Too bad you didn't
It may not arise too often in practice, but suppose you want to find the derivative of z = y2e4x with respect to s, where x = 2s − 3t and y = 7s − t2. It's a perfectly good problem for testing the differentiation skills of one's calculus students and their understanding of the chain rules for partial derivatives. As is often the case, there is a good way and a bad way to work out the correct answer. The bad way involves brute force substitution, replacing each occurrence of x and y and then differentiating with respect to s. Even in a problem as elementary as this one, finding ∂z/∂s by substitution and differentiation adds additional steps (like the product rule, for example). It is a daunting task to tackle the direct differentiation of z = (7s − t2)2 exp(4(2s − 3t)).
This could be avoided with a little thought, which reminds the attentive student about the chain rule. The individual derivatives in the chain rule formula are relatively easy to compute: ∂z/∂s = (∂z/∂x)(∂x/(∂s) + (∂z/∂y)(∂y/∂s). Most of my students remembered this rule and applied themselves to the necessary calculations. With a little care, one can obtain the correct answer in a few steps.
One of my students, however, decided to obtain the answer in one step. He typed the problem into his calculator, entering the slightly grotesque version that results from substituting for x and y in terms of s and t. He could never have been expected to compute the result successfully in the time available, but his TI-89 quickly spat out an answer that he merely transcribed on his paper.
He had done this before. He had merely written down some calculator results and I had given him only a handful of points for setting up the problems—but none for the answers. But now he was doing it again. Had he not learned his lesson? Perhaps he had learned a lesson other than the one I intended: He could either skip the problem because he could not do it, or he could give it to his calculator and hope to get a few points for the setup. Good thinking!
Too bad the points derived in that manner aren't enough to produce a passing grade.
I've long been concerned about the phenomenon of calculator dependency. I've seen its impact at all levels of mathematical instruction, but the example from my multivariate calculus class is particularly disturbing. The student who can't live without his TI-89 has somehow survived till calculus III. He doesn't know how to apply the basic rules of differentiation. How did he earn passing grades in I and II? Maybe we should give a math degree to his calculator.
But not to him.