Thirty-five individuals from within and outside the mathematical sciences community, of whom a large majority constituted a broad, representative cross section of the community's leadership, participated in the May 1996 workshop "Actions for the Mathematical Sciences in the Changed Environment." In addition, 10 invited speakers presented their views to workshop participants regarding the general themes of new opportunities, the Washington environment, the view of scientists, perspectives on education, the roles and implications of national needs for the mathematical sciences, and major trends at research universities. A list of participants and the workshop agenda appear in Appendix A. Drafts of most speakers' papers (nine of which are presented above in this summary report) were provided to participants in advance of the workshop, along with a number of participants' individual position papers. Also, a copy of BMS's 1992 report *Educating Mathematical Scientists* (the "Douglas" report; NRC, 1992; available for reading on the World Wide Web at http://www.nap.edu/bookstore/pod/POD355.html), a summary report of NSF's recent workshop report *Graduate Education and Postdoctoral Training in the Mathematical and Physical Sciences* (NSF, 1996), a summary of the 1995 Committee on Science, Engineering and Public Policy (COSEPUP) report *Reshaping the Graduate Education of Scientists and Engineers* (NAS/NAE/IOM, 1995; also available online at http://www.nap.edu/readingroom/books/grad/index.html), and a photocopy of the (out of print) 1990 BMS report *Actions for Renewing U.S. Mathematical Sciences Departments* (available on the World Wide Web at http://www2.nas.edu/bms/215e.html) were sent to each participant as preparatory background for the event. Most of the speakers participated in subsequent workshop break-out group and plenary discussions.

From the large number of issues raised in speakers' presentations, background information, and extended workshop discussion (see, for example, Small Group Discussion background in Appendix A), participants focused on several broad topics for much of their considerations, including (1) defining departmental responsibilities and setting departmental goals, (2) broadening and improving education, (3) making connections with other disciplines, (4) rethinking faculty evaluation, (5) framing funding strategies, and (6) improving communication, and offered (7) closing observations.

In those contexts, workshop discussions addressed a number of themes.

Note: This summary reflects the workshop discussions as refined by further consultation with workshop participants, and subsequent NRC review.

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SUMMARY
Background and Overview
Thirty-five individuals from within and outside the mathematical sciences community, of whom a large majority constituted a broad, representative cross section of the community's leadership, participated in the May 1996 workshop "Actions for the Mathematical Sciences in the Changed Environment." In addition, 10 invited speakers presented their views to workshop participants regarding the general themes of new opportunities, the Washington environment, the view of scientists, perspectives on education, the roles and implications of national needs for the mathematical sciences, and major trends at research universities. A list of participants and the workshop agenda appear in Appendix A. Drafts of most speakers' papers (nine of which are presented above in this summary report) were provided to participants in advance of the workshop, along with a number of participants' individual position papers. Also, a copy of BMS's 1992 report Educating Mathematical Scientists (the "Douglas" report; NRC, 1992; available for reading on the World Wide Web at http://www.nap.edu/bookstore/pod/POD355.html), a summary report of NSF's recent workshop report Graduate Education and Postdoctoral Training in the Mathematical and Physical Sciences (NSF, 1996), a summary of the 1995 Committee on Science, Engineering and Public Policy (COSEPUP) report Reshaping the Graduate Education of Scientists and Engineers (NAS/NAE/IOM, 1995; also available online at http://www.nap.edu/readingroom/books/grad/index.html), and a photocopy of the (out of print) 1990 BMS report Actions for Renewing U.S. Mathematical Sciences Departments (available on the World Wide Web at http://www2.nas.edu/bms/215e.html) were sent to each participant as preparatory background for the event. Most of the speakers participated in subsequent workshop break-out group and plenary discussions.
Highlights
From the large number of issues raised in speakers' presentations, background information, and extended workshop discussion (see, for example, Small Group Discussion background in Appendix A), participants focused on several broad topics for much of their considerations, including (1) defining departmental responsibilities and setting departmental goals, (2) broadening and improving education, (3) making connections with other disciplines, (4) rethinking faculty evaluation, (5) framing funding strategies, and (6) improving communication, and offered (7) closing observations.
In those contexts, workshop discussions addressed a number of themes.
Note: This summary reflects the workshop discussions as refined by further consultation with workshop participants, and subsequent NRC review.

