For The Instruction Of

Project-Based Math Courses

Not everything that must be learned can be taught. It is likely that not everything that must be learned can even be identified, much less articulated by an instructor. But one can learn through experience many things unidentified, unarticulated and untaught. For these reasons, we attempt to teach so that practice occurs in the context of an experience whenever possible. Requirements, then, include activities/projects and technology. The activities/projects must be designed to motivate all course concepts and to produce maximum efficiency so that course content can be covered in the time allotted. If, as is true for this course, the study of functions is of primary importance, the activities must involve carefully chosen data which result from (or are described by) an underlying function in the list of course objectives. Technology must be available for student use to obviate prolonged calculations, to make possible real and realistic investigations and to produce reports and presentations.

We are no different than other long-time mathematics teachers. We are accustomed to being an important resource, often the first resource, and at times the sole resource for the student. But this perspective made no more sense when we were taught than it does today, and may well be a major factor contributing to the popular view that mathematics is an esoteric subject. How often have your students asked, "When are we going to use this?" We are required, then, to be willing to give up some control of that which is learned so that more than we try to teach may be learned. And that means promoting other resources - the student's own intellect, that of his or her peers, reference texts, articles, etc. - as key components of the total learning experience. The student should be asked to read, think, understand and apply without excessive examples from the instructor.

To accomplish this, the role of the instructor must change. Many (perhaps most) instructors today teach by example. A concept is introduced via an example which the student is then asked to emulate - perhaps in a context that requires the student to modify the logic of the example somewhat. Then another example follows, expanding on the original concept, and another emulation follows. The scenario continues until the course is complete. In this way we produce patterning, not thinking, and it shows up in our student's subsequent performance. If one is to allow students to learn to think, one is required to provide experiences which require thinking and then be willing to allow that thinking to take place. Writing assignments involving critical thinking are helpful in this regard.

Student thinking will often differ from instructor thinking, and the instructor must be knowledgeable enough and flexible enough to recognize logical thought - even though it may differ from his or her own - and be willing to affirm and support that logical thought. Equally important is the ability to recognize illogical thought as it is displayed in conversation and written communication and to redirect that effort tactfully so as not to alienate the thinker. A requirement, then, is significant preparation and reflection by the instructor to anticipate many more modes of thinking than just those exemplified in a text.

Two-way communication, then, becomes much more a part of the instructor's charge. The instructor must be aware of the quality of student thinking as the student practices thinking. And that is not possible unless that thinking is displayed verbally - orally or in writing. That is not to say that every writing assignment must be graded any more than to say a coach needs to respond to every practice kick his punter takes. But the instructor must be aware of the quality of thinking at all times. This requires the instructor to be attentive at in-class cooperative group sessions and available after hours for consultation in the office, by phone or by e-mail.

One might say that the facilitator of project-based instruction in mathematics teaches students, or perhaps more accurately, helps students learn. The subject he or she teaches/facilitates happens to be mathematics. This student focus requires a different set of skills and perhaps more content knowledge than does "teaching mathematics" from a textbook. But the students' rewards for having an instructor who cultivates the required knowledge and skills appear to be increased learning, the ability to apply that learning and the prolonged retention of that learning.

Your comments and suggestions are important to us. Help us improve Project-Based Instruction in Mathematics for the Liberal Arts. Please e-mail M.B. Ulmer at mbulmer@vm.sc.edu or write him at The University of South Carolina Spartanburg, 800 University Way, Spartanburg, SC 29303.