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HUMAN
RESO URCES
Evidence from many sources points to an impending
shortage of mathematically trained personnel at every aca-
demic level, from high school graduates to Ph.D.'s. Too few
students enter college prepared to undertake the study of
mathematics necessary for their degree programs; too few
graduate with mathematics degrees to meet the needs of sec-
ondary school teaching or industrial employment; and too
few enter or complete graduate study to sustain the math-
ematical strength of universities and the research needs of
our nation. Moreover, many segments of the American pop-
ulation are significantly underrepresented at every stage in
the mathematics pipeline.
Fad
~ ~ ~ ~ e · e ~ e e e e e e · ~ ~ ~ - - · - - - - - · ~ - · ~ ~ - ·
~ o meet tomorrow's needs, we must invest
today in our nation's intellectual capital.
The underrepresentation of minorities and women in sci-
entific careers is well documented and widely known. Less
widely known is the general underrepresentation of Amer-
ican students in all mathematically based graduate pro-
grams. Evidence of disinterest in mathematics permeates all
racial, socioeconomic, and educational categories, although
the level of disinterest varies greatly among different groups.
Young Americans' avoidance of mathematics courses and
careers arises from immersion in a culture that provides
more alternatives than stimulants to the study of mathemat-
ics. Without motivation and elective opportunity to learn,
few students of any background are likely to persevere in the
study of mathematics.
As we enter a decade of decline in the number of college
graduates, concerns of equity join common cause with those
of economic need. Mathematical illiteracy both impedes so-
cioeconomic equality and diminishes national productivity.
it is a significant handicap from which neither the individ-
ual nor our nation can easily recover. The very magnitude
of the problem has forged a new coalition for change based
...investing in intellectual capital
Intended Mathematics Majors
of Top High School Seniors
PERCENT
7
6 \
2
o 1 1 1 1 1 1,,
1976 1978 1980 1982 1984 1986
Since 1975, the percentage of top high
school seniors who expressed an interest
in majoring in mathematics or statistics
has declined by over SO percent, even
as the corresponding percentage for sci-
ence and engineering remained relatively
constant.
17

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Human Resources
Shifting Student Interests
PERCENT
90
Develop a Meaningful
80 ~ph~losophy of
70 \
60 \
/ ~
50
40 ~
Be Very
30 We-Off
Financially
1967 1971 1975 1979 1983 1987
Data about life goals of college fresh-
men for the past quarter century show a
consistent trend away from philosophical
and scientific pursuits toward those that
over a promise of financial security.
"White males, though'
of only a generatiorl ago
as the mainstays of the
economy, will comprise
only IS percent of the
net additions to the labor
force between 19SS and
2000."
Workforce 2000
18
on shared interest between those who currently enjoy the
benefits of mathematical power and those who do not.
M athematical illiteracy Is both a
personal loss and a national debt.
Developing more mathematical talent for the nation wit
require fundamental chance in education.
~-
~. ~.
Our national
prOUlem IS not only now JO nurture talent once it surfaces,
but also how to make more talent rise to the surface. Al-
though more must be done, the United States is reasonably
successful in tapping and channeling the highly visible talent
springs which develop without special support from formal
schooling. But these sources are inadequate to our national
need. We must, in addition, raise the entire water table.
This is a much more massive problem, one that cannot be
attacked successfully through thousands of disconnected lit-
tle programs, beneficial as they may be for the individuals
affected. Although small local programs do sometimes sug-
gest directions for systemic change, too often their effects
remain strictly local. To raise the water table of mathemat-
ical talent, we must understand and change the system as a
whole.
Demographic Trends
During the next two decades, the number of 20- to 30-year-
olds in the United States will decline by about 25 percent,
even as the number of school-age children increases by a
similar amount. Hence, the demand for mathematics teach-
ers will rise just as the pool from which new teachers can be
drawn will be shrinking.
in the long run, the most important factor affecting educa-
tion is the changing profile of students. By the year 2000, one
in every three American students will be minority; by 2020,

