Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.

Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 31

MA THEME TICS
Mathematics reveals hidden patterns that help us under-
stand the world around us. Now much more than arithmetic
and geometry, mathematics today is a diverse discipline that
deals with data, measurements, and observations from sci-
ence; with inference, deduction, and proof; and with math-
ematical models of natural phenomena, of human behavior,
and of social systems.
The cycle from data to deduction to application recurs
everywhere mathematics is used, from everyday household
tasks such as planning a long automobile trip to major man-
agement problems such as scheduling airline traffic or man-
aging investment portfolios. The process of "doing" math-
ematics is far more than just calculation or deduction; it
involves observation of patterns, testing of conjectures, and
estimation of results.
As a practical matter, mathematics is a science of pattern
and order. its domain is not molecules or cells, but num-
bers, chance, form, algorithms, and change. AS a science
of abstract objects, mathematics relies on logic rather than
on observation as its standard of truth, yet employs obser-
vation, simulation, and even experimentation as means of
discovering truth.
M:
~ ~ ~ athematics is a science of pattern and
order.
The special role of mathematics in education is a con-
sequence of its universal applicability. The results of
mathematics theorems and theories-are both significant
and useful; the best results are also elegant and deep.
Through its theorems, mathematics offers science both a
foundation of truth and a standard of certainty.
In addition to theorems and theories, mathematics of-
fers distinctive modes of thought which are both versatile
and powerful, including modeling, abstraction, optimiza-
tion, logical analysis, inference from data, and use of sym-
bols. Experience with mathematical modes of thought builds
searching for patterns
Mathematical Modes of Thought
Modeling Representing worldly
phenomena by mental constructs,
often visual or symbolic, that
capture important and useful fea
tures.
Optimization Finding the best
solution (least expensive or most
efficient) by asking "what if' and
exploring all possibilities.
Symbolism- Extending natural
language to symbolic represen-
tation of abstract concepts in an
economical form that makes pos-
sible both communication and
computation.
Inference Reasoning from data,
from premises, from graphs,
from incomplete and inconsistent
sources.
Logical Analysis Seeking impli-
cations of premises and searching
for first principles to explain ob-
served phenomena.
Abstraction Singling out for spe-
cial study certain properties com-
mon to many different phenom-
ena.
31

OCR for page 31

Mathematics
Back to School
Design a dog house that can be
made from a single 4 ft. by ~
ft. sheet of plywood. Make the
dog house as large as possible and
show how the pieces can be laid
out on the plywood before cut-
ting.
32
mathematical power a capacity of mind of increasing value
in this technological age that enables one to read critically,
to identify fallacies, to detect bias, to assess risk, and to sug-
gest alternatives. Mathematics empowers us to understand
better the information-laden world in which we live.
Our Invisible Culture
Mathematics is the invisible culture of our age. Although
frequently hidden from public view, mathematical and sta-
tistical ideas are embedded in the environment of technology
that permeates our lives as citizens. The ideas of mathemat-
ics influence the way we live and the way we work on many
different levels:
· Practical knowledge that can be put to immediate use in
improving basic living standards. The ability to compare
loans, to calculate risks, to figure unit prices, to understand
scale drawings, and to appreciate the effects of various
rates of inflation brings immediate real benefit. This kind
of basic applied mathematics is one objective of universal
elementary education.
· Civic concepts that enhance understanding of public pol-
icy issues. Major public debates on nuclear deterrence,
tax rates, and public health frequently center on scien-
tific issues expressed in numeric terms. Inferences drawn
from data about crime, projections concerning population
growth, and interactions among factors affecting interest
rates involve issues with essentially mathematical content.
A public afraid or unable to reason with figures is unable to
discriminate between rational and reckless claims in pub-
lic policy. Ideally, secondary school mathematics should
help create the "enlightened citizenry" that Thomas lef-
ferson called the only proper foundation for democracy.
· Professional skill and power necessary to use mathemat-
ics as a tool. Science and industry depend increasingly
on mathematics as a language of communication and as a
methodology of investigation, in applications ranging from
theoretical physics to business management. The principal

