Bernie-Bot math

Examine your assumptions

Did you know that Bernie Sanders is all but assured of a contested Democratic convention? The Bernie Bots tell us so by a startlingly naive application of simple math. One example is embodied in a picture post that the senator's fans have been passing around and sharing. Its most basic claim is that “In order to prevent a contested convention with Bernie Sanders, Hillary needs to win 65% or more of the vote in every future state.” While I agree that Sanders has enjoyed a remarkable degree of success in the campaign to this point, the supposed reversal of fortune for Clinton seems a bit of a stretch. And it is.

The New York Times keeps a running tally of the delegates in each candidate's camp. As of today, the Times reports that Clinton has 1,307 pledged delegates and 469 unpledged delegates for a total of 1,776 (a nice patriotic number). Sanders by contrast has 1,087 pledged delegates and 31 unpledged delegates for a total of 1,118. The successful candidate will need 2,383 delegate votes to secure the Democratic nomination for president. Thus it's a simple matter to determine that Clinton needs 2,383 − 1,776 = 607 more delegates while Sanders needs 2,383 − 1,087 = 1,296.

Per the Times, there are evidently 1,959 delegates up for grabs in future primary elections and caucuses. Therefore, Clinton must secure 607/1,959 = 0.310 = 31% of the remaining delegates while Sanders has the more formidable task of rounding up 1,296/1,959 = 0.662 = 66.2% of them. How can the Bernie-Bot picture post be so wrong? Easy!

First, assume that all unpledged delegates (the notorious “superdelegates”) are undecided free agents. After complaining bitterly for months that superdelegates are Clinton minions who have “rigged” the nomination contest, Bernie's supporters are now pretending they can be ignored and omitted from Clinton's delegate count. If one insists that the 1,307 formally pledged delegates are all she has, then she needs 2,383 − 1,307 = 1,076. That's a whopping 1,076/1,959 = 0.549 = 54.9%. That's not as dramatic as the Bernie-Bot claim that she needs 65%, but it would still indicate that Hillary needs a majority of outstanding delegates to win the nomination! She's at a disadvantage! Or so we can pretend.

Remember how shocked Mitt Romney was when he lost the election in 2012? He and his campaign team had been taking all too seriously the “corrected” polling data from partisans who insisted that professional pollsters had biased their population samples against the Republican nominee. If you assumed that there would be a lot more GOP voters at the polls than the national pollsters were finding in their sampling, the results for Mitt were great! But wrong.

The power of an unwarranted assumption is great. And it gets even greater when you can't do the math.

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.”

The Ritualists

A new strain of tardiness

The old pattern was very familiar, especially since I tend to give my students lots of short quizzes, often at the beginning of a class period: A student arrives late, sees a quiz in progress, and leaps into action, yanking a pencil out of the old book bag, snatching a quiz off the table in the front of the room, and scribbling quickly in a desperate attempt to catch up. That's the old pattern and it's not a surprising one.

Lately, however, I've seen several instances of a new pattern that is, frankly, utterly bewildering. In over thirty years of teaching, I had never seen this behavior until the last few semesters. A few of my tardy students have an unprecedented sang froid. They arrive late, see a quiz under way, and then progress casually to their desks. They never rush up to the front of the room to pick up a quiz. Their leisurely saunter gives me plenty of time to stroll over and hand them one. (Service with a smile!)

This new breed of tardy student is calm and generally unruffled, except sometimes a small moue telegraphs the unspoken thought, “Oh, here we go again!” The serene latecomer positions the water bottle or energy drink or Starbuck cup on a corner of the desk,  carefully tucks away the cell phone or iPod, peels off the coat and rolls it up to tuck in the book cage under the desk, rummages about in the book bag for a pencil or pen (sometimes deliberating over his or her choice of several writing implements—mustn't pick the wrong one!), digs out a calculator and places it precisely in the corner opposite the beverage (whether or not the quiz requires number-crunching), and then finally (as if in surprise) takes note of the quiz sitting atop the desk and begins to ponder it.

This settling-in ritual, in its various versions, eats up at least two minutes, sometimes three. Sometimes there is a lengthy interlude with the smartphone, scrolling through messages and tweets received in the interval between breaking eye contact with the screen upon arriving at the classroom door and arriving at the desk, occasionally extended by the imperative of replying to urgent missives. I imagine most of them are in the vein of

L8 agin
prof :(
lol

Strangely enough, the explanation does not appear to be the simple one: Such casually late students are the class's losers, doomed to fail, and have fatalistically accepted their fate. Nope. That describes very few of them. My unruffled tardies are mostly C students mired in mediocrity. Perhaps they've figured out that they're doing enough to survive and it would be too much trouble to put in the work necessary to rise to the B level. I really don't know.

One thing, however, has not changed. After arriving ten minutes late and getting only five minutes to work on a fifteen-minute quiz, many chronically tardy students are quick (for a change) to complain: “I didn't have enough time!”

“Yes, you did,” I explain. “You just chose to use most of it for something else.”

