Solution by coincidence

Lucky Larry asks for full credit

My introductory calculus class had arrived at the section on finding critical numbers for given functions. These numbers, as you recall, provide the candidates for possible maxima and minima for the functions. They come in two forms: (1) numbers for which the first derivative is zero and (2) numbers for which the first derivative does not exist. After some practice and review of the topic, I gave my students a quiz containing the following function f(x) = x^5/3 + 5x^2/3.

The fractional exponents are a dead giveaway that mischief is afoot. The derivative is straightforward:

The factored form makes clear that there are two critical numbers. The factor x + 2 yields x = −2 as a number for which the first derivative is zero. The factor x^−1/3 shows that x = 0 is also a critical number, since division by zero is undefined and the negative exponent is an indicator of an implied division.

It's a nice little problem that provides a critical number of each type. But one of my students was miffed when I didn't give him full credit for having successfully winkled out the two numbers. When he protested, I gently explained to him that his work was invalid. As a clever student who fully believes in his cleverness, he was certain that an injustice had been committed. “This isn't over,” he muttered. “I can prove that I'm right.”

“Go right ahead,” I said in my most agreeable tone of voice.

We huddled over his paper as he explained his solution to me. Instead of factoring the derivative after setting it equal to zero, he had divided both sides of the equation by 5/3, obtaining

x^2/3 + 2x^−1/3 = 0

“Then I applied the quadratic formula,” he proclaimed. “I had a = 1, b = 2, and c = 0.”

“You could have factored,” I pointed out.

“Yeah, well, factoring and the quadratic formula give the same result,” he said.

“Um, sure,” I agreed, “but only if you're applying them to a quadratic equation. This equation is not quadratic in form.”

“Hold on a second,” he persisted. “Look at what I get.”

“See?” he concluded triumphantly. “When I simplify, I get the critical numbers 0 and −2. I'm right!”

“The numbers agreed with the correct answers, but it's a coincidence. The quadratic formula doesn't apply.”

He was not pleased.

“What do you mean?”

“Look at your equation,” I said. “Your lead term contains x to the two-thirds power and your second term contains x to the negative one-third power. The former is not the square of the latter, which is the necessary condition for treating an equation as a quadratic.”

His face fell.

“But I got the right answers!”

“It's a coincidence.”

He fussed over it a bit more.

“But it'll work every time, won't it?”

“In a problem of exactly this kind? Yes, because the derivative-does-not-exist critical number got converted into a derivative-equals-zero critical number.”

“So can I use this? I just showed that it works.”

“No, you just showed that you lucked out. A coincidentally correct result from an invalid process is still invalid.”

“But if it works—”

“I'll tell you one more time: No credit for lucky accidents. If you use quadratic techniques when they don't apply, you don't get credit. Besides, factoring is easier and gives correct results. Try to remember that.”

He still thinks he was cheated of full credit by a hard-nosed teacher.

He's half right.

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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.”

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Blowing Riemann bubbles

They pop in your face

My student was frantic. She was hyperventilating. It was the evening before our calculus exam and she had called me at home.

“I really, really need your help! This has me totally confused, Mr. Z!” (This was before I earned my doctorate in truthology.)

I tried to calm her down.

“I have no idea why you're so worried, Monica. You've been doing fine all semester. There's no reason to panic.”

She wasn't buying it. The words tumbled out.

“Yes, I know, and I was feeling okay until this afternoon. But then I talked to Jay. I had a doctor's appointment this morning and missed your review session, so I asked Jay what you had covered. He told me you introduced a whole new way of doing Riemann sums and that it would be on the exam!”

I sniffed a rat. Jay was a good student, but also a high-spirited prankster and class clown. I suspected the worst.

“Okay. Well, what exactly did Jay say?”

Monica had just about caught her breath. She paused a couple of seconds and then reported her conversation with Jay.

“He said that we didn't have to do Riemann sums with rectangles when we're trying to approximate area. He said we can use different shapes. He said you were going to ask us to do Riemann sums with circles. Because, you know, you can fill up a space with circles and add up their area, just like with rectangles. And then take a limit, I guess.”

I was glad we weren't face to face, because I had a huge grin on mine. Jay was a little bastard, but he was a clever one.

“Okay, you can calm down, Monica. There will be no Riemann circles on the exam. No such thing, actually. He made it all up. He probably thought you would call him on it, but I guess he made it sound realistic enough that you fell for it. We'll have a Riemann sum with rectangles, but no other shapes. Okay?”


There was silence at the other end of the phone for several seconds.

“For real? That was his idea of a joke?”

“Well, I guess so. Though I doubt you find it all that funny.”

“I'm going to kill the little creep the next time I see him! I swear!”

“No, Monica, don't do that. It would be bad if you killed a classmate on the day of an exam. It would probably rattle the other students. Tell you what: Don't say anything to him, okay? Leave it to me.”

“What are you going to do, Mr. Z?”

“It'll be a surprise. Okay?”

Monica agreed not to kill, abuse, or otherwise assail Jay in class the next morning. Her initial outrage had already faded and she was almost giddy with relief that she didn't have to learn something entirely new on the eve of the exam. Besides, I had told her to leave things to me. An authority figure had stepped in.

There was the usual amount of pre-exam anxiety in the classroom the next morning. Student attendance was high and most of them arrived early. Jay was sporting a big grin as he sat at his desk, but Monica refused to let him catch her eye, although he kept looking across the room at her. He had not confessed to his crime and Monica had not confronted him. There was a bit of buzzing in Monica's neighborhood and I figured her friends in the class were aware of the scare he had given her, but the murmuring died away as I pulled the stack of calculus exams from my briefcase.

I delivered my usual patter as I strolled down the aisles and dropped an exam face-down on each desk. (Please read each problem. Check your solutions for reasonableness. Don't dawdle over any particular problem.) I reached Jay's desk, but he didn't notice that I dealt his exam from the bottom of the deck.

Everyone had an exam now. I returned to the front of the room.

“Okay, everybody. Turn your exam over and please fill in your name right now. Then please check that you have all five pages.”

Students scribbled their names and began to riffle the pages. Jay turned the pages of his exam until his eye fell on the Riemann sum problem. He froze.

They say that people's eyes can bulge out of their sockets when they're shocked, but I never thought I'd see it outside of a Tex Avery cartoon. Jay, however, did his best impression. “Oh, my God!”

All heads swiveled in his direction. Jay brandished his exam at me.

“Mr. Z! How I am supposed to do this problem?”


I smiled at him.

“I don't know, Jay. But since you told Monica last night that we were doing Riemann circles, I thought you'd like to demonstrate the technique.”

The class burst into laughter. Jay shot a guilty look at Monica, who was clapping her hands, and a sickly smile formed on his face. The class settled down and got to work, punctuated by the occasional chuckle, as I walked over to Jay's desk and swapped his bogus exam for the real one.

I think it was one of those teachable moments.

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