Absolute confusion

(Don't) [Fence] {Me In}

My prealgebra students were confused. It is their natural state. On this occasion, they were having an unhappy time with notation. What was the deal with all of these grouping symbols—parentheses, brackets, braces—and what are they for? I pointed out that it was customary in writing mathematical expressions to use difference “fences” when indicating groupings within groupings. If an expression already contains parentheses, one might choose to use brackets to indicate any higher level of grouping; for example,

[4(x + 5)]²,

in preference to

(4(x + 5))².

I wrote the usual symbols up on the board: ordinary curved parentheses (), square brackets [], and the less common (except for set notation) curly braces {}.

Not all of the students were satisfied. One wanted to know the difference between −|−5| and −(−5). Why had I simplified the first expression as −5 and the second as 5? I reminded the class that the vertical bars were the absolute value symbol and had to be taken into account, while the parentheses were used for clarity rather than for functional purposes in the second expression (not, of course, that I phrased it in quite that way).

My perplexed student seemed to grasp the point, which she tried to confirm by rephrasing my remark, thus feeding me an irresistible gag line:

“So it's different when you're inside straight bars?”

“Well, certainly different from being inside gay bars.”

Ha!

A qualified success

Know thyself

Two brothers took the math placement exam last fall and received scores that suggested they were qualified to take prealgebra. However, the scores were at the low end of the prealgebra range and the boys decided to enroll in arithmetic instead. It seemed like a prudent decision. Unfortunately, they didn't take the arithmetic class seriously, their attendance was spotty, and they ended up with bad grades (one D, one F).

How do I know all this? Because they told me. And because I was their arithmetic teacher. On a number of occasions I suggested that they ought to come to class more regularly and earn a few more points. They thanked me effusively for my good advice and then ignored it. I don't think anyone was surprised at the outcome, although the boys seemed just a little bit disappointed, as if they had retained some hope that lightning would strike and they'd get passing grades. They weren't nonplussed (unless “nonplussed” means not sure what to do with an addition symbol).

It was a small surprise to see the brothers in my prealgebra class on the first day of the new semester. I sought them out and reminded them about course prerequisites. They were not taken aback. Instead, they proceeded to fill me in on their experience with the math placement test:

“We could have taken prealgebra last semester based on the results, but we thought we should probably take arithmetic first, even though we got a good score on the arithmetic part.”

“So why didn't you get a good score when you actually took an arithmetic class?”

“That's a good question, Dr. Z.”

But they didn't have a good answer.

“What do you plan to do, since you didn't get a C in arithmetic?”

“Uh, we're going to take your prealgebra class. We still have our placement scores from last semester and that qualifies us to take prealgebra.”

Perhaps they weren't nonplussed, but I was.

“And you think that will work?”

“Oh, yeah,” said one, while the other brother nodded his head.

Oh, boy. It's going to be a fun semester.

Running numbers in Reno

Counting on the caucus

PiD decided to participate in the Nevada caucuses last weekend. He and his wife went to support John Edwards. The experience was, in a manner of speaking, quite educational. PiD phoned me Sunday to recount their adventures. Innumeracy was a big part of it.

The organizers soon found themselves facing a challenge: counting the participants. The leaders of the Clinton, Obama, and Edwards factions decided to go about the room pointing at people, who were supposed to count themselves off. Complications arose when the triumvirate did not always point at the same person. One trouble-maker suggested having each table of participants report its number to the leaders, who could then add up the results. This was deemed too complicated, as well as relying on the honesty of each table. The leaders persisted in the point-and-count technique. After three tries, they finally decided there were 146 attendees.

Then came a new problem, also numeric. The threshold for caucus participation by a candidate was 15% support among the attendees. Most people were in the Clinton or Obama camps, but the Edwards people faced possible elimination of their candidate. But how to compute 15% of 146? While some folks yelled out “twenty-two,” the leaders conferred in nervous agitation, trying to puzzle out the computation algorithm included in the caucus guidelines. “No, we have to be sure,” they said.


