List of Tables
- Table 1: Distribution across five major industry sectors of participants in telephone survey of graduates, broken down by terminal degree.
- Table 2: Average perceived importance of mathematics in respondents' overall organizations.
- Table 3: Percentages of managers' graduate degrees in various areas.
- Table 4: Mathematical areas and industrial applications encountered during site visits.
- Table 5: Work group missions of mathematicians surveyed.
- Table 6: Average percentages of major disciplines in work groups of mathematicians surveyed.
- Table 7: Average numbers and percentages of mathematicians in work group, by industry sector.
- Table 8: Percentages of mathematicians surveyed who mentioned mathematical specialties as a primary technical requirement of their positions; multiple mentions were permitted.
- Table 9: Percentage of mathematicians surveyed who mentioned other disciplines as a primary technical requirement of their jobs; multiple mentions were permitted.
- Table 10: Perceived importance of advanced computation.
- Table 11: Ratings of effectiveness of graduate education in mathematics.
- Table 12: Skills for which respondents' graduate education was rated "excellent" or "very good".
- Table 13: Skills for which respondents' graduate education in mathematics was rated "less than good".
- Table 14: Areas in graduate education suggested by managers as needing improvement. More than one area could be named by each manager.
- Table 15: Importance of changes in graduate mathematics education.
- Table 16: Ph.D. and master's graduates, 1988--1992.
Mathematics within nonacademic organizations
Table 1: The 203 mathematicians (102 master's and 101 doctoral graduates from 1988--1992) and 75 managers who participated in the telephone surveys represent a reasonably broad spectrum of nonacademic organizations. (See the Appendix for more information about the survey sample.) Table 1 shows the distribution of graduates surveyed in five major sectors of industry, based on the Standard Industry Classification codes of the United States Office of Management and Budget. | ||
Nonacademic sector | Ph.D. | Master's |
---|---|---|
Government | 28% | 22% |
Engineering research, computer services, software | 19% | 18% |
Electronic, computers, aerospace, transportation equipment | 17% | 12% |
Services (financial, communications, transportation) | 13% | 22% |
Chemical, pharmaceutical, petroleum-related | 6% | 2% |
Table 2: Mathematicians and their managers were asked in the telephone survey about the status of advanced mathematics in their overall organizations, where "advanced" means at the level of the respondent's highest degree. Those responses are summarized in Table 2 and show the consistent importance of mathematics not only for its practitioners, but also for their managers. | |||
Importance of advanced mathematics | Ph.D. | Master's | Managers |
---|---|---|---|
Primary | 43% | 28% | 51% |
Secondary | 43% | 40% | 37% |
Only for general utility | 11% | 32% | 12% |
The managers interviewed by telephone offered a range of descriptions of the role of mathematics in their groups. Nearly half (49%) characterized mathematics as an underlying requirement or tool for their groups' work. Three main functional roles for mathematics were mentioned by managers: development of algorithms and numerical methods (27%); modeling and simulation (23%); and statistical analysis (15%).
Table 3 shows the diverse educational backgrounds of the managers who participated in the telephone survey, and prompts two immediate observations: the managers' favorable perception of the importance of mathematics does not arise because they are predominately mathematicians; and mathematicians often report to, and hence must communicate effectively with, nonmathematicians. | ||
Area of manager's degrees | Ph.D. | Master's |
---|---|---|
Mathematics | 16% | 11% |
Engineering | 13% | 6% |
Physics | 13% | 3% |
Statistics/biostatistics | 9% | 5% |
Business/management | 0% | 11% |
Computer Science | 0% | 6% |
Chemistry/biology | 0% | 3% |
Applications of mathematics
The site visits, telephone surveys, and experiences of steering committee members in industry build a picture in which mathematics participates in many ways in the overall enterprise of industrial and government organizations.Table 4 indicates selected associations between areas of mathematics and applications encountered in the site visits. | ||
Mathematical Area | Application | |
---|---|---|
Algebra and number theory | Cryptography | |
Computational fluid dynamics | Aircraft and automobile design | |
Differential equations | Aerodynamics, porous media, finance | |
Discrete mathematics | Communication and informatio security | |
Formal systems and logic | Computer security, verification | |
Geometry | Computer-aided engineering and design | |
Nonlinear control | Operation of mechanical and electrical systems | |
Numerical analysis | Essentially all applications | |
Optmization | Asset allocation, shape and system design | |
Parallel algorithms | Weather modeling and prediction, crash simulation | |
Statistics | Design of experiments, analysis of large data sets | |
Stochastic processes | Signal analysis |
Mathematics is a key player in numerous success stories heard during site visits. Common themes are the technical advantages and cost savings that accrue from clever modeling, analysis, and computation by mathematicians working with other professionals. The mathematician's logical, problem-solving approach is widely seen to provide a noticeable competitive edge.
