[BUG] Trusting "use what you know" vs. an algorithm

[Tevian Dray]

> However, I've also come across a few situations where doing things
> without dr vec seems to "not make much sense" to the students.

Am I correctly understanding that it's the method of Lagrange
multipliers which does "not make much sense", rather than any direct use
of drvec?  This wouldn't surprise me; my experience is that students
view Lagrange multipliers as a mysterious algorithm to be followed
blindly, even when I've tried to motivate it geometrically.

> I find it interesting that students followed someone else's algorithm
> (Lagrange multipliers), even as they were confused by it, over a solution
> method they had previously used comfortably.
>  
> (In all fairness, from my students perspective, though, I'm sure they found it
> confusing that we even discussed the method of Lagrange multipliers in the
> first place, when we had previously just "used what we knew" about the 
> constraint, dr, the master formula, and df = 0.)

Perhaps the answer is not to teach these as disjoint topics.  Surely
drvec can be part of the argument used to justify the method of Lagrange
multipliers?  I'm not sure I've ever actually tried this, and wonder if
it would work better than the traditional pictures of level curves
tangent to the given curve.

Best wishes,
	Tevian