Points of View: The Interface of Mathematics and Biology

Avariety of recent reports, notably Bio2010 from the National Research Council (NRC, 2003; but see also CUPM Curriculum Guide [Mathematical Association of America, 2004], Salem and Dilts (2004), Meeting the Challenges (2003), and the collection of teaching material compiled by the Society for Mathematical Biology at http://www.smb.org/teaching/), recommend that undergraduate biology preparation become more interdisciplinary. This is based on the realization that modern biology research requires a breadth of skills that go well beyond the limited set of experiences that undergraduate biologists are exposed to in their traditional biology courses. This is not limited to biology, of course, for physical scientists have long been aware of the interconnections between fields and the reliance on quantitative approaches that go well beyond the basic undergraduate mathematics to which most physical science students are exposed.

In many respects, we have it worse in biology, though, because of the great diversity of topics from all areas of science that impinge on biology today. The days when bench scientists could claim that their students could lead fruitful research careers by learning appropriate skills and having good hands and lab technique are long gone. Similarly, despite the laments of E. O. Wilson about the lack of people being trained in taxonomy, it is next to impossible for those who are skilled field observers to make substantial new contributions to our understanding of natural system function without utilizing methods and approaches that natural historians of the past would never have dreamed of, let alone be skilled enough to use.

So if we take it as a given that modern biology research requires a diversity of perspectives and skills from areas of study outside the typical formal training that students receive in their life science courses, how should we proceed? The difficulty of this is exacerbated by the nationwide push to limit an undergraduate to 120 semester credit hours. How in the world do we squeeze into this limited time all that we think is necessary to prepare students to go on to successful graduate education or fruitful careers that don't require further formal education?

All experienced educators have heard, during discussions of curricular changes, the anxiety of colleagues about what has to be “left out.” Central to this is the notion that students are empty vessels that information can be pumped into, that if this information isn't imbibed in your class they'll never be able to get it, and that they'll be hampered for life by this lack of exposure to the details of ribosome function or cell signaling. I am continually amazed that this naive view of students is prevalent, since virtually all educators argue as well that our goal should be developing students' capacity for critical thinking and problem-solving. If we were successful at that, surely students would then be able to ascertain both what knowledge they lack in order to investigate a particular problem and how to learn about the area (or collaborate with an expert in it).

SOLUTIONS, SOLUTIONS, AT LEAST FOR QUANTITATIVE TRAINING

Bio2010 is of two faces with regard to what might be called the“ interdisciplinary gap”—the fact that students see the courses they take as discrete packages with few interconnections. On the one hand, the report recommends, as I have been arguing for years, that concepts from biology should be integrated within the quantitative courses that life science students take, and quantitative concepts should be emphasized throughout the life science curriculum. This provides what I have called“ multiple routes to quantitative literacy,” a process we have implemented in part at the University of Tennessee (for information see www.tiem.utk.edu/bioed/). One lesson from our experience is that it is possible to enhance a student's comprehension of quantitative topics by emphasizing these skills within the introductory general biology course. We have also redesigned the math sequence taken by many biology students to introduce a diversity of mathematical concepts, not just calculus, choosing the topics based on their utility in the broad life sciences.

The second face of the Bio2010 report on this matter concerns the addition of numerous courses that students could take. As part of the math and CS recommendations, the report provides a list of specific concepts, offered presumably as a suggested template for topics to be included in the life science curriculum. On showing this list to mathematical colleagues, they all agree that they'd be overjoyed to have undergraduate math majors exposed to these and can't quite imagine how a life science undergrad would ever find the time to take all the courses covering these concepts. Indeed, none of the“ proposed curricula” provided in Bio2010 come close to including the variety of mathematical courses that would be necessary to cover these topics.

Happily, the math/CS section of Bio2010 also includes a list of general quantitative principles useful to biology students, including rates of change, modeling, equilibria, stability, and stochasticity. My experience in developing our entry-level math for the life sciences course sequence is that it is indeed feasible (though the students certainly do have to work hard) to provide students with problems related to essentially all these general quantitative principles in a two-semester, three-credit hour sequence. Indeed, this sequence was developed by starting with a set of quantitative principles (not dissimilar to the list in Bio2010) developed in a workshop I organized in 1992 (see the report at http://www.tiem.utk.edu/~gross/Workshop92.recommendations.txt). This course sequence (see www.tiem.utk.edu/~gross/math151.html for the latest course materials) provides a rapid entrée to descriptive statistics, matrix algebra (including eigenvalues and eigenvectors), discrete modeling, and probability. It also provides the first experience our students generally have with the concept of an algorithm, through computer-based projects requiring the use of an appropriate mathematical software package (Matlab). Based on my experiences, integrating key quantitative concepts with life science examples within math courses, and repeating these concepts within life sciences courses in a variety of formats, is an approach that is feasible and fruitful.

So I urge us to specify the concepts, not the facts, and integrate these concepts throughout the curriculum in numerous formats. As just one example, when introducing maximization and minimization problems in our math sequence, I provide a quick summary of evolution by natural selection, mentioning Fisher's fundamental theorem. I don't expect the students to follow this in detail (they are first-year students who have had very few biology courses) but I do hope that in their later biology courses they will recall that evolutionary processes are somehow related to calculus and maximization problems. (Yes, I do mention constraints of history and urge them to read some of Gould's books too.)

ARE TRULY INTERDISCIPLINARY COURSES FEASIBLE?

