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OPPOR TUNITY
Mathematics is the key to opportunity. No longer
just the language of science, mathematics now contributes
in direct and fundamental ways to business, finance, health,
and defense. For students, it opens doors to careers. For
citizens, it enables informed decisions. For nations, it pro-
vides knowledge to compete in a technological economy. To
participate fully in the world of the future, America must
tap the power of mathematics.
Communication has created a world economy in which
working smarter is more important than merely working
harder. Jobs that contribute to this world economy require
workers who are mentally fit workers who are prepared to
absorb new ideas, to adapt to change, to cope with ambiguity,
to perceive patterns, and to solve unconventional problems.
It is these needs, not just the need for calculation (which is
now done mostly by machines), that make mathematics a
prerequisite to so many jobs. More than ever before, Amer-
icans need to think for a living; more than ever before, they
need to think mathematically.
~ ~ uaTitv mathematics education for n11
students Is essential for a healthy economy.
Yet, for lack of mathematical power, many of today's stu-
dents are not prepared for tomorrow's jobs. In fact, many
are not even prepared for today's jobs. Current mathe-
matical achievement of U.S. students is nowhere near what
is required to sustain our nation's leadership in a global
technological society. AS technology has ~rIlathematlcized''
the workplace and as mathematics has permeated society,
a complacent America has tolerated underachievement as
the norm for mathematics education. We have inherited a
mathematics curriculum conforming to the past, blind to the
future, and bound by a tradition of minimum expectations.
Wake up, America! Your children are at risk. Three
of every four Americans stop studying mathematics before
...tapping the power of mathematics
"How can students com-
pete in a mathematical
society whet' they leave
school knowi1'g so little
mathematics?"
- rester Thurow*
* People, reports, and dlata cited! in the
margins are identified on pages 102-103.
1

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Opportunity
"The fastest growing
jobs require much higher
math, language, and rea-
soning capabilities than
current jobs, while slowly
· e ~ ~
growlngJoos require
less.',
Workforce 2000
2
completing career or job prerequisites. Most students leave
school without sufficient preparation in mathematics to cope
either with on-thejob demands for problem-solving or with
college requirements for mathematical literacy. Thus, in-
dustry, universities, and the armed forces are burdened by
extensive and costly demands for remedial education.
Today's mathematics opens doors to tomorrow's jobs. As
successive waves of immigrants have used this country's ed-
ucational system to secure better lives for themselves and
their children, so today's children the world over are using
mathematical training as a platform on which to build up
their lives. America's children deserve the same chance.
Children can succeed in mathematics. Many do so in other
countries and some do so in this country. The evidence
from other nations shows overwhelmingly that if more is
expected in mathematics education, more will be achieved.
Clear expectations of success by parents, by schools, and by
society can promote success by students.
Children can succeed in mathematics. If
more is expected, more will be achieved.
In today's world, the security and wealth of nations de-
pend on their human resources. So does the prosperity
of individuals and businesses. As competitors get smarter,
our problems get harder. Long-term investment in science
and technology both for businesses and for our nation
requires serious commitment to revitalizing mathematics ed-
ucation. It is time to act, to ensure that ad Americans benefit
from the power of mathematics.
Context for Change
In 1983, A Nation at Risk awoke a sleeping nation to
alarming problems in our educational system. Since then,

