I've noticed an interesting phenomenon in my math classes over the last couple of years, which has become more pronounced as the subject matter becomes (to some people) more difficult: the Desperate Calculator Jockey. This is the guy who never studies, is too lazy to do his homework, but somehow manages to struggle his way from one class to the next. He can usually be seen furiously punching formulae into a top-of-the-line TI-89 or its near-equivalent, expecting some magical insight into the problem from the resulting graph. While this approach may actually work in some instances, such as with a simple "differentiate this function" problem, the approach is an utter - and deserved - failure when the student is confronted with the dreaded applications problem (the so-called "word problem.")
Oh, my... this is where thinking skills are really tested, and where numerical prosthesis is nearly worthless. I love these problems: exercises in aircraft fuel vs range optimization, predictions of least-time photon transit through a non-vacuum medium (Snell's Law, an application of the Brachistochrone curve), calculations of marginal cost & maximization of profit, how to make the largest rectangular horse corral with a fixed length of expensive fencing... the good stuff. The Calculator Jockeys rarely know where to start decomposing a problem to its tractable constituents, instead, as usual, attempting to invoke an answer from the aether by keypunch.
I can pretty much tell from looking around a room who's faking it and who's getting it: those who are actually learning the calculus are learning with their pencils, which are constantly moving on paper. The fakers, on the other hand, are constantly pushing buttons. The joke's on the latter group: with the exception of actual symbolic manipulation packages such as Wolfram's Mathematica, hand calculators - to date - depend entirely on methods such as linear approximation. This means that the calculator jockey can actually get the wrong answer for a derivative of a function at a point on a curve, since the curve may not actually be differentiable at that point, but the linear approximation around that point of a secant line connecting points on either side of the original point may exist. Suckers.