Brain pain

Lesson unlearned

My students were not happy with me and they weren't keeping it a secret. After a unit on scientific notation, I gave them a quiz containing a question they deemed terribly unfair:

The mass of a proton is 1.7 × 10^–27 kilograms. What is the total mass of 7.2 × 10³³ protons? (Write your answer in scientific notation and include the units.)

I was told, with exquisite care and patronizing precision, that it was wrong of me not to tell them which arithmetic operation was expected. Addition? Multiplication? Subtraction? Division? How dared I give them numbers without specific instructions!

With professional patience, I waited out their lengthy complaints. Then, without saying a word, I turned back to the chalkboard and wrote out a brand-new problem:

The mass of a nickel is 5 grams. What is the total mass of 6 nickels?

With frowns still on their faces, they blurted out, “Thirty grams!”

Another long silence as I waited for their reactions. The faces went neutral. One brave soul ventured a comment: “Were we supposed to know that?”

“Sure,” I replied. “All of you know that you multiply to solve problems like this. You just yelled out the answer to the nickel problem because it was so easy. What I'm trying to get across is that numbers written in scientific notation are still just numbers. You work with them just like you work with other numbers. You're letting your minds shut down because they look different, but you actually already know what to do.”

A smug expression is bad pedagogy, so I maintained a mild and neutral mien. I was quietly satisfied that I had gotten an important point across. My self-congratulation was just a little premature. (You'd think I would know better by now.)

A student in the back row grunted in dissatisfaction and posed a question in an irritated tone: “So on the next exam are you going to tell us what to do with the numbers?”

My spirits fell a notch.

“What do you think?” I asked.

I hope indeed that they do.

Plus or minus

Rather missing the point 

One of my favorite negative reviews on RateMyProfessors.com is the following:

I don't understand why people say he is a good instructor. Many students in his class struggle to get a good grade. yes he is clear but his tests are extremely difficult. And expect a ton of repetitive homework assignments.

Let's deconstruct my student's complaint piece by piece:

Many students in his class struggle to get a good grade.

Yes? You mean they don't get good grades automatically? The student in question was enrolled in a calculus class. Such classes are notorious for easy grades, right? Yeah, right. More to the point: In a typical college class you can expect a distribution of grades, most of which are C's. Not what I would call “a good grade.” Good grades are A's and B's, earned only by those students who put in the effort.

[E]xpect a ton of repetitive homework assignments.

I checked. The syllabus contained homework assignments for each section with, typically, 12 to 20 problems. There were 33 sections that we covered, so students were expected to solve approximately 500 problems over the course of a 16-week semester, or a little over 30 exercises per week. (My bleeding heart weeps for them.)

Funny thing: There is a remarkably high correlation between doing the homework and getting one of those good grades. There were thirty students in the class. I note that only one student in the top half of homework performance was not earning an A or a B (and that one student was pulling a solid C). Of the fifteen students in the bottom half of homework performance, only four had “good” grades (three B's and one A [there's one in every crowd]). Conclusion: Do the work, get a good grade.

[H]is tests are extremely difficult.

Evidently not the case for those who work at it by doing the “repetitive” assignments. (Average scores were actually in the low eighties.)

yes he is clear

Thank you very much. Clarity is something I strive for and I am pleased that you noticed.

I don't understand why people say he is a good instructor.

Indeed you don't.

My brother's keeper

Cain and Abel?

A student was talking to a friend. He sounded a bit irked.

“My brother is enrolled in a college in Oakland. He's having a really bad time in his math class.”

His friend nodded her head in sympathy. The young man continued his tale of woe.

“Yeah, a really bad time. You know, I took the placement test for him so that he could get into the class in the first place, but it's really kicking his butt!”

Strange to say, the boy sounded exasperated. Here he had done his brother this great big favor, helping him enroll in a class for which he was not prepared, and nevertheless his brother was squandering this golden opportunity by flunking the class. No doubt the brother was insufficiently grateful, too.

At least the young man has a great future before him. He'd be a natural as a Republican campaign consultant.

Self-validation

Oops! ... I did it again

It was an accident.

I gave my students a take-home quiz, due at the beginning of our next class period. This doesn't happen too often, but it's a nice opportunity for them to score maximum points by working together and carefully comparing notes before submitting their results. With a few exceptions (the handful of students who prefer to keep their work as secret as possible), my students spring at the chance to cooperate and rack up the points.

This time was no exception. However, one student e-mailed me with a concern. “Abe” had transportation issues and was afraid he might be late to class or even miss it entirely. As a precaution, he had scanned his solution to the quiz and attached the image to his message. I wrote back to put him at ease, confirming my receipt of his work, and wishing him good luck in making it to class the next day.

As it turned out, Abe was in class that next morning and handed in the original version of his quiz. I slipped it into my binder along with all of the others. Like the absent-minded professor I am, I quite forgot that I had printed out his scan and already had that in my quiz folder. During my grading session that afternoon, I inadvertently graded Abe's quiz twice, marking up both the original and the scan.

I noticed my oversight while sorting the quizzes into alphabetical order for purposes of entering the scores in my gradebook. I placed the two versions of Abe's quiz side by side and discovered that they were still identical: My red-ink marks on the two quizzes were identically placed, the corrections were a perfect match, and both quizzes bore the exact same score.

