# National Policy Statement

**The American Mathematical Society**

Founded in 1888 to further mathematical research and scholarship, the 30,000-member American Mathematical Society (AMS) fulfills its aims through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and everyday life.

**Purposes of the National Policy Statement**

- To articulate public policy issues of significance for the mathematical sciences.
- To inform public policy makers and the public about these issues.
- To help formulate goals at the national level and set priorities for their accomplishment.

**The Mathematical Sciences**

Mathematics, the study of measurement, forms, patterns, and change, evolved from efforts to describe and understand the natural world. Over the course of time it has developed a rich and sophisticated intrinsic culture that feeds back into the natural sciences and technology, often in unexpected ways. Mathematics now reaches far beyond the physical sciences and engineering into medicine, business, and the life and social sciences. Its influence has been vastly enlarged by the advent of modern computers, whose use in problem-solving, simulation, and decision-making relies on powerful computational algorithms derived from continuing new developments in fundamental mathematical theory.

Research in the mathematical sciences is typically conducted by individuals working alone or in small, collaborative groups. Taken together, these individuals and groups constitute a human and intellectual resource of national significance. The AMS therefore considers the professional community of mathematicians a proper subject of national science policy.

Most researchers in the mathematical sciences are also educators: of scientists and engineers, of future mathematicians, of teachers at all levels, and, indeed, of nearly every post-secondary student. Mathematics is the most teaching-intensive of the sciences, reflecting its fundamental enabling role. Education has historically been a high professional calling for mathematicians, one that is now being reinvigorated as the nation's educational system is reformed at all levels.

Research within mathematics, the application of mathematics in other disciplines, and the teaching of mathematics are interdependent - nourishing each other with ideas, methods and inspiration. Individual mathematicians are typically involved in several of these activities. To be properly understood, mathematics must be viewed as a synergistic system in which none of these components can be neglected without weakening the others.

**Goals**

- Maintain the highest level of excellence in mathematical sciences research.
- Connect the power of mathematics and mathematical thinking to problems in science, technology and society.
- Attain excellence at all levels of mathematics education, giving particular attention to the professional development of teachers.
- Communicate the nature of the mathematical sciences and how mathematics contributes to society.

## I. Research in the Mathematical Sciences.

Substantial attention has been devoted recently to re-examining the rationale for federal support of scientific research. Much of the discussion stresses that federally funded scientific research should contribute to the improvement of society and to the pursuit of national goals. The mathematical sciences play such a fundamental role in all fields of science and engineering that their vitality is both essential to and dependent on a thriving atmosphere for research that addresses U.S. national needs.

Maintaining research strength across the breadth of the mathematical sciences, with the support necessary for clear world leadership, must be an integral component of federal science policy.

As the primary U.S. professional organization dedicated to the advancement of basic research in the mathematical sciences,

The American Mathematical Society has as its principal goal the highest level of excellence in mathematical sciences research.

Developments in the mathematical sciences often have major, multiplicative effects, direct and indirect, on other areas of science and technology. However, some parts of mathematics research do not produce an immediate, visible impact; in addition, it is impossible to predict reliably which mathematical underpinnings will be critical in the future. For these reasons, there is a serious danger of neglecting fundamental research in primary areas in favor of more directed payoffs. Such a policy would damage the long-term health of our nation's science and technology. The mathematics that underlies quantum mechanics, relativity theory, modern computers and the communications and information revolution, the CAT scan, modern economic theory, the mathematical analysis of DNA replication, the large scale computation of fluid dynamics or wave propagation, and other major achievements, was in many of these instances developed separately and well in advance of the application.

The AMS urges Congress and those federal agencies that support research and development to nurture the fundamental enabling role of the mathematical sciences by strong support of basic investigator-driven research.

Mathematics aims not only to solve specific problems, but also to find global and synthesizing structures that unify apparently disparate phenomena. The power and perspective afforded by such structures often contribute to the solution of problems previously viewed as unrelated and intractable.

Since major advances in mathematics typically involve sustained effort over a number of years, the AMS advocates that stable, reliable funding be provided for outstanding mathematical sciences research.

