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Mathematics today is approaching a state of cnSIS. As the demands of science and society for mathematical literacy increase, the percentage of American college students intending to major in mathematics plummets and achievement scores of entering college students continue thelt unremit­ ting decline. As research in core mathematics reaches unprecedented heights of power and sophistication, the growth of diverse applied special­ ties threatens to fragment mathematics into distinct and frequently hostile mathematical sciences. These crises in mathematics presage difficulties for science and engineer­ ing, and alarms are beginning to sound in the scientific and even in the political communities. Citing a trend towards "virtual scientific and techno­ logical illiteracy" and a "shrinking of our national commitment to excel­ lence . . . in science, mathematics and technology," a recent study con­ ducted for the President by the U. S. National Science Foundation and Department of Education warns of serious impending shortcomings in public understanding of science. "Today people in a wide range of non­ scientific . . . professions must have a greater understanding of technology than at any time in our history. Yet our educational system does not now provide such understanding. " The study goes on to conclude that present trends pose great risk of manpower shortages in the mathematical and engineering sciences. "The pool from which our future scientific and engineering personnel can be drawn is . . . in danger of becoming smaller, even as the need for such personnel is increasing. " It is time to take a serious look at mathematics tomorrow.

Inhaltsverzeichnis

Frontmatter

Introduction

Introduction

Abstract
Mathematics today is approaching a state of crisis. As the demands of science and society for mathematical literacy increase, the percentage of American college students intending to major in mathematics plummets and achievement scores of entering college students continue their unremitting decline. As research in core mathematics reaches unprecedented heights of power and sophistication, the growth of diverse applied specialties threatens to fragment mathematics into distinct and frequently hostile mathematical sciences.
Lynn Arthur Steen

What Is Mathematics?

Frontmatter

Applied Mathematics is Bad Mathematics

Abstract
It isn’t really (applied mathematics, that is, isn’t really bad mathematics), but it’s different.
Paul R. Halmos

Solving Equations is Not Solving Problems

Abstract
In 1955 I was a Ph.D. candidate at the University of Chicago completing a dissertation in Topology under Shiing-Shen Chern. Although I had been trained exclusively in pure mathematics, jobs in industry were beginning to become available to mathematicians and I was interested. Some of my teachers expressed disapproval of such notions, but Professor Chern did not. He told me he believed that working on physical problems was interesting and difficult, and he encouraged me to keep an open mind. I found that I was curious to learn more about the applications of the mathematics I had studied, so upon graduation I took an industrial rather than a teaching position.
Jerome Spanier

The Unexpected Art of Mathematics

Abstract
There are moments so rare as to bring with them a different kind of time. An event occurs so charged with emotion and intensity that one’s biological clock stops, the background fades away, and the thing itself is seen frozen and close-up as through a zoom lens. You can hear the turn of the key as the scene locks itself deep inside your brain.
Jerry P. King

Redefining the Mathematics Major

Abstract
Mathematical methods and thinking permeate virtually all aspects of business, government, and academia today to an extent few could have imagined a generation ago. This growth is manifested in the omnipresent role of computers in today’s world and in the use of mathematical models and statistical analysis to plan everything from medical treatment to political speeches. Mathematical terms, such as “parameter,” “hypothesis,” and “unknown variables,” have become part of the business jargon. It used to be that most college students took mathematics courses solely for the intellectual discipline of studying mathematics. (Many schools required every student to take either two years of mathematics or two years of a foreign language.) Today, courses in the mathematical sciences are an integral part of most college majors. Business students, for example, are frequently required to study statistics, linear programming, and calculus because this mathematics and its associated modes of reasoning are widely used in management.
Alan Tucker

Purity in Applications

Abstract
When I was seventeen I regarded myself as a Pure Mathematician. There was perhaps arrogance in this—mathematically my academic record was not prodigious, and my purity was open to all sorts of doubt—but it is worth considering what I meant by it.
Tim Poston

Growth and New Intuitions: Can We Meet the Challenge?

Abstract
The mathematical sciences have changed significantly during the past few decades. The most obvious change is the enormous growth of mathematics. However, the most exciting and potentially beneficial movement may well be the extensive mathematization of many traditional as well as newly emerging disciplines. The consequent influx of rich new ideas and alternative intuitional sources can greatly rejuvenate and invigorate mathematics itself. It would thus appear that mathematics should, for some time into the future, exhibit a truly great scientific advance that would bring reasonable prosperity to both its individual practitioners and its supporting institutions. Nevertheless, there is ample evidence to indicate that these possibilities are not being realized, and the prospects for the future are much less encouraging. The most obvious illustration of this is the gross mismatch between the mathematics that students are currently being taught and the skills that are marketable to most current users of the subject.
William F. Lucas

