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.
We use TI-86 graphing calculators, and the software Derive, to help students visualize formulas. There's a nifty little piece of software called Cyclone that draws three-dimensional implicit surfaces; the student can change coefficients of the equation by changing a slider, and the surface changes real-time as the student moves his or her mouse. We use this in calc III.