This is the fourth edition of a very gentle introduction to mathematical modeling. Earlier editions of this book were designed for students who had taken or were taking engineering or business calculus. The new edition has been modified to make part of the text accessible to students without calculus. The authors’ mission is to introduce modeling as early as possible in a student’s program. The consequences of this are reflected throughout the book: in its pace, arrangement of topics, selection of applications and breadth of supporting materials.

The book is divided into two parts. Chapters 1 through 9 consider discrete modeling, while chapters 10 through 13 take up continuous modeling. The balance in pages is roughly 60% to 40%, discrete to continuous. The authors have taken pains to make this text very well-attuned to classroom use. An instructor could use the book for a discrete modeling course that used no calculus whatsoever, for a course that moved quickly through basic discrete modeling to continuous modeling with differential equations, or for examples to supplement a traditional calculus course.

Various levels of technology usage are possible with this book, including paper-and-pencil, graphing calculator, spreadsheet, and *Maple*, *Matlab* or *Mathematica*. Practically speaking, at least *Excel*-type software would be necessary to support interesting projects.

A companion CD includes 53 UMAP modules as well as supporting material for students using one of the third-party software packages. The UMAP modules contain descriptions of an application area together with associated activities and exercises. The authors use the UMAP modules as supplementary projects, further reading, and sources for additional problems. In addition, the CD has links to problems from the various mathematical modeling contests and to Interdisciplinary Lively Application Projects (ILAPs) produced by COMAP. All in all, there are a huge number of applications and projects available to the instructor and students.

The book begins with applications that use simple finite difference equations to model change. Then, having provided some experience with several concrete examples, the authors take up a broader discussion of the modeling process. They classify models, analyze the modeling process and construct some proportionality models that are pursued further in the next chapter. It is good see a treatment of these basic proportionality models, as well as the attention that’s given to dimensional analysis later in the book. These topics are not often treated in books on modeling, yet they can be powerful. They are also one component of the process of teaching students to check that their modeling results make sense.

The new edition of this book has a chapter on graph theory. This acknowledges the increasing use of graph theory in the modeling world. The chapter is relatively short but it does a good job of introducing the modeling capabilities that graph theory offers. It includes maximum flow and matching problems as well as a discussion of social networks.

This is a clearly written, very versatile introduction to modeling that should appeal to students. My only two reservations are that the pace for stronger students is too slow and that the book is too expensive. The first concern could be dealt with by challenging strong students with more complex modeling problems. As for the second, I note (with dismay) that Stewart’s *Calculus* is even more expensive.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.