Naturally I was pleased. Consistent grading is one of the most important factors in treating students equitably. Here I had evidence that my correction process was rigorously—even rigidly—consistent. I have achieved the gold standard in the potentially capricious and subject process of grading!

Either that, or I'm a robot.

Too cool for school

No royal road to algebra

Although its hours have been trimmed by the current state budget crisis, my college's Tutoring Center continues to serve as a lifeline for many of our students. Each semester, therefore, I make a point of ensuring that my math students are aware of the facility's existence. I don't just tell them, I show them. Thus it was once again that, during the second week of the semester, I gathered up the entire class and took them on a “field trip.”

It puzzled my students when I announced it, of course. I told them to leave their books and papers behind in the classroom, which I would lock behind them. We would take a few minutes to stroll down the sidewalk to the Tutoring Center, which was only a couple of buildings over. Short field trip. Once I mentioned where we were going, some students nodded their heads in comprehension, grasping my purpose. Other students, however, had a different reaction.

One came up to me, backpack in hand, clearly ready to make a break for it.

“Is this required?” he inquired.

“We're all going to the Tutoring Center,” I said, in oblique response.

“Yeah, but do we have to? Is it an assignment?” He was nothing if not persistent.

“We're all going to the Tutoring Center and we'll be back in a few minutes to start on the next topic,” I said, demonstrating a charming obtuseness.

I don't think my student was charmed. He got to the point.

“Does this affect our grade? Are we getting participation points?” he asked.

I looked right at him, allowing my surprise to show.

“‘Participation points’? In a college class?”

He fell silent but unrepentant. He wanted points if he was going to go to the Tutoring Center with his classmates. It was finally obvious I wasn't giving any. He trailed along behind the rest of the group and I expected him to lag increasingly until he took a “wrong turn” and vanished toward the parking lot.  I was thus mildly surprised and pleased to see that instead he stuck it out and hung at the periphery of the group as I introduced them to the instructional assistant who managed the math tutors in the Center and walked everyone over to the area where drop-in tutoring occurred. Now that my students had been physically present in the facility and had met the key personnel, I figured it was much more likely that they would feel comfortable about returning to it when they needed help.

We returned to our classroom and launched into the lesson for the second half of the class period. The point-grubbing student sat quietly in the back, apparently ruing his decision not to skip out. At least I assume so, since in the next few days he developed a habit of nonattendance or early departure. When our first exam came along, he achieved the class's low score, missing a D by several points. (In fact, his score in the thirties might reasonably be characterized as an F-minus-minus.) He had never come to my office hours and he had never darkened the door of the Tutoring Center again.

I guess he really needed those participation points.

Not lost in space

The Infinite Tides

The traumatized astronaut is not a new theme in literature. In nonfiction, we have the example of Buzz Aldrin's Return to Earth, which deals with the alcoholism and depression of the second man on the moon, and Brian O'Leary's The Making of an Ex-Astronaut, which chronicles the less dramatic frustrations of a scientist-astronaut who never made it into space. Science-fiction author Barry N. Malzberg penned The Falling Astronauts, in which astronaut Richard Martin gets bundled up by his crewmates after his breakdown and hauled back to earth as a basket case. More famously, Arthur C. Clarke created some extremely stressed astronauts in 2001: A Space Odyssey.

So ... been there and done that. Besides, the Space Age is old news and these days no one interrupts regularly scheduled programming to report on rocket launches or spacecraft landings. Therefore it might seem just a little surprising that a new author should choose a distressed astronaut as the protagonist of his first novel. What was Christian Kiefer thinking when he wrote The Infinite Tides?

The author shared some of his thought process during the SummerWords conference, which he and his English department colleagues at the American River College organized last month. (Yours truly attended and was most likely the only mathematician in the crowd.) At a session on researching one's story, Kiefer mocked the “write what you know” straitjacket, preferring instead the “write what you can find out” approach. Thus he plunged into astronautics and mathematics, dredging up the information that would give his high-flying protagonist substance and credibility.

Kiefer also talked with the Sacramento Bee, explaining the genesis of his novel to reporter Allen Pierleoni:

Part of it was listening to the news and beginning to feel I might be the only man in America who still had a job. Then sitting at Starbucks (grading papers), watching other men at other tables looking through the want ads, then drifting to the sports pages, then to the funnies, then finally to the front page. Basically using the hunt for a job as a way to fill the endless hours of their otherwise vacant days.

It's pertinent to note that people who have academic jobs teaching math and English—subjects deemed indispensable at college—have a security that is rare in the modern world. We are a privileged few. Who else is so lucky?

Astronauts, perhaps. The men and women of the space program comprise an elite corps of over-achievers. They have reached a literal apex of accomplishment as they leave the earth on their missions. What would happen to an astronaut if he were to find himself grounded, his life and career in ruins? That was the question that Kiefer asked himself and he explores the answers in The Infinite Tides.

Astronaut Keith Corcoran is a genius at math and engineering. His goal is to go into space. Corcoran's entire life is devoted to achieving his goal, even to the point of estranging his wife and daughter. Corcoran notices and regrets the increasing distance between himself and his family, but can't see a way to resolve it without jeopardizing his career. He's actually rather irritated with his wife, who once seemed so supportive, but he plunges ahead regardless.

