A Window of Opportunity for Mathematics DepartmentsNovember 20, 2005
James P. Keener
The undergraduate mathematics major is in crisis. The number of math majors is declining; our departments are becoming storehouses of marvelous knowledge, with too few students to pass it on to. This is the premise both of this essay and of a mid-August National Science Foundation-sponsored workshop.*
We usually measure the success of our programs (at least in our hallway conversations) by the number of students who go on to prestigious graduate programs. The reality is that most of our graduates do not end up in careers in which mathematics is the main activity. Employers who do hire our graduates to do mathematics understand that they must be retrained. Furthermore, these math graduates will most likely work as part of teams, on problems that are not well formulated, with people who have greatly different training, background, vocabulary, and expectations; the skills needed in such a setting were barely mentioned, let alone developed, during their undergraduate studies. In short, there is an impedance mismatch between what we teach our undergraduates and what they need to succeed.
A similar difficulty arises in graduate education, as is well summarized by the Carnegie Graduate Initiative (http://www.carnegiefoundation.org/CID/). Here again, the apparent goal of the training we offer is self-replication for the academic market, without consideration of the fact that the majority of our graduates end up on different career paths. Another problem is that traditional graduate mathematics education is often elitist; we provide very little motivation, especially in the early years, requiring students to jump through a number of difficult hoops before they are allowed to engage in "the good stuff."
It is no wonder that many bright students choose to pursue graduate studies in other disciplines. As to undergraduates, there, too, it is not difficult to believe that many students choose majors other than mathematics because they think that their future careers will be better served elsewhere. If we paid more attention to these issues, we might attract more students.
A Mathematics Department's 25-Year Experiment
I believe that one way to revitalize mathematics and mathematics degree programs is via an increased emphasis on interdisciplinarity. Interdisciplinary mathematics has (at least) three distinguishing features. First, it visibly expands the usefulness of mathematics. Second, it puts mathematics at the cutting edge of science, making it more attractive and more relevant to the younger generation. Third, it requires a mode of operation and, hence, a mode of training different from those of traditional education.
One of the hotter areas of interdisciplinary mathematics these days is mathematical biology. Given the tremendous increases in computing power and in the size and number of databases of the past two decades, there is a huge need for mathematically trained individuals to attack problems in the biological and life sciences. This need has been documented by two recent publications, one focused on undergraduate education  and the other on research directions .
It has been suggested that biology will be to mathematics in the 21st century what physics was in the 19th and 20th centuries . Indeed, we have already seen a virtual explosion in the number of quantitative scientists--including mathematicians, physicists, and engineers--whose research is focused on problems of a biological nature.
At the University of Utah, we have been able to use mathematical biology to experiment with modes of research and training. Our experiment began well over 25 years ago, with the formation of a small group of (then young) faculty, and has been evolving ever since. At present, the group consists of 23 graduate students, nine postdocs, and five faculty. We do not (yet) count undergraduate students, although their numbers are growing as well. (An applied mathematics major, now being created, will offer life sciences as one of several tracks.)
Our graduate training program has several unusual features. First, our beginning students participate in "journal clubs." This has them reading and discussing papers and giving talks in a highly interactive environment from the outset of their graduate studies. Second, beginning students take courses in both mathematics and biology/life science. Third, each beginning student does a laboratory rotation or internship in a biology/life science laboratory, in this way gaining exposure to the real-world challenges of data collection and experimentation. Fourth, the students work on research projects that have extra-departmental components, with significant participation of outside faculty members. (By "significant," I mean that the goal is genuine interaction and collaboration, not a perfunctory mentoring meeting that takes place once a semester.) Ideally (although this does not always happen), the student becomes an active participant in the outside mentor's research group/lab meeting.
