## Elementary Math Models Approach and Philosophy

APPROACH.  This course is different from most college general education math courses. For one thing, this course emphasizes a single theme that runs throughout - mathematical models using difference equations and functions. What is a mathematical model? It is any mathematical formulation that approximates or closely resembles something in the real world. Typically in a mathematical model there are variables representing something we can measure: the temperature in a building, the amount of some drug in the blood stream, or the number of new AIDS cases in a month. The model also has equations that describe how the variables are related to each other, or how they change over time. Almost always the entire apparatus of variables and equations is only an approximation to what is actually going on in the real world, and has been deliberately simplified. That is why it is referred to as a model.

A main focus of the course is one method of developing and studying mathematical models. We will see repeatedly how patterns in data can be approximated using difference equations. We will also see that the difference equations often lead to functions, which are easier to use than the difference equations. By becoming familiar with the properties of functions, we will be able to learn a great deal about the patterns we started with. What is more, it is often possible to use a simple model for one aspect of a problem as a foundation for a more complicated model. This more complicated model might give a more accurate approximation to the same aspects included in the original model, or it might provide approximations for additional aspects, or both. The progression from data, to difference equations, to functions, to more complicated models is widely used in very many genuine applications of mathematics.

CONTEXT.  There is a second major difference between this course and most general education math courses. In developing this course, I made a concerted effort to introduce each mathematical operation and procedure in some applied context. In many math books, the subject is presented in complete isolation from any application. First you are told that a polynomial is such and such a thing. Then you are told that it can be written in such and such a way, that it can be factored, and how to solve equations containing polynomials. You are not given much chance to find out why anyone wants to use polynomials, and how factoring and solving equations are useful. My goal is to start with the context, so that there will be no need to ask, What is this stuff good for, anyway?

TEXT.  In order to provide the context, I have written a book that we will use as a text. The book grew out of materials I developed for students in this course in earlier semesters, and has been published by the Mathematical Association of America. The textbook and the course emphasize reading and writing. It is of paramount importance that you read each chapter carefully. It will not work to start on the exercises and look for examples. The first batch of exercises are reading comprehension questions. Reading the chapter is the best preparation for this kind of exercise.

TECHNOLOGY.   In addition to the text, you are expected to learn about models by experimentation.  Technology will help with this.  I will be distributing several excel spreadsheets that are designed for experimenting with difference equations.  We will also use calculators.  The point of using these tools is to focus on the ideas behind calculations rather than on doing the calculations by hand.

LEARNING OBJECTIVES.  Students in this course should

• Develop a general understanding of how mathematical models are developed and used
• Learn specific methods for one modeling methodology, using difference equations
• Experience the progression from simpler to more complex models
• Observe how traditional mathematical operations and functions arise out of the models we study
• Learn the organizational strategy of grouping functions into families defined in terms of parameters
• Learn the core concepts of chaos as a significant limitation of the discrete mathematical modeling methodology.