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

[Wangberg, Aaron]

I've introduced the gradient and dr vector to my students quite early in multivariable calculus this semester, putting a large emphasis on their geometric meaning.  I believe that it will be very helpful to use dr vector when covering double and triple integrals.  However, I've also come across a few situations where doing things without dr vec seems to "not make much sense" to the students.  For example:
 
When asked to find the maximum value of a function  f  along a "path", many students had seemed to understand how to use the master formula, dr vector, and df = 0 to find the solution to this problem.  This was before we learned the method of Lagrange multipliers.  Afterward covering this material, not surprisingly, many students used Lagrange multipliers to find the maximum value of a function  f  given a constraint.  Now, however, they stumbled over certain steps, and many couldn't see the purpose behind the various steps.
 
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.)
 
Aaron Wangberg