As one alternative to the options discussed above, Bialek and Botstein (2004) argue for the development of truly interdisciplinary courses for biology students interested in a research career. They propose the development of courses that link math, the physical sciences, and biology as a substitute for separate disciplinary courses. Leaving aside for the moment the fact that few entering undergraduates have much of a clue what research is about, let alone whether they want to pursue it as a career, can we expect such interdisciplinary courses to be successful? Here the record is unfortunately quite discouraging. The National Science Foundation, as a follow-up to the Calculus-Reform initiative, in the mid-1990s supported the Mathematics Across the Curriculum program with large awards to several consortia. One goal of many of these consortia was to develop courses similar to the ones suggested by Bialek and Botstein (2004), though typically just integrating math and physics, math and chemistry, or some other combination. Although a few of these courses are still extant, there has been essentially no spread of them to other institutions, as there was with the calculus-reform courses. The potential reasons are numerous and not the focus of my comments here but include the “visionary burnout” that occurs when the persons most involved are no longer able to maintain the enthusiasm for interdisciplinarity that, to be successful, must have a strong, continual institutional commitment.

Frankly, I have a quite different view of undergraduates than do Bialek and Botstein (2004). It is a rarity for me to encounter an entering student who I can identify as a future researcher. Although this may be easy at certain institutions, I would argue strongly against focusing curricular reforms on the elite. I believe our objective in entry-level courses should be to entice students to see the connections between fields and to open their eyes to the facts that there are numerous open problems, that we (the entire scientific community) actually don't know that much about many areas of science, and that with hard work they can pursue a career in science that is tremendously satisfying and financially rewarding.

GETTING STUDENTS INTO RESEARCH

In Sung and coworkers' (2003) discussion of the educational implications arising from interdisciplinary projects across the biological and physical sciences, several suggestions are provided to catalyze interactions between research groups. Their focus is strictly on graduate training and beyond, and it would be appropriate to supplement this with interdisciplinary projects for undergraduates.

What better mechanism could there be to aid bright young scientists to negotiate the barriers to interdisciplinary research than to offer opportunities for such research early in their education rather than waiting for graduate school? While such opportunities are enhanced through a variety of U.S. government-funded programs (the National Science Foundation's Research Experiences for Undergraduates initiative and a variety of minority education programs of the National Institutes of Health), these directly impact a very small fraction of the undergraduate pool. Suggestions for linking the science in programs such as these to educational research and details on a variety of case studies are given by in Avila (2003).

Classic undergraduate scientific training, providing a depth of experience in one disciplinary department (a “major”), feeds the associated desire of most graduate programs to obtain entering graduate students sufficiently versed in a field to be able to step directly into their graduate courses. There is a definite tension between this view of undergraduate education and the desire for breadth of experience, exposing students to the connections between fields in order to be able to appreciate (and carry out) modern science.

Bright students with sufficient gumption are now “speaking with their feet” and choosing double majors as the easiest means around the compartmentalization of knowledge that is the hallmark of much of undergraduate education. Multiple majors provide the quickest entrée to interdisciplinary fields. For over a decade, the students I find best prepared for graduate work in mathematical and computational ecology have backgrounds in both math and biology. Given the lethargic response of undergraduate curricula to the need for interdisciplinary training, motivated students are making the best of what the bureaucracy allows. Colleges that limit these options are living in the past, and many forward-thinking institutions are now growing double-degree and 5-year B.S.-M.S. programs specifically to meet the desires of their best students for breadth as well as depth.

While such double-degree programs are not the answer for all students, I have long argued (Gross, 1997) the utility of enhancing the interdisciplinary aspects of standard undergraduate disciplinary courses. A readily implemented method to enhance students' perceptions of the interconnections between fields and the unity of the scientific enterprise is the inclusion of biological case studies in mathematics courses and of more quantitative topics in biology courses (Gross, 2000). Even partial steps such as these should ease the transition to the types of interdisciplinary training that Sung et al. (2003) encourage at the graduate level and beyond. Still more effective are true undergraduate research experiences that cross disciplinary lines, not merely getting hands-on experience in a lab, but linking data collection, statistical analysis, and modeling. While few large institutions have the faculty resources to offer such experiences to the majority of biology majors, even a brief exposure to this as a case study within a quantitative or life science course is a positive step.

SOME NOT-SO-RADICAL PROPOSALS

Coming back to the issue of “fitting it all into 120 credits,” the above suggestions provide at least a few mechanisms to assist educators. From a more global perspective, though, we have ourselves to blame. Over the past decades, there has been a strong push toward semester systems and away from quarter systems at U.S. institutions. The argument for this has often been to encourage “depth,” and administrators also claim reduced costs through fewer registrations, though the evidence at my own institution does not indicate any significant financial savings arise. In the process we have sacrificed students options to branch out and investigate topics outside their major that they could not devote 15 weeks to but could spend 10 weeks on.

In developing future researchers, particularly in biology, the implication of Bio2010 is that breadth of exposure to concepts from various fields should take precedence over depth. So it is past time to initiate a“ back to quarters movement.” At the least, discussion of such an option encourages our colleagues to acknowledge the importance of interdisciplinarity. To encourage this even further, we might urge our institutions to place tenure at the college or university level, rather than in a department, potentially easing the acceptance of colleagues who don't quite fit the mold of a single discipline yet are the best educators for a future generation of researchers.