Highlights of a few of those success stories are summarized next, with company names removed and certain proprietary details omitted. (Throughout this report, displayed quotations are taken from site visit reports or focus group discussions.)
Beginning in the mid-1970s, a chemical manufacturer began developing models of atmospheric reactions and transport. A team of mathematicians and atmospheric physicists used state-of-the-art techniques for stiff ordinary differential equations that allowed integration to a dynamic steady state that no one else could achieve. This advance provided the manufacturer with scientific credibility and a voice in the debate with regulatory agencies. Management developed sufficient confidence in the modeling results that it broke ranks with its industrial colleagues and became the first to cease manufacture of the products shown to be harmful to the environment.Nearly every manager interviewed by telephone cited a particular combination of application and mathematics in which mathematics had made a significant contribution; in fact, 13% agreed that "We couldn't have done it without a mathematician". The following list gives a subset of the cited applications:
A manufacturer of large industrial equipment developed a software system that provides a functional representation of surfaces so that "design data can be quickly moved from computer-aided design to numerically controlled machining and prototype production", thus cutting the cost of design by shortening the prototype design cycle time.
Safety testing of its product is a critical issue for one transportation manufacturer, which routinely uses nonlinear finite element models and large-scale computing to replace a "million-dollar prototype with a ten-thousand-dollar computer run".
One consulting organization contracted with a paper manufacturer to develop a scheduling system for paper production. The initial stages of this contract involved mathematical modeling of the production process, which eventually led to a turn-key system with a sophisticated user interface. The initial application of the modeling-based production system produced a 4% increase in revenue for the paper company, resulting in 6 million dollars per year in increased profit.
Device simulation is important to the semiconductor industry because it is very expensive to design and prototype next-generation devices. One chip manufacturer has been so successful with simulation and modeling that "we wouldn't build a chip without modeling it first".
Rising production costs threatened the profitability of one company's key product. Developing a process optimization methodology cut manufacturing costs so much that the product remained competitive and the company stayed financially viable.
- wavelets in the analysis of brain processes;
- Brownian motion algebra in modeling "limit" orders for financial products;
- representation and manipulation of complex geometry in computer-aided design of aircraft;
- analysis and modeling in turbulence studies and global warming;
- a numerical method for quantifying ultrasonic Doppler readings to allow analysis of regurgitative flow in heart valves, spinal anesthetic fluid migration, and thermal increases in a growing fetus;
- modeling of satellites and algorithms for centimeter-accurate orbit determination;
- matrix algebra applied to optimize managed portfolios and determine an "accuracy quotient";
- algorithms for classification of blocking and its costs in a railroad system.
- modeling and simulation;
- mathematical formulation of problems;
- algorithm and software development;
- problem-solving;
- statistical analysis;
- verifying correctness;
- analysis of accuracy and reliability.
Despite such favorable results, mathematics is often invisible outside the technical work group because its role in a successful project is not highlighted or publicized, especially to higher management within the organization. In some instances, mathematicians and managers commented that higher management was not interested in or would not understand the mathematical details. Others suggested that managers could not be expected to appreciate the contributions of all the disciplines reporting to them. In any case, the word "mathematics" is often disguised, or mathematics is described in nonmathematical terms. For example, one mathematician commented, "We never present anything to management below the level of modeling and simulation".
The contributions of mathematics as a separate, disjoint discipline are also difficult to discern because scientists and engineers in nonacademic environments necessarily join together to produce a single result. As we shall see in Section 3, industrial mathematicians tend to work in groups not entirely devoted to mathematics, and to collaborate with scientists and engineers from other disciplines. Thus, although mathematics is often a basic and crucial ingredient in industrial products and decisions, its role as such may not be explicitly recognized or understood. As expressed during one of the site visits, "Mathematics is alive and well, but living under different names".
