John Goulet,PhD Department of Mathematics Worcester Polytechnic Institute Worcester, MA 01609
Introduction and Philosophy
These projects were developed primarily in the period 1993-1997 to complement an introductory linear algebra course for non math majors. Students have done as many as 6 of them in parallel with studying traditional course material.
The primary purpose for using many of the projects is to have students see how mathematics, in this case linear algebra, may be used to study meaningful problems from the world. These problems are drawn from the areas of business, economics, society and the environment. The mathematical models are seen as a means to an end, that end being to gain increased insight into a problem and to be able to make a more informed decision regarding it.
Aside from the obvious benefits, a major result of adoption of such projects is that students are substantially motivated to work on them. They find them meaningful and they respond to the idea of making a decision or recommendation, as opposed to a calculation.
Most projects use often mathematics from 3 basic models:
Markov chains quickly work into contemporary business applications, Leslie population models relate to both human population issues and environmental concerns, and graphs have immense flexibility for application as well as a rich history of challenging mathematics.
In many of the projects, long term stability of the system is of interest. Thus, from a conceptual point of view, students experience the role of the dominant eigenvalue of a matrix. Once the material on eigenvalues is reached, the project on dominant eigenvalue calculation may be undertaken, providing numerical experience that many instructors desire. Material on the Perron-Frobenius Theorem, which provides the theoretical foundation for results on dominant eigenvalues, has been included for completeness.
An * indicates calculus is involved.
I have used a team approach to these; 3 or 4 students producing one final document. In a semester setting, 2 weeks provides ample time for a well written, printed paper. The team approach allows students to cooperatively work on mathematical and conceptual issues. Further it is generally accepted that writing is now a beneficial part of mathematics. In these projects, the action of properly explaining and supporting ones conclusions is seen as most valuable.
Software is needed. Almost anything that will solve a system of equations and multiply large matrices (10 x 10) will suffice. I take the approach in this course that software is a means to an end; making calculations quickly and accurately so time may be spent on more important issues. At WPI, we use MapleV, Matlab and Mathcad.
The following considerations have worked well:
When time permits, reviewing a first draft has substantial benefits. However, this may be a great deal of work, and the review must be returned very quickly.