**National Science Foundation**

In the years following World War II, our national leadership wisely recognized the need to create an institutional refuge where basic science could be nurtured and sheltered from the winds of political change. That refuge is the National Science Foundation (NSF), the only federal agency primarily charged with sustaining the quality and vitality of basic research across all of mathematics, science, and engineering. The NSF is a relatively small, but absolutely vital and highly cost-effective investment that has served our nation well. The integrity of its enlightened founding mission is particularly important for the mathematical sciences.

The AMS urges the federal government to preserve the support of basic research and effective education of future scientists and engineers as the central missions of the National Science Foundation.

While it is appropriate for the NSF to encourage some thematic research and strategic initiatives, it is important to strike a proper balance. This is impossible to achieve when the budget process systematically shelters only strategic programs. If the economic environment imposes austerity, then it is all the more important that the foundation have the flexibility to manage programs without unhealthy distortion.

The AMS urges the federal government to promote a science policy that allows NSF flexibility to determine the appropriate balance among research programs.

**Mission Agencies**

The mission agencies have for many years supported a mixture of basic and focused research. The long-term effectiveness of these mission efforts depends on the health and continuing enrichment of the basic mathematical culture. The continued well-being of mathematics is therefore an indirect, but significant, concern of the mission agencies.

The AMS will encourage increased dialog with the mission agencies in exploring opportunities for broader participation by mathematical scientists, and in supporting the recruitment and training of talented young people in the mathematical sciences.

**Human Resources**

Many reports have recommended an increase in postdoctoral positions in the mathematical sciences. The 1992 report by the National Research Council (NRC), "Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States," documents the lack of such postdoctoral positions and describes them "as the logical step after completion of the doctorate for the good student, not as a highly competitive prize for a select few." The report concludes that postdoctoral fellowships "could form a bridge to future careers in which teaching or applications are important."

A suitably expanded postdoctoral program will contribute to the profession and to society by allowing more mathematics Ph.D.'s to establish productive careers in academic or industrial research, teaching, education, or the application of mathematics to other disciplines.

The AMS urges federal agencies to support expanded postdoctoral programs in the mathematical sciences to provide continuing professional development in research, education, and the applications of mathematics.

A significant human resource issue for the mathematical sciences is the continuing underrepresentation of women and minorities. A complex set of factors, taking place at all stages of the educational ladder, serves to discourage their participation and continuation.

The AMS is committed to working with other organizations and with federal agencies to seek and support programs and activities to increase the representation and advancement of women and underrepresented minorities in the mathematical sciences.

At present, two conditions have created an oversupply of Ph.D.'s in portions of the mathematical sciences. One is the retrenchment in educational institutions caused by the weak U.S. economy. The second is the large influx of foreign mathematicians produced by the events following the end of the Cold War and more open relations with China. The resulting difficult job market, coupled with the reduction in the number of supported investigators in core disciplinary areas, is having a profoundly depressing effect on young mathematicians.

The talent of the nation's young mathematicians is an important national asset. **We must find productive ways of using it.**

**Communication Networks**

A revolution in electronic communications and information retrieval has begun. As communications networks and technologies grow larger and more versatile, they will become an increasingly significant portion of the infrastructure for research and education. The AMS supports federal programs to increase the capabilities of electronic networks for transmission of text, video and sound, to develop more powerful tools for data retrieval, and to expand access to the networks to all portions of the research and education communities.

The AMS supports federal programs and policies that will make access to modes of electronic communication available to all researchers, educators and students and that will speed development of the electronic networks.

## II. Applications to Science and Technology

**Initiatives**

Federal initiatives, often encompassing support by several agencies, are a growing feature of science funding. The strategy of constructing broad research programs focused on major national problems has several obvious advantages: initiatives draw attention to problems, encourage multidisciplinary collaborations, and produce opportunities for intellectual cross-fertilization. Although the mathematical sciences are important in essentially all existing initiatives, active research mathematicians are often not involved in the initial design and planning. Participation by mathematical scientists at the early stages will contribute to the success of these initiatives.

The AMS urges federal agencies to ensure that active research mathematicians are included in the creation and planning of federal science initiatives.

The AMS, through its programs, publications, and meetings, will seek to facilitate the engagement of mathematical scientists in federal science and technology initiatives.