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...tapping the power oimathematics
dozens of reports have analyzed virtually every aspect of
this enormous problem. Some call for changes in curricu-
lum, others for changes in the structure of schools; some cite
deficiencies in the ways teachers are educated, while others
examine signs of decay in the social and economic structures
of society. All agree that the present system must change.
Mathematics education takes place in the context of
schools. Like other subjects, mathematics is constrained by
limits of school and society, of texts and tests. Much that
needs improvement must be accomplished by systemic reme-
dies that affect all subjects and all schools. Much is being
done by school districts and community organizations, by
legislatures and universities, by corporations and teachers.
Nevertheless, much remains to be done.
Although mathematics is not unique in its importance to
education, mathematics education tends to magnify many
problems of the schools. Literature, history, science, and
other subjects contribute in essential ways to a well-balanced
education; no one can say that one field is intrinsically of
greater worth than another. It is clear, however, that math-
ematics plays a special role in school education, one that is
especially sensitive to deficiencies in the effectiveness of the
educational system.
Since mathematics is the foundation of science and tech
nology, it serves as a key to opportunity and careers. More-
over, mathematics contributes to literacy certain distinctive
habits of mind that are of increasing importance to an in-
formed citizenry in a technological age. Because of the
fundamental importance of literacy and numerary, English
and mathematics are the only subjects taught continuously
throughout the school years.
Education in any discipline helps students learn to think,
but education also must help students take responsibility for
their thoughts. While this objective applies to all subjects, it
is particularly apt in mathematics education because math-
ematics is an area in which even young children can solve a
problem and have confidence that the solution is correct-
not because the teacher says it is, but because its inner logic
· ~
IS SO clear.
"Reading, 'Riting, and "
· --
Mathematics, of course, is not
the only cornerstone of opportu-
nity in today's world. Reading is
even more fundamental as a basis
for learning and for life. What
is different today is the great in-
crease in the importance of math-
ematics to so many areas of edu-
cation, citizenship, and careers.
3

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Opportunity
Myth: "There is no algebra in
my future."
Reality: Just because students do
not use algebra anywhere except
in algebra class does not mean
that they will not need mathemat-
ics in the future. Over 75 percent
of all jobs require proficiency in
simple algebra and geometry, ei-
ther as a prerequisite to a training
program or as part of a licensure
examination.
4
M athematics is a key to opportunity and
careers.
Educators no longer argue that mathematics trains the
mind for clear thinking in other subjects. Mathematics does,
however, provide one of the few disciplines in which the
growing student can, by exercising only the power inherent
in his or her own mind, reach conclusions with full assur-
ance. More than most other school subjects, mathematics
offers special opportunities for children to learn the power
of thought as distinct from the power of authority. This
is a very important lesson to learn, an essential step in the
emergence of independent thinking.
Mathematics for Tomorrow
The increasing importance of mathematics to society is
only one of many factors that compel special examination
of mathematics education. Mathematics itself is now signif-
icantly more diverse than it was several decades ago when
today's leaders and educators went to school. The arith-
metic, algebra, geometry, and calculus taught nowadays are
mere shadow images of modern mathematics. The mathe-
matical sciences of today blend deep new results from these
traditional areas with methods from such applied fields as
statistics, operations research, and computer science. Mod-
ern mathematics provides a powerful instrument for under-
standing the world in which we live.
Several factors growth of technology, increased applica-
tions, impact of computers, and expansion of mathematics
itself-have combined in the past quarter century to extend
greatly both the scope and the application of the mathemati-
cal sciences. Together, these forces have created a revolution
in the nature and role of mathematics a revolution that
must be reflected in the schools if our students are to be well
prepared for tomorrow's world. Education reflecting only

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...tapping the power of mathematics
the mathematics of the distant past is no longer adequate
for present needs.
From the accountant who explores the consequences of
changes in tax law to the engineer who designs a new air-
craft, the practitioner of mathematics in the computer age is
more likely to solve equations by computer-generated graphs
and calculations than by manual algebraic manipulations.
Mathematics today involves far more than calculation; cIar-
ification of the problem, deduction of consequences, formu-
lation of alternatives, and development of appropriate tools
are as much a part of the modern mathematician's craft as
are solving equations or providing answers.
Statistics, the science of data, has blossomed from roots
in agriculture and genetics into a rich mathematical science
that provides essential tools both for analyses of uncertainty
and for forecasts of future events. From clinical research to
market surveys, from enhancement of digital photographs
to stock market models, statistical methods permeate policy
analysis in every area of human affairs.
Challenging problems in such diverse fields as computer
science and social science have invigorated the discipline of
discrete mathematics, a field that reflects both computer logic
and human ambivalence. Moreover, new mathematical tools
such as game theory and decision theory are being applied to
the human sciences where one seeks to make choices, deci-
sions, and coalitions on some rational and systematic basis.
A
pplications, computers, and new
discoveries have extended greatly the landscape
of mathematics.
Today's mathematics is a creative counterpoint of com-
putation and deduction, rooted in data while unfolding in
abstraction. Mathematics today is being continually cre-
ated and adapted to meet new needs. Frequent interactions

