Points of View: The Interface of Mathematics and Biology
It's true. If you live long enough, you will see good ideas (and bad) come, go, and return again. So if you're old enough to remember the early 1960s songster/musical prankster, Tom Lehrer, you might remember his clever satire called “New Math,” which lampooned a popular curriculum reform of the same name. New math's proponents championed the teaching of set theory, Boolean algebra, vectors, matrices and Markov chains, combinatorics, and a little splash of game theory. The motivating factor behind this revolutionary outlook was that these skills would come in handy in the physical sciences, social sciences, and—yes—life sciences. Fast-forward four decades to 2004 and, sure enough, such skills are even more useful now than in those post-Sputnik driven days of science education. But today, most of our students don't possess them—and most of them didn't back then, either. What happened? Did the new math movement wither away out of neglect or no respect? No, not completely. In fact, new math did get integrated in bits and pieces into undergraduate and even into high school curricula, surviving the scorn of“ back to basics” movements. Various components of new math, in the form of “finite” or “discrete” mathematics courses, were “grafted” into undergraduate courses, especially those substituting for or supplementing the (still) traditional year of calculus, particularly in courses for non-science majors. So is there a problem here? Now, in the age of the “new biology,” especially, systems biology, I think there is. Our biology students still tend to be math-averse and many of us instructors could stand a brushing-up, if not outright resuscitation, of our own math skills, which may have atrophied from disuse.
Part of the problem is that the old new math is still taught largely as a math or computer science course by math professors, and although examples from applied disciplines provide ample problem set fodder, students don't seem to transfer those skills into their biology courses as facilely as we might hope. Students need to relearn matrix operations anew, although admittedly, cognitive “savings” may make the (re)learning curve less daunting a climb. It might be useful to take a quick glimpse at curriculum reform in mathematics and see how it has fared since, say, 1960.
Dartmouth College has long been a “hot spot” for mathematics curriculum innovations. The Dartmouth mathematician John G. Kemeny was the lead author of a remarkable textbook, Introduction to Finite Mathematics (IFN), first printed in 1957. This book began with symbolic logic and truth tables, then introduced the reader to set theory. This was followed by combinatorial math (“partitions and counting”) that led directly to probability theory. The final section took up vectors and matrices, which led to game theory and linear programming. The last chapter integrated all this nice math into applications in genetics (Markov chains), economics (game theory), and even some graph theory (communication networks). The prerequiste for IFN was 2.5 years of high school math (1950s vintage) because it was aimed at college freshmen. The book was revolutionary and it went through several editions before ending up in “remainder heaven” by the 1980s, out of print; it had become unfashionable, sadly. Nonetheless, finite math morphed into discrete math in subsequent decades and, with it, a heavier dose of graph theory. Calculus is not a prerequisite for discrete math and its techniques are highly relevant for today's life sciences. I would never deny the utility and necessity of calculus for natural science students, including biology, but discrete math can be learned independently of calculus and, I believe, may draw on cognitive skills on the continuum of mathematical literacy other than calculus—perhaps more user-friendly to the practical mind of a biologist. To finish my digression into history, John Kemeny later became president of Dartmouth College, but even then he continued to contribute to undergraduate math education by inventing the well-known computer language, BASIC, which was intended to be more a means to teach and learn programming than a programmer's working language, such as FORTRAN or C; BASIC is still around and useful in an object-oriented incarnation (e.g., Visual Basic). In the 1990s, Dartmouth mathematicians, led by Dorothy Wallace, proposed a visionary math-based curriculum reform called “Mathematics Across the Curriculum” (MATC; http://www.math.dartmouth.edu/~matc/). MATC was enthusiastically supported by the National Science Foundation's Department of Undergraduate Education. So Dartmouth, in collaboration with other colleges and universities, formed a large consortium group and dedicated themselves to the goal of providing model curricula and supporting textbooks (from Key College Publishing). Thus, mathematics was integrated into courses in both physical and biological sciences, certainly, but also in humanities (English, art history, music, drama) and the social sciences. An impressive set of textbooks and curricula was developed and published (Wilson, 2000; National Research Council, 2003; Doyle, 2004). However, in spite of great dedication and enthusiasm by both students and faculty at Dartmouth, the part of MATC that most strongly engaged the physical science and engineering with math—the Integrated Math and Physics Program—was canceled by Dartmouth in 2002 ( http://www.thedartmouth.com/article.php?aid=200204260103). Among the reasons given: lack of institutional funding to participating departments for additional faculty to provide for the added teaching load resulting from the creation of new, albeit innovative and even popular, courses. Perhaps there is a lesson here, albeit a sad one, for visionary curricular reform efforts. Clearly, the institution's administration must stand behind curriculum reform, a matter I'll return to at the end.
