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Best Practices for Teaching Tricky Topics - Math
Calculus as a subject is different than other courses. Calculus is full of choices that teachers (and authors) must make as they teach (or write). This is the beauty of the subject. Its problems can often be worked in multiple ways, leading to different, but equivalent, answers. It allows multiple options for an instructor when deciding on organization and presentation. Here is an example:
Question: In approximating the value of a definite integral, why not introduce the Trapezoidal Rule as a technique instead of doing it later?
Answer: An interesting question. While it has been traditional in calculus texts to handle the Trapezoidal Rule in the chapter on Techniques of Integration, that answer would be unsatisfactory.
From a mathematics standpoint, there is no reason the TR could not be taught after Riemann sums and the definite integral have been defined. Here are my thoughts.
The definition of a definite integral is complicated and requires careful reading. The early examples ask for an approximation of an integral, using Left Riemann Sums, Right Riemann Sums, and Midpoint Riemann Sums. Examples that ask for the exact value are algebraically challenging. Students are panicking. Yet lurking ahead is the Fundamental Theorem of Calculus that provides a relatively straightforward way of finding the exact value of an integral. So, after defining the definite integral, should we continue with other ways of approximating an integral--leaving the student wondering why are we so interested in this thing, 'the integral,’ when we can't do any better than approximate it?
The answer is clear. Give the student the FTC: if you can find an antiderivative of the integrand of a definite integral, then its exact value is a simple computation. Now the anxiety is reduced and confidence is restored. Now we can show all the wonderful applications of the integral (Chapter 6), demonstrating its usefulness in a variety of settings. All that is required is to find that antiderivative! And that provides the rationale for Chapter 7, Techniques of Integration, or ways to find antiderivatives.
But the story is not that easy because we discover that there are integrands for which there is no antiderivative. The silver bullet called the FTC cannot always be used. And that brings us back to approximating integrals and the Trapezoidal Rule.