When it all comes apart in tragedy and illness, Corcoran finds himself alone. Stripped of flight status and living alone in an empty house in a half-built suburban development, he has nothing but time—and nothing to fill it. Having lived all his life with a keen sense of his mathematical trajectory through spacetime, Corcoran struggles to reassess the axioms of his existence. Vectors are mathematical entities possessing both length and direction, telling you both where and what. They beautifully model things like velocities, expressing both where you are going and how fast you're going to get there. For Corcoran, they were real and gave shape to the way he moved through life. Vectors were both tools he used in his engineering work and dynamic forces that drew him through reality.

Tragically, his sense of mathematized reality was one that he had in common with his daughter Quinn, but which also estranged them. While Corcoran lived within the coordinate grid of spacetime, his daughter was not embedded in the same way. While Quinn perceived the same personalities and characteristics of numbers that her father saw, she was nevertheless a different person. She had the gift of being able to live among the mortals, to be popular and social. Instead of jumping at the opportunity to enroll in an elite school to hone her extraordinary gifts, Quinn preferred to stay in a regular high school and join the cheerleading squad. Thus she became a disappointment to her father, who had already mapped out the inevitable trajectory of her life and could not come to terms with her deliberate violation of deterministic fate.

The Infinite Tides is an engrossing book. I read the entire thing over a single weekend, rarely putting it down. Keith Corcoran is a fascinating character, often maddening, whose sense of place and purpose is wobbling out of control. When he starts interacting with the neighbor woman whose daughter reminds him slightly of Quinn, you expect certain things to occur, and some of them do—but never in quite the way you were anticipating. The surprises keep you off balance and make you all the more sympathetic to Corcoran's disorientation. You begin to wonder how the author can possibly bring the book to a satisfactory resolution.

And yet he does. In fact, the final pages of The Infinite Tides bring Corcoran's story to a cusp, where many different things become possible. There is no pat happy ending, but rather a blossoming of choices. The man who lived in a mathematical framework that had become a deterministic cage begins to grasp the key that his daughter had found.

A divergent coda

Having traversed the trajectory of my review, I find myself left with notes and observations that did not fit into the flow. I offer them here as a collection of tangent vectors.

The Infinite Tides is a stunning accomplishment and I exhort people to read it and watch for future works by Christian Kiefer. The man has staying power. What's more, his capacity for assimilation of background research is prodigious. He admits to being relatively innocent of mathematical knowledge, yet he absorbed what he needed and magisterially portrayed the life of a brilliantly obsessive-compulsive mathematician.

I suspect that people who disdain math might occasionally recoil from Keith Corcoran, who tries even a mathematician's patience as he relentlessly invokes “equations” (one of the book's most frequently appearing words) and their solutions. Everything to him is a math problem, but that idée fixe is the protagonist's defining characteristic, the leitmotif of his life.

There is one bobble in the discussion of Hilbert's hotel, a warm scene where father and daughter are sharing a joyous discovery about the paradoxical nature of infinity. Suppose you have a hotel with infinitely many rooms: Room 1, Room 2, Room 3, and so on, going forever. Suppose the hotel has no vacancies, infinitely many guests being in residence. Suppose infinitely many new people show up, all wanting rooms. What is one to do? Quinn suggests a solution to her father:

“They ask every other guest to move down one room.... If n is a room with a guest the n moves to n plus one and then—”

Her father quickly understands. Unfortunately, Quinn should have said that n moves to 2n, not to n + 1. If the occupant of Room 1 moves to Room 2, and the occupant of Room 2 moves to Room 4, and the occupant of Room 3 moves to Room 6, etcetera, then all of the original guests end up in even-numbered rooms, leaving the infinitely many odd-numbered rooms vacant to accommodate the infinitely many new arrivals.

I doubt the matter will cause much distress among Kiefer's readers, but the mathematically inclined may wrinkle their brows.

My other reservation relates to astronomy. For an astronaut, Corcoran is remarkably ignorant of elementary astronomy when he converses with his Ukrainian neighbor. When Peter explains that he likes to look at Messier objects, Corcoran says, “I don't know what that means.” But Messier objects are an Astronomy 101 topic, a catalog of celestial objects that could be mistaken for comets when viewed through a telescope. The Andromeda galaxy is M31 in the Messier catalog. Even more unlikely is Corcoran's ignorance of the W-shaped constellation Cassiopeia. One expects astronauts to know such configurations for purposes of stellar navigation if the computers fritz out and the sextant has to be dragged out. (This was actually a consideration during the Apollo 13 mission and a factor in the crippled spacecraft's safe return to earth.)

Perhaps Corcoran was exempt from such lessons since he was an engineer-astronaut instead of a pilot-astronaut, but it struck me as unlikely.

I mentioned 2001 in the opening paragraphs of this article. It appears that Kiefer included a related joke to amuse close readers of his novel. At one point, a man named Campbell says to Corcoran, “I'm a busy man. I have the whole day scheduled to sit here on my bony ass and listen to Frank Poole bullshit about the good old days. Let's get out of here before that old windbag shows up.”

