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Exercising the Quads

I needed to review the quadratic formula before addressing it in class, mostly to make sure I could work my way through it smoothly, but partially I must admit to be sure I could do it at all.

Turns out I can. No sweat. Compared to some of the gyrations of differential equations or network optimization, the math is straightforward as long as you grasp the method in the first place. Again, compared to more advanced math, the formula itself looks pretty simple.

But what a frankensteinian monster it appears to students still getting used to algebra! In order to isolate x in a quadratic, you must reduce a quadratic polynomial to a linear one. To do this, you must employ a square root, since dividing by x just leaves you with a troublesome 1/x floating about somewhere else. In order to employ a square root, you must have a proper square to work with. And since a general quadratic isn’t a proper square, you have to build one from spare parts. You do this by hacking off any bits that are too big to fit (the c), sewing on new bits (b squared over 4a) to fill in the missing quantities—quantities that seem arbitrary until you’ve peeked ahead at the answer—taking the square root of all the junk that’s accumulated in balancing the equation, subtracting, dividing, simplifying, and running electricity through the whole mess until it rises to terrorize the peasantry. Small wonder the final formula is so messy.

Not difficult, if you know where you’re going and keep a firm grip on why you’re going through all these operations: to reduce the quadratic to a linear expression with a square root. But arcane, even ridiculous, if you don’t really know where you’re going and why, which is all too often the case with math students.

2011 01 31 mdlake Teaching Comments (0) Permalink

Tunis Envy

Watching the pro-democracy upheaval in the Middle East is an uplifting experience. When Tunisia alone was convulsed with populist protests, the story seemed quaint and remote. When the protestors succeeded in toppling the reigning despot, it was hard to suppress a little “Good for them!” cheer. When neighboring countries’ leaders, employing virtually identical systems for staying in power, began looking about nervously, the story began to get interesting. And now that Egypt is grappling with the same democratic protests, well, “Friends, it may look like a movement,” to quote Arlo Guthrie.

Inspiring though it is, I find myself envying these countries—just a little bit. As the US, shining beacon of democracy, surrenders its freedoms to corporate interests and the terrorist witch-hunt, nations we traditionally look down upon as tin-pot dictatorships are putting us to shame.

Take Al Jazeera, for example. Initially, it sounded to me like a shill for Arab nationalism, but the more I listen, the more it seems to offer good journalism, sounding nationalistic only in the same way and to the same degree that the New York Times offers a Amero-centric (or New York-centric, or even Manhattan-centric) perspective. It takes its adversarial role towards government seriously, seemingly more seriously than American news, which increasingly merely reports he-said/she-said press releases from conservatives and liberals, because that’s cheaper and easier than finding out which side is telling the truth. Our journalists may still aspire to be Woodward and Bernstein, but Al Jazeera is actually going out and doing it, while the bylines of our press increasingly read Murdoch and Zell.

Or take voter turn-out. During the Bush administration, we read about plucky Afghanis traveling hundreds of miles to vote, risking murder as they did so, while here at home, we had stories of citizens (cough) complaining “I don’t like either of ’em, so I’m not voting.” And, disturbingly, stories of our own brown-shirt wannabes hanging around polling booths and intimidating minority voters.

Or take the way reform-minded protestors of Middle Eastern countries are pushing for education: teachers independent of the regime, professorships awarded on merit rather than cronyism, wider availability of public education. As we use our budget crisis as an excuse to dismantle public education in America.

The envy can only go so far: one mustn’t lose sight of the relative state of affairs in the excitement of witnessing the direction in which they’re changing. On a scale of zero to ten, where zero is the kind of cultish Big Brother hell North Korea resembles and ten is a full participatory democracy such as can hardly exist outside of small frontier communities, the US may be in a sixty-year slide from 8 to 6, while democratic protestors in the Mideast might be struggling to work from a 4 up to a 5.

Still, it’s hard not to wish we could share in some of that fervor for democracy that is supposed to be the American national hallmark. Institutions, like a free press and a universal franchise and public education, that underpin a viable democracy are precious elsewhere because they can’t be taken for granted. Peoples that only recently won the vote will fight to retain it. Peoples expressing a new-found freedom of speech will fight to exercise it. Sadly, here in the US, we take such freedoms—and the responsibilities that attach thereto—for granted. And we’re losing them in consequence.

