Saturday, July 25, 2009

Don't lie to your students!

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.”

13 comments:

Eamon Knight said...

Terry Pratchett has a term for this: Lies To Children -- the inevitable simplifications a teacher (or parent) must make when explaining something to a child who (this week) lacks the background or mental equipment to grasp all the nuances.

In a way, of course, the Lies never stop unless and until you get to the Ph.D level in a particular subject, ie. really understand the current state of the art, with all the exceptions, raw edges and terra incognito.

Sili said...

I'm sure that somewhere out there is a whole book of pathological functions - I'm not even gonna google it.

Thanks for the graph - I don't thing I've ever actually *seen* it.

Oded Shimon said...

Shame on that instructor for not knowing that counter example!

I was blown away by that example when our instructor mentioned it as just barely a passing comment, as we had not learned Taylor series yet.

Anyway, you don't need to go so far with such an example, there are much easier ones! It gets no easier than:
1/(1-x)

With power series coefficients of just "1"....

The most amazing smooth and non analytic function is arctan() .. Such a calm function with explosive effects...

Our instructor also "insulted" power series, as being a very bad aproximator of functions - requiring so much of the function - smoothness. And even THAT's not enough, and it doesn't even work everywhere when it does work...

That was right before the introductory phase to Fourier Series. :) Which of course require nearly nothing of their functions - just integrability...

(I am a first year undergraduate student, and I'm enjoying every minute!)

Oded Shimon said...

Sili - There's a whole article on that function in Wikipedia. Not to mention that it is a very easy function to imagine :)
http://en.wikipedia.org/wiki/Non-analytic_smooth_function

I liked the description of our instructor - ALL the derivatives are zero, yet STILL the function managed to "escape", "break free" of the zero x axis, and get non zero values :)

Zeno said...

Oded Shimon, it's always great to see a student tearing into new material with real gusto.

I think there may be two lessons in this particular post. The obvious one is to be wary of over-confident experts, since they may run into trouble even while preaching to others the importance of avoiding it. The other lesson, somewhat more in the background, is that years of experience can actually dull one's acumen. It depends on the experience. The more years elapse since you learned a lesson, the more your grasp of the details may fade. It's been a long time since I earned my math degrees and Mike's story really rattled my cage as a reminder of how we fussed over such fine points way back when -- but may lose the habit of caution after years of being largely unchallenged by students in elementary classes.

Oded Shimon said...

I wholeheartedly say that the best practice to material is to teach it...

When you are bombarded with questions, and really try to realize why is it you know what you know in order for you to teach it, then you truly regain your understanding of the subject...

I remember every step of the proof of the Fourier Series given during our calculus course, because I had to teach it to a friend of mine that missed those lessons.. Almost no other subject during that course I remember as well. Having to go over the material, thinking how I will teach it, forced me to understand it perfectly as well.

As usual, Zeno, thank you for your interesting and amusing posts! :)

unapologetic said...

Oded, that example of yours doesn't work. Yes, the power series coefficients are all 1, but that power series very well does converge to the original function within a nonzero radius of convergence. It's the geometric series, and it is, indeed, analytic everywhere.

Now, if you want to use this forum to show off, come back with a function which is everywhere-smooth, yet nowhere-analytic.

Billy C said...

Sili:

At least one such book exists, and helped me get through my undergrad Advanced Calc courses. It's Gelbaum & Olmsted's "Counterexamples in analaysis."

The function in Zeno's story appears in chapter 6, Example 23 with the title, "A function whose Maclaurin series converges everywhere but represents the function at only one point."

Unapologetic's challenge is nearly met by Ch. 6, Ex. 24, "A function whose Maclaurin series converges at only one point."

Zeno said...

Yes, Gelbaum & Olmsted! I have that book around here somewhere. It was a prized possession when I was a math grad thirty-some years ago.

If I take it to class, I could frighten my calculus students with it.

William said...

My first-grade teacher, an ignorant nun, tried to tell me there were no such things as negative numbers. I say "ignorant", but I assume she actually did know about negative numbers, and her ignorance lay in how to deal with students who knew things that we weren't supposed to yet. But I'm not sure.

Anonymous said...

This story does not surprise me, except for your friend's shocked reaction. I think that most high school math teachers don't even take math courses like real analysis or topology or algebra in college, so it's very easy to think they could make that mistake. I think it's at least as likely the other teacher had never seen an example of a non-analytic smooth function, than that he merely forgot.

plam said...

I agree with Zeno about teaching: you certainly get to know a subset of the material very well when you teach it. But you don't necessarily get to the deep questions about it, depending on your audience.

For instance I taught a course on software testing last term. I'm not an expert on software testing, and I'm still not an expert after teaching the course. Certainly the material in the course I know pretty well now, but there's lots more. I suspect the students would also benefit if I had a broader grasp of the field.

High school students aren't going to quiz you about analyticity. (I used to know some real and complex analysis, but that was 10 years ago, and I always liked algebra better anyhow.)

Zeno said...

Mike was just a bit shocked by the teaching seminar, but I don't think he was amazed to discover that high-school calculus teachers are rusty on their real analysis. He was rather more bemused (amused? irritated?) that the supposed expert leading the seminar had (a) made such a big fuss about accuracy (with which Mike basically agreed) and then (b) ventured into some advanced math where he was a little unclear on the concepts. It was a good example of not following one's own warnings.

When Mike gets to power series in his high school calculus class, I'll bet he doesn't try to impress his students with discussions of analytic functions. He knows better (even though he clearly remembers the subject well).