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Opportu1'ity
"The nafionaispotligh' is
turning on mathematics
as we appreciate its cen-
tral role in the economic
growth of this country.
. . . It must become a pump
instead of abetter in the
pipeline."
Mathematics Pipeline
NUMBER OF STUDENTS
1 0.000.000
Robert M. White
1,000,000 \
1 00,000
1 0,000
Ninth Graders
\ Freshmen
\ B.S. Degrees
\`M.S. Degrees
1,000 ~
Ph.D. Degrees
100 1 1 1 1 1
1972 1976 1980 1984 1988
Data from enrollments in mathematics
courses reveal that roughly half the stu-
dents leave the pipeline each year.
6
among different specialties reveal deep connections and hid-
den unity, strongly suggesting that the different mathematical
sciences are merely facets of a single science of patterns.
A Pump, Nof a Filter
The revolution in the way that mathematics is practiced
is mirrored by a similar but unfulfilled-revolution in our
understanding of how students actually learn mathematics.
Research on learning shows that most students cannot learn
mathematics effectively by only listening and imitating; yet
most teachers teach mathematics just this way. Most teach-
ers teach as they were taught, not as they were taught to
teach.
Research in learning shows that students actually construct
their own understanding based on new experiences that en-
large the intellectual framework in which ideas can be cre-
ated. Consequently, each individual's knowledge of math-
ematics is uniquely personal. Mathematics becomes useful
to a student only when it has been developed through a per-
sonal intellectual engagement that creates new understand-
ing. Much of the failure in school mathematics is due to a
tradition of teaching that is inappropriate to the way most
students learn.
We require students to study mathematics for many rea-
sons: to learn practical skills for daily lives, to understand
quantitative aspects of public policy, to develop problem-
solving skills, and to prepare for careers. None of these goals
is being achieved; not only do we face a shortage of person-
nel with mathematical preparation suitable to scientific and
technological jobs, but also the level of mathematical literacy
(or "numerary") of the general public is completely inade-
quate to reach either our personal or national aspirations.
Mathematics education is an immense enterprise, involv-
ing over 10 percent of the nation's educational resources-
about $25 billion annually. Each year, 25 million elementary
school children, 10 million secondary school students, and
3 million undergraduates study mathematics; indeed, about
60 percent of school-level and 30 percent of college-level sci-
ence and technology education are devoted to mathematics.

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· -
M.
· ~
~ ~ Mathematics must become a pump
rather than a filter in the pipeline of American
education.
Everyone depends on the success of mathematics education;
everyone is hurt when it fails.
More than any other subject, mathematics filters students
out of programs leading to scientific and professional careers.
From high school through graduate school, the half-life of
students in the mathematics pipeline is about one year; on
average, we lose half the students from mathematics each
year, although various requirements hold some students in
class temporarily for an extra term or a year. Mathematics
is the worst curricular villain in driving students to failure in
school. When mathematics acts as a filter, it not only filters
students out of careers, but frequently out of school itself.
Low expectations and limited opportunity to learn have
helped drive dropout rates among Blacks and Hispanics
much higher- unacceptably high for a society committed to
equality of opportunity. It is vitally important for society
that ad citizens benefit equally from high-quality mathemat-
ics education.
Numeracy
To function in today's society, mathematical literacy-
what the British call "numerary"- is as essential as verbal
literacy. These two kinds of literacy, although different, are
not unrelated. Without the ability to read and understand,
no one can become mathematically literate. Increasingly, the
reverse is also true: without the ability to understand basic
mathematical ideas, one cannot fully comprehend modern
writing such as that which appears in the daily newspapers.
Numeracy requires more than just familiarity with num-
bers. To cope confidently with the demands of today's so-
ciety, one must be able to grasp the implications of many
..tapping the power of mathematics
"Numeracy is the abil-
ify to cope confidently
with the mathematical
demands of adult life."
- Mathematics Counts
7

