The template student

When thinking is too much trouble

My exams seldom contain surprises, but my students' answers do. Since I'm a firm believer in keeping track of student progress with frequent quizzes, I telegraph my punches. Students have plenty of opportunity to discern what facts, techniques, procedures, and calculations I deem the most important. (They also—most of them—learn the importance of regular attendance so as not to miss these pre-exam rehearsals.)

Of course, some students take it too far. These are the students who have had the unfortunate educational experience of intensely patterned teaching to “the test.” These are also the students who badger me for “practice tests” in advance of each exam. What do they want? Problems that are exactly like the ones they'll encounter on the exam.(I presume I'm permitted to change the numbers a little bit.)

What surprised me most this past school year was the discovery of this tendency among my calculus students. I was used to seeing it in my lower-level classes like algebra, but in multivariate calculus? An example of the behavior of the template-driven student will suffice. You'll see the problem, even if the terms are mysterious.

On a quiz I asked multiple questions about the gradient of a function of two variables. In part (a) I asked them to compute the gradient and evaluate it at a given point. In part (b) I asked them to use the gradient to compute the directional derivative in a given direction. In part (c) I asked them to calculate the greatest possible value of the directional derivative. In part (d) I asked them to find the direction in which the greatest possible directional derivative would occur. Pretty standard stuff.

On an exam I asked my students to (a) compute the gradient of a function of two variables and evaluate it at a given point. No problem. In part (b) I asked them how large the directional derivative could be? Several students were thrown for a loss. They wanted to compute a specific directional derivative, but instead I was asking them for its maximum possible value. There was no way they could do what they wanted to do because I had not provided a direction, so they made one up. They had memorized the pattern in the quiz and insisted on replicating it exactly on the exam. Since I had, in effect, swapped (b) and (c), they were deeply perplexed and forged ahead with the moves they had learned by rote.

Embarrassing! It wasn't a very large number of students, but I had been hoping they had been weaned away from this tendency by the time they arrived in the calculus III class. I learned otherwise.

Cheerfully innumerate

Happily ignorant

I'm sure we all remember in Season 17 when Marge Simpson expressed to Lisa her regrets about blowing off the calculus final in order to party with her boyfriend Homer: “Since then, I haven't been able to do any of the calculus I've encountered in my daily life.” Ah, yes. Thus do our mistakes return to haunt us, and—as we all know, a working knowledge of calculus is crucial for success as a homemaker.

The obvious basis for the humor is the effective disjunction between calculus and housekeeping. The more subtle reason is perhaps more significant: a sense of relief in the viewer. “Ha, ha! Thank goodness it doesn't really matter that I didn't learn any of that useless stuff!” It salves their guilty consciences over their collegiate screw-ups and omissions. “Math! Who needs it? Only nerds! (And I'm not one. Hurray!)”

 Brian O'Neill seized the opportunity to write a semi-humorous article for the Pittsburgh Post-Gazette after discovering that Google's Laszlo Bock found no significant connection between college grades and job qualifications. He cites Bock as saying in a New York Times interview that “G.P.A.’s are worthless as a criteria for hiring, and test scores are worthless.”

What? Algebra grades don't predict job performance? Grades in English composition don't correlate with corporate success? Shocking!

And why should they? I concede that the classroom is an artificial environment that does not in general (and is not intended to) emulate future work experience. The integration of the knowledge you acquire in the classroom is a separate skill, as is the selection of the right tools for doing a particular job. Why are heads exploding (or pretending to explode) over these “revelations”? People don't begin entry-level jobs with all of their skills and knowledge pre-melded into a unitary capability. Who knew?

A degree really signifies that you are able to achieve a goal, which is why many companies care more about your persistence in achievement than they are in the grade point average you attained. This, however, is the point at which people bewail the math classes that prevent attainment of a degree: “I can't do the math required for a college degree, so math shouldn't be required.” But college degrees are a sign of a range of qualitative and quantitative skills, so this argument suggests a watered-down college degree is okay. Should it have an asterisk on it? Should it be labeled “college degree lite”. Does everyone deserve a college degree even if he or she is illiterate or innumerate? Note how readily the argument generalizes:

“I can't do the _____ required for a college degree, so _____ shouldn't be required.”

Student success would soar! And student job options would correspondingly shrink.

Oh, but Google says academic success doesn't correlate with occupational success. Please pause to consider that Bock was describing what they discovered in the people they hired. Go ahead and visit Google's job opportunity site. They need account managers and executives more than anything else (at least during this summer of 2013). Minimum qualifications? Looking at today's listings in order, I see BA/BS (MBA preferred), BA/BS, Bachelor's (MBA preferred), BA/BS, BA/BS, BA/BS, BA/BS, BA/BS, BA/BS, BA/BS (and that's just page one). You get the idea.

Shall we do what Google says and ignore college attainments, or as Google does? While Google may not ask for GPAs and specific college majors, it still wants to know you can complete a certain level of education. If you can't, they're less interested in you (although they will in some instances accept “4 years relevant work experience” in lieu of the bachelor's).