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·.
today's minorities will become the majority of students in
the United States. Of those under iS, the proportion of
minorities is already nearing 40 percent, almost three times
what it was just after World War IT. Already, the ten largest
school districts in the United States are 70 percent Black
and Hispanic. Because of high birth rates and regular im-
migration, the Hispanic population in the United States is
growing at five times the national average. For as far ahead
as we can reliably project, the percentage of minority chilL-
dren in America will continue to grow.
In addition, according to Census Bureau estimates, 60 per-
cent of children born in the ~ 980's will, before reaching the
age of ~ 8, live in a home with only one parent. More than
one child in four comes from a family that lives in poverty;
nearly one in five comes from a home in which English is
not spoken; and one in three comes home to an empty house,
with no adult to greet the child and encourage attention to
homework. The children of today's high school dropouts are
the high school students of the first decade of the twenty-first
century.
The work force, too, is changing character. Whereas for-
merly 90 percent of the work force was White, between now
and the beginning of the twenty-first century one in three
new workers will be minority. Because of the general aging
of the population as the postwar baby boom reaches retire-
ment age, the United States will soon reach an all-time low
in the number of workers per retiree; instead of ~ 5 workers
supporting every retiree, as there were in 1950, there will be
only three.
Currently, ~ percent of the labor force consists of scientists
or engineers; the overwhelming majority are White males.
But by the end of the century, only ~5 percent of net new
entrants to the labor force will be White males. Changing de-
mographics have raised the stakes for all Americans. Never
before have we been forced to provide true equality in op-
portunity to learn. The challenge we face today is to achieve
what we believe.
tin vesting in intellectual capital
"Every clay in America,
40 teenage girls give
birth to their third
child. "
Harold L. Hodigkinson
19

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Human Resources
Stand and Deliver
~ ~.?~ 11~ ~--1111~_ -
A:_-1
ail - e 1
Jaime Escalante, a teacher at
Garfield High School in Los An-
geles, shows what students can
do when properly motivated. "I
believe potential is everywhere."
During the last ten years, hun-
dreds of Escalante's students have
passed the Advanced Placement
calculus examination.
Voice of Experience
"Black engineering students were
invited to join the Black Hon-
ors Calculus Society, where they
were challenged with difficult
problems. Where before they
worked individually-not always
successfully- now they meet and
tackle much harder problems,
working in teams of two or three."
-Uri Treisman
20
Minorities
Non-Asian minorities (Blacks, Hispanics, and Native
Americans) are significantly underrepresented in all scien-
tific, engineering, and professional fields. The extent of un-
derrepresentation is in direct proportion to the amount of
mathematics employed in the field. For lack of proper foun-
dation in mathematics, Blacks, Hispanics, and Native Amer-
icans are shut out of many scientific and business careers.
Despite the growing size of the Black and Hispanic pop-
ulations, comparatively few individuals in these communi-
ties take degrees in fields that require advanced mathemat-
ics. Of Americans who receive bachelor's, master's, and doc-
toral degrees in the physical sciences (including mathemat-
~cs, physics, and engineering), 95 percent are Whites and
Asians. During the last fifteen years, the total annual num-
ber of American Blacks and Hispanics receiving a doctoral
degree in the mathematical sciences has averaged less than
ten.
Although demographic factors influence all aspects of edu-
cation, they affect mathematics education in especially trou-
blesome ways. Among the many subjects taught in school,
mathematics is probably the most universal, depending least
on a student's background and culture. As a result, math-
ematics education has, with few exceptions, been generally
exempt from public controversy based on religious or social
views. Indeed, mathematics has benefited from widespread
support of its value in general education. Yet at the same
time, precisely because mathematics has few links to issues
of belief, mathematical ideas are not transmitted in our cul-
ture in the same way as are theories of evolution or standards
of ethics.
School mathematics should, therefore, transcend the cul-
tural diversity of our nation. In fact, it does just the oppo-
site. In the United States, mathematics is primarily part of
upper- and middIe-cIass male culture. Except for shopkeeper
arithmetic of a bygone age taught in elementary school, few
parts of mathematics are embedded in the family or cultural
traditions of members of the many large "developing coun-
tries" that make up the American mosaic.

