Title: Symbols, Numbers, & Pictures: Calculus Beyond the Mechanics
Isaac Newton once described people who can do some calculus—but don’t really understand what it’s about—as vulgar mechanicks. When new or different problems arise, said Newton, such people are “at a standstill.”
Calculus students’ difficulties are sometimes attributed to lack of traditional rigor in, say, developing the fundamental limit concept. My diagnosis is different. Calculus students, in my view, are often kept “at a standstill” by seeing the subject presented too narrowly, as a collection of symbolic recipes applied to symbolic formulas. Numerical and geometric points of view, by comparison, may be given short shrift, or be sealed away in obscure textbook corners.
Symbols and symbolic operations are important, but so are geometric and numerical insights. Sometimes the latter are downright indispensable, even for handling the most straightforward-seeming problems. With modest help from technology, graphical and numerical methods become practically feasible and educationally valuable. I’ll illustrate these claims with concrete examples, drawn from derivatives, integrals, and infinite series.
Systematically combining symbolic, numerical, and graphical points of view on calculus can help assure, I’ll argue, that students take Newton’s advice, to remain not “at a standstill” but “never at rest.”