Students without math skills may nevertheless thrive in the many occupations that minimize the need for numeracy, but those students dramatically constrict their options and straiten the path to success. And it's too late to have Euclid himself as an instructor: “Give him a coin, since he must profit from what he learns.”

I threw them a curve

Rote versus reason

Most math teachers would agree that we want two things from our students: (1) correct solutions to math problems and (2) an understanding of those solutions. Of course, some students are perfectly happy with mere technical facility: Please teach us the algorithm so that we can turn the crank on it, generate correct answers, get our college credit, and get the hell out of here. They balk when we probe for conceptual understanding. Other students, naturally, claim a profound knowledge of the conceptual underpinnings of the subject matter but lament their difficulty with the merely technical and computational aspects. Will the twain ever meet?

Course grades in math classes tend to be based mostly on the demonstrated ability to compute accurate results. It's more difficult to probe for evidence of their conceptual grasp. Occasionally, however, I give it the good old college try. Here's a graph I presented to one of my calculus classes. I asked my students to look at each of the points indicated by the red dots and make some judgments about the function and its first two derivatives.

My students had a little table to fill in. The instructions said, “Fill in the table, using +, –, 0, or DNE (for positive, negative, zero, and “does not exist,” respectively) for f(x), f ʹ(x), and f ʺ(x) at the indicated values of x.”

A small panic ensued. “Where's the formula for the function, Dr. Z?” “How can I compute derivatives if I don't have the formula?” I counseled them to calm down and consider that I wasn't asking for numerical values—yes, quibblers, except for 0—and that actual computations were unnecessary.

Consider, for example, the point corresponding to x = −1. The value of f(−1) is pretty clearly 5, hence positive. The point is also a local maximum, so a tangent line at that point would be horizontal; the slope is therefore 0 and that's the value of f ʹ(−1). Finally, the curve is concave down in the vicinity of a maximum, so f ʺ(−1) is necessarily negative.

No need to panic.

The trickiest case (if “tricky” is even the right word) is probably x = 3.2 (or thereabouts). It's approximately midway between a local maximum and a local minimum, suggesting that it must be at or near a point of inflection, where the concavity changes and the second derivative must be zero (or nonexistent). That takes a little discernment. In most cases, however, the answers should be evident to any first-year calculus student with a genuine understanding of the significance of the first and second derivative.

At the class's post-exam discussion of the results, the reviews for this problem were decidedly mixed. When pressed slightly, there was a grudging consensus that, “Oh, yes, it's clear now,” but my more computation-driven students remained unmollified. They preferred to demonstrate their differentiation chops on actual formulas using the rules they'd memorized.

The experience triggered an odd recollection with me. I remembered my grandfather at the dinner table, finishing off a meal my grandmother had prepared with a recipe she had never used before. She was eager for his verdict:

“Was it good?” she asked. “Did you like it?”

My grandfather nodded his head.

“Yes, thank you. It was very good. But don't make it again.”

A few of my students may despair, but I'm keeping that calculus problem in my recipe box.

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.

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.”

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.

Notes on calculus

Cris de coeur

I think I gave my students too much time on last session's calculus final. Some of them lingered over their exams and wrote me notes. That's a bit unusual. I guess they wanted to share.

R wrote the shortest and starkest note: “BAD DAY!”

I'm not entirely sure what that was all about. He ended up earning an A in the class. It probably was related to his having done some of the problems three times, as best as I could tell from his multiple erasures and scribblings. (Like I said: too much time.) He had ample opportunity to question his answers and drive himself to distraction. Fortunately, he managed to hang on to his sanity and his grade.

J jammed his tongue into his cheek and wrote a blurb for the multivariate extremum problem:

If you like computation, you'll love problem #6. Come on down and try one on for size, this Sunday only, all problem 6's must go!

---

J's remark strongly reminded me of a marginal note penned by a university student who worked with me years ago on a solutions manual. On a page of business applications problems, he wrote

Hey, kids! Why not create your own system of economics! (Hint: Try to use functions and derivatives!)

---

I figure that leaked out of his brain after a night-long session of problem-checking. (And now it's stuck in my brain.)

I was especially charmed by H's lengthy effort, which ran all the way to the bottom of the sheet of paper where she started it and spilled over for a few additional lines on the back of the page. She had been busy studying the night before and preparing the notecard I permit them to use on tests. H discovered that she had achieved self-knowledge, always a disconcerting thing during an exam (especially a math exam):

P.S. I thought it was kind of funny— i was going through my notes while making my notecard to see if I had left myself any great pearls of wisdom & i found all these random sayings like “furlongs per fortnight“ & recalcitrant ... figures those will be the only 2 things i remember from calc 3 in 20 yrs... Well, maybe not... You've beaten the cone thing into our head pretty well... baskin & robbins trips are forever tainted... i'm going to catch myself yellin @ the ice cream scooper guy to get with it, locate the centroid of my beloved sugar cone, & gently place my choc chip cookie dough smack where it belongs...

Great... i'm one of THOSE people...

---

Come. Join us! (Don't be afraid.) You will be assimilated. (Resistance is useless!)

You are getting sleepy!