OCR for page 31

...searchi1lg for patterns
M athematics is a profound and powerful
part of human culture.
goal of most college mathematics courses is to provide stu-
dents with the mathematical prerequisites for their future
careers.
· Leisure- disposition to enjoy mathematical and logical
challenges. The popularity of games of strategy, puzzles,
lotteries, and sport wagers reveals a deep vein of amateur
mathematics lying just beneath the public's surface indif-
ference. Although few seem eager to admit it, for a lot of
people mathematics is really fun.
· Cultural the role of mathematics as a major intellectual
tradition, as a subject appreciated as much for its beauty
as for its power. The enduring qualities of such abstract
concepts as symmetry, proof, and change have been devel-
oped through 3,000 years of intellectual effort. They can
be understood best as part of the legacy of human culture
which we must pass on to future generations. indeed, it
is only when mathematics is viewed as part of the human
quest that lay persons can appreciate the esoteric research
of twentieth-century mathematics. Like language, religion,
and music, mathematics is a universal part of human cul-
ture.
These layers of mathematical experience form a matrix of
mathematical literacy for the economic and political fabric
of society. Although this matrix is generally hidden from
public view, it changes regularly in response to challenges
arising in science and society. We are now in one of the
periods of most active change.
From Abstraction to Application
During the first half of the twentieth century, mathe-
matical growth was stimulated primarily by the power of
l
``Ifyou want to under-
stand nature, you must be
conversant with the lan-
guage in which nature
speaks to users
- Richard Feynman
33

OCR for page 31

Mathematics
Strictly Speaking
MATHEMATICAL SCIENCES is a
term that refers to disciplines
that are inherently mathematical
(for example, statistics, logic, ac-
tuarial science), not to the many
natural sciences (for example,
physics) that employ mathemat-
ics extensively. For economy of
language, the word "mathemat-
ics" is often used these days as a
synonym for "mathematical sci-
ences," as the term "science" is
often used as a summary term for
mathematics, science, engineer-
ing, and technology.
34
abstraction and deduction, climaxing more than two cen-
turies of effort to extract full benefit from the mathematical
principles of physical science formulated by Isaac Newton.
Now, as the century closes, the historic alliances of mathe-
matics with science are expanding rapidly; the highly devel-
oped legacy of classical mathematical theory is being put to
broad and often stunning use in a vast mathematical land-
scape.
Several particular events triggered periods of explosive
growth. The Second World War forced development of
many new and powerful methods of applied mathematics.
Postwar government investment in mathematics, fueled by
Sputnik, accelerated growth in both education and research
Then the development of electronic computing moved math-
ematics toward an algorithmic perspective even as it pro-
vided mathematicians with a powerful too! for exploring
patterns and testing conjectures.
At the end of the nineteenth century, the axiomatization
of mathematics on a foundation of logic and sets made pos-
sible grand theories of algebra, analysis, and topology whose
synthesis dominated mathematics research and teaching for
the first two thirds of the twentieth century.
. · . ~. ~
These tradi-
onal areas nave now oeen supplemented oy major develop-
ments in other mathematical sciences in number theory,
logic, statistics, operations research, probability, computa-
tion, geometry, and combinatorics.
In each of these subdiscinTines~
__ 7 applications parallel
theory. Even the most esoteric and abstract parts of
mathematics number theory and Tocic. for example are
now used routinely in applications (for example, in com-
puter science and cryptography). Fifty years ago, the leading
British mathematician G. H. Hardy could boast that number
theory was the most pure and least useful part of mathemat-
ics. Today, Hardy's mathematics is studied as an essential
prerequisite to many applications, including control of au-
tomated systems, data transmission from remote satellites,
protection of financial records, and efficient algorithms for
computation.
., .