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Defining Departmental Responsibilities
Diversity in the focus of different mathematical sciences departments is desirable and healthy, but each department will necessarily define its unique role. An individual department is the best body to decide what it could or should do to meet its societal responsibilities and respond to the challenges that face the community.
This is a good time for a department to re-examine its responsibilities and goals within its local university framework. In so doing, it might delineate for itself what challenges it needs to address, and devise strategies for addressing them. Added to the omnipresent challenge of preserving the nation's pre-eminent strength in research and discovery in all mathematical sciences areas, there now are new challenges confronting mathematical sciences departments. They include educating large numbers of non-mathematical sciences majors; making the mathematical sciences more accessible to all students; training mathematical sciences majors in a manner that promotes career development in teaching, business and industry, government, or advanced study and research; and preparing mathematical sciences graduate students for diverse career options. Provided with the necessary resources, many departments might be even more successful in meeting these challenges than previously. Those departments that wish to have a way of demonstrating their progress, both for their own benefit and for communicating their accomplishments to others, might identify ways to assess performance or progress toward meeting these challenges.
Mathematical sciences professional societies are the best stewards to develop an effective means for accumulating, centrally depositing, and making available for general use (say, through the World Wide Web) valuable data of interest to many departments, as well as to the whole profession. In recent years, a number of mathematical sciences departments have developed creative or more effective ways of performing customary tasks and implementing innovative directions or methods.1 However, it takes considerable time for the larger mathematical sciences community to learn what works well, what has been or is being explored, and what changes are under way at various different institutions. Information about innovative departmental strategies—for instance, in curriculum and educational programs, as well as about unusually successful traditional ones—might be made more widely available as examples to emulate and analyze. A coordinated database of departmental strategies that are submitted in a standard format could be created at a single site on the World Wide Web, and availability of this database made known at national professional society meetings. Having such information posted in a consistent format on the World Wide Web would permit searches, an accumulation of data on mathematical sciences departments for continuing analysis, and a repository for archiving and accessing the experiences of different departments concerning innovation and experimental approaches in research, scholarship, and education. Such postings might also be
1
Examples include the University of Chicago mathematics department's new master's degree program in financial mathematics, and the University of Notre Dame mathematics department's new policy to have senior faculty who are the most effective lecturers teach the freshman-level courses for non-mathematics majors.

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attractive advertising for a department, and if available in a universal format could be very useful for departments in other institutions and the whole discipline.
Broadening and Improving Education
Key tasks for mathematical sciences departments are to provide undergraduate-level mathematical sciences courses that give all students an understanding of both the historical and current role of the mathematical sciences in society, that supply the mathematical understanding and skills students need to make informed decisions and to function in modern technological society, and that enhance students' appreciation of mathematics. Mathematical literacy has become increasingly necessary in many areas of society, most especially in industry and business.2 Since the mathematical sciences are broadly used throughout the economy, their importance continues to grow for an ever larger cross section of the population.3 Raising mathematical skills and awareness is a long-term effort to be pursued at all educational levels, including K-12, undergraduate, and graduate. Areas here include improving the mathematical preparation of school teachers by, for instance, working with the K-12 community in designing mathematical sciences courses for potential teachers. Courses can also be designed for students who are not mathematical sciences majors that convey the importance, power, and value of the discipline in modem society. All students benefit from instruction in quantitative reasoning and problem solving, and by learning what roles the mathematical sciences play in daily events as well as in modern science, medicine, and technology. Citizens who learn about and understand the mathematical sciences enhance their own and the nation's economic capabilities. "General education" courses are a quite effective avenue for communicating examples to the educated general public that show why the mathematical sciences are crucial for the nation, how they contribute in vital ways, and what highly leveraged and far-reaching societal benefits accrue from them.
Employment difficulties that many advanced mathematical sciences degree recipients have recently encountered (see, for example, Fulton (1996) and Benkoski et al. (1996)) suggest it might in some cases be beneficial to adjust the postbaccalaureate education program. Several studies and reports have raised concerns as to whether the median time to degree has become too long, and have called for adjustments in the way mathematical sciences graduate programs prepare students. For instance, the National Research Council's "Douglas" report, Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States (NRC, 1992), called for broad academic preparation of mathematical scientists, and (in an appendix) advised potential graduate students in the mathematical sciences to learn, among other
2
For example, in the communications industry, computing, much of the managerial sector, and the scientific and technological professions.
3
Technicians and manufacturing production supervisors, for instance, now increasingly deal in quantitative issues, logical thinking, and problem solving, and so need to be more mathematically prepared and numerically literate than in the past.