2011 01 28 mdlake Politics Comments (0) Permalink

Hornblower

We’ve been watching a series of made-for-TV movies based on C. S. Forester’s Horatio Hornblower. The stories have a reputation as ripping good yarns, which are deserved. They also have a reputation for historical accuracy and capturing the flavor of naval life in the Napoleonic era, with which I have to quarrel—although it took me several days to realize why.

Hornblower is a young British naval officer who rises quickly from midshipman to admiral, managing to see just about everything worth telling a tale about in the course of the Napoleonic wars: sea battle, boarding action, landward action, duels (two), mutinies (also several), abusive superior officers in an inflexible military bureaucracy (many), sinkings, rescues (both ends), and that’s just the first two episodes. He is cut from the same cloth as Horatio Nelson: eager to get to grips with the enemy, even at insanely dangerous odds and despite forbidding complications. In a just world, he would be killed by his third action, yet always manages to come out on top, often through a deus ex machina, or even several in rapid sequence. His adventures and misadventures are all plausible; indeed, I suspect Forester didn’t so much make them up as copy them from history; all these remarkable events, with a few names changed, actually happened at some time or another, to someone or another.

But not all to the same person!

Any individual segment of the Hornblower tales is plausible, and backed by the attention to naval detail that can only come from a lover of ships and everything connected thereto. It carries the weight of history. Collectively, the events that comprise a Hornblower tale grossly exaggerate both the excitement of naval life and the self-evident superiority of Nelsonian aggressiveness over caution. (A lot of officers seeking to emulate Nelson got shot up and lost their ships and crews in the bargain. For all he was a brilliant seaman, Nelson was also damned lucky.) In doing so, the stories do the reader a historical disservice, even in the good cause of telling a ripping yarn. The Hornblower stories are trivially accurate—in the telling of details like how watches are counted, or why sand is sprinkled on deck before a battle—but substantially every bit as fabulous and fantastical as your typical swords-and-sorcery tale, which is also typically a ripping good yarn.

2011 01 27 mdlake Uncategorized Comments (0) Permalink

Snow Day

To my surprise, today was declared a snow day at my school. Between NPR grossly underestimating last night’s snowfall and the fact that we had no snow day a week ago, with nearly as much snowfall, I expected snow warnings to prove a false alarm.

So I missed the opportunity to sleep in. Not a total loss; I didn’t need to shovel out at the crack of dawn this morning, and of course it’s nice to sit around in my pajamas through most of the morning.

There’s a price, though, one that isn’t worth the free time. Dad taught biology, and he surprised me once by complaining about a snow day he considered unnecessary. Didn’t he get the day off with the students? I asked. Well, yes, he admitted, but a snow day still meant more work for him, not less: shaken out of his well-established rhythm, he needed to figure out how to to cover all the same material in less time, which is what he spent the day doing. My brother and I shoveled.

And y’know, he was right. I would rather have gone in to work today than have to rejigger my lesson plans. And—insult to injury—I have no kids, so I had to do my own shoveling. Pfah.

2011 01 27 mdlake Teaching Comments (0) Permalink

First Order Probability

I’m a strong believer in motivation when it comes to instruction. I do not mean here the importance of motivation as a general desire to learn; that is indeed critically important, and the techniques for instilling a desire to learn are discussed endlessly in education. Here I mean something much smaller: the importance of explaining why a particular technique is useful before drilling a student in the technique.

Too much math instruction begins with considering an Abelian group A and its many properties, or the volume swept out by a cone rotating about its vertex, or some equally arbitrary exercise. A student keen on math might perform the exercise for the fun of it, but most will do so simply for fear of a bad grade, thinking all the while, “So what?” Keep that up long enough, and kids become convinced that math isn’t worth the effort. They may recognize that some egghead somewhere uses math to make better mousetraps, but will come to conclude that that kind of math must lie immeasurably far beyond what they’re asked to do, that useful math is unattainable to all but a select few. That’s a toxic attitude. Besides, homework is a lot less enjoyable when it feels like mere busywork.

Factorial math is important to an understanding of probability. For a positive integer N, the factorial of N (written as “N!”) equals the product of N times N-1 times N-2 times…all the way down to 1. So 6! equals 6 x 5 x 4 x 3 x 2 x 1, which equals 720. Factorials are important to probability because they often count the number of possible cases of a given event. Find the total number of cases, and find the number of cases satisfying a given condition—also often a factorial—and you can calculate the odds of the condition being true.

