Suggestions and StrategiesThe suggestions that follow are made by the MII steering committee, based on detailed comments from telephone interviews and site visits and on our own experiences. Our intent is to embody guiding principles and values in the form of numerous specific actions ranging from small-scale to long-term, from local to institutional. They are in no sense claimed to be exhaustive; some of them, in fact, immediately suggest other possibilities. The principles guiding the educational suggestions are exposure to applications in other disciplines, real-world problem-solving, computation, and communication and teamwork.
This section lists and in some cases briefly sketches approaches that have been tried successfully, but details and references are not given here. More information can be obtained from SIAM in several ways: via Internet from SIAM's home page http://www.siam.org, following the links to "Career Information" and "Mathematics in Industry" (where you are now); by sending electronic mail to email@example.com; or by contacting SIAM at the address given at the front of this report.
The steering committee is aware that implementation of these suggestions will require time, dedication, and persistence. But we believe strongly that the effort will be worthwhile.
- Graduate education
- Suggestions for faculty
- Suggestions for students
- Suggestions for business, industrial, and government organizations
Graduate educationGraduate education in mathematics serves its students well today in providing the skills from Table 12: thinking logically, dealing with complexity, conceptualizing, formulating problems, modeling, and creating new ideas. Section 3.2, Section 3.4, and Section 3.6 indicate that the special value-added of mathematicians hinges on these same traits; training in mathematics, more than in other fields, is perceived to allow analysis at a high, system-wide level, thereby revealing underlying patterns and structure. Hence graduate education in mathematics is highly successful in preparing students for many key elements of nonacademic jobs.
In contrast, industrial mathematicians and their managers consistently mentioned several qualities, shown in Table 13 and Table 14, to which greater attention could be given in mathematics graduate education:
- substantive exposure to applications of mathematics in the sciences—physical, biological, medical, and social—and in engineering;
- experience, both inside and outside the classroom, in formulating and solving open-ended real-world problems, preferably involving a variety of disciplines;
- communication and teamwork.
Familiarity with applications as well as interdisciplinary problem-solving skills can be generated through courses devised in cooperation with other departments and nonacademic mathematicians. A 1995 National Research Council report, Mathematical Challenges from Theoretical/Computational Chemistry [NRC-Chem], makes several suggestions about graduate education. Some of its recommendations, suitably generalized, have been included in our suggestions.
- Join with other departments to create special "interdisciplinary tracks"
of graduate courses that combine the most important advanced topics in
several disciplines. Nonacademic mathematicians should be involved in
designing and teaching such courses.
- Encourage graduate degrees that involve dual mentoring by mathematics and
- Form a mathematical consulting group involving mathematics graduate students (
and perhaps faculty) to be available to other researchers on
- Allow and encourage mathematics graduate students to minor in another
- Join in teaching, preferably with a nonacademic colleague,
mathematically-based specialty courses already offered in other departments.
It seems inefficient and ill advised to allow a serious divergence between
advanced mathematical applications and advanced mathematics.
- Create new mathematics courses focusing on the techniques most needed in
certain applications, and include substantial course material on those
applications. It is natural to share teaching in such courses with at least
one other department and with nonacademic mathematicians.
- Develop general courses on industrial mathematics. Several books are
available that offer suggestions and guidelines.
- Incorporate nonacademic internships or other on-site problem-solving
experiences into degree requirements.
- Participate in or initiate programs in computational science, which stress a
combination of mathematics, computer science, and applications. Existing
computational science programs typically involve nonacademic scientists from
- Invite speakers from other disciplines, including scientists from industry,
to speak in research seminars or colloquia. Such interdisciplinary gatherings
can also be organized in cooperation with other departments.
- Maintain a departmental database, with links to national databases supported by government agencies, of summer positions or internships in industry or government laboratories.
Mathematics problems assigned in academic contexts almost never include the work already invested to strip away all but the essentials and to describe the problem in a form that suggests how it might be solved. The skills needed to perform those unseen preliminary steps are crucial for nonacademic mathematicians (see Section 3.4), and they cannot be taught except by direct experience.
The interdisciplinary and applications-oriented courses suggested in Section 5.1.1 are an obvious milieu for assignments based on open-ended, real-world problems. A growing selection of publications on problem-solving and industrial problems is available to help mathematics faculty in building course content and materials. Several relevant books, some based on joint university–industry mathematics programs, are listed in the References section.