PiD and other participants were getting exasperated, but it appeared that computing a percentage was a major problem. Some people tried to explain: “Ten percent of 146 is 14.6. Add half again to get to fifteen percent: 14.6 + 7.3. You get 21.9, so the result is 22!”

The explanation didn't help. The leaders finally obtained a calculator and worked through the math several times. It was finally announced that 15% of 146 was 22. There was much rejoicing. (Well, not really.)

Although their ranks had been reinforced by a few Kucinich stragglers, the Edwards cohort did not reach the 15% threshold. The Clinton and Obama leaders pounced quickly: “Okay, you have to choose sides. You have to pick Clinton or Obama.”

The edict was not well received. PiD said, “What if we don't want to?”

“But you have to.”

Diplomacy was not the order of the day, but the Edwards people did understand that their candidate was out of the running in the day's caucus and that they were up for grabs.

“Okay, so why should we pick one candidate over the other?”

Although that might have sounded like an invitation to offer some persuasive arguments to attract the Edwards supporters, it was instead taken as an excuse to engage in mindless chanting. The Clinton forces yelled “Hillary!” over and over while the Obama troops shouted “Obama!” The Edwards cohort resisted the compelling pitches. Eventually the leaders realized it was time to talk. PiD listened to their paeans to their preferred candidates. The Clinton leader was not doing her candidate a lot of good. PiD had met her the day before when she was walking precincts to drum up caucus attendees. One of her arguments was that Hillary was not “as liberal” as her husband. That was sure to confirm PiD in his support for Edwards. Fortunately, the Clinton leader did not offer her opinion about Clinton's relative liberality again and the Obama leader was a weak advocate for his candidate.

PiD ended up moving to the Clinton side of the room while his wife took up with the Obama forces.

It was time for the calculator again. The leaders discovered that a couple of caucus attendees had slipped away. They counted up the the tally anew and determined that the Clinton forces comprised 59% of the remaining caucus participants. Someone pointed out that this meant Obama had 41%. The leaders balked. No, they had to be sure. They computed the numbers for Obama and seemed surprised to get 41%. Imagine that!

I asked PiD whether he had found the three-hour caucus experience rewarding. He told me that his first caucus would definitely be his last.

Believing impossible things

It's an act of faith!

A colleague who teaches physical anthropology was telling me about her recent experiences with creationist students. She's a senior faculty member and has seen just about everything. Recently, however, she's noticed something new:

“I have always told them it's their right and privilege to believe what they want to believe. To pass my class, however, they have to learn about natural selection and be able to answer the questions on the chapter test with the scientific evidence for evolution. There's always a few who insist on writing ‘This is not true and I do not believe it,’ but now I'm finding that some students actually skip class during the unit on evolution and simply take a zero on the exam. When they return to class and I ask them why they cut class, they say, ‘I believe the Bible and my religion is very important to me.’”

Having been raised Catholic, I can't help remembering my childhood catechism classes and the emphasis on avoiding “the near occasion of sin.” While the Protestant fundies in my colleague's physical anthropology class would recoil in horror at the comparison, surely they were trying to spare their delicate faith from the slings and arrows of actual scientific evidence. For them, the chapter of evolution is a “near occasion of sin.”

I asked the anthropology professor, “They think it's sinful to learn about the fossil record and transitional forms?”

“They don't believe in them,” she said, “so they think it's better to just take the zero.”

I could have added, “and just avoid temptation.” They're like the prelates of Catholic legend who feared to look into Galileo's telescope lest their eyes lead them astray. They know the truth, please not to molest them with inconvenient facts.

Earlier this week (Tuesday, January 15, 2008) I heard an installment of Southwest Radio Church's weekday program. The last few minutes of the broadcast featured a question-and-answer exchange between announcer Jerry Guiltner and resident apologist Larry Spargimino. The Q&A opened with a criticism of rival Christian exegete Hank Hanegraaff, who had claimed it was absurd to take too seriously the notion that stars will actually fall from the skies in the last days before the second coming of Christ. Hanegraaff called it “allegorical language”:

To suppose that stars are literally going to fall from the sky is nonsense. One star alone would obliterate the earth.

