Book Reviews

A Course in Mathematical Modeling

by Rich West

A Course in Mathematical Modeling 
by Douglas Mooney and Randall Swift
Series: Classroom Resource Materials
The Mathematical Association of America
$41.95
ISBN 0-88385-712-X

The essence of the mathematics reform of the 1980s and 1990s has been the integration into our mathematics curricula of themes.  Three of these themes are:  the use of computer and calculator technology to delete archaic topics and enhance access to difficult concepts and problems; the increase in understanding by requiring concept descriptions and justifications in words; and requiring original mathematical models.  To me the key ingredient to making mathematics come alive is the use of motivating examples that require the student to struggle with the set up of models in realistic problem-solving situations.  The authors' purpose for A Course in Mathematical Modeling is to provide a guide for a stand-alone mathematical modeling course that they offer as an elective to calculus-trained sophomores.  This is an excellent text to accomplish this goal.  At the same time, this text provides the opportunity to integrate the three themes I described above. In addition, I will use this book as a resource of math models to be used throughout the other mathematics courses I teach.  Further, I will use it as a text to guide a semester-long faculty development seminar.  More later.

Here's a quick description of the book.  There are six chapters, each with numerous and diverse examples.  There are very few exercises:  five to ten for each chapter.  There are numerous projects: five to thirteen for each chapter.  Each chapter has a list of supplemental material for further examples and possibly out-of-class research projects.  There are four appendices:  three probability distributions and all the Mathematica code and text for the examples in the book.  There are no answers to the exercises or solutions to the projects.  Again, more later.

The obvious and stated intention of the authors' math modeling course is that it will be project-based.  The authors state that they have tried many mixes of assessment instruments.  Their current mode is to give a midterm exam, final exam, and all the rest are projects.  My guess is that they would do one project for each chapter.  So a semester would have six projects, if all six chapters were covered.  The authors further intend that different projects are used each semester.  However, the authors do not address whether they use groups for the projects or not.  Like many other things in this course, I think that is left to the instructor.  Because I anticipate that this course would require a lot of preparation the first time through, I will add my opinion on groups.  I have been using projects (three to four each term) for nine years and I prefer to use pairs.  If a larger-sized group were desired, I would assign specific roles and responsibilities to individuals within the group.  The only times I have used individual projects is when there is no modeling involved.

The six chapters as presented have very scary titles for students.  Yet, I like the organization.  Discrete modeling is covered first, because it is the more intuitive and quite often more robust.  The first chapter is on discrete dynamical systems (or difference equations), a fairly common choice of modern texts, because it only requires high school algebra skills.  These are deterministic models, in contrast to the second chapter where discrete stochastic models are covered.  The third chapter is a very clever development and use of Markov chains.  The fourth chapter is on interpolation and approximation of empirical models.  Finally, in the fifth and sixth chapters, the calculus student gets his or her wish and sees the deterministic and stochastic continuous models, respectively.  I find this organization logical and plenty for a one-semester course.  In fact, for each chapter or topic, I would choose, or write, a project first, and then maybe delete sections not needed.

All the models in this text are limited to the disciplines of ecology, biology, sociology, and finance.  I find these disciplines accessible and highly appropriate for the authors' mathematical modeling course.  Models must be accessible to motivate all students.  More technical models would require further development and may not be accessible or of interest to all students.  Besides, the references provided at the end of each chapter cover many of these more technical models, and some outside research might be more appropriate for these specialized students.

Since I am writing this review for a Mathematica journal, I would be remiss in not mentioning that all the examples in the book are coded with text explanations in Appendix D.  This provides a nice insight into the world of technology and provides some opportunities for experimentation and sensitivity analysis.  Without technology we tend to limit our investigations.  I also would like to address the issue of answers:  I agree with the authors that they are inappropriate for a modeling text.  Many problems have multiple solutions and models.  If a student's model can be defended, it should be explored.  Along these same lines of exploration, the authors have provided a modeling website that can be accessed through the MAA website (www.maa.org).

Teaching a project-based course the first time can be a high-risk proposition.  The bottom line is that there will be more preparation required, but that the added learning and motivation generated are well worth the effort.  In agreement with the authors, I would encourage the first-time instructor to change projects each term.  If a project is not new and interesting to the instructor, it will be less effective in motivating the students.

In short, I really like this text.  I would love to use it to teach a modeling course.  I will definitely use it as a reference to integrate modeling in my other courses, and I plan to use it as a guide to motivate my colleagues to use projects through the faculty development seminar I mentioned above.  I think there is enough diverse material to interest our entire faculty.  My proposed seminar environment would be a great way to pool our expertise, learn something from one another, and hopefully empower our teaching with new and interesting modeling projects.  I thank the authors for this resource and the impetus.  I encourage the reader to get this book for a course, a reference or as a generator of collegial discussion.

Rich West
Department of Mathematics
Francis Marion University
Florence, SC 29501
Email: rwest@fmarion.edu

Mathematica in Action (Second Edition)

Mark McClure

Mathematica in Action (Second Edition)
by Stan Wagon
TELOS-Springer Verlag
$49.95
ISBN: 0-387-98252-3

In 1991, I was a graduate student teaching in Ohio State's fledgling Calculus&Mathematica program. After the course, I found myself searching for information on uses of Mathematica beyond calculus.  The book I found was Stan Wagon's Mathematica in Action.  This book has had a strong impact on how I work with Mathematica and it is a pleasure to write a review of the second edition [Wagon 1999].

