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, 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

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.
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. 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.

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.

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.

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:

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.
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