One the finest of Monty Python's skits is the "Albatross sketch." John Cleese, clad in a cute little lacey pink number, stands in front of a listless theatre-going audience, attempting to entice their appetites with a giant dead albatross piled uncomfortably in the box hanging around his neck. "Albatross! Albatross!" Michael Palin enters, stage-right, and politely asks for two choc-ices. "I 'aven't got any choc-ices, all I've got is this bleedin' 'uge sea bird!" responds Cleese. Palin pleads futilely for some other sort of treat, but cannot get around the fact that if all Cleese has got to sell is that damned bird, then that's all Palin's going to get.
Teaching college mathematics (or any other subject, I'm sure) can feel like this sometimes: constrained by some generally agreed-upon list of "must-know" facts, I've got an inventory of information I've got to unload on my students, and whether they like it or not, they're not going to get any substitutions. But not everyone cares about real vector spaces and complex eigenvalues for their own sakes. No one gets hot under the collar about mind-numbingly boring change-of-basis transformations and rank equations. And why should they? For crying out loud, I don't get excited about these sorts of things...that's why I hated linear algebra when I first had it! I didn't give a rat's rear end about it until I had to use it.
I've always found that the most difficult point to get across to college math students is that there really is meaning behind the mad methods of mathematics. Mathematics, from calculus on up, and from calculus on down, plays a pivotal role in thousands of different applications, many of which involve the students in their everyday lives, and which thus can be made instantly tangible to them. There's really no need to shout in their faces, "you oughta buy this, this means something!" (an only slightly more articulate rendering of "albatross") when there are more civilized means of impressing them with the relevance of the mathematical tools they pick up in any given course. It's not always easy to keep this in mind, the way many math courses have traditionally been taught: definition, theorem, proof, definition, theorem, proof, definition, theorem, proof...with an occasional example thrown in to break the monotony.
For the past few years I've toyed with the idea of teaching a course from a purely problem-based point of view: I'd hand the students some real-world posers when they first walk in the door, let 'em tinker, let 'em play, let 'em experiment. I'd let 'em suffer a bit, if the suffering is a useful one, but gradually direct them towards more sophisticated experimentation and understanding by slowly supplementing their bag o' mathematical tricks. Ideally, all questions of relevance will have been answered by the course's end: every student will have played a part in solving at least one substantial realistic problem by applying the techniques learned in the course.
Wow!
But I've been afraid to take this pedagogical plunge, and why shouldn't I be? I've always gotten good reviews from my students when I teach the way I do, so I'm obviously not doing anything truly heinous. And gosh darn it, I'm comfortable in the way I teach: it's a method that's worked for me for years. I've simply never had the courage to try something radically new until now.
For the past few weeks, I've joined several of my colleagues at the University of North Carolina, Asheville in a Learning Circle focussed upon the book Creating Significant Learning Experiences, by L. Dee Fink. This man challenges the university instructor to take that last step off the high-dive by designing and implementing a college course which will provide students with a meaningful learning experience, one whose effects they will feel years after the course is completed. It's not sink or swim, though: besides laying out a coherent structure for successful course design, Fink gives a lot of hints as to how one might create such a meaningful course. The book is replete with helpful suggestions and illuminating case studies, and lively discussions with my colleagues have supplemented the reading with oodles of ideas, particularly when it comes to "UNCA-specific" issues. The past month that I've spent with my colleagues has given me the little bit of courage (and a good deal of the practical know-how) I was wanting in order to make my "dream course" a reality.
Beginning in mid-August, I'll be leading thirty-odd students (or, just as likely, "thirty odd students") through a problem-based study of introductory linear algebra. From the outset they will approach the topic from a practical point of view, learning linear algebra by working through realistic applications to chemistry, network design, crystallography, cryptography, thermodynamics, economics, aeronautical engineering, traffic flow...whew! Working in small teams, they'll wrestle with unprettified problems and learn advanced mathematical techniques by playing with them and applying them to the questions that arise in their research. They'll take notes on their research activities and keep track of the progress both of their teams and of themselves as individual thinkers. They'll write preliminary reports and academic articles. They'll present their findings to each other in class, and to the department as a whole at an end-of-semester "symposium."
And, I hope, they'll have fun. Why shouldn't math be fun?
From now until the end of the course, I'll be posting all sorts of muck on this site: thoughts, meditations, ideas, course material...I'll likely often use this blog as a laundry list for myself, so I warn the reader in advance that some of what I write will probably either be boring as hell or intelligible only to me. I'll do my best to keep it clean and coherent, though, and in the spirit of fair play, unless otherwise instructed to do so I will betray no one's secret identity save my own: the names have been changed to protect the innocent. (Oh, except Fink's...that one's real...)
I invite all readers to respond to my ramblings, and to offer me their own thoughts, comments, and suggestions. The course (as yet in the planning stages) is now and, until it's over, will be, a work in progress, and I will be delighted to hear from my students, colleagues, and assorted other interested parties, as that progress is made. I'm excited to find out what we can make of the course!
I'll conclude this first post by giving hearty thanks to my colleagues in the UNCA Learning Circle, and in particular to its esteemed facilitator: you know who you are! Without you all, I'm certain I would not have found the derring-do to carry this out.
