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.
Plus
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.