---

But the Bible scholars of Southwest Radio Church are made of sterner stuff:

Guiltner: Brother Larry, do you agree that belief that stars are going to fall from the sky is nonsense?

Spargimino: No, Jerry, I don't believe that it is nonsense. We have to beware of basing what we believe on what is possible.

In Joshua chapter 10, verses 12 and 13, we are told that Joshua commanded the sun and the moon to stand still and they actually did. Are we to say that this never really happened, and that it is a lot of nonsense because of the drastic tidal and climatological effects that would ensue?

Are we to deny that there ever was a literal talking snake in the Garden of Eden who addressed Eve, because snakes don't have vocal organs and therefore can't speak?

Are we to argue against the virgin birth of Christ because virgins don't normally conceive?

And what about the star followed by the wise men from the east in Mathew 2:2? Was that really not a star, but simply a reflection off of the Sea of Galilee or something of the sort?

On the basis of such reasoning, Mr. Hanegraaff would have to call the bodily resurrection of Jesus Christ from the dead nonsense.

---

Precisely. Except that Hanegraaff is as much a believer in the resurrection of Christ as anyone at Southwest Radio Church. Spargimino has, however, put his finger right on the key problem. If you pick and choose what to believe, instead of believing it all, you run the risk of becoming a rational person. Although Hanegraaff is a miracle-believing, evolution-denying Christian, he has set himself the task of trying to separate the grain from the chaff. In the battle for the survival of the fittest Christian, he will always be at a disadvantage in the religious competition with those whose faith has no vulnerable chinks through which rational thought could seep in.

My colleague's creationist students would undoubtedly applaud Dr. Spargimino and jeer at Mr. Hanegraaff. If, that is, they dared listen to someone who at least tries to apply reason.

Before and after Bradley

Courting the racism vote

Hillary Clinton's upset victory over Barack Obama in the New Hampshire primary has some pundits muttering about the Bradley effect, the possibility that a significant number of white voters lied to pollsters about their willingness to vote for a black candidate. It's named after Tom Bradley, the former mayor of Los Angeles, who was pipped at the political post in the California gubernatorial election in 1982, despite having led his opponent by a significant margin in the polls immediately preceding the balloting. Thus did California miss its chance to elect the first post-Reconstruction African-American governor in history.

Since it was a surge in Clinton's vote share rather than a big drop in Obama's pre-election strength that led to the New York senator's triumph, it seems unlikely that the Bradley effect was at work in the Granite State. Nevertheless, racism sometimes lurks under the radar and in especially egregious cases politicians try to appeal to it.

That was Mayor Sam Yorty's ploy in 1969, when City Councilman Tom Bradley nearly defeated him outright in the mayoral primary election. Yorty pulled out all the stops in the run-off, warning the white citizens of Los Angeles that black power was threatening to take over city hall. The smear campaign worked and Yorty won one more term. Defeated but not despairing, Bradley regrouped and came back to oust Yorty in 1973, inaugurating a distinguished two-decade tenure as mayor of Los Angeles.

It's possible, therefore, that Bradley experienced the baneful impact of the so-called Bradley effect in two separate elections. That first defeat in 1969 perhaps suggested to some people that California's white voters were still sufficiently prejudiced against minority candidates to be exploited for political advantage. This lesson was taken to heart the very next year when the state superintendent of public instruction ran for re-election.

Max Rafferty was a harbinger of the future right-wing era. He had been elected in 1962 to California's highest elective educational post by harping on the evils of progressive education and espousing the “back to basics” movement. He spent his time in office railing against liberals and trying to ban textbooks and reference materials of which he disapproved. Rafferty also harbored further political ambitions. In 1968 he ran for the U.S. Senate, denying incumbent Thomas Kuchel renomination in the Republican primary. However, the abrasive and combative campaign he had run in the GOP primary did not work for Rafferty in the general election, which he lost to Alan Cranston.

