Friday, October 15, 2010

The Antiphany

I'd like to coin the term "antiphany" or anti-epiphany. It represents a sudden insight or understanding which is not in any way transcendent, but in fact horrible and gut wrenching. A moment in which you put the pieces together, and feel not enlightened, but as though you understand how much you've missed, and that things are much worse than you ever thought. Not just when you feel your stomach drop, but when you know it just entered the express elevator to hell.

I've had two this week.

The lesser was a simple matter of realizing how little success my precalculus students ever seemed to have in understanding transformations of graphs. I thought about the fact that none but the best ever understood this topic very well, and that perhaps a tiny percentage of those going on to calculus would understand or remember virtually any of this. And I thought about episodes in my calculus classes when I would casually, as an aside, mention how some particular result could be viewed in terms of transformations of graphs. Calculus is filled with lovely insights which blend the tools of geometry and the tools of algebra. In many ways, calculus is the blending of geometry and algebra. Analytic geometry makes calculus possible. So I try to share these insights with my students whenever possible. I try to point out the wonderful connections which seemed so magical and delightful to me when I first studied calculus (and which I still find magical and delightful, truth be told). And I thought about how little my precalculus students understood about transformations, just one semester before they might enter my calculus course.

And then... Antiphany. My calculus students don't have the foggiest idea what I'm talking about when I share those insights, do they? They're missing a basic facility with graphs and functions to be able to hear anything other than an impenetrable wall of words, and without that basic facility, there is no way they can hear it. Unless they are able to grasp the basic facts of analytic geometry intuitively, they can't hear an insight about those facts.

The second antiphany was an odder moment, and much worse. Without going into detail, it was the sudden understanding, in a brief revelation, that one project I've been working on for a few years was doomed from the beginning. The effort had generated stress and frustration for me (and for others), and I hadn't been able to succeed. I'd had to deal with the effects of the failure, and keep struggling to try to make it succeed, and then in a single moment, I saw why success was impossible. I had in fact been sabotaged before I ever started, without knowing it. And I found out how I'd been sabotaged, all at once. I could see the past few years flushed down the toilet, wasted effort worrying about trying to push the damned rock up the hill when it kept being shoved back down from the other side.

Antiphany. It's a good word, even if I did make it up.

(By the way, a brief Google search turns up some previous coinages of antiphany, based on the same idea of an anti-epiphany. But I intend a somewhat different meaning.)

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