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Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, Sep 2002 by Wallace, Dorothy

ABSTRACT: This paper describes a successful course in mathematical biology at Dartmouth College. The course targets premedical students and biology majors, rather than mathematics majors, and requires only one semester of calculus as prerequisite. Real world problems form the basis of student work.

KEYWORDS: This paper describes a successful course in mathematical biology at Dartmouth College. The course targets premedical students and biology majors, rather than mathematics majors, and requires only one semester of calculus as prerequisite. Real world problems form the basis of student work.

BACKGROUND

The Mathematics Across the Curriculum Project at Dartmouth College is perhaps best known for its innovative courses combining mathematics and the humanities, yet the project made a serious effort to reach out to the sciences as well. One of the largest majors at Dartmouth is biology, serving around 300 students per year. As is typical at other places, little mathematics is required for the biology major. Our students are not at all incapable of doing mathematics; in fact many of the biology majors can avoid taking mathematics because they have already passed out of the calculus requirement by obtaining a suitable score on the advanced placement test. Yet, the Biology Department requires nothing more and the students typically did not take more mathematics voluntarily.

As a mathematician with an abiding interest in applied questions, I am quite aware of the vast scope of modeling problems that are now coming out of the biological sciences. Gene sequencing may be the hot topic in the press, but problems of the classic dynamical systems and control theoretic kinds abound across all biological fields. Medicine takes mathematics very seriously indeed, and it is difficult to imagine tomorrow's MD's doing any research at all without a basic understanding of dynamic modeling.

Conversations with the advisor to biology majors at Dartmouth led me to some understanding of how biology faculty perceive the needs of their students. A sizeable minority of the biology faculty at Dartmouth are ecologists, whose research papers sometimes consist entirely of mathematics. The advisor to majors was an ecologist, and expressed great frustration with our undergraduate offerings. Listening to my own department members talking about their colleagues, I had expected to be told that biology students needed more estimation skills, more statistics, more data analysis techniques. Instead, the biologist I was talking to expressed his desire for students to know how to read a phase portrait. He also wanted students to understand what an eigenvector is. He asserted furthermore that the primary problem was one of attitude: the biology majors do not believe that mathematics has any relevance to their studies. He concluded by stating that he would surely advise the majors to take a sequel to introductory calculus if we offered a class that met their needs in these areas. As an aside, he added that most of the biology students were "just premeds" and "not really interested in biology".

This last comment ran an interesting parallel to the attitudes one often hears expressed by mathematicians about the students in their "service" courses. Biology, no longer even a single research area, has become a sort of "service" major. If we mathematicians think that too small a proportion of our majors go into mathematical research, we should be glad we are not biologists. In any case, the advisor's comment raises an interesting question. If we are going to build the perfect sequel to introductory calculus with biology majors in mind, how shall we imagine the typical major? Do we envision him or her to have a precocious inclination toward ecology and its models? Or are we better off imagining this person as interested in medical applications and perhaps even pressing social needs?

In either case, the criticism of our existing mathematics classes rang true. The linear algebra class, where undergraduates would learn about eigenvectors, is a relatively advanced class requiring three quarters of calculus as prerequisite. It is considered the gateway to the mathematics major in my department, and great emphasis is placed on proof. It clearly would not help the biology major at all. The dynamical systems class, where one would naturally study phase portraits, is worse. Requiring calculus through differential equations and some understanding of real analysis just to prove any theorem at all, it is a challenge to mathematics majors and completely out of the question for anybody else. The elementary differential equations course also requires three quarters of calculus. Although one could include phase portraits in this class, the biologist I spoke to was not enthusiastic about it. He pointed out that the emphasis of the course was on systems of linear equations and their explicit solutions. Such equations are almost nonexistent in the biologies.

So, our task was to build a course that did several jobs. First of all, it would be accessible to students with only the first quarter of calculus behind them. It would introduce the students to modeling such as is done in ecology and parts of medicine. We did not attempt to include eigenvectors, but concentrated on creating a course that would teach the students to read and interpret graphical output, especially phase portraits. In fact, "interpret" is too weak a word; we wanted students to think critically about the output of their system. The course would have to appeal to biology majors whose primary interest is in medicine, as well as the rest; therefore it could not be primarily ecological in topic. We would address the attitudinal issue by selecting applications of particular relevance or interest and attempting to answer important questions about these systems. We wanted them to see how powerful a model could be. We wanted to make a special effort to appeal to African American students, as most of those who major in the sciences are studying biology with an eye towards medicine. My own personal priority, which I took pains to inject into this mix when my turn came to teach the course, was to insist that the students design the actual equations describing the situations we studied. I hoped that this activity would simultaneously strengthen their understanding of both the elementary calculus they already knew and also the mechanisms driving the systems they would study. In short, I wanted to turn them into researchers.