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things, what is the average length of time to a doctorate for a prospective mathematical sciences doctoral program. The report of a 1995 National Science Foundation (NSF) workshop, Graduate Education and Postdoctoral Training in the Mathematical and Physical Sciences (NSF, 1996), recommended that graduate departments broaden their educational programs by introducing more interdisciplinary courses and off-campus experience, and discussed the time to degree issue. The Mathematics in Industry report produced by the Society for Industrial and Applied Mathematics (SIAM, 1995) includes a number of other useful references.
The strength of America's graduate mathematical sciences enterprise is the envy of the world, and there is widespread, strong agreement that this strength must be carefully safeguarded whenever adjustments are made to it. Melding breadth—which reasonably involves the general skills used in employment settings (be they academic, commercial, industrial, or educational) where most of a program's graduates secure employment—with that strength, while striving to reduce the time required to earn a Ph.D. degree, is a difficult and major issue. Resolving the breadth-vs.-length-of-program issue might depend on additional resources being available or different infrastructure being put into place in order to not diminish a program's existing strengths. One possible time to address this issue might be when a department with a doctoral program undergoes external review. One approach might be offering mathematical sciences degrees other than the Ph.D., for instance, professionally respected master's degrees that business and industry value.4 Many successful master's degree and internship programs take advantage of geographically neighboring opportunities and local resources that are particular to an individual mathematical sciences department. A professional master's degree program may be appropriate, for instance, when a clear career path exists in the local industrial or business community for recipients of such a degree, and when the requisite departmental resources and infrastructure are available. Such an approach might help counter a potential national contraction of mathematical sciences graduate programs that informal indications suggest may be under way. Yet another possibility is to move toward mentored postdoctoral/internship experiences as a normal part of the professional development of mathematical sciences Ph.D.s.
One responsibility of the mathematical sciences community is to preserve and strengthen the mathematical sciences enterprise, including the research enterprise. However, serious concerns have been expressed that, while university mathematical sciences departments are and have been doing an outstanding job of discharging all their many responsibilities, they are now being expected to do even more while not being provided the additional resources or infrastructure needed. In response to these difficult circumstances, some departments are, for example, investigating the economic costs of swapping, two-for-one, graduate teaching assistants (TAs) for mentored postdoctoral positions, where the postdoctoral teaching load would equal that of two graduate TAs. Others are determining what kinds of employment their
4
For practical information on starting an industrial master's degree program—and how in general to develop involvement with industry—see, for instance, Friedman and Lavery (1993) and SIAM (1995).