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Opportunity
8
mathematical concepts-for example, chance, logic, and
graphs" that permeate daily news and routine decisions.
Literacy is a moving target, increasing in level with the ris-
ing technological demands of society. Indeed, there is some
evidence that the decline in reading comprehension scores
over the last several decades is due in part to the growing
mathematical content of what one is required to read. It is
not just computer manuals or financial reports that require
an understanding of mathematical ideas; so do reports of
political polls, debates about AIDS testing, and arguments
over the federal deficit. Even Supreme Court decisions re-
semble mathematical arguments whose subject matter is law
rather than numbers; often, legal cases rest as much on prob-
abilistic inferences (for example, DNA fingerprinting, fiber
analysis) as on direct evidence.
M ...............................
athematical literacy is essential as a
foundation for democracy in a technological
age.
Functional literacy in all of its manifestations verbal,
mathematical, scientific, and cultural provides a common
fabric of communication indispensable for modern civilized
society. Mathematical literacy is especially crucial because
mathematics is the language of science and technology. Dis-
cussion of important health and environmental issues (acid
rain, waste management, greenhouse effect) is impossible
without using the [Language of mathematics; solutions to
these problems will require a public consensus built on the
social fabric of literacy.
The study of mathematics can help develop critical habits
of mind to distinguish evidence from anecdote, to recog-
nize nonsense, to understand chance, and to value proof.
Citizens in a democracy must recognize that change is a pro-
cess with expected regularities; that order can beget disorder

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...tapping the power oimuthematics
(as in turbulence) and vice versa (as in statistical experi-
ments); that similar mathematical models can represent dif-
ferent phenomena (for example, patterns of growth in biol-
ogy, economics, and chemistry); and that simple models can
clarify complex systems (as in linear models of economic
systems), even though simplistic analysis can result in mis-
leading interpretations. Citizens who are bombarded daily
with conflicting quantitative information need to be aware
of both the power and the limitations of mathematics.
The great majority of American children spend most of
their school mathematics time learning only practical arith-
metic. Few retain much of what they learn about geometry;
fewer still learn anything about chance. Secondary education
is particularly devoid of exposure to modes of mathematical
thought required for intelligent citizenship. Even colleges
and universities seem unable to infuse appropriate mathe-
matical ideas into liberal education. There is no consen-
sus whatsoever on a collegiate mathematics curriculum for
students outside the preprofessional programs where math-
ematics serves a well-defined yet strictly utilitarian purpose.
Rarely does mathematics contribute as it should to liberal
education, to the honing of values, and to effective citizen-
ship.
Attitudes
One of the more disturbing conclusions of recent stud-
ies of mathematics education is that the American public
tends to assume that differences
in accomplishment in school
mathematics are due primarily to differences in innate ability
rather than to differences in individual effort or in opportu-
nity to learn. These beliefs surface in many ways in studies
of parents' views, in common self-deprecating remarks ("I
never could do math"), and in public infatuation with early
tracking as a strategy for mathematics education.
One consequence of such beliefs is that parents often
accept and sometimes even expect their children's poor
performance in mathematics. Another consequence is that
adults who determine policy in mathematics education often
measure the mathematical needs of today's students by their
"We've inherited a woe-
fully limited se! of expec-
tations of what schools
can accomplish and what
children can learnt."
William R. Graham
9

