An Innovative Modeling Approach at the Freshman/Sophomore Level

Abstract

The course begins by introducing students to the notion of functions, and to familiarize them with polynomials, exponential, logistic and sine functions. Characteristics of these functions are explored and used to decide what type of function to fit to given real-world data. This segment provides the first example of how a model can be created. It is followed by an overview of the modeling process beyond just fitting data, namely translating assumptions into mathematical equations. Examples are used to illustrate the need to test a model against data to verify its validity, and if necessary to refine the model.

The second part of the course concentrates on discrete models, both single equations and systems. Equilibrium points and their stability are introduced and discussed through examples. Eigenvectors and Eigenvalues are explained geometrically using Mathematica in order to perform a qualitative analysis based on the form of the general solution for systems. The last part of the course introduces the dual notions of derivative as rate of change and as slope of the tangent line. The first is used for model formulation, whereas the second one is the basis for a qualitative analysis of long-term behavior and stability of equilibria through slope fields.

In order to harness the power of Mathematica for advanced calculations and simulations without getting students frustrated, the author developed a user-friendly interface based on the palette feature in version 3.0. This allows students to adjust model parameters interactively to answer "What If?" questions, and to see the influence of parameters on the model outcome. Overall, the goal of this course is to make the student a critical consumer of models, to understand what goes into a model as well as its possible limitations.