Mathematicians as part of their organization
The value of mathematicians to a nonacademic institution depends on their contributions to the institution's mission. Mathematicians are part of the infrastructure; mathematics cannot be viewed as an end in itself. Managers evaluate their people by what they contribute to the company.Table 5: Within nonacademic organizations, mathematicians frequently work in groups whose primary missions include, but are not limited to, mathematics. Here are the five major work group missions mentioned by the surveyed mathematicians: | ||
Mission | Ph.D. | Master's |
---|---|---|
Mathematical specialty (such as modeling) | 40% | 24% |
Computing, computer services, software | 35% | 42% |
Research, research and development | 23% | 11% |
Consulting | 9% | 6% |
Engineering, risk analysis | 8% | 9% |
A small number of nonacademic mathematicians–most in government laboratories, some in large corporate laboratories—spend part, or occasionally all, of their time performing basic mathematical research similar to academic research. But even groups with a research charter are increasingly called upon to make a business case for their work.
Research often has a serious difficulty: too much understanding and too little transfer. It needs examples of success to justify continued support by the business part of the company.
Mathematicians in industry and government are almost always part of an interdisciplinary group. One site visit participant commented,
"Although a few mathematicians are clustered in one group that has a mathematical charter, most are scattered among engineers, physicists, and computer scientists, where they often function as ``hunter-gatherers'', seeking a share of their support from mission-oriented project groups. Mathematicians here must extract the mathematics from the projects that need it."
The data in Table 6 illustrate the interdisciplinary character of the groups in which industrial mathematicians work. Both Table 6 and Table 7 show that industrial mathematicians seldom hold a majority of the positions in their immediate work groups. Mathematicians also blend in with their colleagues because their titles rarely reflect the presence of mathematics in their jobs; only 20% of the graduates interviewed by telephone hold positions with mathematical titles. In addition, many mathematicians hold positions that do not require a degree in mathematics and hence could be filled by graduates of another discipline. Among Ph.D.'s surveyed by telephone, only 31% stated that an advanced degree in mathematics was required for their position; among master's graduates, the analogous figure was 14%. Once in an industrial position, many mathematicians find that meeting the demands of their organizations' missions takes precedence over their disciplinary identities. | ||
Discipline | Ph.D. | Master's |
---|---|---|
Mathematics | 25% | 16% |
Computer Science | 27% | 24% |
Engineering | 23% | 15% |
Physical Sciences | 8% | 2% |
Table 6: Average percentages of major disciplines in work groups of mathematicians surveyed. |
Industry sector | Ph.D. | Master's |
---|---|---|
Government | 6.6 37% | 5.3 10% |
Services (financial, communications, transportation) | 4.1 49% | 5.5 38% |
Chemical, pharmaceutical, petroleum-related | 4.0 25% | 1.5 50% |
Engineering research, computer services, software | 4.2 15% | 2.0 16% |
Electronic, computers, aerospace, transportation equipment | 2.6 15% | 2.0 24% |
Other | 2.7 18% | 2.5 13% |
Table 7: Average numbers and percentages of mathematicians in work group, by industry sector. |
Depth and breadth in mathematics
Not surprisingly, wide variations occur in mathematical specialization within nonacademic working environments. Mathematicians working in large departments with a specific mathematical function are almost by definition required to be experts in that area. Similarly, mathematicians' work is highly specialized if the institution's core business is closely linked to a mathematical area like computational fluid dynamics.For a few Ph.D.'s, their area of expertise in industry remains that of their dissertation. However, managers often view completion of a Ph.D. dissertation as evidence of ability to exert sustained effort to solve a difficult problem rather than as training in a particular specialty that will occupy a professional lifetime.
If mathematicians are functioning largely as consultants or if the demands of the organization's mission lead to shifts in technical requirements, it may be impossible or professionally undesirable for a mathematician to work on only a single specialty. Many of the mathematicians interviewed indicated that, soon after starting work, they were shifted to projects much different in mathematical content from those for which they were originally hired. Several speakers at the 1994 SIAM Forum [Davis94] cited instances in which responsibilities were changed not only by management, but also by the employees themselves, to improve career prospects or job security.
Telephone survey responses regarding needed mathematical specialties are summarized in Table 8; also see Table 9. It is evident that many of these positions require a variety of mathematical knowledge.
Mathematical specialty | Ph.D. | Master's |
---|---|---|
Modeling and simulation | 73% | 68% |
Numerical methods/analysis | 65% | 47% |
Statistics | 55% | 61% |
Probability | 50% | 55% |
Engineering analysis/differential equations | 50% | 28% |
Operations research/optimization | 38% | 42% |
Discrete mathematics | 26% | 24% |
Table 8: Percentages of mathematicians surveyed who mentioned mathematical specialties as a primary technical requirement of their positions; multiple mentions were permitted. |
We never know what kind of mathematics is the right kind, so an "algebraist for life" is not the right kind of mathematician.Furthermore, regardless of their areas of expertise, mathematicians may be seen by colleagues and managers as a resource to answer general mathematical questions. One industrial mathematician noted, "If you are a mathematician, everyone expects you to know statistics and operations research".