A spiral approach to math

The quiz topic was Maclaurin polynomials and the reaction of one of my students was utter blankness. He had no idea. He was lagging behind and hadn't gotten to that section yet. Some students in this predicament begin writing apologetic essays containing various excuses about the sorry state of their lives. Not this boy! He resorted to mesmerism.


No, it didn't work. He did, however, do much better on the second quiz on Maclaurin polynomials. So I guess one of us got through.

Holy definite integral, Batman!

Applied math

Even the smartest students occasionally get stumped during an exam. A physics problem concerning the trajectory of a proton caused one young scholar to give up and use the remaining time to devise (and solve!) an original problem. There's no sign that the student was rewarded for his or her endeavor. The fruits of this student's labors, however, have been launched into the tubes of the Internet, where it may find a kind of electronic immortality. This morning it was forwarded to my campus mail box. This afternoon, I post it on my blog. Who knows where else it has appeared? Where will it appear next?

Behold as calculus is used to unmask the Dark Knight! (Observe that the diligent student did not omit the crucial d(bat) from the integral.)

The asportual male

The only one?

Growing up in the rustic isolation of California's great Central Valley, I knew I was the only one. Perhaps in the big cities I might find others of my own kind, but in my formative years I was utterly isolated.

Fortunately, I didn't care all that much. That certainly made it easier to endure my uniqueness.

All of my experiences in school confirmed that I was growing up alone. No one else shared my predilections. I was confident, however, that college would contain other birds of my feather. In fact, I was sure when I received my acceptance letter from Caltech that my isolation was about to end.

Wrong again.

It turned out that my brainiac classmates at Tech were just about as fascinated by sports as the students I had known in elementary and high school. This discovery stunned me. I had been so certain that it was just a matter of time before I encountered my peer group. Generalizing from a sample of size one (apparently not a good idea), I had assumed that other really smart, clever, and literate boys would be as disdainful of sports as I was (and am). Nope. My smarty-pants peers clustered around the televisions in the rec rooms and commons just as happily as the denser types of my high school cohort. Damn.

I still recall one small glimmer of light, which I glimpsed in a newspaper during my teen years (as best as I can remember). It was an article published in the Fresno Bee, and I believe it was a story plucked off one of the wire services. I might even have clipped it out, in which case it is lost in the bundled bales of ephemera from my youth (tucked in various boxes and drawers both in my home and my parents'). The title was “The asportual male.” The article reported that the non-sports-obsessed male was more common than generally assumed, but such males were often invisible because they chose to “pass” as sports fans by dutifully sitting through game broadcasts, perusing sports pages, and occasionally traveling with sports-minded buddies to local sports stadia. The horror! Thus I learned that I was not alone, but that most others of my inclination had gone into hiding.

Rats.

Sui generis

I must not have been adequately socialized during my youth, because I can't imagine enduring long hours of tedious sports viewing just because it's the cultural norm for American males. Perhaps it would help if I had developed a taste for beer, but that remedy doesn't appeal to me. (You can imagine how attractive I find sports bars.) Perhaps the oddest thing about me is that I never felt compelled to pretend an interest, although I can see in retrospect why other young males would prefer to conform. We are a mostly gregarious species.

In my entire life, I have never watched a football or baseball or basketball or soccer game on television. Sure, I've seen plenty of snippets. I have a father and a younger brother who never miss a chance to watch whatever organized sport is going on after the holiday feasts at Thanksgiving and Christmas. As Dad gets older and deafer, I have to retreat ever greater distances to escape the blaring boob tube and breathless babbling of the commentators. One works out accommodations.

I have, however, managed to sit through two high school football games, one of which was the junior varsity coach debut of my college roommate, who had just taken a math teaching job at a local secondary school. I sat with his parents in the bleachers and replied honestly to his father's question about the team's prospects by saying that (a) I had no idea, (b) I had never seen one of their games before, and (c) that I had never seen a football game before. The man's eyes bugged out. He turned to his wife in utter astonishment and said, “Did you hear that? Can you believe it?” I was in my twenties at the time. The experience of those games was more than enough to sate my virtually nonexistent curiosity.

A few years later I was working in downtown Sacramento, an aide in state government. I had an office in a big office building that housed several state agencies. My agency was a small operation and my few colleagues soon learned not to bother to ask me if I had “seen the game last night” or what I thought about some recent newspaper article about Sacramento's perpetual quest for respectability through acquisition of a national sports franchise. (I think they eventually got one. Like I care.) Others in the building were slower to learn, many an elevator ride lapsing into an uncomfortable silence after my typical response, “Oh, was there a game last night?”

At least the couple of years when I had a college roommate, before I could swing a deal for a single apartment, had fewer complications. My roomie was delighted to discover that the newspaper's sport pages were instantly available to him. The subscriptions were mine, but those sections were always pulled out and tossed to one side, where he would eagerly pounce upon them. Of course, it was occasionally a bit awkward when I'd decide to grab lunch at a local lunch counter. I'd be there browsing the daily news and someone would ask me if I was done with the sports page. You betcha! I'm so done with it that it's not even here. It's back at the apartment where my roommate is probably chewing the pages in rapt bliss.

A glance at any other newspaper-browsing patron would usually reveal a guy poring over a sports section. Sometimes the business section, but that was a distant second.