**Connections with Other Disciplines**

The research of many mathematical scientists is linked to other fields. For these mathematicians, cross-disciplinary contributions and mathematical achievements are inseparable. But the outreach of mathematics is not limited to settings where the interdisciplinary context is known in advance. A common experience of mathematical scientists is the discovery that some research in mathematics has been absorbed by other scientific disciplines to such an extent that the mathematical foundations are obscured. In the other direction, insights and techniques from many areas have inspired mathematical research that has taken on a life of its own, independent of the field of origin. It is the view of the AMS that many of the nation's problems could benefit from increased attention by mathematical scientists. However, a long history of collaborative experience indicates that substantial groundwork is often necessary to understand and define common issues. In the belief that mathematics research will continue to offer opportunities for productive and effective interdisciplinary activities,

the AMS will cooperate in working with federal agencies and policy-makers, and with university mathematical sciences departments, to enlarge the scope and extent of interdisciplinary research connecting mathematics and other fields. Such connections should also find expression in the mathematical training of both undergraduate and graduate students.

**Industrial Mathematics**

The needs of U.S. industry are increasingly cited as one of the primary justifications for federal support of scientific research. However, effective mechanisms for direct technology transfer are far from simple. The AMS favors increased interaction between mathematical sciences research and industry, and will work with other societies such as the Society for Industrial and Applied Mathematics (SIAM) to enhance these opportunities. Federal support for such interactions might take several forms. In particular,

the AMS supports programs that facilitate the creation of formal liaisons between industry and academic mathematical scientists, including industrial internships and postdocs.

## III. Mathematics Education

American education in mathematics and science is in a state of reform at all levels, kindergarten through graduate school. In the technological economy to which this country aspires, many citizens will require substantial technical knowledge and reasoning skills, and the flexibility to adapt to different jobs and even different careers. In particular, such competence must be broadly achieved by women and underrepresented minorities, the very populations for whom mathematics and science education has historically been least successful. This situation poses a challenge that must be addressed by the entire community of teachers, including those in colleges and universities.

A comprehensive and integrated reform -- of curricula, pedagogy, assessment methods, teacher professional development, and the cultural value placed on education -- is required. In scientifically oriented education, where mathematics is foundational and pervasive, mathematics educators have a special, even primary, responsibility in meeting this challenge. In this effort the AMS will join forces with other more educationally focused organizations, including the Mathematical Association of America (MAA), the American Mathematical Association of Two Year Colleges (AMATYC), the National Council of Teachers of Mathematics (NCTM), and the Mathematical Sciences Education Board (MSEB).

Mathematicians can play an important role in educational reform by furnishing disciplinary expertise that informs the development of curricula, assessment materials and disciplinary teacher training. They can also help to communicate the power and creative nature of mathematics, and to enliven the classroom experience with issues of contemporary research.

The AMS will encourage increased participation of research mathematicians in the comprehensive reform of mathematics education at all levels, and will support federal programs to facilitate that participation.

**Graduate Education**

American universities have led the world in training mathematicians in the core disciplines for careers as research-scholars in universities and research labs. However, the traditional academic job market is contracting. Mathematically trained students are increasingly finding employment in colleges without graduate programs, in two-year colleges, or in non-academic environments where mathematics is not the primary focus. Graduate programs in mathematics should, accordingly, provide students with more versatile professional skills, in forms that may vary among institutions and regions. Building upon the recent NRC report "Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States,"

the AMS will promote the enrichment of graduate programs in mathematics in order to provide students with a more versatile range of mathematically based professional skills.

This broadened graduate training might typically include courses in probability and statistics, theoretical computer science, and especially in pedagogy and communication. Such enhancement of core subjects could furnish a range and depth of skills at the Masters level for which there are increasing employment opportunities in a variety of fields.

The AMS encourages the development of strong Masters Degree Programs designed for diverse professional preparation, including teaching careers, based on sound training in the core mathematical disciplines.

**Undergraduate Education**

The changes in career opportunities and in technology that are affecting graduate education have prompted a rethinking of undergraduate pedagogy as well. The first efforts at undergraduate education reform have focused on calculus. Various pilot projects have already produced calculus curricula that are available for wider trial and development. More systematic review of the whole undergraduate program, for the general mathematics student as well as the major, is now being initiated.

The AMS, in cooperation with the MAA, SIAM, and other professional organizations in the mathematical sciences, supports efforts to review and reform the undergraduate mathematics curriculum, in response to changing student and national needs.

