Teaching The Fundamentental Theorem

On February 2, (remember groundhog day when we treated the wobbly table theorem?), an interesting article appeared in the American Mathematical Monthly. It is by David Bressoud and titled "Historical Reflections on Teaching the fundamental Theorem of Integral Calculus". You find the article below.

Bressoud concludes his article with:

Many years ago, my eyes were opened when I dared to probe what students really take away from their experience of calculus. I was particularly horrified by one undergraduate who had completed two full years of calculus yet, despite all my efforts to evoke something more, was able to remember nothing about the meaning of calculus beyond recalling that differentiation is a process of turning functions into "simpler" functions--in the sense that quadratic polynomials become linear and cubic polynomials become quadratic--and that integration is the reverse of this process. As we think about how we should teach the Fundamental Theorem of Integral Calculus, we must keep in mind what it is that we want students to remember from this course, and then we must work hard to ensure that they do.

Before that, Bressoud writes:

Patrick Thompson, Marilyn Carlson, and others have explored the pedagogical obstacles that students must overcome before they can comprehend the dynamic FTIC. The very first step is to understand functions as descriptions of covarying relationships. Most students think of functions as static objects, either algebraic expressions--in Thompson's words, as "a short expression on the left and a long expression on the right, separated by an equal sign" --or as the geometric object that is the graph of the function. We can not expect students to comprehend the dynamic view of the FTIC unless they are able to see functions as describing a dynamic relationship between covarying quantities. Understanding covariation is only the first step toward the FTIC. Next comes accumulation. We would do well to require our students to formulate their own explanations of the Mertonian rule. Accumulation is hard. A recent article in Science describes a survey of graduate students at MIT, students with undergraduate degrees in economics or the sciences and thus who had studied calculus, who were asked to explain the behavior of an accumulation function when a nonconstant rate of accumulation was specified. Even these students had difficulty with this task. The next step is to consider the rate of change of the accumulation function. Rate of change is already a difficult concept. Compounding this with an accumulator makes it all the more difficult. But once all of these pieces are in place and are understood, the FTIC is virtually self-evident. Newton never proves it. He simply observes that, of course, the rate of change of the accumulated quantity is the rate at which that quantity is accumulating. Calculus emerges from the awareness of the power of this observation.