Fifth, all students and postdocs participate in one or more of the group/lab meetings around which our research activity is organized. At a group meeting, five or more individuals with common research interests discuss their recent results, describe papers they have read and recent seminars they have attended, practice poster presentations, and the like. The meetings are highly interactive, collaborative, and stimulating. Sixth (but not finally), we hold annual retreats, and sponsor student-run workshops and meetings. In summary, we are deliberately attempting to change graduate education and the research culture by being serious about mentoring, building a strong sense of community, and engendering interaction, collaboration, and communication.
Preliminary results of our experiment are very encouraging (although some of these features were introduced quite recently). Students are well trained in applied mathematics, they are broadly exposed to many areas of modern biology, and they are in high demand on the employment market. One intangible benefit comes from working on problems that are "hot"--problems that the students can talk about at a party--an added source of motivation, purpose, and value in their education.
Broadening the Definition of "Good" Mathematics
Of course, it would be irresponsible to suggest that interdisciplinary training is without problems and challenges. Our students experience the usual difficulties: in passing qualifying exams, identifying interest areas, and finding advisers. Interdisciplinary work has the added difficulty imposed by the enormous language and cultural barriers that must be overcome. As we all know, it is difficult enough to become an expert in one discipline. Learning two is extremely difficult, and as to meeting the expectations of two traditions--impossible.
It is also difficult to find committed outside advisers: The idea of "sharing" a student is unfamiliar to most. Outside advisers usually find it difficult to work seriously with students who are not directly involved in one of their specific funded research projects. Most life scientists have no clue as to what to do with a math student.
But the challenges presented by the outside scientific community are easily matched by those arising within the mathematical community. Topics of (sometimes heated) discussion relate to what requirements can be dropped (you can't add requirements without dropping some), what the preliminary exam structure should be and what should be required in the exams, and what to do about (and how to fund) non-traditional students whose backgrounds are different from that of a typical undergraduate math major.
But the question heard most often relates to definitions. It takes a variety of forms, such as "Where's the math?" or "Is this person a mathematician?" or "Isn't there something that all math PhDs should know?"
How we as a community choose to answer these questions will determine a great deal about the future of our discipline and our departments. The answer that I favor requires that we broaden our definition of "good mathematics." The currently reigning definition favors the axiomatic method, in which rigorous establishment of facts is the holy grail. However, other endeavors, such as problem identification and problem formulation (modeling), also require sophisticated mathematical reasoning skills, and both identification and development of problem-solving tools and mathematical experimentation require mathematical creativity and skill.
The answer that I do not favor is to retain the current definition of "good" mathematics and, in so doing, decide that the exploration and lack of rigor that characterize interdisciplinary endeavors do not make them worthy of being called mathematics. Adoption of this answer will result in the formation of new administrative structures, or in the departure of the best people for departments in which their craft is embraced. Either way, there would be a substantial loss for both parties.
I personally cherish the environment of a mathematics department, in which clarity, precision, and generalization are ingrained in the culture; I find working with students who lack an appreciation for these values difficult. Simultaneously, mathematics loses when people on the edge (especially students and junior faculty) are told that their work falls outside rather than inside the bounds of what is good and respectable.
The future of interdisciplinary mathematics is very bright. More and more students will choose to study it, and it will flourish and prosper. What is unclear at this point is what it will be called and where it will be housed. If it is nurtured within mathematics departments, interdisciplinary mathematics has the potential to help revitalize the discipline of mathematics, our degree programs, and our departments. Mathematics departments must decide now how to take advantage of this growth opportunity; otherwise, it will be lost. I fail to see how that can benefit anyone.
 Lynn Arthur Steen, ed., Math and Bio 2010, MAA, Washington, DC, 2005.
 Mathematics and 21st Century Biology, National Academies Press, Washington, DC, 2005.
*"Engaging Young Mathematicians," Washington, DC, August 12–13, 2005. This essay is a summary of the talk the author gave at the workshop.
James P. Keener is Distinguished Professor of Mathematics and an adjunct professor of bioengineering at the University of Utah.