OCR for page 31

...searching for patterns
Mathematics is the foundation of
science and technology. Without strong
mathematics, there can be no strong science.
in 1960, at a time when theoretical physics was the central
jewel in the crown of applied mathematics, Eugene Wigner
wrote about the "unreasonable effectiveness" of mathematics
in the natural sciences: "The miracle of the appropriateness
of the language of mathematics for the formulation of the
laws of physics is a wonderful gift which we neither under-
stand nor deserve." Theoretical physics has continued to
adopt (and occasionally invent) increasingly abstract math-
ematical models as the foundation for current theories. For
example, Lie groups and gauge theories exotic expressions
of symmetry are fundamental tools in the nhv~icist's search
for a unified theory of forces.
~, _ ~
,
During this same period, however, striking applications
of mathematics have emerged across the entire landscape
of natural, behavioral, and social sciences. All advances in
design, control, and efficiency of modern airliners depend
on sophisticated mathematical models that simulate perfor-
mance before prototypes are built. From medical technology
(CAT scanners) to economic planning (input/output models
of economic behavior), from genetics (decoding of DNA)
to geology (locating of! reserves), mathematics has made an
indelible imprint on every part of modern science, even as
science itself has stimulated the growth of many branches of
mathematics.
Applications of one part of mathematics to another of
geometry to analysis, of probability to number theory-
provide renewed evidence of the fundamental unity of math-
ematics. Despite frequent connections among problems in
science and mathematics, the constant discovery of new al-
liances retains a surprising degree of unpredictability and
serendipity. Whether planned or unplanned, the cross-
fertilization between science and mathematics in problems,
"Equations are just
the boring par' of
mathematics. reattempt
to see thi1'gs it' terms of
geometry."
- Stephen Hawking

OCR for page 31

Mathematics
Myth: As computers become
more powerful, the need for
mathematics will decline.
Reality: Far from diminishing the
importance of mathematics, the
pervasive role of computers in
science and society contributes to
a greatly increased role for math-
ematical ideas, both in research
and in civic responsibility. Be-
cause of computers, mathematical
ideas play central roles in impor-
tant decisions on the job, in the
home, and in the voting booth.
36
theories, and concepts has rarely been greater than it is now,
in this last quarter of the twentieth century.
Computers
Alongside the growing power of applications of mathemat-
ics has been the phenomenal impact of computers. Even
mathematicians who never use computers may devote their
entire research careers to problems arising from use of com-
puters. Across all parts of mathematics, computers have
posed new problems for research, supplied new tools to solve
old problems, and introduced new research strategies.
Although the public often views computers as a replace-
ment for mathematics, each is in reality an important too}
for the other. Indeed, just as computers afford new opportu-
nities for mathematics, so also it is mathematics that makes
computers incredibly elective. Mathematics provides ab-
stract models for natural phenomena as well as algorithms
for implementing these models in computer languages. Ap-
plications, computers, and mathematics form a tightly cou-
pled system producing results never before possible and ideas
never before imagined.
Computers influence mathematics both directly- through
stimulation of mathematical research and indirectly by
their effect on scientific and engineering practice. Comput-
ers are now an essential too} in many parts of science and
engineering, from weather prediction to protein engineer-
ing, from aircraft design to analysis of DNA. In every case,
a mathematical mode! mediates between phenomena of sci-
ence and simulation provided by the computer.
Scientific computation has become so much a part of the
everyday experience of scientific and engineering practice
that it can be considered a third fundamental methodology
of science-parallel to the more established paradigms of
experimental and theoretical science. Computer models of
natural, technological, or social systems employ mathemati-
cally expressed principles to unfold scenarios under diverse
conditions scenarios that formerly could be studied only
through lengthy (and often risky) experiments or prototypes.
The methodology of scientific computation embeds mathe