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Ph.D. graduates are finding, and making available within their doctoral program opportunities for persons pursuing those careers.
Is the quality of education affected by the practice of having mathematical sciences courses in two- and four-year academic departments taught by individuals other than full-time mathematical sciences faculty? There is at this point only anecdotal evidence that economic pressures have prompted some mathematical sciences departments to staff courses with larger numbers of adjuncts or other individuals who are not full-time faculty. One result could be an increase in the departmental tasks and responsibilities that full-time faculty members must shoulder—advising students, serving on committees, developing curriculum, handling department administration, mentoring majors, and the like. The causes and effects associated with increased reliance on non-full-time faculty, especially the effects on quality of instruction, are unclear. It is worth investigating how widespread it is, and how to best address concerns associated with the practice.
Because the mathematical sciences are more important for society in a wider variety of ways than at any previous time, and because the mathematical sciences profession needs to be continuously strengthened and replenished by drawing well-educated individuals from the widest possible pool of talent, it is essential to ensure that women, underrepresented minorities,5 and students from educationally deprived backgrounds are encouraged both to study the mathematical sciences at all levels and to become part of the mathematical sciences community.6 Considerable progress has been made in recent years in attracting, retaining, matriculating, and employing a broader cross section of society in the mathematical sciences compared to former years. For example (Fulton, 1996), there are more women faculty in mathematical sciences departments than ever before (although their distribution among institutions is not yet uniform). To ensure continuing progress, particularly with respect to underrepresented minorities and educationally deprived students, approaches that work, both those with proven success and fresh approaches, need to be sustained, expanded, and replicated in other locations. One way to help propagate success might be for mathematical sciences professional societies to make broadly available from a central repository, for example, at a single site on the World Wide Web, information on what has succeeded.
Making Connections with Other Disciplines
While there has been a steady increase in mathematical sciences work linked to most major national needs as well as to almost every area of science, technology, and medicine (see, for instance, NRC (1991, 1993, 1995, 1996)), there is not yet sufficient professional recognition of these achievements, and the community is not well enough
5
These include, for example, African Americans, Native Americans, and Hispanic Americans.
6
Workshop participants noted that, in the near future, a separate workshop might propose explicit mechanisms addressing the issue of participation in the mathematical sciences.

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informed about them. Sometimes the mathematical ideas involved in interdisciplinary research are very recent mathematical achievements, such as the concepts of group theory in coding and communication, or the theory of wavelets. Other times they evolve out of applied or interdisciplinary work that feeds back into the mathematical sciences and becomes general mathematical knowledge. This has recently occurred in four-dimensional geometry and mathematical physics, between topology or statistics and the Human Genome Project, and between various mathematical areas (including topology, complex analysis, probability and Statistics, and measure theory) and fractals in analysis, compression of data, and nonlinear dynamics. Mathematical sciences students at all levels should be as aware of such interdisciplinary progress as they are of efforts within their own discipline. Although a number of mathematical sciences programs offer interdisciplinary courses at all levels cooperatively with other departments, many more could do so. To involve existing mathematical sciences faculty in these courses, professional societies could periodically offer appropriate short courses to broaden the capacity of such faculty (analogous to what was done in the 1970s to address a need for faculty with computer science training). Mathematical sciences degree holders will be much better ambassadors for their discipline if they are acquainted with an understanding of what interdisciplinary work involves and an appreciation of the value their discipline brings to other areas. Mathematical sciences professional societies might establish awards and prizes to recognize interdisciplinary achievement by mathematical scientists, and encourage mathematical scientists who participate in outstanding interdisciplinary research to publish articles on their work for the mathematical sciences community.
Rethinking Faculty Evaluation
A pivotal issue that strongly affects the future of the mathematical sciences community is evaluation of its academic members. The activities, attitudes, and attributes deemed valuable in an academic setting set a tone that is reflected throughout the community. Since most mathematical sciences departments now function in circumstances quite different from those of 30, 20, or even 10 years ago, faculty evaluation criteria may need to evolve for a department to appropriately and successfully address current and future expectations, responsibilities, and goals. Due to the changing environment, some departments may change their hiring practices, for example, hiring faculty who pursue interdisciplinary research or mathematicians who combine an interest in curriculum development and innovative instruction with research in the mathematical sciences. Some workshop participants felt that there may be a need to rethink the ways in which mathematical sciences faculty are evaluated for promotion, tenure, and salary increases, echoing issues raised in the Joint Policy Board for Mathematics' report Recognition and Rewards in the Mathematical Sciences (JPBM, 1994). If new interdisciplinary areas are incorporated into a department's goals, then that department needs to adopt criteria suitable for these areas. With an increased emphasis on quality of instruction, and the importance of mentoring minority and women students, reliable ways to assess teaching

