences.
In response to this crucial situation the National Research Council has established the Committee
on the Mathematical Sciences in the Year 2000, under the direction of the Mathematical Sciences
Education Board and the Board on Mathematical Sciences, to assess the present state of collegiate
education in the mathematical sciences, to identify existing problems, and to recommend action that
can remedy this situation. Itis hoped that this analysis by the committee will lead to a rational agenda
for renewal and revitalization of the mathematical sciences and to a strategy for implementation that
will stimulate all sectors of our society into action.
The present descriptive report, A Challenge of Numbers, is one of three planned by the committee.
It describes the circumstances and issues related to the human resources in the mathematical sciences,
principally students and teachers at U.S. colleges and universities. The mathematical sciences, based
largely in academia, are crucial not only for the development of new knowledge but also as a needed
resource in the education of our technological work force. As with other scientific professions, the
challenge of meeting the growing need for workers in the face of shrinking supplies will require a
program of national action, but the situation in the mathematical sciences is particularly severe.
Stetson trom the educational pipeline In mathematics has given rise to low degree production at all
levels in colleges and universities, and socioeconomic forces have resulted in unprecedentedly low
interest in mathematics as a major among entering freshmen. At the same time, growing enrollments
in lower-level mathematics courses with small or no increases in staffing levels have created less than
. . .
vail

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an ideal environment for faculty in the mathematical sciences. Because of the fundamental role of
mathematics in support of science and engineering, the problems in the U.S. educational system will
not be corrected unless the problems in mathematics education are corrected. On the other hand, a
revitalization of the mathematical sciences could lead the way in effecting change across the
educational spectrum.
A Challenge of Numbers contains data from a large number of existing sources. It provides in
one place a comprehensive set of data describing the demographic situation in the mathematical
sciences. This report does not provide answers, nor does it draw conclusions. Rather, it serves as
a reference. As such, it will be an important source of data for the Committee on the Mathematical
Sciences in the Year 2000 in formulating its recommendations. Also, it is hoped that the report will
be used by educators and administrators in industry, government, and education to alert policymakers
about the urgent need for revitalization of the mathematical sciences.
This is the second report of the MS 2000 project. The first, Everybody Counts, was published
jointly with the Mathematical Sciences Education Board and the Board on Mathematical Sciences
and provided a general analysis of the trends and needs in mathematics education. The next review
paper, to be published in late spring 1990, will assess the college and university curriculum, and the
final paper, due in summer 1990, will report on resources available for instruction in the mathemati-
cal sciences. Following the completion of the review papers, the Committee on the Mathematical
Sciences in the Year 2000 will be in a position to issue its final recommendations in a report to the
nation scheduled for late ~ 990.
On behalf of the committee, ~ want to express our gratitude to Bernard L. Madison and Therese
A. Hart for their work in compiling the necessary background information and for writing A
Challenge of Numbers. They have provided an important service to the members of the Committee
on the Mathematical Sciences in the Year 2000 and to the mathematical sciences community by
making these data available in a convenient and accessible format.
HE
in,
William E. Kirwan
Chairman, Committee on the Mathematical Sciences
in the Year 2000
President, University of Maryland at College Park
1X

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Contents
Preface viii
List of Figures
List of Tables
List of Boxes
. ~
X11
XlV
XV
Introduction and Historical Perspective
The MS 2000 Project and the Scope of This Report 1
Three Roller Coaster Decades 2
National Efforts Toward Renewal
Contents of This Report ~
2 The U.S. Labor Force and Higher Education
Introduction
More Skills and Greater Adaptabiliry 10
Growth in Science-Based Occupations 11
Higher Education in the United States 12
The Pool of Potential Students and Workers 13
Persistence in College Enrollment 14
Shifting Interests of College Students 15
Natural Sciences and Engineering 16
The Challenges and the Responsibility 1 ~
3 College and University Mathematical Sciences
Introduction 19
Strong at the Top 20
Mixed Precollege Indicators 22
Troublesome Transitions from High School to College 24
Remediation in College 28
Service Courses 30
Mathematics as an Academic Competency and Subject 32
x
1
9
19

