Myths in Math

	Charles E. Mannix, Jr.
	Ken Ross

There are many myths about mathematics. Some people believe that only gifted people can learn mathematics, that mathematics is only for boys, etc. But this article is concerned with myths in mathematics. The December 5, 1994, issue of Newsweek included an article titled, "No Ph.D.'s Need Apply." It discusses The Myth, which took hold in the 1980s, that the nation would face a shortage of scientists in the 1990s. Notices readers have seen several articles in 1994 and a recent February 1995 editorial on the Myth's practical impact on today's young mathematicians seeking career employment.

A related myth in mathematics, which we hear every so often, goes something like this: "Jobs were tight in the early 1970s and then the market improved. It's a cyclic business and the market will get better again soon." Many of us no longer have faith in this myth, for reasons we will explain below, and we believe that mathematics departments should reconsider their missions. In particular, they should consider "downsizing" their graduate programs and re-examine the education provided in graduate school so that it more closely fits the reality of what our graduates will be doing in the future. Some group I universities, such as the University of California at Berkeley and the University of Michigan, have already started this process.

Many long-term economic, political, academic, historical, and technical issues indicate that the current downturn in full-time tenured employment of new young mathematicians is not likely to be reversed in the next decade. Even though you are aware of them all individually, it may be useful to consider them in totality and ponder their impact on mathematics. Our purpose is to state our reasons for our views without claiming to own a crystal ball.

First, the abrupt end of the cold war eliminated many compelling requirements for advanced R&D along with the organizations and staff supporting weapons development. Sizable rollbacks now exist at national labs and high-tech aerospace, electronic and design companies which for decades welcomed and employed many mathematicians, engineers and scientists. Displaced, highly qualified, mid-career individuals are entering the civilian economy on both sides of the (former) iron curtain. For thousands of them, their option will be to compete with new graduates for teaching positions at all educational levels. Overall, this is a healthy development because mathematics has always been a world-wide activity that has largely ignored artificial national boundaries, but there's no denying the impact on the current and future U.S. job market.

Our world is increasingly international. World-wide economic competition is forcing downsizing on most high-tech and even traditional American employers. Much detailed design, which creates new openings for U.S. mathematicians and engineers, has followed offshore manufacturing. More scientific cooperative efforts, another good thing, also lead to fewer technologists in any one country. The fiscal, political, and scientific pressures to collaborate are rising in areas from space research to high-energy physics.

A technological productivity and efficiency revolution is affecting routine mathematical and scientific work just as the industrial revolution affected manual labor. Many common time-consuming analytical tasks, which gave employment, especially at the entry level, now are accomplished using very powerful and efficient utility software developed to a mature state. Consider the numerous symbol manipulators, numerical analysis algorithms, statistical data packages, and graphical products on display at the Joint San Francisco meeting in January 1995. These greatly speed lengthy calculations yet most do not require their ordinary user to possess unusual mathematical talent. Note too, fewer mathematicians are "needed" to write and develop classical applications programs from scratch. Packaged or easily modified codes now exist in fields ranging from orbit mechanics to finite element modeling. This software maturity results in completion of complex designs like the Boeing 777 with a smaller but more efficient technical workforce. This touches another myth, namely that business and industry can absorb unlimited excess Ph.D. production. While there is much mathematics to be done out there, industry traditionally hires people from other disciplines to do this mathematics. Until we educate large numbers of mathematics students appropriately (as scientists, and not just thinkers) for R&D positions in industry, those hiring patterns won't change.

State college systems are largely built according to optimistic expansion plans of the 1960's and 70's. Thus, few or no new sites or major expansions at old campuses can be expected. Worse, the equivalent of corporate downsizing is occurring in academia. Budget cutbacks to state college systems have removed fat; ongoing annual reductions are having a serious impact on vital programs. In many places, present faculty and staff positions are more likely to be cut, not replaced, when vacated. Another aspect is a growing backlog of campus-wide deferred projects that will need funding when, and if, budgets ever increase. These are all arguments for a long lag before any substantial hiring occurs in mathematics departments.

Most tenured faculty today are in mid-career or older, but not eager to retire. It is unlikely that they will transfer elsewhere and leave vacancies which ripple down to create tenure-track slots for new Ph.D.'s. It has been suggested that their anticipated retirements will create a wave of new positions adequate to eliminate unemployment if only we wait. We agree that the projected retirements will have some influence, but it will fall far short of a one-for-one replacement of a tenured retiree with a tenure-track hire. Present and projected numbers of undergraduates annually obtaining mathematics diplomas are substantially below the plateaus established two or three decades ago. Since large numbers of math majors and first-year graduate students help in justifying math department staffing levels, there will be fewer actual tenure-track openings created than some people expect.

None of this is helped by the shrinking percentage of the American student body who elect to take upper-division majors in the analytical disciplines. Related to this is the trend to reduce analytical course work requirements in non-math degree programs. Many academic scientists and engineers feel that advanced mathematics is best learned in the context of their discipline; thus much of what was traditionally provided by mathematics departments has gravitated to those disciplines. Where other degree programs do need math, more mathematics is being taught within those programs. Many business schools, engineering schools, and other math-utilizing departments now have their own calculus, statistics, quantitative methods, and applied mathematics courses.