Staying the course

While the sports-loving tendencies of my Caltech classmates had been a big surprise (and, frankly, a huge disappointment), surprise #2 was waiting in the wings. Years later, now out of government service and working as a teacher, one of my faculty colleagues invited me to his birthday party. It was a convivial event, with plenty of soft drinks for teetotalers like me. My colleague and his partner had a circle of interesting friends, many of whom were gay. And several were eager to chat about recent sports news. As the clueless idiot I was (and, probably, mostly still am), I had assumed that sports talk was the province of straight guys. It was a neat explanation for why many asportual males nevertheless felt it necessary to pass as sports fans. As with most neat explanations, it was much too simple-minded to be universally true.

People like sports because they find them entertaining. I don't find them entertaining, so I have no obligation to pay attention to them. Sports fans find the world full of instant friends and instant opponents. It's probably more interesting as a topic of conversation between strangers than, say, the weather. While I'll never go through a conversion experience that makes me sit through a game again, I suppose I have a clue why others do. I can continue to use the many hours I save by not watching sports to improve my life in other ways. Better ways. Such as reading books. Appreciating fine music. Constructive ways.

Blogging?

Anyway, I'm not really alone. I hear that Russell Baker wrote a column in which he came out as an “asportual male.” His example gives courage to others to similarly declare themselves. The Seattle Post-Intelligencer ran several column-inches of comments from readers who were apparently reacting to some game or another that is scheduled for this weekend. (Honestly, I can't tell you at this moment who is playing in the Superbowl, although folks are always prattling about it. None of that registers with me.) I enjoyed the frank admissions of those who said they were going to find something else to do tomorrow, as will I. It appears that I am not alone, even if I don't know these people.

Of course, there will always be some bozo like this:

It's not necessary to have an obsession with sports but an interest is definitely needed. Not every American man needs to be fanatical, but if you're watching “Queer Eye for the Straight Guy” on a day classically known as one of the biggest sports days of the year, you're wandering into feminine territory. I believe it is important for men to be physical to be considered masculine. Whether it's participating in sports (or other physical activity) or living vicariously through the athletes on TV, I do believe it's necessary to enjoy sports every once in a while.

—BS

---

Thanks, BS. Love those initials, man!

Thinking inside the box

A hands-on math problem

Everyone who's taken elementary calculus has met the box problem: Suppose you have a rectangular piece of material. If you cut squares out of the four corners and fold up the sides, you create a box. What is the size of the squares you should cut out if you want the volume of your box to be a maximum?

If the rectangular piece of material is a square, the answer turns out to be the side of each corner square should be one-sixth of the original square's side. If the rectangle is not a square, it gets more interesting.

Since the problem is easy to model and visualize, it's a classroom favorite. You can present the problem to a class in many different ways: computer models, spreadsheet calculations, or stand-and-deliver symbol manipulation. I have come to prefer a very mundane approach that has sparked lots of student participation. You'll need paper (heavier stock is better), scissors, and tape. Yes, we're going to actually build some boxes by hand.

My first step is to print out a bunch of full-page half-inch grids on card stock. You can use the table feature in Word or WordPerfect or a spreadsheet grid to force horizontal and vertical rules spaced half an inch apart. Since printers can't print to the very edge of a page, the grid won't be quite complete. I turn off the outermost borders on the outermost grid boxes and tweak their size in conjunction with the print margins so that the spacing remains half an inch all the way across. Then I cut each 8½ by 11 sheet in two, creating a pair of rectangles marked off in a square grid, each one 11 by 17 (using half-inch units, of course).


It's very easy to hand out rectangles, scissors, and tape and tell your students to start cutting, folding, and taping up boxes. However, most of them will immediately pick 2 by 2 or 3 by 3 squares, so you end up with big stacks of duplicates. Here's what I do to avoid this problem:

I break the class up into groups of four. Each student has a rectangle and each group has a pair of scissors and a roll of tape. The student to whom I hand the scissors in each group is designated the Recorder and is given a tally sheet on which to enter the results of the group's work. The Recorder gets to make the first box, choosing how big the corner squares should be in his or her box construction. When the Recorder has finished using the scissors, they are handed off to another student in the group, who is not allowed to use the same size cuts as the Recorder. In fact, each student is required to choose cuts of different dimensions, so that the group will produce four distinct boxes.

Here are some of the directions on an instruction sheet that each student has:

When it’s your turn, take your cardboard rectangle, which is 17 units long by 11 units wide, and choose how big a square to cut out of the corners. You cannot choose the same size as anyone else in your group. Don’t actually remove the squares, just make one cut in each corner so that you can fold up the sides and make a rectangular box. Tape the corners.

Write your name on your box, along with the length, width, height, and volume. Make sure the recorder has all of your information.

---

The pressure to choose different sizes for the square corners produces interesting results. Instead of choosing small integers like 2 or 3, some announce that they will use e or π in addition to more conventional “risky” numbers like 1.75 or 2.5.

The Recorders take their tally sheets to the board and write down the cut sizes, each accompanied by the volume of the resulting box. I plot these on a Cartesian coordinate board at the side of the room. A few minutes are spent tracking down and correcting errant results (usually given away by failing to fall on the curve that is forming on the coordinate board).

