An interview with Lou Gross, University of Tennessee
Based on your many years of experience with the Interdisciplinary Quantitative Curriculum Development project, what do you see as the main opportunities and primary challenges for mathematics and biology faculty--both in curriculum and in pedagogy?
First, seeing the educational paradigm shift from purely disciplinary to interdisciplinary. This implies that mathematics courses--not just those for life science students, but all general-audience undergraduate courses--should be infused not with old hokey examples, but with real ones. This means incorporating real-world data in mathematics courses. It also means that to be effective teachers, all mathematics instructors need to learn about subjects outside of mathematics.
For biology faculty, this means challenging students to use the quantitative skills they have already developed, and helping those who have not developed them to do so. Biology faculty therefore would need to be much more pro-active in encouraging students to get to the point of being able to read the literature in the field. Essentially all biological literature, even the more experimental and field-observation types, involves a great deal of quantitative components.
Second, mathematicians and biologists must work together to encourage students to think about problems scientifically, in an hypothesis-formulation and evaluation framework. This can be done just as readily in mathematics courses as in biology courses. Students can be encouraged to form hypotheses based on data regarding the behavior of mathematical objects gathered through graphical or numerical experiments. As part of this effort, biologists need to emphasize much more the importance of theory to the scientific enterprise and how mathematical tools serve to elucidate theory.
Just how far would you like the educational paradigm to shift? Is the extensive use of real problems with real data enough to make a mathematics course interdisciplinary? Doesn't real interdisciplinary teaching require a melding of goals and methods of two or more disciplines?
While I agree that truly interdisciplinary courses require either two or more instructors with complementary expertise or one instructor whose background is itself interdisciplinary, courses that really explore the linkages between different fields (or focus on a topic that intrinsically requires viewpoints and methods from different fields) are rare in the structured academic system in which most students find themselves. At most institutions, academic boundaries are not going to change soon. I believe in practical approaches, and thus trying to overthrow the current structure in universities is foolhardy. We have developed a system of higher education that, although far from perfect, is most likely the best on the planet. The availability, diversity, and quality of higher education in the U.S. is one of the major achievements of our nation in the latter half of this century--comparable to the magnificent achievement of very high literacy rates that were achieved by the early 1900's relative to the very low rates virtually everywhere else on the planet.
No, just throwing examples into a mathematics course doesn't come close to making it interdisciplinary. In general I think it's an unrealistic goal to develop truly interdisciplinary mathematics courses at the entry level. Most institutions do not have the emphasis on teaching, the interest of faculty, or the resources needed to do this.
What we can do is motivate students by including real data and examples. We can develop a few truly interdisciplinary courses so that at some point in their undergraduate experience all students are exposed to such courses. We can encourage more linkages between faculty in different fields to share expertise in an undergraduate setting. We can encourage graduate students with interdisciplinary interests to be involved in undergraduate education. We can help non-quantitative departments add more quantitative components to their own courses. We can encourage departments hiring new faculty to think about breadth rather than just specialization.
I am well aware that I am taking the large-institution view and that there are many educational institutions (and sometimes programs within larger institutions) that strive to be highly interdisciplinary throughout the undergraduate training they offer. However, relative to the total number of undergraduates in the U.S., the number of students involved in these programs is small.
Has the widespread use of computers for modeling and graphical analysis changed the kinds of mathematics (in school or college) that students need in order to pursue the study of biology?
My project has dealt extensively with the issue of what quantitative skills life science students should be exposed to as undergraduates. These go far beyond the traditional topics covered in a standard calculus course for students in the social and biological sciences. They include statistics, probability, linear algebra, discrete-time modeling (e.g. difference equations), and differential equations.
The more extensive use of computers merely reinforces the need for an appreciation of the diversity of mathematical tools that are applied in hosts of areas in the life sciences. More than that, the use of computers actually makes it feasible to expose life science students to such a breadth of topics, both within quantitative courses and within biology courses.
So the use of computers and modeling increases students' need for mathematics. Many think just the opposite! Are you comfortable with students knowing primarily computer mathematics (e.g., solving equations on a graphing calculator) rather than traditional pencil-and-paper methods?
I am comfortable with students being able to solve problems and think about real-world issues in a quantitative manner. I would argue that to be successful at this in today's world requires both the ability to use computers as a tool and the ability to conceptualize how to use them as tools. Thus the ability to set up a problem in terms of some underlying set of variables and determine the relationship between these variables is much more important than how to solve the resultant set of equations using either pencil and paper or a computer.
Because so many problems (particularly real-world ones) do not lend themselves to easy pencil-and-paper solutions, the ability to use computers to visualize and analyze solutions is critical. No matter which way a solution is obtained, a person's intuition as to whether the result makes sense is also important. Doing this effectively no doubt requires some pencil-and-paper ability.
You have written about the "CPA" approach to developing quantitative interdisciplinary curriculum. Can you explain what that is?
CPA stands for "Constraints, Prioritize, Aid" and was designed to help mathematics faculty who wish to develop an interdisciplinary approach with colleagues in other disciplines. First, it is important for mathematics faculty to understand the constraints imposed on a curriculum, constraints which limit the amount of time any student can devote to a particular subject (or group of subjects) and which limit particular courses that must serve a varied clientele. Second, due to these constraints, faculty must work together to prioritize the key quantitative concepts essential to fields outside of mathematics and the level of skill development students need in order to apply those concepts and skills in courses outside of mathematics. Third, mathematics faculty need to aid colleagues in other disciplines to include directly within their courses regular use of such quantitative concepts and skills.
This "CPA" approach is suited especially to typical institutions with disciplinary compartmentalization. An entirely different approach, less often possible, involves complete revision of course requirements to create real interdisciplinary courses that allow students to automatically see connections between various subfields.
Mathematics teachers sometimes worry that the context-rich environment of an interdisciplinary course will impede learning rather than enhance it, since it will be harder for students to sort out the mathematics principles from the surrounding context. What has been your experience in this regard?
This is silly. It is an excuse offered by instructors who don't feel comfortable dealing with real applications. In my experience, students are not side-tracked by real examples unless those examples are carried to such an extreme that the context is lost. What is most important is the ability of an instructor to provide motivation by giving conceptual, intuitive explanations of how the mathematics is useful. Unfortunately, in my experience, few mathematics faculty have any desire to develop sufficient intuition in other fields necessary to motivate the mathematics with examples.
In college, especially after the first year, students know pretty well what area they want to pursue. This makes it feasible to plan special courses for biology students. But what about grades 10-13, when mathematics classes cannot assume a singular scientific or vocational focus for the students? How can mathematics teachers with heterogeneous classes provide the kind of instructional environment that your IQCD project hopes to accomplish in more specialized contexts?
As I have said, I do not believe specialized courses are the only option. I do think they are appropriate and efficient in disciplines such as biology (there are typically more life science majors at an institution than chemistry, physics and mathematics majors combined). However, what is important is the use of motivating examples tied to real data. Biology provides a host of these for virtually every concept within the 10-13 curriculum that, with appropriate assistance from instructors, can readily help students see the connection between the mathematical abstractions covered in the courses and their utility in the real world.
Lou Gross is
Professor of Ecology and Evolutionary Biology and
Mathematics at the Institute for Environmental Modeling at the University
of Tennessee in Knoxville. Gross is editor of the on-line Mathematics Archives for Life
Sciences and can be reached by e-mail at firstname.lastname@example.org.
Last Update: June 17, 1997