Teaching and Learning Mathematics

Frontmatter

Avoiding Math Avoidance

Abstract
The 1970’s could well be described as the decade of anxiety. The action of the OPEC countries in quadrupling the price of oil in 1973 signalled the beginning of an era of profound economic uncertainty for the advanced industrial nations of the world; and the decade closed with 52 American citizens held hostage in the United States embassy in Teheran and more than 50,000 Soviet troops deployed in Afghanistan in support of a puppet regime. Politically and economically, the decade of anxiety could be the precursor of the decade of despair.
Peter J. Hilton

Learning Mathematics

Abstract
A year ago we undertook the task of designing a mathematics course for freshmen who are admitted to a liberal arts college, but who are seriously limited by deficiencies in their pre-college mathematics education. We are not alone; many of our colleagues throughout the nation are involved in similar attempts and presumably are as naive as we are in matters related to this new assignment, such as the psychology of learning, education in elementary and secondary schools, and methods of evaluating students, teachers and courses.
Anneli Lax, Giuliana Groat

Teaching Mathematics

Abstract
Each person has at any given time a private reality, a private realm of the meaningful. To teach someone is to extend this reality, to enlarge it.
Abe Shenitzer

Read the Masters!

Abstract
It is as good an idea to read the masters now as it was in Abel’s time. The best mathematicians know this and do it all the time. Unfortunately, students of mathematics normally spend their early years using textbooks (which may be, but usually aren’t, written by masters) and taking lecture courses which are self-contained and make little or no reference to the primary literature of the subject. The students are left to discover on their own the wisdom of Abel’s advice. In this they are being cheated.
Harold M. Edwards

Mathematics as Propaganda

Abstract
One night several years ago while watching TV, I was surprised to see a mathematical equation make an appearance on the Tonight Show. The occasion was an interview with Paul Ehrlich, author of The Population Bomb and popularizer of population control as a solution to the world’s problems. At that time the ecology movement had just started to capture the attention of the public, and Mr. Ehrlich was arguing that the solution, as always, was in population control.
Neal Koblitz

Mathematicians Love Books

Abstract
Many of us remember certain books, often specific pages, their layout, the arrangement of definitions, theorems and proofs. These were the passages that influenced our development as mathematicians. Such experiences make us identify with yellow, green or blue books for the rest of our lives as mathematicians; they form our ideas of style and typography.
Walter Kaufmann-Bühler, Alice Peters, Klaus Peters

A Faculty in Limbo

Abstract
Have our still very young two-year colleges changed too rapidly during the last 20 years? Could it be that their remarkably fast enrollment increases and greatly broadened program focuses have produced change faster than most faculty can cope with? Has this growth and change been so swift that mathematics faculty of two-year colleges are in limbo and becoming recluses?
Donald J. Albers

Junior’s All Grown Up Now

Abstract
Exactly ten years ago this month I was helping to write an editorial for the Journal of the New York State Mathematics Association of Two-Year Colleges on essentially the same topic as this article. Entitled “University Dominated,” the editorial expressed the frustration of a growing number of junior or two-year college mathematics educators with the lack of an honest effort by established associations (The Mathematical Association of America and the National Council of Teachers of Mathematics) to recognize and deal with the unique problems of two-year college mathematics teachers.
George M. Miller

NSF Support for Mathematics Education

Abstract
The National Science Foundation Act of 1950 created a new Federal agency with two principal missions: providing needed support for basic scientific research and increasing both the quality of instruction in science and the number of scientists trained. The first mission was to encourage broad classes of fundamental research comparable to that which various agencies had used to accomplish military objectives in World War II, which ended five years earlier. War experience had shown the frequent dependence of applied research and development upon previous basic research often done by scientists or mathematicians when no definite applications were apparent. The mission in science education resulted in part from the rapid growth of education following the return of the veterans and the support of the GI bill. This growth was imposed upon a base of depleted faculty, many of very marginal qualifications, and outmoded equipment and curricula in the sciences. Also the booming postwar economy and rapid advances in science and technology had accelerated the need for well trained scientists and engineers.
E. P. Miles

Issues of Equality

Frontmatter

The Real Energy Crisis

Abstract
She stood at the podium wearing a pastel dress stiffened by yards of crinolines, hands shaking a little, internal butterflies fluttering. The audience of about 300 of her classmates and 1,000 parents and teachers was hushed and awaiting the beginning of the annual ninth grade promotion exercises. She took a deep breath and began her speech:
The launching of the first Sputnik by the Russians and the sudden awareness of Soviet superiority in one area of science has had the effect of a stunning surprise on the American public... . Our schools will be called upon to supply an increasing number of persons with a wide range of skills, many in new fields resulting from the growth of technology. The welfare and security of our people as a whole may well depend on the extent to which we are able to educate each young man and woman to his or her full capacity... .
Eileen L. Poiani