Traditionally, the first lessons in probability are preceded by a discussion of factorials. Also traditionally, but very foolishly, lessons and exercises in factorials are rarely offered with the motivation for learning them at all: factorials allow you to count certain kinds of things easily, and that lets you find probabilities. Anyone from high-powered stockbroker to street punk shooting craps in a back alley can appreciate the value of understanding probability. Only math geeks canappreciate the value of understanding factorials as an exercise in itself. It makes factorials seem pointless. Ultimately, of course, the week passes and the class reaches probability, and the students begin using factorial math for concrete purposes. But by then, the damage is done.

For this reason, I’m breaking with tradition. I’m opening my lectures on factorials with the first, basic problems in probability. The kids won’t be able to solve those problems right away, but they will be able to see somewhere on the conceptual horizon that we’re engaged in this bizarre form of counting for a reason. We’ll end up skipping back and forth between concepts in probability and concepts in arithmetic, but I think they can handle it, and their homework will make a lot more sense.

Wish me luck. Sometimes traditions exist for good reasons, and I may be about to find out why this one exists.

2011 01 25 mdlake Teaching Comments (0) Permalink

Training Wheels Off

Like virtually all schools, Barringer practices “tracking,” the segregation of students by speed and quality. Tracking is the subject of a lot of educational debate, and arguments can be made for and against it, but nearly every school in the country has an honors track, a standard track, and a remedial track by some name or another.

None of my classes is the fast track. For a high school math teacher, this means fighting a lot of mental blocks: “I’m just no good at math.” Well, it’s true. Lots of people aren’t good at math. But then, you don’t need to be good at math to handle 90% or more of high school mathematics. Almost all of what we do is conceptually simple, the meaning tragically lost in arcane symbols. As a tutor back in college, I repeatedly reached the point where a student who had been struggling with a concept finally heard the penny drop; he or she would invariably sit up straight and say, “That’s all there is to it?” And I would say, “That’s all there is to it.” High school math is easy, if only you can keep a grip on what the symbols mean..

So a high school math teacher often labors to get this point across. “You can do this. You really can. Trust me.”

Case in point: long division of polynomials. Someone who works with math often has need to “find the zeroes” of a polynomial, a process of determining which of a very few values a variable, traditionally X, might take so as to satisfy a given equality, such as 2x^3 +5 x^2 – 14x – 8 = 0. Generally speaking, x can take a number of values equal to power of the polynomial—in this case, three—and to find them, the solver must tease the polynomial apart into separate linear factors, in this case (x-2)(x+4)(2x+1). Since the product of these three factors is zero, one of them must be zero, meaning X must equal 2, -4, or -1/2, according to which of these three factors equals zero. The solver can then test which of his several solutions satisfies the needs of the actual, material problem to which he is applying the algebra.

(See what I mean? The idea is easy. If x=2, then x-2=0, so any product including x-2 must be zero. And conversely, if a product containing x-2 is zero, there’s a good chance that x=2. But talking in jargon like “linear factors of a third-degree polynomial” makes the process seem incomprehensible.)

Often, a solver can spot one “zero,” one possible value of X that satisfies the equation, through a combination of luck and intuition and basic arithmetic. He can then remove one factor from the polynomial and continue working with a simpler problem. Here is where long division comes in. Had we, for example, realized that x=2 is one of the “zeroes,” we could divide 2x^3 +5 x^2 – 14x – 8 by x-2, set aside x=2 as one of our “zeroes,” and find the remaining two “zeroes” for 2x^2 + 9x + 4 = 0, a much simpler prospect. (Really. It’s easier if you can see it done; an online journal entry is a poor place to demonstrate the manipulation of polynomials.)

The process of removing the factor of x-2 from the original polynomial is exactly analogous to long division, the process by which we all learned to divide 155771 by 17. Drawing out that long bracket shape, putting the thing to be divided beneath it, putting the thing by which it is to be divided to its left, and, column by column, constructing the quotient, taking care to keep our columns tidy and watch our signs. The process is exactly the same, except that our columns now represent powers of X instead of powers of ten.

And so I was able to introduce the subject to my students by saying “You can do this,” and follow up with “I know you can do this because you already do it. It’s so easy that you’ve been doing it since you were in grade school. Just nobody told you at the time that you were already doing algebra.”

I think it worked. Most of my kids in that class caught on quickly. Not that the idea was difficult to grasp—again, high school math rarely is, and I was sincere when I told the students they already knew how to do it—but it’s easy to panic when confronted by X’s and Y’s and square roots. To their credit, most of my students didn’t panic.

