Modeling and Computation Seminar
Teaching Dynamics to Biology Freshmen: a Modeling Approach
There is a great need to reform how we introduce math to beginning students in Life Science. The usual “Calculus for Life Sciences” leaves students with two overwhelming impressions, as they have indicated in survey after survey: (1) “I hate math” and (2) “math has no application to biology”. Even worse, the math gateway courses into the life sciences serve as powerful filters keeping women and under-represented minorities out of the life sciences and medicine. Recently, there have been calls, from all the leading voices in US biology and medicine, for a new approach to mathematics for biology. We designed, and are currently teaching, a course like this. The course introduces students, on day 1, to the concept of modeling a system that has multiple interacting variables and nonlinear relations. The student quickly learns that models give rise to differential equations, and that differential equations can always be “solved” (that is, simulated numerically) using Euler’s method. They learn to program their own code for Euler’s method in a Python-like environment. We found that the key concept is the idea of a vector field, assigning “change arrows” to every point in state space. This 20th century concept has proven to be superior pedagogically to the 19th century math of differential equations as ‘expressions of the form. Students learn the typical sorts of behaviors that nonlinear differential equations exhibit, like equilibria, oscillations and even chaotic behavior. The major concepts of calculus, derivatives and integrals, are developed, as well as an introduction to matrices and eigenvalues and eigenvectors. Throughout, there is an emphasis on biological applications of these concepts, such as homeostatic (equilibrium) behavior in physiology and in ecological systems, multiple equilibria causing switch-like behavior, oscillation in insulin and glucose levels as well as in biological populations, etc.