Abstract: There is a need to reform how we introduce math to beginning students in Life Science. The usual “Calculus for Life Sciences”, which is a watered down version of Calculus I, possibly including some trivial biological examples, has failed to inspire students. 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 such a course, and are currently teaching it. 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 change equations called 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.

Our key concept is the idea of a vector field, assigning “change vectors” to every point in the system’s “state space”. This 20th century concept has proven to be pedagogically superior to what we usually teach (differential equations as ‘expressions of the form …’) which is 19th century math.

Students then learn the typical sorts of behaviors that nonlinear differential equations exhibit. A discussion of systems with multiple equilibria leads to an understanding of switch-like behavior in biological systems. We also study nonlinear phenomena like robust 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, oscillations in insulin and glucose levels as well as in biological populations, etc.