No wonder HAL 9000 killed Frank Poole when he went outside the Discovery to repair the AE-35 communications gyro.

Fill in the blanks

Template tests

I was flummoxed. Under normal circumstances, algebra students abandon the complete-the-square technique for solving quadratic equations as soon as they meet the quadratic formula. It is by a significant margin the least-favored of the solution techniques, trailing badly after formula and factoring.

Why, therefore, were so many of my students diligently completing the square when they didn't have to? Even worse, they were doing it on an exam problem, when time is at a premium. Worst of all, they were completing the square to solve a quadratic equation where its use was clearly contraindicated! I was at a loss.

As you may know, the solution of the quadratic equation is the great pinnacle and climax of your traditional introductory algebra class. The end of the semester wraps up with the astonishing revelation that one can now solve any quadratic equation. No exceptions! Such universality is rare, and I try to engender a little appreciation in my students for so powerful a conclusion, the big finish of Algebra 1.

Of course, I also try to get them to approach quadratic equations thoughtfully and methodically. First of all, does the equation factor easily? Then go for it! Is it (or does it appear to be) prime? Then one can apply the never-failing quadratic formula or—in certain specific cases—resort to completing the square. The specific case, naturally, is one in which the quadratic polynomial in question is monic (has a lead coefficient of one) and possesses a first-degree coefficient that is even (making it easy to take half of it and square the result, as required for completing the square).

Otherwise, don't even think of completing the square.

The problem that was puzzling me was monic, all right, but its middle term had an odd coefficient, making it a quite unsuitable candidate for square completion. Why, then, did so many of my students plow right in and start juggling fractions and slogging through more and more complicated expressions? They didn't know and couldn't tell me why they had done it.

The reason finally came to light while I was paging through my collection of quiz keys. I paused to consider the quiz containing the combined-work problem (or “joint effort”—computing the time a job takes if two or more people pitch in and you know how long it takes each person to do the job alone). This was exactly the kind of problem that had caused so much square-completion grief on the exam.

I noticed that I had solved the resulting quadratic equation on the quiz's solution key by completing the square. The polynomial had been monic with an even linear coefficient, so completing the square gave a quick and easy solution ...

... and my students had learned the lesson that combined-work problems are solved by completing the square! After all, the teacher had demonstrated this in a quiz solution key that he had posted on the course website. Did he not constantly encourage them to emulate his example? Follow his lead? Write solutions like he did? Indeed! Indubitably!

Damn.

They learned a lesson I wasn't teaching. They had studied my solution to a particular combined-work problem and then followed it slavishly when next they encountered a problem of the same type—even though the resulting quadratic equation had different characteristics and argued for a different solution technique.

I failed to banish the template problem. My fault!

You know what a “template problem” is, don't you? I'm sure you do. Lots of books are full of them. It occurs when a section of the text presents a carefully worked-out problem in Example 1, you turn to the homework section, and Exercises 1 through n follow the prompt “See Example 1.” And then all of the problems are exactly like Example 1 except that the numbers got tweaked a little. Or maybe Example 1 was a word problem about Sally and Exercise 1 is about Sam. Trivial changes. You can copy the solution of Example 1 as a template and go through filling in the old numbers with the new numbers.

Hardly any thought necessary.

I don't want to be too harsh. Routine drill problems are useful for building basic skills. They are, however, too bland for a steady diet and do not do much (if anything) for building conceptual understanding. Students, however, often prize them for their dull predictability and lack of challenge. They even ask for more, as when they beg for a “practice test” before a big exam. The most favored practice tests are those full of templates for the real thing. Woe betide the instructor who gives in to the pleas for a practice test and then changes the problems too much in the actual exam! Students will feel betrayed.

I refuse to give practice tests. I decline to channel my students' attention too narrowly to specific kinds of problems solved in specific kinds of ways. I want them to consider each problem independently, with a minimum of prompting, examining their knowledge of solution tools and picking the most appropriate one to apply.

The complete-the-square affair demonstrates, I'm afraid, that I have discouraged template thinking less than I had hoped. Perhaps I should ask my colleagues how they avoid it and then do exactly what they do....

It figures

Or perhaps it doesn't

I'm still disappointed when it occurs, but I'm no longer surprised. Sometimes, such as when I give an exam on the last day before spring break, I send out a grade update via e-mail so that my students don't have to wait till school resumes to find out their status in the class. My report, which pops up in student e-mail, presents the latest grade distribution in descending order. The closer to the top you find your secret student ID number, the better off you are.

I also provide the weighted components that go into computing each semester score (and grade): homework, quizzes, and exams. I present averages rather than individual scores, and therein lies the rub. Students write back when they receive the grade report and ask, “What was my score on Exam 5?”

Let us consider this. What does the student have in hand?

The student has his average exam score: the grades on Exams 1 through 5 all added together and divided by 5. The student has his old exams, numbers 1 through 4.

How on earth is an algebra student supposed to figure out the unknown value of his score on Exam 5? It is a puzzlement, is it not? If only they had a better teacher, perhaps they could do it for themselves, but I'm afraid the classroom door is a portal to real life, beyond which nothing in the classroom has any relevance. It's not as though the stuff I teach them can actually be used for anything! (Not even for classroom-related applications!)

