The goal of this workshop is two-fold. Primarily we seek to increase the exposure of undergraduates across the nation to the subject of Linear Optimization (LO, a.k.a. Linear Programming). Secondarily we encourage the use of discovery- and inquiry-based methods in the classroom as particularly effective in this context.

First Goal

With regard to the first goal we take the position that LO should be a standard course option in every mathematics major in the country. From the point of view of impact, few algorithms have equalled the Simplex algorithm's affect on modern mathematics, economics, engineering, and business in the last 50 years. SIAM News and Computing in Science and Engineering both listed it among the top ten algorithms of the 20th century (for having the greatest influence on the development and practice of science and engineering), along with the Monte Carlo method, Quicksort, decompositional matrix methods, the Fast Fourier Transform, and others. Students who learn linear optimization prepare themselves for graduate school in mathematics, optimization, bioinformatics, and even business, economics, and finance, for careers in industry and government (national labs, NSA, etc.), and also for the teaching profession (LO is now finding its way into many national and state high school mathematics standards). Very few courses offer the integration of topics and skills that this subject does --- linear algebra, geometry, probability, combinatorics, algorithms, computing, game theory, economics, graph theory, and modeling, for example.

The 2000 CUPM Guide suggests that one of the areas in which Mathematics departments lag behind those of, say, Physics and, most notably these days, Biology, is in the delivery of up-to-date discoveries in the subject (galaxies, big bang, genomics, medicine, etc.). One can easily check their own school's archived catalogs to see that today's college mathematics curriculum is virtually unchanged from 50 years ago and that, in order to compete for majors, departments need to be perceived by students as far more relevant to the modern world. According to the 2000 CBMS Survey, the percentage of departments offering a course in linear optimization in 2000--1 was 13\%, down from 24\% in 1995--6. Such courses, however, are often service courses to engineering or business colleges, suggesting that the percentage is much lower among liberal arts colleges. In addition, the truth may be much lower due to courses still on the books that are no longer taught or are taught infrequently. The point here is that LO is important, relevant, useful, (might we add beautiful) and vastly undertaught. A PREP workshop is one way to begin making colleagues, and therefore students, more aware of the subject and this disparity. We are therefore more interested in colleagues from schools that do not offer LO than from those that do.

Second Goal

Regarding the second goal we have been moved by both research literature and personal experience. Born of the Socratic method and made famous in mathematical circles as the Moore method, inquiry-based and guided discovery learning add the layer of discovery to the foundation of holding students responsible for the proofs of results. The mathematics education research done in the last 10-15 years by Alan Schoenfeld, Annie Selden, Ed Parker, and Jennifer Smith (especially Smith's work on Michael Starbird's use of the modified Moore method in his transition to proofs course at the University of Texas), among others, has begun to shed light on the effectiveness of the method in developing students' skills in exploring, inventing, discovering, solving, generalizing, and articulating mathematics. While one reason for requiring students to study mathematics is because of it's utility, another is because it trains one to solve problems and make arguments in a complex world. While lecture style teaching and repetition may be great tools for the former purpose, more active learning styles may be more appropriate for the latter.

I do not intend to force these alternative methods on anyone or argue that they are inherently better in some sense. Instead I hope to expose people to these methods (if they don't already know them) in hopes that they might afterwards consider attending workshops and conferences that focus on them (such as those given by the Educational Advancement Foundation, Project NExT, and others). I do believe that we all need a large bag of teaching tools in order to meet the needs of students of varied backgrounds in different programs with wide ranging goals.


One interesting benefit of attending a workshop in LO is that faculty who are unable to offer a new course at their home institution can still plug in pieces of subject (in the form of modules, honors projects, senior theses, extra credit assignments, etc) into existing linear algebra, discrete mathematics, geometry, graph theory, and transition to proofs courses.