Since he was in the midst of his second term as state superintendent of public instruction when he lost the U.S. Senate race, Rafferty has time to lick his wounds and prepare to seek the consolation prize of another term as education chief. He began the 1970 campaign as a strong favorite. However, Rafferty's senate campaign had damaged his image as an educator in a nonpartisan elective office. He drew opposition in the primary election, one of them his deputy superintendent. Wilson Riles decided to seek the top post himself, as did educator Julian Nava. Neither one came close to defeating Rafferty in the primary, but their combined vote managed to deny Rafferty a majority vote by the thinnest of margins. The election would have to be decided with a run-off vote in November between the top two candidates, Rafferty and Riles.

The aftermath of the 1970 primary election was one of the most racist campaigns in the latter half of the 20th century. It lacked only the overt, explicit race-baiting rhetoric of such politicians as George Wallace, Lester Maddox, or Orval Faubus. It was a discreet sort of racism, but it was scarcely subtle. Rafferty's campaigns paid for a series of newspaper advertisements that contrasted his record with his rival's, much to the incumbent's advantage. That, of course, was to be expected. The unusual part was that every Rafferty ad featured pictures of both the incumbent superintendent and his opponent. Rafferty appeared as a smiling, open-faced white man, looking as avuncular as could be. Wilson Riles, by contrast, was difficult to see. His portrait was rendered in such dark tones, exaggerating his actual complexion, that only gleaming teeth and spots of white in his eyes could be distinguished from the black blob that was his face. The message was completely obvious: “Vote for whitey and beware the darkie!” Rafferty's campaign people were too genteel to include the actual words, but the words would have been redundant.

Despite the incumbent's best (or worst) efforts, each California Poll published between the primary election and the run-off showed Riles building his share of the vote while Rafferty continued to fall short of a majority. On election day, Wilson Riles was elected as the new state superintendent of public instruction by a vote of 54.1% to 45.9%. It was the end of Rafferty's career in elective office.

The rejection of Max Rafferty's unsubtle appeal to white racism in 1970 was a hopeful sign that the state electorate was outgrowing old prejudices. It was a moment of triumph and satisfaction, but it was also premature. The election of 1982, for which the term Bradley effect was coined, still lay in the future. And we still have plenty to learn.

Creative parasitism

Answers raises more questions

Answers in Genesis is still basking in the afterglow of the successful launch of its Creation Museum, attendance records having exceeded the creation ministry's original projections. (These people seem to be wrong about everything.) How best to take advantage of AiG's higher profile? Ken Ham knows what to do: As long as AiG can profit from pretending to be a scientific organization, why not embellish the charade with a “professional peer-reviewed technical journal”?

I'm sure the key word in that phrase is “peer.” Creation “scientists” tend to be kind to each other. The review process will undoubtedly be extremely gentle.

Funny thing about the new Answers Research Journal: It's Ken Ham's second rip-off of Creation Ministries International, the Australia-based organization with which Ham's Answers in Genesis used to be affiliated. A couple of years ago Ham bamboozled CMI by using its Creation magazine subscriber list to solicit sign-ups for AiG's new Answers magazine. With a breezy disregard for the commandment about not bearing false witness, Ham told Creation subscribers in the U.S. that the magazine would no longer be available (implying it was ceasing publication), and the slick new Answers magazine was being launched to replace it. Nowhere were Creation's subscribers told that CMI was continuing to publish the magazine, because that would have defeated Ham's plan to seize Creation's entire U.S. subscriber base as he detached AiG from the CMI parent organization.

You can anticipate the sequel: Yes, CMI also had a “professional peer-reviewed technical journal.” Originally called the Creation Technical Journal, CMI's “scientific” publication is now published under the name Journal of Creation. Since Ken Ham presumably no longer has access to CMI's subscriber database, his launch of the rival Answers Research Journal will perforce be a more legitimately competitive move than his sly double-cross in the case of Creation versus Answers. Perhaps he would like to be more underhanded, but the opportunity has slipped away.

Pythagoras gets pulled over

Applications of mathematics

Jason arrived several minutes late for our precalculus class with a smug grin on his face. It wasn't unusual for him to be late to class. It wasn't unusual to see him with a grin on his face. The combination, however, was decidedly peculiar. Jason's preferred mode of late arrival involved ducking into the back of the classroom as surreptitiously as possible, a rueful expression on his face to show he harbored dutiful feelings of guilt over his tardiness. This day, by contrast, he seemed quite pleased with himself.