The cutting and taping and tallying takes about twenty minutes with a class of thirty. The students soon agree that the 182 cubic units corresponding to a cut of 2 must be the maximum possible volume (after which there's often some grumbling among those who claim they wanted to choose 2 but were prevented from doing so by the rules of the exercise). That sets up phase two of the experiment.

I hand out worksheets on which the groups are to replicate their construction-paper experiments, but this time with x playing the role of the cut. The students collaborate in figuring out that the dimensions of the resulting box would have to be 17 − 2x by 11 − 2x by x, after which they compute and simplify an expression for the volume of the box as a function of x:

V(x) = 4x³ − 56x² + 187x.

Rather to their surprise, as they work through finding the first derivative and setting it equal to zero to find critical values, the students discover that x = 2 is not the value corresponding to a maximum. It turns out to be x = 2.18. (That's an approximation, of course. The exact result contains a radical.) A volume of almost 183 cubic units is actually possible.

The classroom exercise in box construction appeals to me in several ways: There is an apparent result which turns out to be not quite correct. There is a an opportunity to work together in tracking down the exact answer. There is the chance to learn by personal construction and computation how a function can be developed to represent a quantity to be optimized. It's a lot better than following a teacher's scratch marks on a board in front of a class. On top of all that, the materials are cheap and easy to come by.

By the way, even though I've always conducted this exercise with college students, I have found it prudent to distribute round-ended safety scissors. I'm just saying.

Addendum

You can find a charming Java applet treatment of the box problem in all of its rectangular generality at Alexander Bogomolny's Interactive Mathematics site.

Calculus for classroom defense

Build your own fort

I'm grateful to PZ Myers for tipping me off to the efforts of Bill Crozier, the Republican nominee for state superintendent of education in Oklahoma, to make our classrooms safer. Crozier has made available an amateur video (a really amateur video) on how textbooks can be used to protect students from armed assault. Under carefully controlled laboratory conditions (actually, it's just Crozier and a bunch of guys standing around in an empty field), the distinguished Republican educator uses an AK-47 and various pistols to shoot at some textbooks.

His exercise in ballistics was reported by KOCO, Channel 5 in Oklahoma City:

“We are doing this as an experiment because at Fort Gibson, many young people were shot in the back,” Crozier said in the videotape, referencing a December 1999 middle school shooting in eastern Oklahoma, in which a student wounded four students with a 9 mm semiautomatic handgun.

---

As a trained observer of the world around him, Crozier had noticed that one of the assailant's bullets had not penetrated a textbook in a student's backpack. His keen mind quickly grasped the possibilities of using textbooks for classroom self-defense.

Crozier's first experiment (after verifying that they hadn't forgotten the AK-47's firing pin by the simple expedient of actually firing it) was to take a shot at a calculus textbook from a distance of 15 feet. As noted in the video, the AK-47's slug made it all the way through the calculus book and through the phone book that was behind the math text. I immediately noticed a problem with the experimental protocols. While the choice of the calculus book by Larson, Hostetler, and Edwards was perfectly defensible, that book being one of the more popular texts in the U.S. market (Stewart's book, however, with its 70% share of the market, would have been even more appropriate), Crozier chose to use the single-variable edition, which is only two-thirds as thick as the standard edition.

But even the comparatively skinny book proved to be more than equal to the task of stopping a 9-mm bullet. In the second experiment, the text that had been pierced by the AK-47 needed only seven of its chapters to stop the 9-mm projectile. (We can clearly see that Crozier opened the single-variable edition of Larson to the beginning of Chapter 8, where we are greeted by a two-page spread on fractals—an eye-catching topic with cool graphics that has extremely little to do with first-year calculus. Benoit Mandelbrot smiles up at us from the lower corner of the right-hand page.) That experimental result will make me feel safer the next time I use one of the thinner editions because I don't want to lug about the significantly heavier complete version.

The standard 1000-page edition of a calculus book has major stopping power, of course, as many former college engineering majors know. (They mostly became liberal arts and business majors.) Calculus books should be just as useful in stopping slugs (I'm not actually talking about students now). Crozier's brilliant insight into classroom defense procedures generalizes naturally to protecting professors in their campus offices. A typical math teacher has scads of books in his or her office. At the first sign of danger, a diligent mathematician should be able to construct a sturdy textbook fort, as shown in the accompanying illustration. (The two embrasures for returning fire are optional, of course, mostly depending on whether your state allows teachers to pack heat.) At our next department meeting, I think I'll propose that we schedule some training drills.

Thanks, Bill Crozier! Who but a Republican candidate for state superintendent of education could have come up with such a brain stroke?

Jesus H. Christ, Ed.D.

We got your Christian calculus right here!

The indefatigable D. James Kennedy never ceases to provide a platform at this annual Reclaiming America for Christ conference to exponents of the Bible-blinkered worldview. On Monday, August 7, 2006, Kennedy's Truths that Transform radio program featured the first part of Dr. Paul Jehle's talk titled “Evaluating your Philosophy of Education.” Jehle is the senior pastor of a church in Plymouth, Massachusetts, as well as the principal of a Christian school and education director of the Plymouth Rock Foundation.