But the results were not uniformly positive. A few student’s homework was nearly blank; either they were so lazy that they didn’t bother to try, or were so at sea that they didn’t know where to begin. When a math teacher extends his figurative hand and says, “You can do this. You really can. Trust me,” he means it. But of course the student ultimately has to trust, or it all breaks down.

And so begin the exercises in building student-teacher trust.

2011 01 24 mdlake Teaching Comments (0) Permalink

Lost Innocence

My first week of student teaching just finished, and already the kids are testing my authority. One quite publicly announced my fly was open—I think I handled that one with aplomb, stage-whispering that none of the students should let Mr. LeBlanc, sitting at the back of the room, know, lest he consider me unprofessional. I’m having a lot more difficulty with the don’t-know-don’t-care-fuck-you-sir attitude I’m getting from too many of the kids, especially in second-year algebra.

Already troubled by the discovery that the classroom is not always a sea of bright faces eager to learn—a truth every teacher-to-be learns but can’t really understand without experiencing it for himself—I sought Martin’s advice. His advice, couched in more euphemistic terms, was for a close relative of triage: help the students you can, and recognize that you simply can’t help every kid, especially not the ones determined not to be helped.

Probably good advice. Barringer is a “troubled school,” officially failing to meet No Child Left Behind requirements, and targeted by a hostile governor indulging the Republican attitude that destroying public education will be good for it. You can’t teach someone unwilling to learn, and a lot of Newark kids are already lost to the system—lost to frustration and despair. Triage makes sense in the brutal reality of the trenches, literal or metaphorical. There isn’t time to reach everyone. A teacher only has so much attention to spread among his students. Like a doctor, a teacher in a tough school will burn himself out quickly unless he learns not to torture himself over statistically inevitable failures, and a new teacher has trouble distinguishing those who are failing but can be rescued from those who are failing and can’t, much less remembering to nurture along the way the students eager to learn.

Yep. Probably good advice. It still felt like staring the devil in the face. Small wonder so many teachers burn out so young.

2011 01 21 mdlake Teaching Comments (0) Permalink

Breakfast in Shanghai

Before arriving in the Philippines, we were delayed overnight in Shanghai, having missed that day’s single connecting flight to Manila. Honestly, coming off the hell trip described earlier, I was glad of the chance for a rest and shower before boarding another jet. But the delay offered another benefit: the chance to see China, however briefly. Because we’d missed our flight, customs officials gave us a 24-hour visa stamp, and six of us braved an expedition into the communist dictatorship.

In truth, there was nothing to see in the space of three hours that can’t be found in any metropolis. Eileene and I spent our time breakfasting in one of Zagat’s highly rated choices, which proved to be a hotel buffet. The food was delicious, but no more exotic than the tiny glimpse we got of the rest of Shanghai. What the buffet did provide was variety: station after station with traditional breakfasts from around the world. One station devoted to Chinese dumplings, another to the British British heart attack special, another to Japanese fish and seaweed, another to “Mongolian” custom noodle bowls, another to French pastries, and so on.

This presented me with something of a dilemma. Should I take this rare opportunity to breakfast in foreign style, feasting on dumplings and noodles and sticky buns, the closest I’d get to experiencing China? Or was I so desperate for a little sanity after—nay, amid—a nightmare trip that I should retreat to familiar foods like the baguette and quite excellent salami?

In the end, I split the difference, though not as a conscious strategy. I opted for Chinese food first, an act of will over immediate desire. I quickly decided I’d rather have baguette and salami, though not before filling up a bit. I soon wished I’d asked for a hot chocolate when it was offered as we were seated, for neither hostess nor waiters approached our table again, but I made due with many glasses of fresh-squeezed grapefruit juice. It was good. Even at the time, I commented to Eileene that this might be the high point of the trip. Looking back, it may indeed have been the high point, though not indisputably so: both of us together without the noise and distraction of a larger family, relatively fresh attitudes toward (not very) foreign experience before homesickness set in, no sign yet of exotic diseases, the temporary high of bed and shower after several days without and all the stressful pre-travel work complete, enjoying some very fine food. I refuse to feel guilty about traveling to the opposite side of the globe only to stuff myself with French bread and salami. If I really want dumplings for breakfast, I can get them just as good in New York’s Chinatown.

