## Embedding Mathematics in Context

An interview with Chris Arney, United States Military Academy

These days biology is the most popular science on most campuses. Based on your years of experience in interdisciplinary curriculum development, what do you see as the main opportunities and primary challenges for mathematics and biology faculty--both in curriculum and in pedagogy?

Biology is the science of life and of living organisms, including their structure, function, growth, origin, evolution, and distribution. Biologists try to find relationships that help us understand and explain the structure of life, which is obviously of great importance to the advancement of society. However, these relationships can be very complex. To facilitate their analysis, biologists often transform these relationships into mathematical constructs. The process of explaining real-world behaviors using mathematical constructs is known as mathematical modeling. Thus biology can be viewed as a form of applied mathematics.

The biologist and the mathematician have different bases of knowledge, however, they have the same goals. Both want to find relationships that provide a better understanding of the universe.

For example, effective modeling of the population of a particular species requires both the knowledge of a biologist and the capabilities of a mathematician. Biologists can provide insight on a species' birth rate and death rates, environmental capacities, migration, and epidemics. Furthermore, biologists can identify the model's simplifying assumptions and understand their implications. However, the processes generated by a mathematical model can be most effectively examined and verified by a mathematician. Biologists are concerned about the relationships involved in population growth, whereas mathematicians are concerned with the process that validates these relations. Both groups, however, are concerned about gaining a better understanding of real-world behavior.

Mathematics and biology faculty have many opportunities to contribute to curriculum and pedagogy. There are many recent textbooks in the areas of biostatistics and mathematical modeling for biology and environmental sciences. Nevertheless, additional opportunities for students to learn statistical principles within a setting of biology would be a positive change. In addition, discrete dynamical models can help students understand population growth (e.g., "predator-prey" models) under many conditions, thus leading to insight into the complex dynamics of nature.

I see this kind of interdisciplinary setting as beneficial. The challenge lies in the dynamics of making it work with faculty and for students. I would recommend an interdisciplinary course in "Mathematical Modeling with Biology," taught by both a biology instructor and a mathematics instructor. Such a course could introduce biological concepts by having students employ scientific experiments, collect data, and perform mathematical analyses. The biggest challenge is time. The departments must provide the extra time necessary for instructors to build, plan, and teach a course that functions as a single course. Course preparation is often much longer for interdisciplinary courses, especially when they are team taught.

Has the widespread use of computers for modeling and graphical analysis changed the kinds of mathematics (in school or college) students need in order to pursue the study of science?

The advent of computers has altered the skills used to apply mathematics in the analysis of a problem. No longer is it necessary to have the skill (or patience) to manipulate numbers in order to transform data into a useful form. Because of the computer, real problems with large data sets are now accessible to everyone. The computer has given us more time and more opportunities to analyze problems and discover relationships. In essence the computer has allowed mathematics, especially at the undergraduate level, to become more relevant.

The widespread use of technology for mathematical modeling and data analysis has provided mathematics students with more visual power over the subject. Thus today's mathematics students have the tools to be more competent problem-solvers. Some students may be less capable of by-hand manipulations, but all students can use computer tools to gain valuable insights into problem. Moreover, today's students are often asked to describe verbally what they see and how to model the data. Students frequently need help in translating the results back into the subject area. Instructors must be prepared to bridge the gap between verbal descriptions and the ability to "make it happen."

Mathematics teachers often worry that the distinctive nature of mathematics will be lost unless it is taught as a separate subject rather than as part of an integrated interdisciplinary program. Do you share this concern?

I do not know the distinctive nature of mathematics. I do know that mathematics is a framework for analysis. This framework can be used effectively without understanding its underpinning. However, we do need to continue to increase and improve the current mathematical framework. Often in interdisciplinary programs where the framework is used, it is not examined and critiqued.

Mathematics teachers also worry that the context-rich environment of an interdisciplinary course will impede rather than enhance learning since it may be harder for students to sort out the mathematics principles from the surrounding context. What has been your experience in this regard?

Interdisciplinary courses motive the study of mathematics! We do not want to sort out the mathematics from the surrounding context. We want the mathematics woven within the context. However, we also want students to understand the mathematics and not apply it blindly.

One must articulate course objectives prior to designing a course. An interdisciplinary course needs to be a blend of science and mathematics designed to enlighten the science. New mathematics can be taught and learned if there is a need. But the scientific subject area should create the motivation to learn the mathematics. The blend must not be weighted more heavily to either science or mathematics. Moreover, you do not want to blend a 3-hour biology course and 3-hour statistics course into a single 3-hour bio-statistics course covering all the former material. This simply cannot be done. An interdisciplinary course works best as a sequel that builds on basic building blocks from previous courses.

Colonel Chris Arney is a member of the Department of Mathematical Sciences at the United States Military Academy at West Point and editor of Interdisciplinary Lively Application Projects (ILAP) published in 1997 by the Mathematical Association of America. He can be reached via e-mail at ad6819@exmail.usma.army.mil.