You complete me

I had a blast in today's Calc I classes: students were engaged and alert, and eager to follow up on some of the missteps made in The Clock Problem. The few students I've talked to who missed the boat on that project were more than happy to make up for it and grateful for being given a chance to revise and resubmit. I look forward to seeing their recovery!

I spent a few minutes at the start of both sections drawing a big fat ol' line between the following two solutions I received to the problem "Find the number of local maxima and local minima the function f(x) = |x| has":

1. "The function f(x) = |x| has no local maxima and one local minimum, at x = 0."

2. "0, 1."

Though technically correct, the second response is dramatically inferior to the first: the first makes sense even when taken completely out of context. More than simply correct, it is complete and perfectly composed. It could appear in a textbook, expressing as it does, in a clear and unambiguous fashion, a true mathematical fact. One doesn't need to know that it was written in response to a homework question, for its truth transcends that humble context. From a practical standpoint, this solution could be used as a useful study tool later on, even without the question which prompted it.

The second solution, on the other hand is practically meaningless.

I asked the students whether they would dare to hand an assignment into their instructor of their LANG 120 (first-year composition course) which did not entirely comprise complete sentences. Predictably, they shook their heads.

So why shouldn't they pay the same respect to their math instructor? After all, aren't they using writing for the same purpose here, namely, to convince, and to communicate?

Moreover, it hit me this evening what the most profound philosophical difference between the two solutions above is: while there's no "movement" in the second solution, the first moves students a great distance, from being consumers, merely reacting to a stimulus given to them by their instructor, to being authors of their own truth and knowledge. The simple act of creating a complete sentence (expressing a concomitantly complete thought) transforms the student's role from a passive one to an active one.

I'm looking forward to tomorrow. I feel like this semester is finally getting into a groove. Stay tuned!