As seen in Section 2, the problems confronting industrial mathematicians arise from the needs of their institution, and cannot be chosen to fit a predetermined palette of mathematical tools. Hence breadth and depth are both important.
You can't isolate yourself in just one area. Even a simple project has many aspects. But you do need to be an expert in one area.
Interest in and knowledge of other areas
Because of both the interdisciplinary and varied natures of their technical problems, nonacademic employers strongly prefer mathematicians with an interest in applications.Mathematicians working here must like to apply real solutions in the real world.Knowledge of technical areas outside mathematics is regarded as extremely helpful in nonacademic positions. Only one of the 203 recent graduates interviewed worked in a position for which a knowledge of mathematics was the sole requirement.
What kind of mathematicians are attractive? Mathematicians with the right mind set who want to solve problems rather than just do mathematics.
A metaphor for success that we heard from more than one manager was the letter T, which meant that a successful mathematician must have depth in an area of specialization but at the same time develop a broad understanding of technical and business issues in the company.The balance between depth and breadth as well as the need for a genuine interest in applications are issues outside mathematics as well; see, for example, [Hans91, NRC-Grad, Natr89].
Table 9 lists the percentages of mathematicians surveyed who mentioned other disciplines as crucial in their jobs. These data demonstrate the importance of interdisciplinary knowledge.
Discipline | Ph.D. | Master's |
---|---|---|
Computer science | 69% | 88% |
Physics | 32% | 28% |
Electrical engineering | 29% | 19% |
Mechanical engineering | 19% | 10% |
Chemistry | 6% | 10% |
Biology | 9% | 6% |
Materials engineering | 38% | 42% |
Chemical engineeriing | 5% | 2% |
Civil engineeriing | 3% | 2% |
Table 9: Percentage of mathematicians surveyed who mentioned other disciplines as a primary technical requirement of their jobs; multiple mentions were permitted. |
It is clear from Table 9 that the most significant second discipline is computer science. Table 10 presents responses to a telephone survey question about computation; its perceived importance by all groups is unmistakable and striking. One manager commented during a site visit, "It's hard to envision a pencil-and-paper mathematician here".
Role of advanced computation | Ph.D. | Master's |
---|---|---|
Essential | 54% | 31% |
Very important | 28% | 24% |
Somewhat important | 12% | 15% |
Not particularly important | 7% | 31% |
Table 10: Perceived importance of advanced computation. |
The most valued computational skills obviously depend on the context and cannot be prescribed in advance. Often, however, software embodies a mathematician's contribution to a problem, and it may be impossible or undesirable to delegate implementation to others. Thus expertise in both programming and numerical analysis is essential. As one manager noted, ``Unless mathematics is put into software, it will never be used''. Computing is also central in managing and analyzing large sets of data, connecting and coordinating all pieces of a problem, and displaying results graphically in a meaningful, compact form.
In groups where computing is important, on average the Ph.D.'s estimate that they spend nearly half their time (44%) on computational tasks, with 47% of those tasks requiring advanced mathematics. The master's graduates estimate that they spend on average 33% of their time computing, with 26% of those tasks requiring advanced mathematics.
Comments from nonacademic mathematicians and other sources (see, for example, [Hold92]) suggest that, in addition to basic sciences and engineering, business and finance are increasingly important disciplines for analysis of technical questions about markets, pricing, and related issues.
Perceptions of Graduate Education
Several recent reports (for example, [NRC-Grad]) have suggested that U.S. graduate education is too focused on producing faculty "clones"---professors at research universities---and hence prepares students poorly for nonacademic careers. One of the aims of the SIAM MII study was to determine the views of industrial mathematicians about how well their graduate education prepared them for their jobs. The findings of our study show an interesting mixture of opinions. Certain aspects of graduate education are almost universally praised, whereas others are widely regarded as needing improvement.As shown in Table 11, industrial mathematicians interviewed by telephone mostly believed that their graduate education had helped them to obtain and perform well in their present positions.