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effectiveness may be needed. Professional societies can be catalysts for this rethinking by identifying and widely communicating case studies and findings of proven practical techniques and approaches. Any evolution in faculty evaluation needs to strengthen a department. All parties involved (including the mathematical sciences faculty—specially new faculty—and university administrators, and the faculties of other departments) have to clearly understand what is changing and the reasons why. Since the issue of faculty evaluation has broad repercussions for the entire community, the various opinions, developments, and associated impacts need to be publicly discussed and widely communicated for the benefit of all community members. For example, panel discussions at annual meetings might be devoted to faculty evaluation in a changing environment, and might involve both faculty and administrators. There could then be follow-up reports on the discussions in professional society news publications.
Framing Funding Strategies
The present outlook for mathematical sciences funding is one of constraint and possibly diminishment. Furthermore, previous National Science Foundation disciplinary advisory boards no longer exist due to a 1993 Executive Order. To ensure the continuing health of the discipline, most workshop participants considered it crucial that a group of representatives of the discipline be designated to work with funding agencies, particularly the National Science Foundation, in judging the adequacy of research support for the mathematical sciences, and in developing strategies to make the best use of available funds. Many participants felt that mathematical sciences professional societies could help the National Science Foundation and other agencies make the case for the mathematical sciences based on national needs by establishing such a balanced, representative, formal group to liaise with funding agencies, and to routinely supply them with high-quality, thoughtful community input for that demanding and delicate decision making. Strategies for innovative funding might be designed to stimulate and support the mathematical sciences community's goals in research (both within the mathematical sciences and in interdisciplinary areas), in education, and in outreach programs. Contributing to this decision-making process will manifestly increase the mathematical sciences community's credibility, recognition, respect, and influence with government leaders.
Improving Communication
The mathematical sciences community needs to convey the excitement and importance of the mathematical sciences to the public. When new results and their potential ramifications are announced within the community, they could also be communicated to the general public at a level that is broadly understandable by nonspecialists. Regular, but carefully prepared and presented, communications could be specifically

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tailored for that general public to impart a modicum of understanding, a sense of appreciation, and an impression of how the mathematical sciences contribute to the benefit of society. The value of the discipline's research and application achievements needs to be expressed to the public in terms the public will understand. Efforts in these directions such as, for example, the National Academy of Sciences' Beyond Discovery series (see its Web page at http:/www2.nas.edu/bsi/) or the American Institute of Physics' WonderScience series (see its Web page at http://www.acs.org/edugen2/education/ws/ws.htm) need to be emulated by the mathematical sciences community. The use of imperfect analogies can be very useful in this. Because of the importance of public communication efforts, mathematical sciences professional societies might identify and publicize possible sources of support for such activities. A ''high-leverage" continuing opportunity is offered by entry-level undergraduate courses, which often provide the sole chance to expose the full panorama of the mathematical sciences to a majority of the general public. To significantly improve the teaching of and the effective communication with non-specialists will require concerted efforts by individuals from all sectors of the mathematical sciences community.
A number of examples were discussed by workshop participants of possible ways to improve the discipline's communication with the public, some of which are already being acted on by certain members of the community. They include:
Having senior mathematical scientists routinely write expository and/or survey articles about accomplishments and breakthroughs in mathematical sciences research or education. These articles could target different sorts of audiences, ranging from junior high school students through mathematical sciences Ph.D.-holders, as well as the public at large.
Having members of the mathematical sciences community collect, present, and publish expository essays, written at appropriate levels, that trace the impact of the mathematical sciences and vignettes on their role in new developments as well as in history.
In all communication venues, emphasizing aspects of the mathematical experience that involve human interest, utilizing television and videos whenever possible, and providing role models from underrepresented groups.
Interacting with the local community, for example, by giving talks at schools, working with teachers in developing classroom materials, initiating a teacher-exchange program, or inviting business and industry representatives to present problems to mathematical sciences groups.
Posting information on the World Wide Web of interest to and presented at an appropriate level for the general public, and publicizing its availability.
Devising mathematical computer games or other captivating or entertaining software based on aspects of the mathematical sciences.