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...inresting in intellectual capital
Indicators of achievement and aptitude support this gen-
eral assessment. Students enrolled in advanced high school
mathematics courses come disproportionately from White
upper- and middle-cIass families. Differences in culture and
parental expectation magnified by differential opportunities
to learn imposed by twelve years of multiply tracked classes
produce vastly different evidence of mathematical power.
T -- - -- - - ---
~ nadequate preparation in mathematics
imposes a special economic handicap on
· . -
minorities.
The long-term effect of minority underrepresentation in
mathematics is magnified because so many mathematics pro-
fessionals are teachers. During the next decade, 30 percent
of public school children, but only 5 percent of their math-
ematics teachers, will be minorities. The inescapable fact
is that two demographic forces increasing Black and His-
panic youth in the classrooms, decreasing Black and His-
panic graduates in mathematics will virtually eliminate
classroom role models for those students who most need mo-
tivation, incentive, and high-quality teaching of mathemat-
ics. The underrepresentation of this generation of minorities
leads to further underrepresentation in the next, yielding an
unending cycle of mathematical poverty.
Women
For reasons that are deeply rooted in culture and tradi-
tion, men significantly outnumber women in mathematics-
based careers. As students progress through the mathemat-
ics curriculum, girls and boys show little difference in abil-
ity, effort, or interest in mathematics until adolescent years
when course and career choices begin influencing school ef-
fort. Then, as social pressure increases and career goals
"The issue of the full
participation of women
in science is at the very
heart of the question of
who will do science in the
years ahead."
Sheila Widnall
21

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Human Resources
Distribution of Ph.D. Degrees
/White Males/
74%
\
White Females
19%
Non-White Males
/ Non-White
Females t%
White males earn three of every four
doctoral degrees in the mathematical
sciences awarded to U.S. citizens.
22
are formed, girls' decisions to reduce effort in the study of
mathematics progressively cut women off from many pro-
fessional careers.
Many women drop mathematics in high school or in the
transition to college. Others drop out later. Women perform
virtually as well as men in college mathematics courses, but
beyond the bachelor's degree women drop out of mathemat-
ics at twice the rate of men and at twice the rate of women in
other scientific disciplines. Women now enter college nearly
as well prepared in mathematics as men, and 46 percent
of mathematics baccalaureates go to women. Despite this
record, only 35 percent of the master's degrees and 17 per-
cent of the Ph.D. degrees in the mathematical sciences are
earned by women.
Overall, women receive approximately one third of uni-
versity degrees in science and engineering. The highest per-
centages of women are found in those sciences with the least
mathematical prerequisites: psychology, biology, and sociol-
ogy. The lowest percentages of women enter fields requiring
the most mathematics, namely, physics, engineering, eco-
nomics, geoscience, and chemistry. Evidence from many
sources suggests that it is differences in course patterns rather
than lack of ability that matter most in limiting women's ac-
cess to careers in mathematically intensive sciences.
Widely reported studies concerning the high percentage of
boys among mathematical prodigies those who at age 12
perform at the level of average college students- often con-
vey the impression that gender differences in mathematics
are biologically determined. But evidence from the vast ma-
jority of students shows almost no difference in performance
among male and female students who have taken equal ad-
vantage of similar opportunities to study mathematics. In-
ferences from very exceptional students child prodigies-
mean little about the performance of the general population.
Cross-national studies of gender differences in mathemat-
ics suggest that most of the differences observed are due to
the accumulating effects of sex-role experiences at home,
in school, and in society. The gender gap in mathematics
widens with increasing exposure to school and society; more-
over, in countries with more rigid curricula where mathemat

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.
ice courses are required and students do more homework,
gender differences are reduced significantly.
G· · · · · . . . . . . . . . . . . . . . . . . . . . . . . · . .
ender differences in mathematics
performance are predominantly due to the
accumulated effects of sex-role stereotypes in
family, school, and society.
Although American society is committed to equality of op-
portunity, public attitudes perpetuate stereotypes that "girls
can't really do math," that "math is unfeminine," and that
"girls don't need much math." As long as these stereotypes
persist, young women will continue to drop out prematurely
from mathematics education, thereby losing opportunities
for future careers. The nation cannot afford this loss, es-
pecially in view of the projected shortfall of mathematically
trained personnel. We must reduce societal factors that com-
promise a student's potential to learn mathematics.
Disabled Persons
Mathematics is a field well suited to offering opportunities
to disabled individuals. As a mental discipline, mathematics
requires only mental acuity for effective performance. Suc-
cess in mathematics depends neither on physical skills nor
on the physical means by which the worker communicates
his or her results.
The special opportunities afforded by mathematics for dis-
abled persons apply at all educational levels, in all fields of
mathematics, and to all areas of employment. In recent
years, the growing use of computers as an aid for disabled
people and as a too} for mathematicians provides yet an-
other effective link to enable disabled persons to succeed in
mathematics-based careers.
..investing in intellectual capital
"Although there are har-
riers for the disabled it'
mathematics, it is my im-
pression that these har-
riers are lower in math-
ematics that' it' many
otherfields."
I. Richard Savage
23

