Paul Davis
Mathematical Sciences Department
Worcester Polytechnic Institute
Worcester, MA 01609
College Mathematics Journal, 25(5) 1994, 394-400
``What's on the test?'' is not a silly question. Its answer defines what students will learn. The problems we pose to our students are the ultimate measure of what we hope to teach. If we aim in the introductory ordinary differential equations course for more than mechanics - for problem solving, modeling, analysis, interpretation, and the exercise of critical judgment - we must pose problems that demand these skills. Indeed, we need a spectrum of problems ranging from mimic-the-example exercises to open ended projects so that students are supported and challenged as their skills and confidence develop.
This article describes some design principles for constructing ranges of such exercises. The next four sections present a variety of strategies; they are illustrated with examples. The last section states the design principles upon which those strategies are based.
The use of interesting problems benefits everyone. Instructors can break away from a dull routine. (What if solving constant coefficient linear equations were like radioactive plutonium? What if exposure beyond a modest maximum lifetime dose were fatal?!) Students rise to the challenge of problems that require thinking. And interesting problems let mathematics present a better face, showing itself much more than a dull concoction of recipes and rules.