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On the Role of Proof in Calculus Courses

I use the word “should” with all its moral overtones because I think that to many this is as much a moral issue as a pedagogical one. On the other hand, I have a hard time distinguishing moral considerations from considerations of taste, where the dictum de gustibus non est disputandum applies. Therefore I will for the most part think of “should” as “need” or “must” or even “does it help”. I do believe these questions are at the root of some of the debate about calculus reform, but I have heard very little thoughtful discussion of the issue. I must, however, apologize that what I say is woefully uninformed by research in mathematical education, and all I intend is to begin a dialogue. Before I proceed I would like to differentiate between two pedagogical uses of proof. The first, which I’ll call Proof I, is part of a process of formalization and organization. In this setting, the student is presumed to have an effective, reliable understanding of concepts and results; the goal is to develop a formal language with which those results can be proved true. I would consider the proof that the limit of a sum is the sum of the limits as Proof I. The second sort of proof, which I’ll call Proof II, is less formal, and is used to answer a question that is in doubt. Students don’t often encounter such questions in a calculus course. Here is a very simple example: if the functions f and g are twice differentiable and their graphs are concave up, must the graph of f+g be concave up? The answer is not obvious if one thinks only pictorially and one could well start looking at examples; but of course, once the question is interpreted in terms of the sign of the second derivative, the question easily yields its answer and a proof.