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Human Resources
Understanding the Universe
Stephen Hawking, a mathematics
professor at Cambridge Univer-
sity who holds the chair once oc-
cupied by Sir Isaac Newton, has
had amyotrophic lateral sclerosis
for the last twenty years. One of
this century's most brilliant sci-
entists, Hawking uses his mind
alone as a tool to explore the se-
crets of the universe.
24
M
athematics offers special opportunities
as a productive vocation for disabled persons.
Despite the potential of mathematics as a natural voca-
tion for persons with disabilities, far too few recognize or
act on this relationship. Many other countries are much
more effective in identifying and developing the mathemat-
ical talent of disabled persons. With as many as 10 percent
of the population disabled in some way, the nation can ill
adorn continued underrepresentation of disabled persons in
mathematical careers.
Graduate Students
Since 1970, the percentage of Americans studying math-
ematics in graduate schools has declined steadily so that
now Americans are frequently a minority in U.S. graduate
schools. As in engineering, fewer than half the mathematics
doctorates awarded by U.S. universities go to U.S. citizens.
Moreover, in the last ten years, the number of U.S. doctor-
ates in the mathematical sciences has dropped by nearly 50
percent.
As Americans drop out of mathematics, international stu-
dents converge on the United States to study mathematics-
based subjects. What our own students see as a burden, stu-
dents from other countries see as an opportunity. The result
is that American graduate schools in mathematically based
fields are enrolling ever higher percentages of international
students. in mathematics and engineering, nearly half the
graduate students are international and over half the gradu-
ate degrees now go to international students. This trend is
even more pronounced among the best graduate students in
the top graduate programs, where as many as three out of
four may come from overseas.
As a nation of immigrants, we should welcome the oppor-
tunity to be the schoolhouse to the world. Excellent graduate

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· -
programs-the best in the world-attract the ablest students
from around the world, especially from Third World nations
where opportunities for advanced study are limited. Many
international students remain in the United States to con-
tribute to our research efforts; others return to help raise
the mathematical standards of their own countries as am-
bassadors of American education. Either way, the United
States benefits from the opportunity to educate international
students; the cause of mathematics is advanced and the in-
sularity of America from the rest of the world is reduced.
T········ · · ·········-
oo few American students pursue graduate
study in the mathematical sciences.
Three aspects of international student enrollments do,
however, pose serious problems for U.S. mathematics. First,
we need to examine why U.S. students are not taking greater
advantage of graduate school opportunities in the mathemat-
ical sciences. The low percentage of American students in
graduate programs is a symptom of grave illness in Amer-
ican pregraduate education. It calls our attention to the
obvious that American mathematics education, especially
undergraduate mathematics, is not healthy.
Second, since international students who enroll in U.S.
graduate schools typically receive concentrated undergradu-
ate training in mathematics, they compete unevenly in grad-
uate courses with American students who often have had
a broad education with less specialization in mathematics.
The better preparation of international students discourages
many Americans who are trying to embark on career prepa-
ration in mathematics, leading to unnecessary dropout from
many graduate mathematics programs by capable American
students.
Finally, and of most concern to the general public, math-
ematics departments in universities more than any other
departments rely on graduate students to teach undergrad-
uates. International graduate students rarely make ideal
..investing in intellectual capital
Decline in Mathematics Ph.D.'s
Coon ~
900
Boo
500
400
200
100
:`
~: ~ ~ it:
:~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ :: ~ ~ ~ ~ ~
I:: :~ ~ :~:: ~ ~ ~ ~ ~ ~ ~ ~ it: :~ ~ :: ~:~:~ ~:~
1973 1975 1977 1979 1981 1983 1985 1987
Since 1970, the number of Ph.D. degrees
in the mathematical sciences earned by
U.S. citizens has declined by nearly SO
percent. The majority of new Ph.D. de-
grees awarded in U.S. universities now
go to foreign citizens.
25