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4 Majors in Mathematics and Statistics
Introduction 35
Unclergraduate Majors 37
Degrees for Secondary School Mathematics Teachers 40
Graduate Students 42
Master's Degree Recipients 45
Doctoral Degree Recipients 48
Patterns and Prospects 52
5 Mathematical Scientists in the Workplace
Introduction 53
General Characteristics and Trends 54
Employment of Recent Graduates 56
Secondary School Mathematics Faculty 58
Characteristics of College and University Faculties 61
What Faculty Members Do 62
Faculty Members by Duties and Credentials 65
The Research Faculty 67
Faculty Salaries 68
Ages of Faculty Members 68
Women ant] Minorities on the Faculty 70
Two-Year College Faculty Mobility 70
Four-Year College ant! University Doctorate Faculty 70
Four-Year College ant! University Nondoctorate Faculty 70
Summary 71
6 Issues and Implications
Bibliography
Appendix Tables
35
53
73
75
87
X1

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List of Figures
1.1 Total number of scientists and engineers.
1.2
Ph.D. degrees in mathematics, 1986-1987.
Left: Total undergraduate enrollments in mathematical sciences departments.
Right: Mathematical sciences faculty at colleges and universities.
Mathematical sciences degrees awarded.
1.3
1.4
2.1
2.2
2.3
2
3
The educational requirements of the work force are increasing.
Percent distribution of undergraduate enrollments by race and ethnic group.
The pool of college students is changing, 18- to 24-year-old population.
4
6
10
13
13
2.4 Percent of 18- and 19-year-olds who are high school dropouts, by ethnic group. 14
2.5 Enrollment in institutions of higher education as a percent of high school
graduates. 14
2.6 Shifting interest in selected majors. 17
3.1 SAT mathematics scores, 1967 to 1987. 22
3.2 ACT mathematics scores, 1973 to 1988. 22
3.3 Percent increase in enrollments in selected mathematics courses in colleges and
universities, 1965 to 1985.
3.4 Undergraduate enrollments in mathematical sciences departments at U.S.
colleges and universities.
4.1
4.2
4.3
30
32
Students in the mathematical sciences pipeline about half are lost each year.
A representation of U.S. students in the mathematics pipeline.
Number of mathematical sciences degrees awarded by U.S. institutions, 1950
to 1986.
36
36
37
4.4 Percentage of entering college freshmen expecting to major in mathematics. 38
4.5 Bachelor's degrees awarded in the mathematical sciences, 1970 to 1986. 38
4.6
Bachelor's degrees awarded in the mathematical sciences, 1970 to 1986.
Number of bachelor's degrees awarded in mathematical and computer sciences,
1970 to 1986.
4.7
4.8
4.9
Expected versus actual number of bachelor's degrees in mathematical sciences.
Left: Interest in mathematics and education among entering college freshmen.
Right: Degrees in mathematics and education among exiting college seniors.
Percent of full-time graduate students in doctorate-granting institutions who
are non-U.S. citizens, 1975 and 1986.
4.10 Mathematical sciences graduate students enrolled full-time in doctorate-
granting institutions, 1975 to 1986.
x~
39
41
44
44