Jehle is also a master of the mock debate and a skilled user of false dichotomy: “There are only two philosophies of life.” Can you guess what they might be? Yes, it's Christianity versus non-Christianity, the latter of which is also known as humanism. (This is going to be a very big surprise to lots of Jews, Muslims, Hindus, etc., etc.) That quote, however, just makes Jehle look like a doofus and one might be concerned that it was taken out of context. Let's allow him to pay out a little more rope, in some extended excerpts in his own words:

You cannot candy-coat paganism and swallow it as godly. And because of that we need to clearly distinguish that which is Christian from that which is not Christian. If we do not, and if we fail to do that, we will often swallow something we think is Christian that will be bitter later on.

That's why there is such a thing as theistic evolution, for instance. For it's the buying of a lie. It's trying to Christianize a pagan religion. In the late 1800s almost every scientist who was a Christian attempted to do that. After Darwin's book came out in 1859, everyone was trying to Christianize paganism.

We have that today in almost every field. You think that's only limited to science? No way! We have christianized pagan jungle music that goes by the terminology of Christian. It's determined to be Christian because the words were Christian or the person singing it is Christian. But the issue is you cannot combine something by its nature which is pagan and built on humanistic principles and make it Christian by a magic wand.

---

So Jehle is putting on notice all of us paganistic, humanistic, jungle-music-loving, evolutionists—even the theistic ones.

I'm not strongly motivated to spring to the defense of theistic evolutionists, who tend to waste a lot of time wringing their hands over how their religious beliefs do not contradict their dedication to the scientific method. (Get over it, guys! Your religious practice and your scientific research are independent of each other. So stipulated!) To me, theistic evolutionists are not much of an issue. Jehle, however, feels obligated to beat up on them because they smear too much gray on his black-and-white world. Perhaps they even listen to jungle music, too. Shocking.

We can, I supposed, give Jehle some points for consistency: He insists on shoving everything through his Christian worldview meat-grinder:

I was taking calculus. I was a mathematics major and I was at a Christian college that was called Christian, but was not Christian....

I asked a question to my calculus professor: “What makes this course distinctly Christian?” He stopped. He said no one has ever asked that question before...

He said, “Okay, I'm a Christian; you're a Christian.”

I said, “That's not what I asked! What makes this calculus course distinctly Christian? What makes this different from the local secular university? Are we using the same text? Yes. Are you teaching it the same way? Yes. Well, then why is this called a Christian college and that one a non-Christian college?”

---

You can imagine my disappointment when Jehle abandoned this topic and never returned to it. Perhaps it was my fault for not listening to the second part of his talk, but my endurance can be tested only so far. If he indeed deigned to reveal the nature of Christian calculus, I didn't get to hear about it. I would imagine that something needs to be done about the godless limit process, wherein x is routinely permitted to go to infinity without asking God's permission. Too bad that Newton and Leibniz were never referred to the Inquisition.

Jehle did wrap up the first part of his talk with a wonderfully stage-managed mock debate, in which he helpfully portrayed both sides—the holy, God-fearing side with truth, right, justice, and Christ on its side, and the evil, godless, humanistic side with its fetters of evolution and that never-to-be-forgotten jungle music. Fortunately for Jehle, the side he favors managed to win when he was in charge of all aspects of the debate:

The answer is not whether religion influences law but which religion should influence law that produces the best liberty.

If I was on a debate show, and a man said to me, “You people would shove Christianity down the throats of every person in the United States of America because you're a fundamentalist right-wing—you know, I mean—out there in the ozone layer Christian.”

And I said, “Thank you for the introduction. I'm really glad that you understand how powerful Christianity is since you shake so much in its power.” I said, “Listen, the answer is not whether Christianity or any religion does that, but then which religion, sir, would you like to introduce as the base of law that would give the greatest of liberty and I'd like you to give me at least five civilizations in history that have proved to be liberty and protecting rights as the result of that religion.”

He's silent as it is now.

“Do you like the New Age movement?”

“Yes.”

“Okay, It's built on Hinduism. Let's look at all the Islamic countries over in Europe. Tell me which one would you like to live in?”

I said, “I'm glad you're not answering because you're living in the United States, and the very Christianity you criticize gives you the liberty to debate me on radio. And that's why! I said, do you understand that we wouldn't have this freedom if we lived in the country where your religion reigns?”

---

The audience went wild with applause as Jehle expertly eviscerated his imaginary debate foe. I love the way his hypothetical narrative quickly became indistinguishable from an account of an actual event (“I said” instead of “I would have said”). A real opponent might have had just a little more success. Perhaps along these lines:

It seems unfair that you ask for five examples. Would you be happy with one really good one? The United States of America was founded by men who were at pains to avoid the European example of religion-influenced government, which is why the U.S. Constitution never mentions God and cites religion only in the Bill of Rights, where it declares that the people have religious freedom. That is the reason that our country is free—because it is not officially Christian, although an aggregate of Christian sects comprise a majority of the population.