Graduate students can experience interactive problem formulation by attending intensive seminars and workshops organized by universities in collaboration with local industrial and government organizations. In one of the most successful formats, industrial scientists with real problems describe their problems; Ph.D. students then work in small groups to understand, formulate and analyze; and both sets of participants meet frequently to explain, redefine if necessary, and evaluate the results. Preliminary reading is assigned to the students as preparation for the technical background of the problem. In addition to teaching problem-solving, activities of this kind necessarily enhance students' awareness of the applications of mathematics.
In evaluating problem-solving experiences, especially internships, it is important to remember that the first priority of an industrial mathematician is problem-solving, not exercising a predetermined set of mathematical ideas. Consequently, it is difficult to guarantee in advance that students working on industrial problems will use a certain kind or level of mathematics. A successful student experience is one that attacks an important problem and produces a solution that is helpful to the user.
Members of the steering committee are convinced that computing as a discipline has enormous intellectual content. The view that "anyone can write computer programs" is just as wrong as the statement that "anyone can write"; the task of transforming a mathematical procedure into a correct computer program involves attention to structure, accuracy, efficiency, convergence, and reliability. Hence time spent learning about computation is time well spent.
The most obvious way to ensure that mathematics students understand and experience computation is to impose course requirements involving computer science, but students should encounter serious computational work in their mathematics courses as well. Almost all computer science departments offer courses in programming, numerical analysis, data structures, and algorithms. Since many of these courses have a high mathematical content, their inclusion in a graduate mathematics curriculum is entirely appropriate.
We know from Section 3.3 that computation involving advanced mathematics is extremely important in nonacademic settings. It may be less widely appreciated that courses combining traditional pure mathematics and computation have been created in several universities to encourage new patterns of mathematical thinking and new mathematics. Even students wishing to prepare entirely for an academic career in the purest of mathematics would, in the steering committee's view, benefit from expanded opportunities for course work that links mathematics and computing.
Communication and teamwork
"Communication" may seem at first to be a skill far removed from the traditional mathematics curriculum, but it is simply too important to be ignored or left for others to teach. Writing and speaking skills are important for all mathematicians, not only those in industry; and careful listening is essential in formulating complex problems. The following suggestions have been applied with proven success to teach communication within mathematics courses.
- Assign term papers in selected advanced mathematics courses, and grade
the papers for exposition and clarity as well as for content. To help
students learn to present advanced material for nonspecialists, two
versions of a paper can be required, one of which must be understandable to
nonexperts. For at least one paper, require an "executive summary" that
justifies the importance of the work in nontechnical terms. Have students
give short (10-minute) presentations explaining the nature of the work and
its (nontechnical) justification; invite nonspecialists, such as faculty or
graduate students from the business school, to critique the
- Require a course (or mini-course) on technical writing. Such courses can
be internal to the mathematics department, or jointly offered by several
departments; excellent textbooks are available. Reinforce the message with
uniformly high expectations for written communication throughout the
- Require at least one "projects" course in which work is entirely
performed and graded in teams, or incorporate team projects with written and
oral presentations into conventional courses. An interdisciplinary course is
well suited to this format, but such a course can also be centered around
open-ended problems mixing several areas of mathematics.
- Hold regular seminars, preferably with other departments, in which
students are required to read a technical paper outside their area and
present an oral report on its contents. Course grading should be based on
evaluation of the presentation by other students. Such programs have a
recognized track record of producing students highly skilled in oral
- Require each student to give a technical presentation that is videotaped and then graded by him/herself.
Suggestions for faculty
It is widely perceived, and there is some evidence, that graduate education in mathematics is inordinately concentrated on preparation for traditional academic research careers; see, for example, the article [Jack95] about the general study [NRC-Grad]. Recent degree recipients in mathematics have commented in print and privately that some faculty do not seem to support or understand career choices outside academia; see, for example, [Lot95]. We therefore offer further suggestions for faculty because committed faculty involvement is an implicit but essential element in broadening graduate education.
As in the NRC report [NRC-Chem], our suggestions are designed to enhance connections between mathematics faculty members, other faculty members, and nonacademic colleagues working in mathematics and related disciplines. Mathematicians of all varieties acknowledge the stimulation resulting from technical discussions; our suggestions focus on ways to create occasions for such discussions.
- Actively encourage mathematicians, scientists, and engineers from
industry to speak in and attend mathematics seminars and
- Invite at least one highly visible nonacademic mathematician to visit for
one or two days during each term; the visit should include presentation of a
talk in your departmental colloquium. Provide ample opportunities for this
visitor to meet with students and faculty. The "visiting lecturer" programs
of scientific societies can help to supply names of outstanding speakers in
- Organize joint colloquia and seminars with other departments, especially
those strongly tied to outside mathematical scientists, and with
mathematics-intensive groups from industry.
