Industrial Mathematics: Modeling in Industry, Science and Government. By Charles R. MacCluer/Applied Mathematical Modeling: A Multidisciplinary Approach. By D. R. Shier and K. T. Wallenius

When I began to peruse two new modeling books to prepare this review, my thoughts wandered onto the subject of undergraduate curriculum reform. I thought of a comment in David Kammler's recent book on Fourier analysis, which I paraphrase: for every hour spent on 18th and 19th century mathematics, that is one less hour not spent explaining exciting new and important topics like wavelets, sampling theory, etc. So, somewhat apologetically, I would like to incorporate in this review some of my own views as an applied mathematician who teaches at a large, state university and who has witnessed both success and failure in curriculum reform. I think Kammler's comment reveals where the review is going. First, most agree upon a general definition of mathematical modeling. It is the process of taking a complex problem arising in empirics, inventing a simple set of equations that describe the phenomena, developing solution methodology, and then interpreting the solutions against the empirics. This last phase may force several modifications before a satisfactory model is found. My colleague, Glenn Ledder, puts it succinctly as, "Mathematical models are mathematical objects based on real phenomena and created in the hope that their behavior will emulate the real behavior on which they are based. " The model's success is measured in its simplicity and in its predictive capability. Traditionally, mathematicians often viewed themselves as only part of the solution process. We thought we could sit in our offices and wait until an engineer knocked and entered with a well-posed differential equation; we would solve it, send them on their way, and then wait for the next knock. Thus, for many years applied mathematics courses were focused around solution methods for integral equations, differential equations, and so on. In the 1970s that general philosophy changed. With the revolution in modeling and industrial mathematics came a new initiative for mathematicians; they must learn to participate in all steps of the modeling process and learn to communicate with applied scientists with a common dialog. In his text, MacCluer makes the point that that most undergraduate students are grounded in fundamentals of mathematics, but not its practice. And when they graduate they have little experience with the mathematics that is useful in industry and little appreciation of the fact that "money" is often an issue. With efforts by COMAP, SIAM, and NSA, that is beginning to change, and courses in mathematical modeling or industrial mathematics are beginning to get firmly established in the curriculum. At some colleges and universities, progress toward an undergraduate curriculum that includes modeling courses has been slow. Frequently, at large institutions, the undergraduate curriculum plays second fiddle to the graduate program, particularly with regard to applications of mathematics. Our department recently introduced a graduate course in mathematical modeling, which has successfully been