While the New Age movement is full of nonsensical ideas, some of them borrowed from Eastern religions, it is foolish to identify it with Hinduism. It is even more foolish—or perhaps simply careless or ignorant—to identify Hinduism with Islam. If you bother to check, you will find many Muslims living in Europe, but precious few Islamic states. It would therefore be difficult for me to pick one.

Finally, the freedom to speak comes not from Christianity, as history clearly attests through the examples of Queen Mary's England, the Inquisition's Spain, and Calvin's Geneva, but from the neutralization of religion through universal religious freedom. Be glad of that, for otherwise you would be a prime candidate to be silenced for your sedition against the liberty of the citizenry.

---

Amen.

Calculation without concept

Attack of the coneheads

My multivariate calculus students are not entirely happy with me. I am not entirely happy with them. The results of this week's exam on multiple integrals were—shall we say?—a trifle disappointing. Perhaps you will not be surprised to learn that I had great expectations for this exam. I had fussed over its preparation at length and I had been rather pleased with the way it progressed through the various coordinate systems (rectangular, polar, cylindrical, and spherical). I knew that some students would nevertheless insist on using suboptimal choices of coordinate system for some problems (and they did), but I keenly anticipated a generally good outcome.

That's what I get for trying to be helpful. My students swore they would have done better on the exam if I had not “tricked” them. They advanced as Exhibit A the “misleading” figure that had accompanied one of the exercises. During the hours of grading, I had marveled over what my students did to that particular problem. Let's work through their complaint and see if we can figure out what's at the bottom of it.

The exam problem in question required a given function to be integrated over a very specific three-dimensional region. Here is how I described it:

Let Q be the cone bounded by z = r and z = 2.

---

Okay, it's a cone with the pointy end at the origin and its axis coincides with the z axis. In the margin of the exercise, I included a generic figure of a cone in the correct orientation. It was not to scale and I left it to my students how best to label it. As shown, the figure was labeled generically: a for the radius of the circular base (is it okay to call it a base if it's on top instead of on the bottom?), h for the height, and (just for the wild ones who want to use spherical coordinates) the semi-vertex angle is labeled α.

To my surprise, most of the answers worked out by my students contained a and h in them. For some reason, they did not realize that they knew (or should know) the dimensions of the cone Q. A few had grasped that z = 2 is a horizontal plane and thus the height of the cone must be h = 2. Even these people, however, couldn't figure out the value of a despite the helpful cone formula z = r. What was going on?

This is what you taught us

A few students looked at the graph of the cone and promptly wrote down a description of the region in terms of cylindrical coordinates (good choice!):

0 ≤ θ ≤ 2π
0 ≤ z ≤ h
0 ≤ r ≤ az/h

That was, of course, fine as far as it went, but I had given them specific equations. Could we perhaps please use those? No! Never! We have a picture to look at and we will use it, damn it! Don't try to stop us!

As I questioned them, I began to understand a bit more about what had occurred. They had (several of them) memorized the cylindrical coordinate definition of a cone, but it was just memorized, not understood. I've seen this many times before, but more typically among algebra students than in the ranks of a multivariate calculus class. Perhaps the recent spring break had caused them to lose their momentum and fall back on computational gimmicks in lieu of actual understanding. But perhaps I should have seen it coming. Students had asked me several time for “formulas” for converting the limits of integration of multiple integrals from one coordinate system to another. This was a bad sign I had insufficiently appreciated. In our discussion of the exam problems, I tried to get them to go back to basic principles.

“I'm afraid that you're hoping for a step-by-step computational procedure for transforming integrals. You can do that to some degree with integrands, where it's mostly a matter of substituting for one set of variables in terms of another, but don't expect that to work for the limits of integration. It's much, much better to rework the description of the region of integration. Go back to basics when you need to. Remember similar triangles?” On the board I sketched a triangular cross-section of a typical cone.

“Remember how we got the formula for a cone in cylindrical coordinates in the first place? z tells you how high you are and r tells you how far you are from the z axis. If you know the cone's radius is a and its height is h, then r and z have to be in the same proportion.”

r/z = a/h

When we solve for z, of course, we get z = hr/a. Since I had given them z = r in the statement of the problem, it should be pretty clear that a/h = 1, so a = h. What's more, since z = 2 provides the height of the cone, as previously noted, we must have a = h = 2. The value α = π/4 falls out immediately from these results, in case anyone cared to look at the isosceles right triangle.

They were right

In a way, my students were right. I have dutifully provided them with the basic tools to describe a region like a cone in cylindrical coordinates, but I had not worked with them sufficiently to instill an understanding of the patterns behind the formulas. When they saw the unidentified a and h in the illustration for the exam problem, they instantly fixated on the formal description that I had previously helped them to master (or at least memorize). Of course, it was their responsibility to work with the tools in their hands until they became adept with them (that's why I give them assignments to do at home and quizzes to take in class), but I could have drawn their attention more strongly to the process by which we built the descriptions and emphasized the flexible ideas that drove the process rather than the formulas on which it was based.

As I said at the beginning, we're unhappy with each other. They're on notice that I expect more from them than ritual formula application. And I'm on notice that once again the ideas in my head don't magically transfer into theirs. The follow-up quiz will probably show that the exam problem was a learning experience for most of my students—at least as far as cones are concerned. If I want more trouble, I could change the region to a sector of a sphere.