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Opportunity
Myth: Learning mathematics re-
quires special ability, which most
students do not have.
Reality: Only in the United
States do people believe that
learning mathematics depends on
special ability. In other countries,
students, parents, and teachers
all expect that most students can
master mathematics if only they
work hard enough. The record
of accomplishment in these
countries and in some inter-
vention programs in the United
States shows that most students
can learn much more mathemat-
ics than is commonly assumed in
this country.
10
own meager and outdated mathematical accomplishments.
From the faulty premise that most students "can't do math"
and the fact that many adults who never learned mathemat-
ics have succeeded without it, they rationalize that official
expectations should be limited to minimal basic levels. The
result is a spiral of lowered expectations in which poor per-
formance in mathematics has become socially acceptable.
~u0~c acceptance of deficient standards
contributes significantly to poor performance
in mathematics education.
Even worse, the unrestricted power of peer pressure often
makes good performance in mathematics socially unaccept-
able. This environment of negative expectation is strongest
among minorities and women-those most at risk during
high school years when students first exercise choice in cur-
ricular goals. Even strong positive support of adults is often
powerless to open the eyes of children who are blinded by
their entertainment-dominated culture.
Public attitudes about mathematics are shaped primar-
ily by adults' childhood school experiences. Consequently,
mathematics is seen not as something that people actually
use, but as a best forgotten (and often painful) requirement
of school. For most members of the public, their lasting
memories of school mathematics are unpleasant since so
often the last mathematics course they took convinced them
to take no more.
Some adults blame the "new math" for their fears, having
become convinced of their own mathematical ineptitude by
instruction steeped in premature abstraction. Others have
been made apprehensive by a teacher's rigid view of mathe-
matics as a string of procedures to be memorized, where right
answers count more than right thinking. Either extreme-
mindless abstraction or mindless calculation yields mind-
less mathematics. These widespread fears reinforce common

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...tapping the power of mathematics
public perceptions that mathematicians are responsible for
making mathematics hard and that only geniuses are capable
of learning mathematics.
Children respond to expectations of their parents and
teachers. It is no wonder that many students stop study-
ing mathematics as soon as it becomes optional, since soci-
ety provides so few hero-images for whom perseverance in
mathematics has paid off. When parents think that ability
supersedes effort, most students never learn the value of ef-
fort. "Hard work pays oh' should be society's message to
those who study mathematics.
Goals
Historically, schools in the United States were designed
with a dual mission: to teach all students basic skills required
for a lifetime of work in an industrial and agricultural econ-
omy and to educate thoroughly a small elite who would go
to college en route to professional careers. As the needs of
society have changed as the fraction of students preparing
to work in factories or on farms has declined- the balance
of these two goals has shifted. Today's schools labor un-
der the legacy of a structure designed for the industrial age
misapplied to educate children for the information age.
Not only in mathematics but in every school subject, edu-
cators are faced with rising expectations for preparing the
kind of work force the country will need in the future.
Tnformation-age technology will continue to grow in impor-
tance; pressed by rising international competition, industry
will demand increased quality and increased productivity.
The world of work in the twenty-first century will be less
manual but more mental; less mechanical but more elec-
tronic; less routine but more verbal; and less static but more
varied.
The changing nature of work will make continuing educa-
tion a lifelong reality for adults. Schools, therefore, will have
to provide all students with a strong foundation for lifelong
learning; colleges and universities will have to educate both
young adults and older workers; and industry will have to
focus its continuing education on areas that extend rather
Back to School
A newspaper interested in a tax
proposal conducted a poll using a
random sample of 100 voters on
each of two successive Mondays.
The first week they found 57 per-
cent in favor; in the second week,
they found 59 percent in favor.
The headline for its story read:
"Support Grows for Tax Hike."
Write a letter to the editor about
this story.
11