For the past 10 or 15 years, the teaching of college calculus has been the subject of a nationwide reform movement in mathematics departments across the country, which continues to the present. It has drawn many enthusiastic supporters and some detractors, for it remains controversial. Its aim is to teach the meaning of calculus concepts so that they can be confidently and successfully applied in the student's own discipline. Framing the concepts of calculus within real-world applications is a strong pedagogical theme. In some courses, in order to bring out the useful and powerful computational aspects of calculus as applied to the sciences and everyday life, instruction may include “crash courses” in useful computer software, such as Mathematica, Matlab, and Maple, and handheld calculators. The“ cost” is to de-emphasize the rigorous but foundational aspects of formal proof in matters of continuity and differentiability, for example. This movement has, as part of its aim to increase the retention of students taking math postcalculus, to make math more attractive to students who might ordinarily be “turned off” or ignore math and to lower the number of students who drop out before completing the calculus sequence. The content of reform calculus is broadened to include examples from many disciplines, including biology. The reform goes beyond how the subject matter of calculus is reorganized; at many institutions, this is paired with new pedagogical practices, such as group learning in the context of small classes. Narrative-based materials such as the history of mathematical ideas and biographies of famous mathematicians supplement the traditional problems sets. Group learning is part of the “active learning” movement, itself a curriculum reform that, although several decades old, has begun to emerge as an important practice in the teaching of the physical and the life sciences.
What particular implications, if any, are there in these developments for addressing the mathematical training of twenty-first century life science undergraduates? Well, the new math, MATC, and calculus reform were centered primarily in mathematics departments. But is that sufficient? Perhaps it's time for biology faculty themselves to take some proactive steps and instruct their students in the necessary mathematics as situationally and pedagocially needed, rather than just leaving math instruction to the math department and hoping that their students will retain their math when they come to their biology courses. This might mean that that some biology faculty might need to shake the rust off their own math skills and learn some new ones. This does NOT mean that biologists should teach the math basics (we have enough to do as it is), but that they should reinforce what their math colleagues had taught their students earlier. How best to do this? Collaborative teaching between math and biology faculty is one way as I discuss later.
NEW BIOLOGY AND THE OLD NEW MATH
This could be an entire article by itself. I'll take just two examples.
Genomics. All areas of biology will feel the impact of genomic and proteomic science. The sheer volume of sequence data poses difficult and interesting challenges to biology researchers at all levels of biological analysis and all size scales, ranging from molecular to eco- and global systems. The wealth of sequences of DNA and RNA nucleotides and amino acids from cellular macromolecules and whole organism genomes and proteomes opens up undreamed of possibilities for comparative biology that boggle the mind. Making sense of sequence comparisons is the province of computational genomics and computational biology and is being implemented by wedding biology with the algorithmic expertise from mathematics, statistics, and computer science. The computations involve pretty fancy combinatorial mathematics and probability and the depiction of relatedness of sequences in the form of lineage and other phyletic trees. This ups the ante for knowledge of statistics and probablility needed by undergraduates. No longer is it sufficient to know the canonical descriptive statistics of normal distributions, such as measures of central tendency and variation, correlational measures, and hypothesis testing. The reams of molecular sequence data available on the Internet databases permit (or require) techniques such as factor and cluster analyses, principal components analysis, and Markov processes, to name a few. These techniques were presaged by educational visionaries like John Kemeny and his Dartmouth colleagues.
Neuroscience. The nervous system is the information superhighway of any animal more complex than a single cell. In humans, information flows as bioelectric currents that propagate throughout the living neural networks in the brain and spinal cord, which connect the body's sensory systems to its motor systems. The building blocks of our neural systems are single neurons that communicate with each other at anatomical and functional specializations called synapses. At synapses another form of information processing takes place that involves a network of chemical reactions occurring within each neuron partaking in communication—these “chemical cascades” are networks nested within each neuron (node) of an expansive interneuronal network that may includes hundreds or thousands of nodes, the totality and configuration of which vary on time scales ranging from micro- and milliseconds to seconds to minutes and longer. The chemical cascades within each neuron may lead to gene activation that may involve many genes, themselves forming a genetic network within the neuron's nucleus, which may ultimately lead to gene expression that leads to a cascade of protein synthesis (again, interacting within yet another network—of protein interactions in the cytoplasm of the neuron). The “end result” may be alterations in the structure and function of synapses, which manifest themselves as familiar behavioral changes such as learning and memory. If this seems baroque and complicated in terms of the different “players” and “pathways” of interaction, there is a branch of mathematics that can be applied to each step and size scale, with a generic set of computational tools to characterize each network. That is the province of graph theory.