OCR for page 31

...searchi1'g for patterns
matical ideas in scientific models of reality as surely as do
axiomatic theories or differential equations.
Computer models enable scientists and engineers to reach
quickly the mathematical limits permitted by their models.
Robotics design, for instance, often encounters limits im-
posed not by engineering details, but by incomplete under-
standing of how geometry controls the degrees of freedom of
robot motions. Models of weather forecasting consistently
reveal uncertainties that suggest intrinsically chaotic behav-
ior. These models also reveal our severely limited knowledge
of the mathematical theory of turbulence. Whenever a sci-
entist or engineer uses a computer mode! to explore the fron-
tiers of knowledge, a new mathematical problem is likely to
appear.
Computer models have extended the
mathematical sciences into every corner of
· . ~ . · · .
sclentl~c anc . engineering practice.
Whereas, traditionally, scientists and engineers who were
engaged primarily in experimental research could get along
with a small subset of mathematical skills uniquely suited to
their field, now even experimentalists need to know a wide
range of mathematical methods. Small errors of approxi-
mation that are intrinsic to all computer models compound,
like interest, with subtle and often devastating results. Only
a person who comprehends the mathematics on which com-
puter models are based can use these models effectively and
efficiently. Moreover, as a consequence of current limits on
computer models, further advances in many areas of sci-
entific and engineering knowledge now depend in essential
ways on advances in mathematical research.
The Mathematical Community
Because of its enormous applicability, mathematics is-
apart from English the most widely studied subject in
37

OCR for page 31

Mathematics
Back to School
Two banks are offering car loans
with monthly payments of $100.
One has an interest rate of 16
percent; the other has a higher
rate of 18 percent together with
a premium of a free color televi-
sion (worth $400~. If you need a
$5,000 loan and would really like
the color TV, which bank should
you choose?
38
school and college. Present educational practice for mathe-
matics requires approximately 1,500,000 elementary school
teachers, 200,000 high school teachers, and 40,000 college
and university teachers. Mathematics education takes place
in each of 16,000 public school districts, in another 25,000
private schools, in 1,300 community colleges, 1,500 colleges,
400 comprehensive universities, and 200 research universi-
ties. Roughly 5,000 mathematicians, principally those on the
faculties of the research universities, are engaged in research.
Only half of the nation's students take more than two
years of high school-level mathematics; only one quarter take
more than three years. That remaining quarter roughly one
million enter colleges and universities with four years of
mathematics. Four years later, about ~ 5,000 students emerge
with majors in mathematics. One quarter of these students
go on to a master's degree, but only 3 percent (about 400)
complete a doctoral degree in the mathematical sciences.
M
athematics is the nation's second-
largest academic discipline.
Just to replace normal retirements and resignations of high
school teachers will require about 7,000 to 8,000 new teach-
ers a year, which is half of the expected pool of ~ 5,000 math-
ematics graduates. Elementary school teachers, in contrast,
are drawn primarily from the three quarters of the popula-
tion who dropped mathematics after two or three courses in
high school. For many prospective elementary school teach-
ers, their high school experiences with mathematics were
probably not positive. Subsequently, teachers' ambivalent
feelings about mathematics are often communicated to chil-
dren they teach.
In sharp contrast to the eroding conditions of mathemat-
ics teaching, one finds enormous vitality and diversity in the