At first glance, Mathematica in Action, is a book about Mathematica.  There is a brief introduction to Mathematica, Chapter 0, and the uses of many built-in functions are illustrated throughout.  More importantly, however, Mathematica in Action is a book about mathematics and this is where the book's strength lies.  While comprehensive reference materials and texts on programming fundamentals are essential to the serious Mathematica user, perhaps the best way to learn to use Mathematica is to play firsthand with code illustrating a variety of mathematical concepts.  This is exactly the approach taken here.

The book has 26 chapters, after the introduction, with a wide variety of topics.  A smattering of the topics includes: basic 2 and 3 dimensional graphics, more complicated graphics including penrose tiles and fractals, numerical mathematics, and number theory including public-key encryption and prime certificates.  The final seven chapters are aptly termed "advanced projects" and investigate problems such as the four-color theorem, the Riemann hypotheses, and the Banach-Tarski paradox.  Even the professional mathematician is likely to learn many things from this book.

The applications range from simple one-liners to fully developed packages.  Here are three examples to illustrate the spectrum.  The choice of examples is somewhat arbitrary and shows my predilection for graphics.  Not all of the book's applications involve graphics.

The first example, from section 1.7, is a one-liner yielding a pretty parametric plot.  The result is a triple orbit, such as the path of a ship in orbit around the moon relative to the sun.

ParametricPlot[
        Evaluate[{Cos[t], Sin[t]} + (1/2) {Cos[7 t], Sin[7 t]} + 
            (1/3) {Cos[-17 t + (π/2)], 
                Sin[-17 t + (π/2)]}], {t, 0, 
          2 π}, 
AspectRatio  ->  Automatic,  
        Frame  ->  True,  Axes  ->  False ];

In spite of the simplicity of this example, it illustrates an important point.  Wagon is a mathematical enthusiast who keeps up with current work.  In fact, the above image was taken from a recent article in Mathematics Magazine [Farris 1996].  The results of the author's readings are incorporated in an extensive bibliography.

The next example, from section 17.4, plots all of the Gaussian primes less than 50 in absolute value.  All the necessary concepts involving Gaussian primes are outlined in the chapter.

\(\(boxes[{x_, \ y_}]\  := \ 
        Rectangle\  /@ 
	{{x, \ y}, \ {y, x}, \ {\(-x\), \ 
              y}, \ {y, \ \(-x\)}, \ {x, \ \(-y\)}, \ {\(-y\), \ 
              x}, \ 
			{\(-x\), \(-y\)}, \ {\(-y\), \(-x\)}};\)
			\)
\(\(GaussianPrimePlot[n_, \ opts___]\  := 
        Show[Graphics[{
	
              Circle[{0, 0}, 
                n + 0.5], 
	{			
		{GrayLevel[0.5], 
                    boxes[{1, 0}]}, 
		\(boxes[{Re[#], Im[#]}] &\)\  /@ \ 
                    Select[Flatten[
							Table[
                          a + \ b\ I, {a, 0, n}, {b, 0, a}]], \ 
                      Abs[#]\  ≤ \ n\  && \ 
                          PrimeQ[#, 
                            GaussianIntegers -> True] &]
			} /. {x_?
                      NumericQ, y_} :> {x, y} - 
                    0.5}], 
		opts, 
          AspectRatio -> Automatic];\)
\)
GaussianPrimePlot[50];

This example is clearly a bit more advanced than the last and underscores the fact that Wagon not only keeps current on recent work, but contributes to it.  This code is further developed later in the chapter to illustrate ideas in an article by Wagon which appeared in the Monthly [Gethner, Wagon, and Wick 1988].

The final example uses a package (on a CD which comes with the book) by Wagon and Rick Mabry to automate a process due to M. C. Escher.

Needs["Escher`"];
EscherTiling[{{1, 1}, {-1, -1}}, 
        Motif -> EscherCurves[24], Size -> {8, 6}];

The Escher package is fully developed and its use requires only a modest understanding of Escher's notation.  This notation and the mathematics behind the package is fully explained chapter 22.  The next two chapters also come  with fully developed packages, the PlaneGeometry package and the GraphColoring package.

If the job of a β-tester is to break software, I suppose that part of a book reviewer's job is to find problems.  My only complaints are minor and seem to stem from Mathematica's limitations as a publishing tool.  For example, there is a font problem in the code at the bottom of page 37 and the output at the bottom of page 49 seems to indicate that 2931374143  |  50!.  Slightly worse are missing curly brackets in the definition of groupExtend on page 557.  Finally, the bibliography, while extensive, has a confusing labeling scheme and several mistakes.  For example, there are two references labeled [GW] half a page apart..

Again, these complaints are minor.  I have been through the book extensively and virtually all of my complaints are listed in the paragraph above.  This should be taken as a compliment for a work of this magnitude.  The bottom line is that Mathematica in Action is an outstanding book containing many examples of real uses of Mathematica for the novice, intermediate, and expert user.

References

Frank Farris 1996.  Wheels on wheels on wheels -- surprising symmetry.  Mathematics Magazine.  69:185-189.

E. Gethner, S. Wagon, and B. Wick  1998.  A stroll through the Gaussian Primes.  American Mathematical Monthly.  104:327-337.

Stan Wagon  1999.  Mathematica in Action (2nd edition).  Springer-Verlag NewYork

Mark McClure
Department of Mathematics
University of North Carolina at Asheville
Asheville, North Carolina 28804
mcmcclur@bulldog.unca.edu