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4.11 Percent of non-U.S. citizens as mathematical sciences graduate students
by type of institution, 1977 to 1986.
4.12 Source of major support for mathematical sciences graduate students
in doctorate-granting institutions, 1986.
4.13 Types of major support for graduate students in doctorate-granting
institutions, 1986.
4.14 Master's degrees awarded, mathematical sciences.
4.15 Master's degrees in mathematical sciences, distribution by subfield.
4.16 Ph.D. degrees in mathematics.
4.17 Doctoral degrees in mathematical sciences, distribution by subfield
and sex.
4.18 Number of doctorate recipients in broadly interpreted mathematical
sciences.
5.1 Percent of recent mathematics degree holders employed in a science
or engineering job, 1976 and 1986.
Field of employment for recent mathematics degree recipients, 1986.
Primary work activities of recent mathematics degree recipients, 1986.
5.4 Median annual salaries of recent science and engineering graduates.
Supply and demand of new elementary and secondary school teachers,
5.5
5.6
5.7
5.8
5.9
1970 to 1992.
Left: Number of full-time mathematical sciences faculty members at colleges
and universities. Right: Number of part-time mathematical sciences
faculty members at colleges and universities.
Mathematical and computer sciences enrollments per FTE of faculty.
Mathematical sciences faculty salaries, 1970 to 1985 (in 1985 dollars).
Age distribution of full-time mathematical sciences faculty in four-year
colleges and universities.
5.10 Estimated number of retirements of full-time college and university
mathematical sciences faculty.
5.11 Top: Source of new hires of two-year college full-time faculty in
mathematical sciences. Bottom: Destination of departing mathematical
sciences two-year college full-time faculty.
5.12 Source of two-year college part-time faculty in mathematical sciences.
S.13 Left: Source of new hires of doctorate faculty in mathematical sciences,
1983 to 1988. Right: Destination of departing mathematical sciences
doctoral faculty, 1983 to 1988.
44
45
46
46
47
48
51
51
55
57
58
58
59
60
61
63
64
65
66
68
69
. . .
X111

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List of Tables
2.1 Attainment rates of advanced degrees for selected fields, 1971 to 1985
4.1 Mathematical sciences bachelor's degrees per 1,000 mathematical sciences
enrollments, 1965 to 1985
Changes in mathematical sciences majors by undergraduate grade point
averages, 1981 freshman cohort
1985 bachelor's degrees awarded in mathematical sciences
Summary of responses on quality and quantity of undergraduate majors
1987 SAT scores by intended college major
1986 GRE scores by undergraduate and intended graduate major
Summary of responses on quality and quantity of graduate students
1985 master's degrees awarded in mathematical sciences programs
4.3
4.4
4.5
4.6
4.7
4.8
4.9 Attainment rates of master's and doctoral degrees
4.10 Ratio of new doctorates in mathematics to new doctorates in selected other
fields, 1970 to 1985
4.11 Mathematics majors going on to doctoral study in other areas of science and
engineering, 1960 to 1985
4.12 Ethnic representations among new mathematical sciences doctorates, U.S.
citizens, 1977 to 1986
Ethnic representation among all new research doctorates, U.S. citizens and
permanent residents, 1977 to 1986
Estimates of the number of mathematical scientists by National Science
Foundation (NSF), Bureau of Labor Statistics (BLS), and Conference Board
for Mathematical Sciences (CBMS)
Type of employer of mathematical scientists by degree level, 1986
Professional activities of four-year college and university mathematical sciences
faculty
5.5
5.6
5.4 Professional activities of two-year college mathematical sciences faculty
Numbers of mathematical sciences faculty members by teaching area and type
of institution, 1987
Age distribution of mathematical sciences faculty members in 105 research
universities, 1980 and 1986
Full-time mathematical sciences faculty by ethnic origin and sex, 1985
5.8 Estimate of average annual net flow into doctoral faculty at four-year colleges
and universities, 1982 to 1987
x~v
18
37
39
40
41
42
42
45
47
48
49
49
50
50
54
56
62
64
67
68
69
71

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List of Boxes
I-1
I.2
{.3
2.!
Computer Science
Sources of Data
Statistics
Degree Programs in Mathematics
2.2 Degree Programs in Statistics
3.1 Professional Organizations
3.2 AMS-MAA Survey Reports
3.3 CBMS Surveys
3.4 Minorities and Women
3.5 Intervention Programs
3.6 The Texas Prefreshman Engineering Program
3.7 Professional Development Program
3.8 The Mathematics, Engineering, Science Achievement Program
Is
8
11
12
21
25
26
27
28
29
31
33
xv

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