I recently responded to an inquiry from a student who was earning a B:

Didn’t you realize you could have computed it yourself? You have your average exam score from the grade distribution I sent out. Multiply your average exam score by 5 and then subtract your scores from Exams 1 through 4. What’s left is your Exam 5 score.

He gave me a cheery reply:

I should of know but thanks I'll make sure I put that in my notes.

I'm thinking of forwarding that to his English teacher.

Boycott Ellen!

In for a Penney, in for a pound

Ellen Degeneres and JC Penney have mortally offended me!

No, I'm not talking about that silly whining from the harpies at One Million Moms (who are no better at counting than they are at living in the 21st century). My objection is to the mathematically inaccurate television ad in which Ellen dons 19th century garb and asks a milliner the price of a hat. When the lady informs her that the hat costs “fourteen pounds and ninety-pence,” Ellen responds with, “Okay, so fifteen pounds.” The lady firmly disagrees, but Ellen persists and finally gets her to admit that the stated price is as good as fifteen pounds.

Not!

The British pound was not divided into 100 pennies (the “new pence” of 1971) until the 20th century. Before that, a pound was divided into 20 shillings, each of which was worth 12 pence. If you do the math, that's 240 pence (old pennies) to the pound. If 19th-century English hat shops had been in the habit of shaving off a penny to make prices look lower, a one-penny reduction in a hat costing 15 pounds would result in a price of 14 pounds, 19 shillings, 11 pence—or £14/19/11 in the notation of the day. My penpal in Birmingham (England) used to send me letters in the 1960s whose stamps were labeled in pence, e.g.,  4d (“d” was reserved to the old penny and was replaced by “p” when the new coinage was introduced).

My trust in Ellen is shattered and I will never again take her advice on matters monetary.

I've half a mind

Bad teacher!

When it's early in the semester, I tend to cut my students a little more slack. Of course, I expect them to pay attention when I explain why I take off points for some calculations that manage to produce correct answers. For example, how many minutes does it take you to travel 12 miles at 18 miles per hour? Here's what one student told me:

Yeah. Well, I'm really not happy with that. Sorry, but 12/18 is simply not equal to 40. Equality is supposed to be a transitive property, folks! Of course, this could be redeemed with the appropriate use of unit conversion:

This I like. Careful use of units is a powerful way to keep one's calculations in order and to make sense of the results. Full marks! But then you get the woefully calculator-dependent student who presents this travesty:

Heck, you can keep your puny old leap-seconds! My students can conjure up a dozen seconds out of the thin air of feckless rounding. This is a particular gripe of mine. You actually need to grab for a calculator to compute two-thirds of sixty? Good grief!

Thoughtless calculations like these were sprinkled throughout the early semester quizzes and exams. But the pièce de résistance came in a different problem. One that had nothing to do with rounding. I gave my students (gave them, mind you) some volume formulas. All of the most popular shapes were there: cone, cylinder, sphere, box (ahem! Sorry. I mean rectangular parallelepiped, of course). The formulas were actually written out on the assignment sheet. I then asked my students to use the formulas to compute the volumes of some specified shapes. One of the shapes was a hemisphere.

Sure enough, several students decided the formula for a sphere was the best match they could make, computed the result, and ended up with an answer that was two times too big. Arrggh! Naturally, I took off points for that mistake. One of my students waxed indignant when he got his paper back and issued a two-part complaint: (a) I had not given them the formula for the volume of a hemisphere and (b) I had not done an example in class where we had to divide a result by 2 to get the correct answer.

I offered a plea of “no contest” to both charges. They were irrelevant. I patiently explained: “I have higher expectations of my students than merely plugging mindlessly into formulas. I want my students to think about what they're doing. This is not just a plug-in and grind class. Sorry.”

But not very.

A grade goeth before a fall

And it's all my fault

While I doubt it registers with my students, I am at pains every semester to explain to them that they earn grades. I do not merely give them. Unfortunately, the students who most need to hear this message seem to be the least likely to retain it.

I recently taught an algebra class in an accelerated format. Students were warned at the outset of the course's brisk pace and the need to work diligently to stay abreast. The faint-hearted quickly folded their tents and stole away. The braver students stuck it out to the end—a bitter end for a few of them. Overall, though, the success rate was over 80 percent. I was happy that so many of my students passed the class.

One of the students was less than enamored with her “success.” Yes, she passed the class, but she passed it with only a C after having spent most of the semester at the B level. She had spectacularly flunked the comprehensive final (earning fewer than half the possible points on it) and her average plummeted. I declined to award a B to a student who couldn't even earn a D on the final exam. She called me up to complain at the injustice of the result.

Her particular complaint focused on what she perceived as the inequity of students getting a C grade with composite semester scores of 68.5 while she was being denied a B despite a composite score of 78.5. Why did I “round up” the scores near the C-D boundary but not hers at the B-C boundary?

