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Symbol manipulation and hand computation

Although we emphasize concepts and use technology freely, we by no means ignore symbol manipulations in general, or hand computations in particular. We take the symbolic points of view of our title fully as seriously as the graphical and the numerical. We cover, for instance, the usual techniques for formal differentiation and antidifferentiation. Why do we do so?

That machines can do calculus manipulations does not render by-hand operations obsolete. Some manipulative practice and skill builds and supports conceptual understanding. Hand computation illustrates concepts concretely, builds ``symbol sense'' (the algebraic counterpart of numerical intuition) and an ability to estimate, and gives students a sense of mastery. It does not follow, though, that harder, more baroque computational problems are necessarily better or more useful; we deemphasize them. Both research results and our own experience suggest that diverting some time and attention to concepts does little, if anything, to reduce students' hand computational facility.

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