OCR for page 31

...searching for patterns
breadth of the mathematics profession. Over 25 different or-
ganizations in the United States support some facet of pro-
fessional work in the mathematical sciences. Approximately
50,000 research papers 20,000 by U.S. mathematicians-
are published each year in 2,000 mathematics journals
around the world. At the school and college level alone,
there are over 25 U.S. publications devoted to students and
teachers of mathematics. Students and faculty participate
in problem-solving activities sponsored by these journals as
well as learn about the ways in which current research can
relate to curricular change.
This massive system of mathematics education has had no
national standards, no global management, and no planned
structure despite the facts that each step in the mathemat-
ics curriculum depends in vital ways on what has been ac-
complished at all earlier stages and that scores of professions
depend on skills acquired by students during their study of
mathematics. Both because it is so massive and because it is
so unstructured, mathematics education in the United States
resists change in spite of the many forces that are revolution-
izing the nature and role of mathematics.
Undergraduate Mathematics
Undergraduate mathematics is the linchpin for revitaliza-
tion of mathematics education. Not only do all the sciences
depend on strong undergraduate mathematics, but also all
students who prepare to teach mathematics acquire attitudes
about mathematics, styles of teaching, and knowledge of
content from their undergraduate experience. No reform of
mathematics education is possible unless it begins with revi-
talization of undergraduate mathematics in both curriculum
and teaching style.
During the last two decades, as undergraduate mathemat-
ics enrollments have doubled, the size of the mathematics
faculty has increased by less than 30 percent. Workloads
are now over 50 percent higher than they were in the post-
Sputnik years and are typically among the highest on many
campuses. Resources generated by the vigorous demand for
undergraduate mathematics are rarely used to improve un
"Between now aids the
year 2000, for the firs t
time in history, a ma-
jority of all new jobs will
require postsecond~ary
education."
Workforce 2000
39

OCR for page 31

Mathematics
"Too many teachers over
mathematics on a take
it-or-leave-it basis in the
universities. The result is
that some of the brightest
mathematical minds elect
to lea ye it."
Edward E. David, Jr.
A Pipeline to Science
The undergraduate mathematics
major not only prepares students
for graduate study in mathemat-
ics, but also for many other sci-
ences. Indeed, nearly twice as
many mathematics majors go on
to receive a Ph.D. in another sci-
entific field rather than in the
mathematical sciences them-
selves.
40
dergraduate mathematics teaching. To administrators wor-
ried about tight budgets, mathematics departments are often
the best bargains on campus, but to students seeking stimu-
lation and opportunity, mathematics departments are often
the Rip Van Winkle of the academic community.
R· · · . . . . . . . . . . . .
form of undergraduate mathematics is
the key to revitalizing mathematics education.
During these same two decades, both the opportunity and
the need for vital innovative mathematics instruction have
increased substantially. The subject moves on, yet the cur-
riculum is stagnant. Only a minority of the nation's colle-
giate faculty maintains a program of significant professional
activity. Even fewer are regularly engaged in mathematical
research, but these few sustain a research enterprise that is
the best in the world. Unfortunately, those who are most
professionally active rarely teach any undergraduate course
related to their scholarly work as mathematicians. Mathe-
maticians seldom teach what they think about and rarely
think deeply about what they teach.
Departments of mathematics in colleges and universities
serve several different constituencies: general education,
teacher education, client departments, and future mathe-
maticians. Very few departments have the intellectual and
financial resources to meet well the needs of all these fre-
quently conflicting groups. Worse still, most departments
fait to meet the needs of any of these constituencies with
energy, effectiveness, or distinction.
Since almost everyone who teaches mathematics is edu-
cated in our colleges and universities, many issues facing
mathematics education hinge on revitalization of undergrad-
uate mathematics. But critical curricular review and revital-
ization take time, energy, and commitment essential in-
gredients that have been stripped from the mathematics fac-
ulLty by two decades of continuous deficits. Rewards of pro-
motion and tenure follow research, not curricular reform;

OCR for page 31

...searching for patterns
neither institutions of higher education nor the professional
community of mathematicians encourages faculty to devote
time and energy to revitalization of undergraduate mathe-
matics.
To improve mathematics education, we must restore in-
tegrity to undergraduate mathematics. This challenge pro-
vides a great opportunity. With approximately 50 percent
of school teachers leaving every seven years, it is feasible to
make significant changes in the way school mathematics is
taught simply by transforming undergraduate mathematics
to reflect the new expectations for mathematics. Undergrad-
uate mathematics is the bridge between research and schools
and holds the power of reform in mathematics education.
41

OCR for page 31