Several factors influenced my decision. First of all, the C-D boundary is basically academic life versus death. A grade of D forces you to repeat the course for credit. I give very close scrutiny to the scores of all students teetering on the precipice of the C-D divide. Furthermore, the three students in question had all beaten my complainant by several points on the final (and the weakest of the three was in the enviable “hammock” position). Unlike my former B student, they had not used the final exam to demonstrate utter confusion and lack of subject-matter retention (a consideration of some significance in a prerequisite course like algebra).

Then, of course, there's the other tiny factor: Among the students with passing grades, the student in question had one of the lowest participation rates in the quizzes that I used throughout the semester to gauge my students' progress. To be fair, it was not chronic absences that caused her to miss so many quizzes (although her attendance did suffer near the end of the semester). No, it was her refusal to submit her paper to me when I collected them, even when I made a point of asking her directly. “No,” she'd say. “It's no good.” Brimming over with sweet reason, I would explain, “Five points on a ten-point quiz may be a little embarrassing, but five points in the grade book is significantly better than zero points!” She'd shove the crumpled quiz into her binder and resolutely refuse: “No, I don't want you to look at it. It's no good.”

In the end, she withheld or missed over twenty percent of her quizzes. A series of truly bad decisions. Those points were not there to reinforce her against a bad result on the final exam, which turned out to be a significant matter in the end. I guess her problems weren't just in algebra.

Even lotteries have winners

Lucky Larry hits a grand slam

It has been noted that state lotteries are basically a tax on innumeracy. You're better off dealing with the house percentage at a casino in Nevada. Nevertheless, even lotteries have winners. Just don't expect it to be you. Lightning is not going to strike you.

Of course, sometimes it strikes near by.

One of my students won the calculus lottery during the exam on integration, beating very long odds indeed. The result was the most bizarre “Lucky Larry” of my years as a math teacher. My colleagues were as flabbergasted as I was when I shared the student's “solutions” with them. Her work was nonsense, yet her answers were correct. Three times in a row. Of course, when that happens one suspects a hidden underlying pattern that produces valid results, contrary to all expectations. In this case, though—no. It was a giant fluke.

Or, rather, three flukes in a row. My flabber, she is as gasted as possible.

The problem on the integration exam was one of my “conceptual” exercises. One of my tasks as a calculus teacher is to clarify the meaning of the definite integral, ensuring that my students grasp its significance. Of course, one of the most common (and visual) interpretations of the definite integral is as the area under a curve. Surely any first-year calculus student must understand at least that much.

Accordingly, I presented my students with the graph of a simple function and asked them to evaluate three definite integrals of that function by inspection of the graph. I did not forbid them to use antidifferentiation and the fundamental theorem of calculus, but I emphasized that the simplest of calculations would suffice.

What, pray tell, is the value of the definite integral of f(x) from x = 1 to x = 2? A cursory examination of the trapezoidal region spanning the space between the x axis and the graph of the function reveals the area (and thus the definite integral) to equal 1.5. Easy! Not satisfied, however, with such a trivial computation, one of my students rolled out the big guns:

Damn! What a coincidence! The calculus is bogus, but the result is accidentally correct. No one expects lightning to strike twice, of course.

Brace yourself.

What if we ask for the definite integral from 1 to 3 instead? We get a little more area now. Take a look at the new graph, in which a second trapezoid now joins the first. We get an additional 2.5 square units which, added to the original 1.5, gives us 4. My student swung into action and unlimbered her surreal calculus calculation again:

I was now quite beside myself, shaking my head in astonishment as my red pen hovered over the page. Twice in a row! (What were the odds?)

Fortunately, I knew that I could count on part (c) to set the record straight and demonstrate to my student the error of her ways. It was, in fact, the simplest part of the problem. A kind of gift to the student possessing a clue. Can you find the area of a rectangle measuring 2 by 4? Of course! The answer must be 8.

My student presented her solution:

Nooooooooo!

Time to hit my head against the desk a few times.

In a million years, this will never happen again. (For one thing, this problem is going straight into the waste can, never to be recycled.)

I need to go lie down for a few minutes.

A moment's reflection

Another conceptual understanding problem

I gave my algebra students a pretty little problem involving the graphs of functions and their inverses. The prompt was fairly simple:

The graph of y = f(x) is shown in the figure. Use the graph to find the following function values and then sketch the graph of the inverse function y = f ⁻¹(x) on the same coordinate grid.

The student was asked to find the values of f(−3), f(1), f ⁻¹(2), and f ⁻¹(10). As you can see from the graph, I conveniently provided my students with several points highlighted on the graph. If one examines the point on the function curve where x = −3, it is fairly easy to discern that y must be 2. Hence f(−3) = 2. Similarly, f(1) = 10. It's elementary graph reading.

After reading the initial two function values, I expected my students to discover the method in my madness, noting that I'm asking them to figure out the value of the inverse function for the input values 2 and 10, which were the initial output. Since the inverse function, by definition, maps in the direction opposite that of the original function, it immediately follows that f ⁻¹(2) = −3 and f ⁻¹(10) = 1. What could be simpler?

Apparently, lots of things. Some of my students were quite irked:

“You didn't give us the function.”

“On the contrary. I certainly did. Its graph is right there before you.”

“No, I mean, you didn't give us the formula. We can't figure out the inverse function without the formula.”

