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Elementary Mathematical Models: Order Aplenty and a Glimpse of Chaos

Dan Kalman
Mathematical Association of America
Publication Date: 
Number of Pages: 
Classroom Resource Materials
[Reviewed by
Kelly Black
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Elementary Mathematical Models offers a gentle introduction into the ideas of discrete mathematical modeling with some basic extensions to continuous functions. The book is well balanced and succeeds in introducing the use of discrete models to students who might view a mathematics class with a weary eye.

The first chapter of the book is written for both teachers and students. It offers a broad overview of how the material is presented and certain expectations of the students. Moreover, an explicit discussion on the use of numerical, theoretical, and graphical approaches is given in the first chapter. This is a nice touch that sets the stage for the following chapters.

The first chapter is quite short and offers a smooth transition to the second chapter. In the second chapter sequences are introduced, and the examples are directly linked to the use of difference equations. The second chapter is also introductory in nature. A variety of difference equations are introduced, and the different aspects of each equation are carefully examined. The author goes to great lengths to insure that the foundation for the rest of the book is firmly laid down in the first two chapters.

The mathematical sophistication that is expected of the students gradually increases in the chapters that follow. By the end of the book, students will have taken part in problem sets that examine linear growth, quadratic growth, the basic shapes and properties of rational functions, logarithms, exponentials, and some nonlinear difference equations (the logistic equation in particular). More importantly, problem sets are given which allow for a diverse set of explorations.

Even as the material increases in complexity, new concepts are introduced in the gentlest and slowest way possible. However, the ramifications of some of the definitions can be developed too quickly or taken for granted. For example, the use of first differences in linear, discrete models is fully explored and motivated in the most cautious and deliberate manner possible while the use of second differences for discrete, quadratic models is given to students quickly with the expectation that they can easily identify with this next layer of abstraction (see page 81).

Moreover, the slow introductions can be confusing at times and some more aggressive students may feel that the narrative is condescending. For example, a discrete logistic equation is introduced (page 298) and the fixed points are discussed. Instead of simply factoring out the zero root of a polynomial that is already factored (dividing both sides of the equation by x), the author treats the remaining linear term as a variable growth rate and relating it to the linear growth models found earlier in the book. Another example, is a Taylor expansion (page 156) for the square root of 1+x. The expansion is given with little motivation except for the qualifier, ``methods of higher mathematics show that...'' Some students who want to look more deeply at the material might not react well to this.

On the whole, though, the author does a superb job of addressing a difficult audience. This is especially true of the problem sets. An excellent mix of reading, simple/short answer, and word problems of varying difficulties are given. Furthermore, complete answers to some of the problems are given in the same chapter (rather than in an appendix).

Kelly Black is an assistant professor of mathematics at the University of New Hampshire.

Overview; Sequences and Differences Equations; Arithmetic Growth; Linear Graphs, Functions and Equations; Quadratic Growth Models; Quadratic Graphs, Functions and Equations; Polynomials and Rational Functions; Fitting a Line to Data; Geometric Growth; Exponential Functions; More on Logarithms; Geometric Sums and Mixed Models: Logistic Growth; and Chaos in Logistic Models.