But that's a different problem and a different story.

David Berlinski vs. Goliath

This time Goliath wins

Intellectuals don't win too many popularity contests. The “smart kid” in class is often the target of bullies. Speaking as a former smart kid, I find it perplexing that we sometimes become bullies ourselves. We should know better, although it may be that some of us cannot resist taking a turn when circumstances change. Perhaps this is what happened to David Berlinski, a peculiar icon of the anti-evolution movement.

Berlinski is an intellectual bully, a trained mathematician who enjoys using his special status to confuse and abuse others. His biographical sketch on the website of the Discovery Institute notes that Berlinski earned a Ph.D. in mathematics philosophy from Princeton, no shabby accomplishment. His book A Tour of the Calculus was a very successful semi-popular account of the work of Newton and Leibniz and its consequences. The hardcover edition's book jacket includes a paragraph about the author on its back flap; it contains this sentence: “Having a tendency to lose academic positions with what he himself describes as an embarrassing urgency, Berlinski now devotes himself entirely to writing.” As is usually the case with someone who has failed to find himself an academic niche, Berlinski's fame rests on matters other than his research papers. These days he is more recognized as an anti-Darwin skeptic than as a mathematician.

Nevertheless, it is as a mathematician that Berlinski rides into combat against his evolutionary targets. What could be more natural for him than to hurl numbers at his opponents?

Could I ask you to give us your best estimate of the number of changes required to take a dog-like mammal to a sea-going whale?

---

The quote is from William F. Buckley's Firing Line on PBS. The installment on December 4, 1997, was devoted to a debate on the proposition, “Resolved: The evolutionists should acknowledge creation.” Berlinski's position among the creationists was ambiguous, since he purports to have no opinion on creation itself; he is simply making common cause with those who attack Darwin and evolution. His question on the evolution of the whale was aimed at Eugenie Scott of the National Center for Science Education, an organization dedicated to promoting sound science education and opposing creationism.

It was a theme to which Berlinski returned several times in his exchanges with the evolutionists on the Firing Line panel. He variously denounced natural selection as “Que sera, sera” and archly demanded to know whether Darwinism comprised any actual theory “that would be recognizable by any physicist or a mathematician.” Berlinski is particularly enamored of physics, which is highly mathematized and fraught with numbers. To the degree that evolution is not numerical, Berlinski appears to argue, it is not really a science. For him, science seems to be an absurdly all-or-nothing proposition. Witness this exchange between Berlinski and Barry Lynn of Americans United for Separation of Church and State. (I have abridged the discussion to remove some of the repetitions and digressions. The full text is available here.)

DB: I do have a scientific question to ask you. Every significant paleontologist says that there are gaps in the fossil record. Do you have a particular reason for demurring?

BL: Of course it has gaps.

DB: Okay, so to that extent the evidence does not support Darwin's theory of evolution.

BL: No, that is absolutely wrong.

DB: It follows as the night the day.

BL: Of course not. How could you have a cell, for example, hundreds of millions of years old, that would leave a fossil record? It would disintegrate. It would quite literally not be able to be found in the fossil record.

DB: I never suggested that there may not be explanations of the gaps. But the fact [is] that the fossil record does not on its face support Darwin's theory of evolution.

BL: No, it does. And once again I say, how many times do we have to find those intermediate fossils?

DB: All I'm asking for is enlightenment on a significant point. Darwin's theory requires a multitude of continuous forms. We do not see that in the fossil record. In fact, major transitions are utterly incomplete. Would you accept that as an empirical fact?

BL: No, you sound like a guy who is writing a story about baseball, comes in in the fourth inning, and says, “Well, you know, I'm going to write about the fourth inning on; the first three innings didn't happen because I wasn't there to see them.”

DB: We can't find any of the major transitions between the fish and the amphibian.

BL: Of course we find them. It's just that when we find them, doctor, you say it's still not enough.

---

The method to Berlinski's madness is easily characterized: Darwin's theory requires continuity in transitional forms. No continuity, no evolution. Simple as that. Q.E.D. While no serious scientist would argue that the absence of mathematical continuity in a fragmentary fossil record would imply there was no continuity in the first place, Berlinski's non-serious approach is perfectly compatible with such an absurd conclusion. It might be helpful at this point to remind everyone that a single gap in a mathematical proof is enough to invalidate it. While this standard is a natural and necessary part of mathematics, where a partial proof is no proof at all, it is completely out of place in the observational sciences. Berlinski either does not know this, or pretends not to.

I watched the Firing Line debate when it was originally broadcast in 1997. Somewhere around here I have a videotape of the whole two hours. Reading the on-line transcript reminded me how much is lost in the translation from video to text. The page does not fully convey Berlinski's supercilious manner as he combines spurious arguments with intellectual disdain. He was going to browbeat his opponents into submission with the immensity of his mathematical knowledge and he oozed contempt for their counter-arguments.

I learned a valuable and frightening lesson watching Berlinski in action. Please never let me lapse into behavior like his! And the bullies who beat the snot out of him in school have a lot to answer for.