“Leave that for a moment. Can you do the first part of the problem? Can you find the value of the original function at x = −3 and x = 1?”

“No, I already told you: You didn't give us a formula to plug into.”

“I recommend you try looking at the graph a little longer.”

In a few variations on the above theme, the querulous student suddenly lit up and rushed back to his or her desk to fill in the answers. In other cases, the student instead sat down, head shaking, and appeared to be muttering sotto voce imprecations at the instructor's expense.

Later, of course, when the exams were returned, I demonstrated what I had expected them to do. Since most of them had memorized the procedure for computing an inverse function—switch x and y in the formula y = f(x) and solve for y—they should have realized that the presence of the point (1, 10) on the graph of the original function implies the presence of (10, 1) on the graph of the inverse. Previously perplexed students rolled their eyes: “Oh, is that all? Why didn't you say so?”

I thought I did.

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.

Garfield does math

And so do we

I always look at the comic sections of the newspapers I read, but I don't necessarily look at all of the comics. “Pearls Before Swine” always gets my attention, as does “Bizarro,” but others need to do something special to draw me in—like sprinkle their panels with numbers. “Garfield” did exactly that yesterday. (Is it true, as Stephan Pastis says, that cartoonists prefer to bury their weakest efforts in their Saturday strips?)

Everyone realizes, of course, that a giant mutant 98-year-old lady would be physically impossible, despite such earlier documentary evidence as Attack of the 50 Foot Woman. Galileo's square-cube law should have put that notion to rest (but Hollywood prefers to honor that law in the breach). But let's allow Garfield the same leeway that movie producers get. Let's accept that a giant 98-year-old lady is driving her 32-story 1965 Bonneville into town, threatening the entire community.

The 1965 Bonneville was a gigantic (in its own way) vehicle over 18 feet in length. Its height was about 4.5 feet (with allowances for tire pressure and passenger load). In the comic strip, the giant old lady's Bonneville is said to be scaled up to 32 stories in height. While architects are allowed quite a bit of variation in what constitutes a “story,” we can use 10 feet as a reasonable mid-range measure. In other words, the giant old lady's car is 320 feet tall, or (divide by 4.5) over 71 times as tall as a regular Bonneville. That's big.

And if your 98-year-old great-grandmother is five foot two, she'd be nearly 370 feet tall if she were scaled up to be the little old lady in the car.

Scary!

Now, about that turn-signal thing. Garfield says it's 16 feet tall (and blinking, of course). A look at the back end of a '65 Bonneville shows us that the rear lights were not quite half as tall as your basic license plate. If we call it 4 inches (being just a little generous—I don't have a Bonneville handy to actually measure), scaling it up by a factor of 71 results in 284 inches—or nearly 24 feet.

But Garfield said 16 feet. Oh, oh! But you know, that's probably good enough for the funny papers. Let's give him this one.

You are right, I guess

And I'm right: you guess

The aftermath of the semester's first exam is often a teachable moment. I frequently assign my students to analyze their results. This usually comes in the form of a two-part prompt, to which I want a written response: (1) What kinds of mistakes did you make? (2) What steps will you take to minimize these mistakes on the next exam?

Most of the responses are dominated by the usual litany of math's most persistent errors and shortcomings:

• I misread the problem.
• I made a stupid mistake.
• I used the wrong formula.
• I made a calculation error.
• I didn't study.
• I didn't do the homework.
• I need to catch up.

The proposed remedies are as predictable: More study. More answer-checking. More diligent attention to homework. More visits to office hours or tutors. All good ideas and apt to be helpful if actually applied.

Occasionally, however, I get the whiny response from someone who is looking to place the blame elsewhere. Why not engage the instructor's sympathies by explaining to him that he is to blame? Most students avoid this approach, but sometimes you get a brave one:

After looking to see if I had done the problem right in which case it was correct but the only thing that I had over-looked was the correct notation.

Ah, yes. Notation. I may be a little stricter about notation than other math teachers, but I refuse to countenance false statements like

4x + 3 = 11 = 4x = 8 = x = 2.

I'm just not crazy about taking the equal sign in vain. Putting an equal sign between things that aren't equal is irksome, sloppy, and—darn it!—untrue.

In the present instance, the student was taking a calculus class and had presented me with solutions that were mostly bits of scratch work and the occasional untrue statement. For example,

6x + 3h − 5 = 6x − 5

is a false statement unless you indicate that you are taking the limit of the left-hand side as h goes to zero (if you would please be so kind). The student got most of the credit for deriving the correct answer, but he lost a few for neglecting correct notation. His tone was a bit pettish, but he came to a correct conclusion in his analysis:

Overall, I think in order to improve myself as a math student in Dr. Z's class, I need to focus on how he wants me to solve or work out the problems so I can meet his expectations. Because it seems to me that I do the work as best as I can but fall short of what is expected of me from him. So my best solution to this dilemma is to find out how he wants things done and pretty much follow his rules in order for me to get an A in his class.

A helpful hint: The best way to find out how I want things done is to watch what I do in class, because I model it in every example I do and in every homework question I solve for the class. And—one more hint—be there when I do it.