I do feel guilty, however, about a suppressed urge. I was sorely tempted that morning to breakfast on corn flakes. Very safe, very familiar, very, very comforting to a guy strung out by four straight days of helpless anxiety. The only thing preventing me from eating corn flakes that morning was the price tag of $35 for that breakfast–$45 if you take the bullet train ticket into account. I’ll be damned if I’m going to spend $45 for a bowl of corn flakes.

2011 01 20 mdlake Uncategorized Comments (0) Permalink

One Dollar Rum

One of the small delights of foreign travel is finding extraordinary bargains.

Not necessarily what’s cheap, mind you. As Ella observed, some of the extremely cheap food available from Manila street vendors comes with a free case of Hep B. Even the sanitary street food sits out under the blanket of Manila’s thick, gray, choking atmosphere. Cheap doesn’t always mean a bargain.

But it’s hard to go wrong with $1 rum. No, not a $1 shot, nor those little cocktail bottles, but a large flask. At the convenience store just outside our hotel, no less. For the relatively captive audience of hotel patrons, prices are traditionally jacked up, and several items in the store were visibly more expensive than what we saw elsewhere in the city. But nowhere did I see 350ml (about 12 oz.) of rum going for 42 pesos, almost exactly one American dollar. I thought it had something to do with the easy availability of sugar cane, but that same store stocked brandy at a mere 47 pesos a few days later. No fan of rum, I couldn’t resist the brandy, which I like to add to my hot chocolate.

I cracked it open last night. It won’t win any awards for fine distillery, but it was entirely adequate. And, since I didn’t go blind from methyl alcohol or other impurities, I’m going to chalk this one up as a terrific bargain. I should have bought the entire shelf.

No, I take that back: beyond a certain point, the weight you have to haul through an airport isn’t worth the savings. But the thought of a cabinet of $1 flasks of brandy is fun to consider.

2011 01 19 mdlake Uncategorized Comments (0) Permalink

Getting to Know You

1/18/11

At one point in our trip to the Philippines, I had a chat with one of Eileene’s granduncles. He was eager to assure me that the Bautista clan—the maternal branch of Eileene’s family—was much larger than was in evidence that day. I knew it to be so; I’d seen far more assembled some ten years earlier. Bu time passes, and families grow, and the common ancestor becomes too distant to tie families together. I don’t meet my great-grandparents’ children any more, and rarely met them in the first place; I never met my great-great-grandparents’ children as family. It’s a natural and inevitable process.

But this bothered the grand-uncle, who expressed a sincere need for these ten-year reunions. “I don’t know them all any more. You live far away, and don’t take the time to meet family. Babs [a nickname for my mother-in-law]…Eileene…if you didn’t come to visit, I wouldn’t know you. Now I know you.”

Well, if it makes him feel better to think so, let him.. But two five-minute conversations spaced a decade apart doesn’t strike me as “knowing me” at all. The exchange had a peculiar, atavistic flavor to me, harkening back to a time when the clan was not only the fundamental social unit but a very real expression of power, political and otherwise. Family wasn’t sought as a natural source of friendship, but cultivated as a resource, and a big clan was always better than a small one. The discussion rather rubbed against my more American sense of frontier independence and lack of concern for the clan as an end unto itself. I didn’t like the sense that we owed the elders of the Bautista clan more attention, quite independent of whether the Bautista elders ever bothered to reciprocate.

The conversation also put me in mind of a bit of anthropology I ran across years ago. I can no longer remember even whether I read it or heard it on the radio or watched it as video, but the story left a strong impression on me, nevertheless. The author or speaker explained that humans have a natural conflict on meeting a stranger between a desire to protect territory and a desire to live and let live. The sharper the competition for resources, the more likely conflict is. In some of the Pacific island chains—I think the specific island related in this story was Papua New Guinea—the protein-deficient environment even drives people to occasional cannibalism. On meeting, strangers immediately settle down to a discussion of their family trees, attempting to establish a common relation. If one can be found, the strangers are now relatives, and can settle down to swapping stories; if none can be found, a duty to protect tribal territory compels them to try to kill one another, so a serious effort is made to cast the net as widely as possible, and remember even the most distant relations.

Even more atavistic than a desire for a large clan as a demonstration of wealth and power, this story, or a social mechanic very like it, seemed to lie behind the statement “Now I know you,” and bothered me even more than the implication that we owed the elders of the Bautista clan more attention.

2011 01 18 mdlake Uncategorized Comments (0) Permalink