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Human Resources
Undergraduate Instruction
The eject of international stu-
dents on undergraduate instruc-
tion continues well beyond gradu-
ate school. In 1987, 40 percent of
all full-time assistant professors
of mathematics in U.S. doctorate-
granting universities received
their baccalaureates from non-
U.S. universities, as compared
with 35 percent for engineering
and 20 percent for science over-
all.
Source: NSF Science Resources.
26
teachers for American freshmen, for reasons of language,
tradition, and background. Good teaching requires ex-
perience, empathy with one's students, and the ability to
communicate qualities that are rather uncommon in stu-
dents only recently arrived in this country. Moreover, stu-
dents from other nations cannot serve easily as role models
for American undergraduates; in many cases, international
students find it hard to understand how American students
can possibly be the way they are.
Were it not for the large numbers of international stu-
dents who study and teach in the United States, the state
of U.S. mathematics and science would be in total dis-
array. Under current circumstances, most universities have
little choice but to employ international graduate students
as teaching assistants.
Their budgets are not nearly suffi-
cient to staff undergraduate classes with Ph.D.'s; even if they
had enough money, the United States does not have enough
mathematics Ph.D.'s to fill all college and university posi-
tions. As long as universities employ graduate students to
teach undergraduates, the best way to improve mathematics
instruction for university undergraduates is to recruit more
well-qualified American students to graduate study in math-
ematics.
Supply and Demand
As science and technology become more mathematical,
demand by industry for mathematically trained persons in-
creases. Moreover, demand for mathematics teachers in-
creases with demand by industry, since most students who
study mathematics do so as background for another subject
rather than for majoring in mathematics.
We are now entering a period when the total number of
students, after a long decline, is beginning a fifteen-year rise.
At the same time, many teachers are nearing retirement, hav-
ing entered the profession in the heyday of the post-Sputnik
era. The consequences of these three factors rising num-
bers of students, rising demand for mathematics, and ris-
ing retirements of mathematics teachers will combine to
produce very strong demand for mathematics graduates well
into the next century. Some projections suggest that the

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...inYesting in intellectual capital
demand for high school mathematics teachers will double in
less than ten years.
The potential supply of mathematics teachers is, however,
rather weak. For nearly two decades, the number of students
receiving degrees in mathematics has been declining, falling
roughly 50 percent from its peak in the early 1970's. Al-
though the number of bachelor's degrees earned in the math-
ematical sciences has begun to rise in recent years, much of
this increase represents new joint mathematics-computer sci-
ence degrees whose holders usually enter computer careers
rather than careers in the mathematical sciences.
D· · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
emend for mathematics teachers at all
levels will exceed the supply of qualified persons
throughout the next decade.
Compounding the decline in the number of students se-
lecting degrees in mathematics is the decrease in absolute
The pool
numbers or Americans in their mlc-twentles.
from which future mathematicians and mathematics teach-
ers must come is shrinking and will continue to shrink until
nearly the end of the century. The number of Americans be-
tween 25 and 30 years of age, the age when individuals typ-
ically finish their graduate studies, will be 30 percent lower
in two decades than it is today.
Many variables influence the national marketplace for
mathematics teachers, including regional issues, licensure
requirements, degree awards, mathematics course require-
ments, economic conditions, and competition from private
industry. Because so many of these issues are specific to par-
ticular states or regions, geographic variation in supply and
demand is quite large; surpluses coexist with shortages in
a patchwork quilt of very different local markets for mathe-
matics teachers. Despite these variations, however, the over-
all pattern points to a decade or more of national shortage
at every educational level.
Bachelor's Degrees
x1 ,000
30
25
20
15
10
5
, ~
I...............
· ~ 1iN
O ~
1950 1956 1962 1968 1974 1980 1986
Between 1970 and 1985, the number of
undergraduates majoring in mathematics
and statistics fell by over SO percent. In
recent years, it has begun to rebound.
27

