In my last post I alluded to the hand-waving we do discussing limits in calculus because students have a imprecise conception of real numbers. The clever article in the current The College Mathematics Journal, The Intermediate Value Theorem is NOT Obvious–and I Am going to Prove It to You by Stephen M. Walk discusses how to handle the situation by defining real numbers rigorously as bounded sequences of rational numbers.
A precise definition of real numbers is the honest way out. The definition and its subsequent use in proofs meets the needs of math majors and satisfies math instructors like me who don’t like loose ends and hand-waving. However most calculus students aren’t math majors. They are science and engineering majors who better respond not to proofs but to physical analogies. That is why they are science and engineering majors. My question would be: How far do you go? Questions that could come up during a discussion of real numbers might be, Why this definition? Are their any other numbers on the real line? Why not? (Hyperreals?) Who needs real numbers anyway? Aren’t rational numbers good enough? I can imagine digging a deeper and deeper hole and finally waving my hands again.
Yet as Dr. Walk wrote, “If I encourage students to accept, unquestioning, whatever their intuition tells them, then I risk teaching them to avoid working on careful arguments or thinking beyond their own experience.” We need to be encouraging this type of conscious thought in all of our students. The question is just where in the calculus curriculum do we face this head-on for all our students. I don’t know the answer.
Another point. A modern calculus student’s intuition is developed mostly with graphical depictions of functions, slopes and areas. I expressed some frustration with this approach in my post on rates of change. It is not enough to say that real numbers are all the numbers on the real line. That is a virtual tautology.
Finally, The Intermediate Value Theorem is NOT Obvious–and I Am going to Prove It to You is both clever and wise. The use of the two different fonts for, in this case, two different realizations of the word continuity is one I hope to adopt in my teaching. “Perfect ceiling” as a euphemism for “least upper bound” is a cool way to start the subject. The tone of the article, the self-reflection and focus on the student’s growth (see quote above) are evidence of Dr. Walk’s insight.