Effectiveness of mathematics graduate education | Ph.D. | Master's |
---|---|---|
Obtaining a highly desirable position | ||
Extremely effective Somewhat effective Not particularly effective |
58% 25% 17% |
57% 23% 20% |
Helping in your work | ||
Extremely effective Somewhat effective Not particularly effective |
59% 32% 9% |
38% 41% 21% |
Providing a superior career path | ||
Extremely effective Somewhat effective Not particularly effective |
27% 51% 22% |
18% 58% 24% |
Table 11: Ratings of effectiveness of graduate education in mathematics. |
The mathematicians surveyed were asked to rate how well their graduate education had provided them with specific skills. There were four categories for educational preparation: "excellent", "very good", "good", and "less than good".
Skills for which preparation was rated as "excellent" or "very good" by more than half the Ph.D.'s are given in Table 12. Except for "creating new ideas or innovative approaches", educational preparation in these qualities was also highly rated by the master's graduates.
Skills very well developed by graduate education | Ph.D. | Master's |
---|---|---|
Thinking logically; dealing with complexity | 87% | 92% |
Problem-solving | 79% | 84% |
Conceptualizing and abstracting | 76% | 84% |
Formulating problems; modeling | 65% | 72% |
Creating new ideas or innovative approaches | 60% | 41% |
Table 12: Skills for which respondents' graduate education was rated "excellent" or "very good." |
Graduate education is perceived by a substantial majority of these mathematicians as having prepared them very well for thinking analytically, dealing with complexity, and conceptualizing. Smaller proportions of the respondents, but still a majority of Ph.D.'s, felt very well prepared in formulating problems, modeling, and creating new ideas. As discussed in Section 3.2, Section 3.4, and Section 3.6, much of the value of industrial mathematicians depends on these skills. Hence graduate education in mathematics is highly successful in preparing students for many key elements of nonacademic jobs.
Table 13 lists the qualities for which more than 30% of the Ph.D. respondents rated their educational preparation as "less than good": working with colleagues; communicating; having broad scientific knowledge; using computer software and systems; and dealing with varied problems.
Skills for which preparation was "less than good" | Ph.D. | Master's |
---|---|---|
Working well with colleagues | 87% | 92% |
Communicating at different levels | 79% | 84% |
Having broad scientific knowledge | 76% | 84% |
Effectively using computer software and systems | 65% | 72% |
Dealing with a wide variety of problems | 60% | 41% |
Table 13: Skills for which preparation was "less than good." |
This set of weaknesses in graduate training is significant because Section 3.1 and Section 3.3--Section 3.5 show that these skills, in addition to those shown in Table 12, are crucial for success in nonacademic careers. The high proportions of both Ph.D.'s and master's graduates who felt inadequately educated in working well with colleagues and in communicating at different levels are notable because of the special importance of these skills in interdisciplinary environments; see Section 3.5. It is also interesting that nearly half the Ph.D.'s felt less than well educated in knowledge of other scientific fields and in use of computer software and systems, in spite of the importance of these attributes.
Each manager in the telephone survey was asked to indicate areas in which graduate education in mathematics needs improvement; the six leading replies are shown in Table 14.
Area for improvement in graduate mathematics education | Percentage of managers |
---|---|
Application of mathematics | 40% |
Knowledge of other disciplines | 23% |
Real-world problem solving | 21% |
Communication; technical writing | 19% |
Computer skills | 13% |
Teamwork | 8% |
Table 14: Areas in graduate education suggested by managers as needing improvement. More than one area could be named by each manager. |
Mathematicians in the telephone survey were asked about the importance that they attach to changing graduate education. Table 15 indicates that a majority of both Ph.D.'s and master's graduates strongly support changes in mathematics graduate education. Only a small proportion of Ph.D.'s or master's graduates believe that change is not important.
Importance of educational change | Ph.D. | Master's |
---|---|---|
Extremely or very important | 60% | 69% |
Somewhat important | 34% | 26% |
Not important | 6% | 5% |
Table 15: Importance of changes in graduate mathematics education. |
To summarize, our inquiries indicate that graduate education in mathematics is seen by recent graduates working in industry as excellent in certain areas, but needing improvement in others. More than 90% of these mathematicians believe that changes in graduate education are important. Furthermore, the areas for improvement highlighted by both the mathematicians and their managers (shown in Table 13 and Table 14) overlap significantly with one another and with attributes of successful industrial mathematicians. Based on these results, Section 5 presents our suggestions for strategies to enhance and expand graduate education in mathematics to strengthen these areas.
Table 16: Ph.D. and master's graduates, 1988--1992. | ||
Category | Ph.D. | Master's |
---|---|---|
Total number of graduates | 3,701 | 17,780 |
Estimated number of graduates in industry | 925 | 7,823 |
Entries in MII database | 335 | 271 |
Telephone interviews | 101 | 102 |