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Human Resources
Certification vs. Qualification
Data on certification show that
most classroom teachers are cer-
tified for the simple reason that
it is generally illegal for a prin-
cipal to employ a teacher with-
out proper credentials. Certifica-
tion comes in many forms- some
solid, some flimsy, some perma-
nent, some temporary. Qualifica-
tion by contemporary standards
means much more; it requires
solid preparation in the mathe-
matical sciences appropriate to
today's curriculum. Even teachers
who received substantial prepara-
tion twenty years ago may today
be unqualified although fully
certified-unless they have kept
up with new topics such as statis-
tics and computing that are now
important parts of school mathe-
matics.
28
Even if supply and demand were balanced, mathematics
teaching would face a serious shortage of teachers for a dif-
ferent reason: deficit financing of intellectual capital. When
demand for mathematics in universities increased sharply
during the last decade, most institutions responded either by
increasing class size or by hiring underqualified temporary
teachers part-time instructors, graduate assistants, and ad-
juncts with minimal qualifications and little continuing sense
of professional commitment. Even with no changes in stu-
dent enrollment or faculty retirements, it would take about
10,000 new faculty positions in mathematics just to restore
the intellectual and pedagogical vitality of our nation's col-
leges and universities.
T.
oo few mathematics teachers are prepared
to teach the mathematics their students need.
High schools, too, have been filling classroom positions
with teachers whose qualifications are substandard by reas-
signing teachers with little preparation in mathematics, and
none at all in modern mathematics, as demand for mathe-
matics increases. Few elementary school teachers are pre-
pared adequately in mathematics; typically, they take only
one of the four courses in mathematics recommended as ap-
propriate preparation for teaching elementary school math-
ematics.
Of the nation's 200,000 secondary school teachers of
mathematics, over half do not meet current professional
standards for teaching mathematics. Probably no more than
10 percent of the nation's elementary school teachers meet
contemporary standards for their mathematics teaching re-
sponsibilities.
Equity and Excellence
Two themes dominate every analysis of American
education the need for equity in opportunity end for

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...investing in intellectual capital
excellence in results. Although the goals of equity and excel-
lence sometimes appear to clash, in mathematics education
they converge on a single focal point: heightened expecta-
tions.
Equity for all students requires a full range of opportu-
nities that can stimulate each person to tap fully his or her
interests and capabilities. As students reveal different levels
of achievement and different rates of learning, the oppor-
tunities and context for the study of mathematics must be
continually adjusted to ensure appropriate stimulation and
reward for each student. Equity for all requires challenge
for all.
Excellence demands that students achieve all that they
are capable of accomplishing. National need underscores
this demand: our future depends as much on a steady flow
of strong and imaginative research leaders as it does on a
quantitatively literate work force. Excellence in mathematics
education demands results that unfold fully every person's
potential.
Many special programs seek to promote equity and excel-
lence. The best of these provide different levels of expec-
tation for students with different levels of need. Such pro-
grams educate all students well not by giving them identical
assignments but by setting for each child individually appro-
priate expectations. In such programs, there is no ceiling on
a child's aspiration, no perfect grade within easy reach.
w a----~---------.----------------~-
~quity for all requires excellence for all:
both thrive when expectations are high.
The range of accomplishment in programs devoted to eq-
uity and excellence demonstrates to students, teachers, and
parents the futility of limited expectations. As educators see
the surprising mathematical achievement of students who
are stimulated by challenges appropriate to their interests,
their entire outlook on appropriate expectations will change.
Rising expectations ensure equity and excellence for all.
Options for Excellence
Across the nation, special pro-
grams abound to enhance school
experience with mathemat-
ics, including special statewide
mathematics-science high schools,
after-school talented youth pro-
grams, urban clubs, magnet
schools, interscholastic problems
contests, summer institutes, and
industry internships. All seek to
expose young scholars to the chal-
lenge and excitement of mathe-
matics. By raising expectations
for all, such programs enhance
both equity and excellence.
" 'Twice as much, twice
as fast, twice as hard ' is
stat at' appropriate pro-
gram for highly tale1'fedF
stud~e1tts."
Harvey Keynes
29

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