Other situations in academia translate to problems for mathematics departments. Within static or shrinking outside income sources, universities and colleges must foster new disciplines: biotechnology, genetic engineering, telecommunications, to name a few. Increasing overhead costs (i.e., support staff, pensions and insurance) encourage the trend to hire postdocs and part-timers. Costs for repairs, materials and labor to maintain the operation and upkeep of the physical plant at publicly-supported institutions are accelerating faster than the tax base and state support. A rising percentage of every dollar allocated to colleges goes to worthwhile compliance costs for accounting to funding agencies, enforcing equal rights laws, enabling the Americans with Disabilities Act, safety laws, etc. It is no wonder tuition fees, room and board, and other student expenses are rising much faster than inflation, which traditionally outpaces middle-income family earnings. As these attendance costs spiral upward, an increasingly larger part of our population will be denied access to a traditional four-year college education, especially if the government cuts back on student support. The trend will be towards ever smaller departmental enrollments and not for substantial numbers of new tenure-track entry level positions.

The unemployment situation facing young people in mathematics is far worse than dismal unemployment statistics for any single year's class suggest. Consider the invisible "unemployed." There is already the equivalent of several years' annual Ph.D. production embedded in the woodwork of U.S. colleges and universities as postdocs, part-time faculty, adjunct faculty, and, of course, the actively unemployed. This accumulation vigorously competes with any current year's graduates for the annual pool of available full-time tenure-track openings. At current hiring levels, it would take some years to absorb this backlog even if all Ph.D. production suddenly ceased.

Another myth is that the situation could be dramatically improved if national attitudes and governmental priorities quickly changed. The feeling that "Science helped win World War II" transferred to the public the notion that science would help win the cold war too and help the country in other ways. Now, the average citizen no longer ranks pure mathematical research as a top national concern. Not only that, a diploma in a technical field, as we understand the term, is viewed by fewer American "families" as a compelling dream to be pursued and a worthy cause on which to sacrifice substantial amounts of money. With truly rare exceptions, such as Al Gore, leaders of national prominence do not possess an agenda of technical excellence in the analytical sciences. In addition, very small numbers of new Ph.D.'s have entered government at any level to become future role models and voices at the table when budgets, priorities, and hires are being established.

Where does this leave us? First, let's acknowledge the accumulating evidence that the present traditional program leading to a B.S., M.S. or Ph.D. in "mathematics" does not produce highly marketable skills central to the "hot" growth disciplines in the peacetime global economy. We have been training students to understand the detailed intricacies in a specific set of problems, but failing to educate them on the potential broad relevance of the contributions our discipline can offer to the solution of those problems. The research and development world seeks creative researchers and implementors with the flexibility to adapt techniques and ideas to new contexts. The sad irony is that sophisticated mathematical skills, but not traditional mathematicians, are often needed in precisely these disciplines.

In both education and the industrial high-tech workforce, people not trained as mathematicians are doing mathematical work, often quite successfully. This phenomenon is the legacy of a long and profound failure of mathematicians to communicate with other groups. For example, many mathematicians believe that engineers and scientists are only interested in the formulas and not the theory of calculus. However, anyone who takes physical chemistry or thermodynamics needs to understand the chain rule and implicit function theorem at a much deeper level than is taught in standard calculus of several variables. The net result is that physicists and chemists are teaching these things more abstractly and thoroughly than most mathematics departments. The future of mathematics may depend on whether the emphasis is on concepts and insight , or on Bourbaki-like formalism and proof. This doesn't mean that proof is dead, just that insight needs to play a more important role. Successful careers in practical life often require conceptualization and abstraction of the essential problem without the usual list of clearly posed questions at the end of the textbook's chapter. The majority of our future graduates must be professionally adroit and flexible over a lifelong career which includes many uncertain conditions of excess, insufficient, or conflicting theories and data with rarely adequate time for contemplation.

Next, let's be honest with our students very early on. Their roles as TA's and RA's, peers, faculty impetus, and the present reward system create a mindset that the only quality careers are in academic teaching and research. Graduate students need to realize that their prospects for a satisfying academic career in a research institution are dim. Their love of mathematics will have to be the main motivation for pursuing an enriching intellectual experience in graduate school. Graduate work in most mathematics departments is no longer an apprenticeship program in which talent and hard work almost surely will lead to a satisfying career in mathematics. Future graduates from our programs will need the breadth and flexibility to assimilate new bodies of knowledge and to attack problems in a wide range of settings.

In short, we need to take professional moral responsibility for the present gap between the 800 or so Ph.D.'s that enter academia yearly and the 500 or so ultimately lucky enough to obtain permanent positions, and take the necessary steps to close the gap.

There is likely no one single answer to this employment problem. A spectrum of changes and reforms will be needed to improve the situation. We doubt that industry can absorb the excess Ph.D. production, and a long time must be allotted for some rooted attitudes to change. Surely, we must encourage all realistic, sensible attempts to increase suitable opportunities in industry, government, and academia. Research scientists and engineers, even investment counselors, increasingly need more and more sophisticated mathematics. They make do now with self instruction, but only mathematics departments can provide an integrated concept-based instruction which produces versatility in use of the knowledge. We must understand that it is insufficient just to say this. We must structure many of our offerings so that non-mathematicians will place sufficient premium on such courses that they routinely become part of their curriculum.

This leads directly to the necessity of re-examining the size and content of our graduate programs. These of course are related, and are determined by our conception of where our students are going. Thus, we badly need to re-examine our goals and purposes, our definitions and requirements. Of course, any "downsizing" and other changes must be done most wisely and humanely. The net result ought to be higher-quality students who really want to be mathematicians and who have an education that meshes with the challenges of the next century.