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Attending and giving intra- and interdepartmental colloquia, which are an essential means of communicating the mathematical sciences to colleagues, to other disciplines, and to the public.
Creating a variety of courses to raise mathematical awareness for non-mathematical sciences majors, as well as other courses addressing mathematical literacy.
Creating and taking advantage of opportunities to serve in policy-making positions or in other positions that influence public policy; and
Visiting local, state, and federal legislators, but only after preparing for such a visit by using (say) the Joint Policy Board for Mathematics' legislator-visit kit.
Closing Observations
To preserve the discipline's strengths while meeting challenges posed by a changed environment, the U.S. mathematical sciences community's major national responsibilities include:
Promoting research and discovery in all areas of the mathematical sciences so that the U.S. will continue to maintain preeminence in these powerful, innovative endeavors;
Supporting the use of mathematical results and acumen in order to increase the capabilities of the various scientific and engineering disciplines and to help advance science and technology in the United States; and
Providing top-quality education and skills in the mathematical sciences at all levels and for all members of the student population to ensure that the United States will continue to have a scientifically and technically trained work force and an informed citizenry.
Fulfilling these responsibilities is doubly challenging in the current changed environment. The end of the Cold War has increased the national emphasis on economic competitiveness and pressing social concerns, health care issues, and environmental needs. One consequence of this shift in national priorities is that a strong scientific base, as an end in itself, is not as high a priority as it was. Another change affecting the environment for science, and for the mathematical sciences, involves the national resolve to balance the federal budget and to cut government spending.
There are several implications of these changes. In the near future there may be less public money (in constant dollars) for support of universities and of scientific research. Also, the scientific community may have to justify much of its research as contributing to societal goals. For example, quality education, especially K-12

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education, for all segments of society remains high on the national agenda and is a goal to which the scientific community is expected to contribute. What support—both political and financial—there now is for the mathematical sciences may well diminish if the community's responsibilities are not met. On the other hand, efforts to meet those responsibilities can provide opportunities for additional resources.
Action might be taken by different cohorts within the mathematical sciences community on various fronts. For example, it is clear that much of the educated public, including the scientifically educated public, is unaware of new discoveries in the mathematical sciences and their importance. The general public also is not made aware of the fundamental contributions of the mathematical sciences to many advances in science and technology. In the general technical community, there is a sense that the mathematical sciences community could do more to accelerate the impact of mathematical sciences research on science, technology, and society in general.
The mathematical sciences community has not been very successful in the important area of attracting underrepresented minority groups, and the disproportionately small number of women at many levels in the mathematical sciences is self-evident. By identifying broader career opportunities in the mathematical sciences, some of the obstacles to attracting these groups to the mathematical sciences might be overcome.
A further area where mathematical sciences involvement could be increased is in the education of all undergraduates. It is important for mathematical sciences departments to strengthen their scientific and educational ties with the university community at large. It is also important that the full mathematical sciences community enhance its scientific and educational ties with the general scientific and technological community. The pervasiveness of the mathematical sciences in those diverse areas makes this both a responsibility and an opportunity.
In light of the present environment for science and the direction that continuing change in that environment will likely take, how the mathematical sciences community now acts could be crucial to its future strength and health, and so deserves urgent attention. The mathematical sciences community now faces serious challenges in a changing and demanding environment that warrant quick and effective responses. The health of the mathematical sciences community and how well it fulfills its national responsibilities depend on its ability to resolve these issues.
Clearly, a successful resolution of these issues requires the active participation of mathematical sciences departments, the professional societies, and the community as a whole

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References
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