Women and Mathematics

Abstract
In any history of mathematics one is hard pressed to find mention of women mathematicians. One wonders if there were any and, if any, why so few? To answer this question, it is perhaps necessary only to consider the lives of some of the few there were. We do this in the first part of the paper, when we look at the lives of six women mathematicians from six different countries, the first born in 1718, the latter dying in 1935.
Alice T. Schafer

Spatial Separation in Family Life: A Mathematician’s Choice

Abstract
The media have recently been reporting some strange stories. Are we witnessing the next step in the Women’s Movement? Married couples are living apart in order to practice their careers. The papers seem to assure us that these couples intend to do this for at most a year or two—and not at all if children are involved. Interviews with husband and wife reveal that both believe they must live apart if they are to succeed professionally; the job market is tight. I sometimes wonder whether these couples realize what might be in store for them.
Marian Boykan Pour-El

Mathematics for Tomorrow

Frontmatter

Applications of Undergraduate Mathematics

Abstract
In recent years there has been a phenomenal growth in the professional use of mathematics, a growth so rapid that it has outstripped the capacity of many courses in our schools and colleges to train people for the mathematical tasks that are expected of them when they take employment. People who take jobs with the civilian government, the military, or industry, or who enter quantitative fields as graduate students or faculty, discover with increasing frequency these days that they lack acquaintance with important mathematical models and experience in modeling. Many of them also find to their distress that they have not been trained to be self-educating in the application of mathematics.
Ross L. Finney

The Decline of Calculus—The Rise of Discrete Mathematics

Abstract
Calculus is one of the great triumphs of the human intellect. For this reason alone no educated person should be without some knowledge of it. When, in addition, you consider the panoply of intellectual and practical conquests of classical analysis, whose foundation is calculus, it is small wonder that calculus has been for so long the basis of all college mathematics study. It may well surprise the reader then that the purpose of this essay is to argue that the position of calculus in the college mathematics curriculum is ripe for change and, to a degree, decline.
Anthony Ralston

Mathematical Software: How to Sell Mathematics

Abstract
When faced with the question, “What does a mathematician do if he doesn’t teach?,” most people have one of three answers: “I don’t know.” “He solves problems.” Or, “He analyzes data.” The average person does not think in terms of mathematical modeling or qualitative analysis, much less in terms of abstract theories. This, coupled with the prevailing view that university mathematics is strictly theoretical (ivory tower), and hence of no practical value, sometimes makes the justification of support for research in mathematics a trying undertaking. Outside of the National Science Foundation, managers in industry and government do not always respond well to vague arguments of “eventual” applicability.
Paul T. Boggs

Physics and Mathematics

Abstract
Mathematics and physics are closely related as disciplines. Their histories are intertwined, sometimes in the person of a single figure such as Newton. Each has helped to give form and emphasis to the other. The practice of physics requires a broad knowledge of mathematics, and the mathematician who seeks wider understanding in his or her own field does well to become familiar with the classical areas of physics.
Hartley Rogers

Readin’, ‘Ritin’, and Statistics

Abstract
Statistics has, for a number of years, been the principle mathematical tool used by scholars in a number of disciplines (e. g., economics, psychology and medicine). For example, a survey of a leading political science journal (see p. xiv of [3]) found that 65 percent of all the articles published between 1968 and 1970 used numerical data. This had grown from 12 percent in the period 1946–48. The uses (and abuses) of statistics are so pervasive that at least one course in statistics is now required by most of the undergraduate programs in the social sciences.
Tim Robertson, Robert V. Hogg

Mathematization in the Sciences

Abstract
The superb pictures of Jupiter and Saturn transmitted back to earth by Pioneer 11 impress me as a marvelous technological achievement. Even more impressive is that this was accomplished without several prior attempts. It is certainly the case that subsystems were tested, and that engineers benefited from their experiences with related systems. Nevertheless, the success of this effort in the absence of the usual testing and refinement is remarkable. Contrast this success with the notorious inaccuracy of economic forecasts—despite the best efforts of knowledgeable people and great (dollar) incentives for accurate predictions. We accept success, especially technological success, in the physical sciences and engineering as a matter of course. We are not surprised—disappointed, perhaps, but not surprised—by the lack of success in areas more closely related to the life and social sciences. It is common to say that we understand (or that someone understands) the science and engineering of space probes, but we do not have comparable understanding of the economy or of many biological or social systems. What is frequently meant is that there are good mathematical models for the physical sciences, but that the models used in the life and social sciences are not nearly as effective. Let us examine this idea in somewhat more detail.
Maynard Thompson
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