He was so pleased with himself, in fact, that he could not resist sharing the explanation with me at the end of the period. The smile was still on his face when he sauntered up to the front of the room:

“Sorry I was late, Dr. Z. I got pulled over on my way to school.”

“Pulled over? Were you speeding?”

“Yeah, well, that's why I got pulled over, but the cop changed his mind.”

“That's a surprise. The CHP isn't known for backing off.”

“For sure true, but he had to. I used math to prove I wasn't speeding.”

“This I have got to hear. What happened, Jason?”

“Well, I was getting close to the freeway exit for the college when a black-and-white starting flashing red at me, so I pulled over. The patrolman came over and asked me if I knew how fast I was going, so I said, yeah, I was going at the limit. He started to tell me I was wrong, but I said, ‘I can prove it!’”

“This is going to be good,” I said to my student. “How on earth did you think you could do that?”

“I used the Pythagorean theorem and vector addition, of course. I told the cop that I was going exactly 55 miles per hour when he flashed me, but he thought I was going faster because I was also moving sideways toward the exit. I drew a speed diagram.”

“You mean a velocity diagram.”

“Whatever. It was good enough for him. I showed him that he thought I was going too fast because he was looking at the long side of the triangle.”

“Hypotenuse.”

“Yeah, right. I said ‘hypotenuse,’ too. I think he was impressed. I told him it was unfair to ticket me for speeding when my forward motion was legal and that's what really counted. He started thinking about it and I asked him if it was okay if I ate some pizza and did he want some. I had takeout in my car to eat for lunch in the parking lot before class and it was just getting cold while we were sitting on the shoulder. He said, yeah, it was time for his lunch break anyway, so we sat there eating pizza and talking about math and speeding.”

I wasn't sure they really talked all that much about math, although I could believe that the patrolman was impressed by an excuse he probably hadn't heard before.

“So no ticket?”

“Yeah, no ticket! This math stuff can be useful! We got to talking and finished the pizza, so I didn't get to school when I expected. Sorry that made me late.”

He would have been late anyway, I'm sure, noshing on his pizza in the parking lot before class, but I was impressed by an excuse I hadn't heard before.

Once upon a time in 1960

48 years

This morning the newspapers are full of headlines and speculation concerning today's political caucuses in the state of Iowa. These meetings mark the first substantive balloting and delegate selection in a presidential campaign that already seems like the longest in history. That's why it's interesting to think about what people were reading in their morning newspapers on January 3, 1960, nearly half a century ago.

The day before a young senator from Massachusetts had announced his candidacy for the Democratic nomination for President of the United States. John F. Kennedy's announcement wasn't exactly a surprise. He had been positioning himself for some time as a contender. Nevertheless, his first overt action as a candidate was his formal declaration of candidacy on January 2, 1960.

Imagine if today's presidential candidates had waited till January 2, 2008, to announce they were in the race. Too late! The selection and election process has grown into the elongated gantlet in which we are now immersed, a political marathon that exhausts the voters as much as it exhausts the candidates. It makes you nostalgic for the good old days.

Of course, those days were not very democratic. Many delegates to the nominating conventions were selected by political bosses and other purveyors of patronage. State primaries were relatively few. In most respects, the system we have today is more open, albeit chaotic and confusing. Still, I think we could do better.

The master of time

Who was I kidding?

My students don't wear wristwatches. They all carry cell phones instead, which display the time and make watches unnecessary. Who needs a watch when you're plugged into the great electronic web of modern life?

I started wearing a watch in high school in the 1960s. It was a gold-colored (though not gold) Bulova with a rectangular face. A separate little dial marked off the seconds. It was easier to watch the clock in school when you could look casually down at your desk instead of craning your neck to peek at the wall clock behind you.

The clocks in my high school were highly synchronized. All of them were wired to the master clock in the administration building. They clicked as one. Perhaps the master clock had a sweep second hand, but the classroom clocks did not. Every sixty seconds, they audibly clicked from one minute mark to the next minute mark. The alarm bells rang when the minute hands clicked into the position that marked the end of the class period. Each click of the clock was preceded by a minuscule backstroke of the big hand, as if the timepiece were anticipating its imminent move. If you were close enough to the clock to see the tiny backstroke, you had an instant's warning ahead of your classmates when it was time to bolt for the door.