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Opportunity
"We canimprove. We
can keep kids in school
longer a1'dF achieve a het-
fer result. We cannot fad'
in education. If we fail,
we fad! our kids and we
fai'ourfuture."
12
I· . . -
n the future, schools and colleges will need
to meet goals that they now believe to be
impossible.
than repeat what schools provide. Education in the future
must build continually from childhood to retirement on a
versatile foundation provided by school education.
Literacy and numeracy are the primary sources of strength
and versatility in school education. As colleges, universities,
and continuing education attract significant fractions of the
population, schools must now prepare all students for some
type of postsecondary study. The level of literacy formerly
associated with the few who entered college must now be a
goal for all.
Quantitative literacy provides the foundation of techno-
logical expertise in the workplace. The facility with math-
ematics formerly required only of those preparing for sci-
entific careers is now an essential ingredient in the founda-
tion for lifelong work in the information age. In tomorrow's
world, the best opportunities for jobs and advancement will
go to those prepared to cope confidently with quantitative,
scientific, and technological issues. Mathematical Dower oro-
vides the key to these opportunities.
Students at Risk
~. . ~
Daunting as the challenge of reform appears, its cost is
insignificant when compared with the consequences of in-
action. Over 25 percent of all high school students drop out
before graduating; although half of these students do eventu-
ally receive an equivalency diploma, their detour is costly for
them and for society. Among Blacks, Hispanics, and Native
Americans, the dropout rate often exceeds 50 percent.
The majority of those who drop out are functionally il-
literate and hardly any of them possess enough mathemati-
cal skills to make productive contributions to the American
economy. Dropouts and illiteracy are destroying individual
Jangle Escalante

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...tappi1lg the power of mathematics
hope and threatening the foundation of this country's econ-
omy. Disparities that divide one third of our nation from
the rest compromise quality of life for all citizens.
Industry spends as much on remedial mathematics educa-
tion for employees as is spent on mathematics education in
schools, colleges, and universities. in addition, 60 percent
of college mathematics enrollments are in courses ordinar-
ily taught in high school. This massive repetition is grossly
inefficient, wasting resources that could be used better to im-
prove rather than to repeat mathematics education.
We are at risk of becoming a nation divided
both economically and racially by knowledge of
mathematics.
Broad-brush attempts to deal with underachievement in
mathematics often aggravate the very problems they are try-
ing to solve. Raising standards for graduation often widens
the gap between those who know mathematics and those who
do not, since increased standards are rarely accompanied by
program changes to provide appropriate courses for students
who are not motivated to study mathematics. Requirements
designed to ensure that all students reach certain minimum
levels often yield results in which most students reach only
the minimum level because rarely are schools given the re-
sources necessary both to achieve minimum standards for
all and to provide unlimited accomplishment for most.
Despite massive effort, relatively little is accomplished by
remediation programs. No one not educators, mathemati-
cians, or researchers- knows how to reverse a consistent
early pattern of low achievement and failure. Repetition
rarely works; more often than not, it simply reinforces pre-
vious failure. The best time to learn mathematics is when it
is first taught; the best way to teach mathematics is to teach
it well the first time.
"America is moving
backward no'
forward in its efforts to
achieve the fullparticipa-
tion of minority citizens
it' the life and prosperity
of the ration."
- One Third of a Nation

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Opportu1'ity
14
Apart from economics, the social and political conse-
quences of mathematical illiteracy provide alarming signals
for the survival of democracy in America. Because mathe-
matics holds the key to leadership in our information-based
society, the widening gap between those who are mathemati-
cally literate and those who are not coincides, to a frightening
degree, with racial and economic categories. We are at risk
of becoming a divided nation in which knowledge of mathe-
matics supports a productive, technologically powerful elite
while a dependent, semiliterate majority, disproportionately
Hispanic and Black, find economic and political power be-
yond reach. Unless corrected, innumeracy and illiteracy will
drive America apart.

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