For teachers of mathematics, the primary model of professional instruction comes from undergraduate mathematics courses. In view of the pressing need for improved disciplinary training of school teachers, it is all the more important that we seek a high level of instructional performance in undergraduate courses.

The AMS encourages increased attention to and professional development of pedagogical performance of both faculty and graduate students.

Mathematical research has a largely unfulfilled contribution to make to undergraduate education. Ideally, all college-educated Americans should be aware of the liveliness of today's mathematical sciences research, which occupies a large international community of scholars and supports the technologies that continue to transform our world. Researchers can enhance this awareness by describing their own work and encouraging more research by students.

The AMS will foster wider understanding and appreciation of mathematics as a creative discipline, in particular through its presentation in undergraduate mathematics instruction.

The AMS endorses the value of undergraduate research experience in the mathematical sciences.

**K-12 Education**

The mathematics teaching community has initiated extensive reforms of mathematics and science education in the schools. This effort is aligned with the NCTM national standards for curriculum and teaching and the emerging standards for assessment.

These standards provide an unprecedented national framework to guide K-12 mathematics education reform; however, they should not be viewed as definitive or fixed. The research community has much to contribute in reviewing these standards, and much to gain by learning about and contributing to the pedagogical and assessment aspects of reform.

The AMS encourages increased participation of the research community in the national standards-based reform of K-12 mathematics education, particularly through strengthening the disciplinary foundation of the standards.

The professional development of teachers, both pre-service and in-service, is the central and most formidable task facing school mathematics reform. Mathematicians, through their disciplinary expertise, have much to contribute to that effort. They can help develop new curricular and assessment materials; they can work cooperatively with teachers and schools in their local communities; they can participate in workshops for teachers; and they can interact with school teachers in shared professional environments, such as the vertically integrated regional geometry institutes.

The AMS will support increased participation of mathematicians in programs for the professional development of teachers of mathematics.

**Valuing Education in the Mathematical Culture**

Just as scholarly research is professionally valued, professionally assessed, and professionally rewarded, so also must educational performance be valued, assessed, and rewarded if we are to achieve the educational quality now called for. Educational performance is here meant to include not only classroom teaching, but also activities such as curriculum development, program design, and educational research. A basic problem is to develop appropriate forms of professional assessment of teaching growth and effectiveness that, for educational performance, can function as do peer-review and archival publication for scholarly research. To this end, and with the goal of ultimately achieving this important change in our professional culture,

the AMS will stimulate discussion and experimentation in the mathematics community leading towards appropriate forms of professional assessment of educational performance.

## IV. Communication

The AMS recognizes the challenges in conveying to other scientists and engineers, to the public, and to policy makers the nature of the mathematical sciences, how they are serving the goals of society, and how in the future they will serve these goals in new and remarkable ways. The potential value of such awareness is illustrated by well known examples of scientists in other disciplines discovering that "the mathematicians have been here before." A significant goal of the AMS is to reduce the time for assimilation of mathematical results into other disciplines.

The AMS will promote understanding of mathematics by encouraging the output of high-quality expositions for students at all levels, the general public, policy makers, and other scientists and engineers.

As a special part of this effort at communication,

the AMS will join with other professional organizations in the mathematical sciences to stimulate the production of expository articles on the nature of the fundamental ideas of mathematics and their pervasiveness in modern life.

A particular challenge will be to convey to the public, and especially to children and to the teachers of children, that mathematics is a creative discipline involving discovery in which they can participate.

Mathematical inquiry is a fundamental mode of human thought. Its roots go too far back in pre-history to trace, but an unbroken chain of such inquiry has continued for more than two thousand years, from Greek civilization at the time of Euclid. The end of this chain, modern mathematics, is not only a subject of tremendous usefulness and a basis for investigations in many fields, scientific and otherwise, but is itself one of the great products of the human intellect.

The AMS will foster public understanding of the beauty and power of mathematics and its role as a fundamental mode of human thought.

**Toward the year 2000**

The passing of a millennium is an extraordinary occasion for reflection and challenge. The International Mathematical Union (IMU) has declared the year 2000 to be World Mathematics Year.

The AMS will join in this international effort to review the achievements of the mathematical sciences in the twentieth century, to celebrate the contributions of mathematics to human society, and to articulate some of the major challenges for mathematics as we enter the 21st century.

*Adopted by the Council in May 1994 so as to speak in the name of the Society*