Suggestions and Strategies

The 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 protected]; 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
  • Exposure to applications of mathematics.
  • Problem-solving.
  • Computation.
  • Communication and teamwork.
• Suggestions for faculty
• Suggestions for students
• Suggestions for business, industrial, and government organizations

Graduate education

Graduate 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:

1. substantive exposure to applications of mathematics in the sciences—physical, biological, medical, and social—and in engineering;
2. experience, both inside and outside the classroom, in formulating and solving open-ended real-world problems, preferably involving a variety of disciplines;
3. computation;
4. communication and teamwork.

We now describe curricular suggestions for developing these skills. Some of these elements can also be incorporated with suitable modifications into an undergraduate mathematics curriculum.

• Exposure to applications of mathematics.
• Problem-solving.
• Computation.
• Communication and teamwork.

Exposure to applications of mathematics.
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 another department.
  
  
• Form a mathematical consulting group involving mathematics graduate students ( and perhaps faculty) to be available to other researchers on campus.
  
  
• Allow and encourage mathematics graduate students to minor in another discipline.
  
  
• 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 several disciplines.
  
  
• 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.

Problem-solving
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.

Computation
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 presentations.
  
  
• 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 curriculum.
  
  
• 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 presentations.
  
  
• 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 colloquia.
  
  
• 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 designated fields.
  
  
• 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 committees.
  
  
• 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 mathematicians.
  
  
• 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 fellow students.
  
  
• 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 scientists.
  
  
• 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- or long-term.
  
  
• 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 organization's.
  
  
• 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 society.
  
  
• Become active in scientific societies; volunteer to serve on committees, particularly organizing committees of meetings; suggest topics for short courses at scientific meetings.