My wristwatch was a wind-up, not electric or self-winding. I had discovered that the tiny sweep second hand would stop moving if I put just enough torque on the winding stem to engage the internal mechanisms (but not quite enough to begin winding the spring). Soon I was using that information each morning to ensure that my wristwatch was precisely in sync with the high school's master clock. Thereafter, for the rest of the day, I knew exactly when each class period would end. I was not lost during the final minute of each session, dependent on the premonitory backstroke to prime me for departure. Soon my neighboring classmates noticed that I was able to give a completely accurate countdown. No one else was in such precise sync and seldom did anyone else think to note in advance the position of the sweep second hand on his or her wristwatch when the classroom clock clicked. Only I could give a countdown with split-second NASA reliability (or tick it off with the fingers of my right hand if the room had grown too quiet to permit murmuring it aloud).

It was, I'm afraid, the coolest thing about me throughout high school.

One afternoon I commented to my classmate Dennis that it would be neat if we could just skip ahead to preferred points in time. Say, Friday afternoons. Dennis shook his head in firm disagreement: “Life would be too short that way.”

I realized that Dennis and I were not looking at things the same way. While I was thinking of operating like the Time Traveller conceived by H.G. Wells, jumping to discrete points in time, Dennis was contemplating the more mundane alternative of squandering the bulk of one's life by being too fussy about when one should “be in the moment.” Do you really want to fast-forward your life?

Dennis later became an insurance salesman. I should have seen that coming.

During my college years several astronauts went tramping about the moon. Well into her seventies at the time, my grandmother shook her head as she watched Apollo crews on television as they drove moon buggies across the lunar surface: “What are they doing up there?” she asked. “I think they've been greasing the world's axle. Every year goes by faster. I'm not sure that was a good idea.”

Grandma was kidding. But not really.

I was a Caltech undergraduate during part of the Apollo program. During my junior year I lived off campus and would walk to school along California Boulevard. As I moved past Sloan Lab toward the student houses, I would come to a metal plate in the sidewalk. For some reason, I fell into a pattern of avoiding it—except on Fridays, when I would step on it with a satisfying clang. I'd think, “It's Friday.” On Fridays I often caught the bus for a weekend trip home. It was convenient to have my laundry done and my meals served by Mom while I also took advantage of having a quiet place away from school where I could do my studying and my homework. But it was also my first year away at school and I was often intensely homesick. In Pasadena you were always surrounded by buildings. At home in Central California, most directions offered you a view of the horizon. Fridays were the day I escaped the oppressive metropolis.

But sometimes the clang of my feet on the metal panel reminded me of Dennis's warning. Do you really want to fast-forward your life?

Add thirty-six years.

Last week, on Christmas morning, I groped through the cabinet in Mom & Dad's bathroom and found the can of shaving cream. It's not Dad's brand, so he never touches it. Nevertheless, it gets pushed aside during the weeks I'm away, so I always have to hunt it down. It has an orange cap, although the one I have at home has a black cap. Same basic brand, though. The can at Mom & Dad's is more of a Time Traveller than I am. Its life has been artificially prolonged beyond its normal expectancy of a month or two of useful service. It can provide perhaps forty shaves. I'm not sure. Since I use it only five or six time a year, its existence could be stretched to as much as six or seven years. Will the can I used on the morning of my nephew's wedding three years ago be the same can I use years from now on the morning of a parent's funeral? Six-year jumps are pretty big leaps in one's life. There's no guarantee that I will even finish the can. My flesh crawls as I shave.

Time whacks us all up alongside the head and chuckles at our pretensions. Or would, if it could. I'm sure I'm just anthropomorphizing. Nevertheless, no one masters it. We may, however, continue a guerrilla campaign of resistance. Next time at Mom & Dad's I'm going to grab that orange-topped can and bring it home with me. And use it up.

That'll teach it.