Do as I say ...
Mike O'Doul was my college roommate back in the seventies. He was working toward a master's degree in teaching. I was working toward a doctorate. It didn't quite work out as planned. Mike ended up earning a Ph.D. long before I did and became a professional mathematician, while I took a detour into state government. It took several more years before I finally ended up back in academia as a teacher and a retread grad student. In the meantime, Mike had racked up teaching experience at the elementary school level (during his master's program), high school (after earning his master's and earning a secondary credential), and college (during his subsequent doctoral program). He had also moved into the consulting business and had jetted about the world, working on U.S. Navy contracts and sailing as part of the civilian complement of carrier groups. He climbed the corporate ladder in the consulting business till he reached the top-level management position of chief information officer. His year-end bonuses were more than half my annual teacher salary.
I was impressed. My old roomie had lapped me on the track several times.
Some good things come to an end. During a period of contraction and corporate acquisitions, Mike's company was purchased by another consulting firm. He found himself working under a manager whom he had once dismissed from the company. His new manager was eager to return the favor and Mike was handed his walking papers.
I've already established that Mike O'Doul was no dummy. He was mathematically acute, articulate, and extremely hard-working. During the fat years, he had tucked away big chunks of his earnings in preparation for possible future lean years. The lean years had arrived and Mike was pleased to discover that his preparations would permit him to retire at a comfortable middle-class level without ever working another day in his life. That prospect, however, did not completely satisfy him. He and his wife had young children, some of whom might actually want to go to college. Mike decided it would be nice to continue earning some wages, both for the satisfaction of staying active and to widen the margin between prosperity and penury.
Dr. O'Doul dusted off his secondary credential and found a job teaching high school math. He enjoyed being back in the classroom, but he was less than delighted with the many hoops he was required to jump through. Even so, he applied himself with his characteristic diligence and established himself as a major resource in the math department. Soon the department chair tapped Mike to teach the AP calculus class in their high school. It would require Mike's enrollment in an orientation and training seminar, but Mike didn't anticipate any problems. He consented to the assignment and put the seminar on his summer calendar.
Mike wasn't surprised on the day of the seminar to discover that it included another series of hoops. In addition to outlining the content of the AP calculus syllabus, the seminar leader was going to tell Mark how to do his job. Perhaps it wouldn't be a problem. Mike would keep his light under a bushel basket and listen quietly. During the preliminary introductions, he didn't mention his doctorate, his previous teaching experience, or his career in research mathematics and consulting; Mike simply said that he was a second-year instructor in the school district who had been assigned his first AP calculus class for fall. He was willing to pick up some tips from more experienced AP calculus instructors.
Mike was encouraged by the way the seminar leader launched his presentation:
“Be very careful not to lie to your students! It's much too easy to offer level-appropriate answers that mislead your students by being stated too definitively. For example, do you tell your beginning algebra students that no one can take the square root of a negative number?”
The teachers smiled appreciatively.
“You need to qualify such statements, mainly by providing the appropriate context. Negative numbers do not have square roots in the real numbers. You don't have to offer your students a premature explanation of the complex plane, but you have discussed the real line and your point is that square roots of negative numbers do not exist there, on the real line.”
So far, so good.
Mike wondered whether he should ask about cautioning students against “distributing exponentiation,” as in the notorious (x + y)2 = x2 + y2. Should we tell them that it never works, except over a field of characteristic 2? Mike decided he didn't need to push the envelope quite that hard, so he keep his question to himself.
The seminar leader moved briskly through the AP calculus topics, offering insights on presentation and cautions on possible overstatements. Mike was pleased at the level of the discussion and ready to concede that this seminar was better than average. Then the discussion move to polynomials and power series.
“Don't hesitate to write polynomials in ascending order. It can significantly raise the comfort level of your students when you get to power series, which are always written in that order, and prepares them to see power series as a natural generalization of polynomials. They already know that polynomials are easy to differentiate as often as you want, so it prepares them to understand the point that functions with derivatives of all orders can be written as power series.”
Mike pricked up his ears at the presenter's fumble and waited to see if the speaker would catch his own mistake and offer a correction.
“Remember that the term for functions with derivatives of all order is analytic.”
Double oops! thought Mike. We're dealing in real variables. He interjected:
“You mean smooth, right?”
The presenter paused, looked at Mike, and blinked.
“No, analytic is the right word. If it has derivatives of all orders you can construct a power series that represents it. A function that can be represented as a power series is called analytic.”
The presenter turned away as if to continue, but Mike was not done.
“Excuse me, but it's not the same thing. Yes, a function that can be represented as a power series is called analytic and it does have derivatives of all orders. However, the converse is not true. Functions that have derivatives of all orders are called smooth”—Mike decided not to mention C ∞—“but it doesn't follow that the function can be represented by a power series.”
The presenter didn't exactly glower as the junior faculty member (an older guy, yes, but a very junior faculty member) who had dared to contradict him, but he did seem a bit piqued. The man who had warned people not to lie to students proceeded to tell a presumably inadvertent untruth:
“You're missing a very obvious point, sir. If you have all the derivatives, you can easily construct a Maclaurin or Taylor series to represent the function.”
“Very true,” agreed Mike. “But the series might not work. Consider the function f(x) = e−1/x2, where we also define f(0) = 0. The function is infinitely differentiable at 0 but the Maclaurin series does not represent the function. The derivatives are identically zero and so is the series, while the function manifestly is not.”
The presenter decided he had encountered a teachable moment. He turned to the board and began to sketch out a derivation of the derivatives of the function Mike had offered as a counterexample. While the audience fidgeted a bit anxiously, the presenter scribbled away. While Mike had been surprised that the presenter had stumbled over the analyticity of real-valued functions, he noted that the fellow was doing a pretty good job of checking the counterexample. With an occasional suggestion from Mike, the presenter was discovering that every derivative of f(x) was indeed equal to 0 at x = 0. Eventually he turned back to the seminar attendees.
With a somewhat awkward smile, he said, “Okay, you see what we have here. It's a definite counterexample to the notion that infinitely many derivatives are sufficient to ensure the existence of a representative power series. The good thing is that you probably shouldn't go quite this far in a high school calculus class. I imagine that I don't have to underscore the lesson here.”
“No, I remember,” said Mike. “Don't lie to your students.”