That's very interesting. In twenty years of teaching calculus, I haven't really noticed a difference in learning of this sort (though I haven't honestly thought about this or looked for it). The modern doctrine for teaching calculus is to show things in several different ways: numerically (tables of numbers), graphically, and finally algebraically. Students are very capable of learning calculational or algebraic dance steps (such as finding a derivative) without having any clear idea of what it means, so new calculus curricula stress intuitive understanding and problem-solving over mechanical algebra skills.
But it's time for me to meet class. First class of the semester, in fact.
Ha! My most recent math professor drilled us in mechanics. He said if he called on the phone at 3AM waking us up we should be able to do an integration by parts, give him the result, and roll over and go back to sleep.
136 - "Students are very capable of learning calculational or algebraic dance steps (such as finding a derivative) without having any clear idea of what it means,"
I agree, but would leave out the 'very'. Rote learning does work, but, since you apparently are a teaccher, and still teaching, and especially since we have computers now which can instantly recalc and redraw, why not try taking a problem and demonstrate graphically, what even very minor changes in different values do to the graphical depiction. I think you would be surprised at how much easier some will catch on to the ideas of what the formulas actually govern, and it may mean far more can understand far more quickly and easily.
In fact, you may find that it will save many of your otherwise 'smart' students who otherwise just don't 'get' calculus, so they give up.