Among systems biologists, it is thought that what is common to the various systems and levels of biology, at virtually any size scale, is the flow of information between and among many nodes. The instantiation or“ snapshot” of a network at a particular place or time within the system can be called a “graph,” not the familiar Cartesian version of orthogonal X and Y axes, but a network of“ hubs” and interconnecting “roads.” Whether the biological instantiation of a network under consideration is the pentose phosphate shunt in relation to glycolysis, or genetic networks and pleiotropic outcomes, or the control of salivation in relation to hearing a bell sound, or the effect of acid rain from the Midwest on the breeding cycle of trout in the Adirondack mountains, the underlying mathematics of graph theory is the same—that's what mathematics is supposed to be: generalizable and content—independent— which is it's power and beauty. But graph theory is also an abstract branch of mathematics. Isn't it already hard enough for biologists to master intracellular metabolic cycles, or the physiology of hippocampal neural systems involved in learning and memory, without climbing the difficult slopes of Mount Mathematics? The question is really whether some elements of graph theory can be taught, albeit without the formal rigor necessary for practicing mathematician, to biologists that can inform their comprehension of biological networks, or better, to enable them to articulate biological problems in a form that would lead to congenial collaborations with mathematicians who so understand graph theory deeply. It is certain that future research in life sciences will become ever more integrative and quantitatively rigorous, and hence necessarily collaborative, requiring the expertise of biologists, engineers, and computational scientists. At the very least, biologists should know enough mathematics to help formulate theory or to provide feedback by articulating biological constraints on the assumptions from which a mathematical model must begin. A good start would be for life scientists and mathematicians to develop synergistic teaching materials from the rich content of biology.
WHAT TO DO?
There are several ways to get this done, but all involve biology professors to refresh or learn anew some new (old) math. Increasingly, interdisciplinary courses are being created that are team-taught, acknowledging the obvious fact that it's getting harder for one person to teach even traditional courses in biology, such as developmental biology, cell biology, or neurobiology, and cover the multidisciplinary nature of modern life science research, which draws on physics, chemistry, computer science, engineering, and mathematics. Teaching teams need to include colleagues from these sister sciences. One of the pleasures of teaching as part of a team is that you receive a fresh look at familiar material—you learn something new, when you sit out there among the students, hearing your colleagues. This is very demanding of faculty time, however, and participation in interdisciplinary team teaching must be recognized as an unalloyed “good” by department chairs and deans. There needs to be a system of computational math workshops that target biology faculty. These could be organized either within one's home institution, if resources were sufficient, or by a consortium drawing from nearby institutions. These could be conducted during intersessions and summers. Again, local administration must acknowledge the payoff for modernizing life science courses (and its faculty) by encouraging and rewarding faculty that participate, by deferring costs to participate in workshops. Even more essentially, if a college is in full-fledged curriculum reform, its administration must support faculty-driven efforts, especially those faculty who risk stepping outside well-prescribed traditional disciplinary ways of scholarship and teaching. Anyone who has engaged in team teaching with faculty from other disciplines knows it takes much more time than teaching a solo course. Even though the number of actual lectures one prepares may be considerably less (e.g., half) than the number given in a solo course, one is obliged to go to every lecture and discussion section because not doing so vitiates any attempt at presenting a seamlessly integrated course. For anyone who's ever done this kind of teaching, it's exhilarating because it enables us to do what we do best—which is to broaden our horizons by learning new material and sharing our gains through teaching. But it takes time—administration should encourage participation in interdisciplinary courses by awarding full teaching credit even though the course “face time” may only be half of that in a solo course.
It is also worthwhile encouraging teaching teams who already have interdisciplinary expertise to develop instructional software in a modular format such that the modules could be the basis of “bootstrap” learning by both students and faculty. Such software should take advantage of the amazing graphical and animation capabilities in commercial development software to design didactic modules that engage dynamic visual approaches, as well as the computational and narrative approaches that are the ways of textbooks. We learn in all sorts of ways, and teaching software should exploit all avenues. The National Science Foundation has for almost two decades, championed multiple approaches toward the end of enhancing the math skills of life scientists, and innovative programs have resulted, including MATC at Dartmouth and its consortium colleges. There, the lead was taken by mathematicians and math departments. That's entirely appropriate and natural, but I'm saying that it's time that biologists and biology departments need to be proactive and work with their math colleagues, to upgrade their courses, computationally speaking. It will also require all biologists to get with the program, themselves.