- Maintain a database of departmental graduates working outside academia.
Ask them for suggestions about the curriculum, and invite them to return and
give a colloquium.
- Include nonacademic mathematicians on departmental advisory
- Appoint a focused nonacademic advisory committee to review curriculum,
meet with students, suggest career opportunities, provide industrial
contacts, and so on.
- Create institutional connections with mathematics-intensive groups in
local industry. Invite appropriate members of such groups to participate in
the intellectual life of your department by, for example, serving as adjunct
faculty or teaching short courses.
- Seek opportunities for interested faculty to consult with or spend time
in industry. Federal programs are in place that support faculty sabbaticals
and summers in industry; many government laboratories will support extended
faculty visits or sabbaticals.
- Arrange extended visits to your department by industrial
- Encourage interested faculty to seek collaborations with colleagues in other departments or industry. If these collaborations develop, it is important that the resulting work be valued in the departmental rewards process. The MII steering committee recognizes that, for many academics, the path to interdisciplinary work is filled with obstacles, often erected by their own departments. (Some of these barriers are described in reports by the Joint Policy Board on Mathematics [JPBM94] and the National Research Council [NRC-Chem].) Nonetheless, we hope that the evident scientific rewards of interdisciplinary, nonacademic mathematics will accelerate the engagement of mathematicians in such problems; positive intervention by deans and university officials also has an important role to play.
Suggestions for students
Many strategies for mathematics graduate students are implicit in our suggestions for graduate education. Section 3 provides clear guidance about skills that are important in nonacademic environments, and students can, to the extent possible, organize their programs to build those skills. Students can also act along lines similar to those indicated for faculty:
- Organize, preferably with graduate students from other departments, a
regular interdisciplinary seminar or colloquium focusing on applications
of mathematics, and invite scientists from industrial or government
organizations to speak.
- Invite nonacademic mathematicians to meet with current students for
informal discussions about their work and to present a high-level
description of an important practical problem.
- Organize technical talks in which students are critiqued by their
- Request videotaping services for student talks from an appropriate
office at your institution.
- Organize visits by students in your department to local industry or
government laboratories to meet with mathematical
- Suggest names of nonacademic mathematicians or scientists to be
invited to speak in your departmental colloquia.
- Pursue opportunities for summer jobs, cooperative employment, or
internships in industry or government laboratories, or for participation
in applications-focused workshops. Besides providing valuable experience,
these activities are likely to be helpful in finding a permanent job. Much
of this information is available online—for example, from the home
pages of major government laboratories.
- Attend talks by nonacademic mathematicians at meetings of scientific societies.
Suggestions for business, industrial, and government organizations
Not surprisingly, the MII steering committee believes that the discipline of mathematics, acting in large part through trained mathematicians, can contribute enormously to solving important problems. The success stories in Section 2.2 and the characterizations in Section 3.6 provide ample evidence of the measurable value added by mathematics and mathematicians. Connections, formal and informal, between nonacademic organizations and academic mathematics departments can build pathways for a two-way flow of both concepts and results. We have therefore included a set of suggestions that can, we believe, help industrial and government organizations to make better use of available mathematical resources. For reasons of convenience, cost, and ease of access, these strategies focus mainly on local academic institutions, but links over a wider area may also be appropriate.
- Become acquainted with academic mathematical scientists by attending
colloquia, meeting with students, offering to give talks, and inviting
students and faculty for informal visits to your organization.
- Invite mathematicians to give technical seminars at your institution and
to meet informally with interested employees.
- Arrange to present a technical problem of interest to a group of
mathematics faculty and graduate students.
- Identify academic mathematicians who specialize in areas of interest to
your organization. A variety of arrangements can then be made in which your
organization might call upon these mathematicians for advice, either short-
- Invite mathematics faculty and scientific societies to present
continuing-education short courses at your organization on mathematical
topics of interest.
- Arrange for visiting lecturers from mathematical societies to visit and
speak to your organization.
- Spend a short or longer-term visit in a university department containing
one or more mathematicians with interests closely connected to your
- Try to find talented students who could participate in a summer
internship and faculty who might wish to establish collegial relations with
members of your organization.
- Encourage interested mathematics graduates to apply for positions, even
those not labeled as "mathematics".
- Volunteer to organize a session at a meeting of a mathematical
- Become active in scientific societies; volunteer to serve on committees, particularly organizing committees of meetings; suggest topics for short courses at scientific meetings.