Tall, Dark, and Mysterious

6/24/2005

College-level counting

File under: Those Who Can't, Meta-Meta, Queen of Sciences. Posted by Moebius Stripper at 1:22 pm.

[Computer’s still a few time zones away, bike-without-adapter is still primary means of transportation - and I have an apartment to move to the mainland. And I’ll be away, again, this time south of the border, for part of July, so don’t expect anything interesting or substantial in this space for quite some time.]

In the meantime, a discussion topic: how mathematics education went so horribly awry that we now teach this stuff in college. Actually, let me flip that one around: why do we bother teaching this material - integers, fractions, areas of plane figures, basic algebra - in elementary school and junior high in the first place? I know why I think students should learn it, but I wonder if educators and curriculum developers at that level can answer that question, given that basic mathematics is so often so poorly retained. I certainly wonder what insight those educators would offer when confronted with the fact that more and more colleges are offering remedial classes, especially in math, than ever before; that in many cases, it is unwise for us college math instructors to assume that our students have any knowledge of prerequisite material. What sort of subject is grade-school math, that it is absorbed so poorly?

If I were to invent a language with counterintuitive syntax and bizarre vocabulary that bore no relation to that of any Western language, and if I were to teach my invention to a classroom full of schoolchildren, some would excel. Others would do decently. In any event, most, I figure, would pass. But a year later, nearly everyone in my class will have forgotten my crazy language, save a few words that they heard in memorable contexts. Why would they remember it, after all? They have no need for it, and the lessons they took were nowhere reinforced in their day-to-day lives. This is how so many college students see math: as something that they need to learn because their teachers said so, but something that is so poorly connected to the rest of their lives that they have no reason to remember (let alone apply, let alone use) any of it.

At the risk of sounding like an obnoxious student, I’m asking those who teach this material the first time: what’s the point of any of this? Why do we have to learn it? What does it have to do with anything?

…and given whatever those answers are…why is math so seldom worked into any aspect of children’s lives outside the mathematics classroom before they graduate from high school?

6/13/2005

What your physics teacher never told you about electronics

File under: Sound And Fury, Meta-Meta, Talking To Strangers. Posted by Moebius Stripper at 11:31 am.

The Gulf Islands were lovely, just lovely. And I have photos! Postcards! Tales of hitchhiking, prawn delivery, epic seasickness, and more! None of which you’ll get to see or read for a long, long time, because my computer is dead.

“Dead?” I said to the guy at Future Shop, in the way that one asks questions when coming to terms with that chasm between one’s formerly-held beliefs and reality. “I thought it was just the adapter. When I left for my vacation, it was working ok, but the battery was discharging even though it was plugged in. Mind you, when I came back, it sputtered for a bit and then just shut off.”

“Adapter’s fine,” said the technician. “It’s giving me nineteen and a half volts. And your computer won’t even turn on.”

“The hell? It’s only a few months old. And -”

“We’ve had a lot of power surges on the Island in the last few weeks,” the technician recited. “And brownouts. It’s the brownouts that’ll really fry ‘em. Lotta machines in in the last couple weeks, completely dead.”

“Dead,” I repeated.

“Is it still under warranty?” he asked me.

“Hell yeah,” I replied. “And the most important files on it are music, so that’s okay. But - a brownout will just kill a computer? Laptops only, or does this happen to desktops too?”

“We’ve had a lot of desktops in here, too.”

“Is there any way to predict brownouts, or power surges?”

He shook his head. “Nothing you could have done about it. This is not your fault.”

And from there, something about serial numbers and warranties and they’ll send you a box to ship it in and should have it or a reasonable facsimile thereof back to you in a few weeks. A few weeks. I’m going to cart the bloody thing down to the mom and pop computer shop at the north end of town - I still have this rental car for another day - for a second opinion, but barring that, a few weeks.

I’m at the library now, and I’ll be back, because my EI reports need to be filed electronically. Should pick up a few books, while I’m here. I hear people used to read those before there were computers.

6/5/2005

Something for you to do while I vacation yet some more

File under: Queen of Sciences, What I Did On My Summer Vacation. Posted by Moebius Stripper at 9:57 pm.

In a few days I’ll be setting out to visit some more of the Gulf Islands. One of them doesn’t allow cars, and is powered entirely by generators. At least two lack bank machines. One is rumoured to have “honesty stands” where local artists leave their work unattended and trust visitors to pay for what they take. Only one has more than a thousand full-time residents. I may at some point pop into a cafe for a few minutes on a rainy day to avail myself of the dial-up connection on the island’s IBM 486 , but don’t hold your breath. I will be making notes on postcards, so holler if you want one.

In the meantime, some of you may be able to help me with a project I’m thinking about working on one of these days book I have no excuse not to work on now that I don’t have a pesky job to worry about. A bit of background: for the last five years, I have worked at an academic summer camp for mathematically gifted high school students. One of my favourite classes to teach - and one of the most popular among the campers - has been one that I call “Calculus Without Calculus”. Those of my readers who know me in real life are well aware that I…well, I don’t hate calculus, so much as I think that calculus doesn’t need me to love it. Calculus gets more than enough attention in the thousands of high schools and universities that inflict it upon every other student that passes through their doors, the overwhelming majority of whom don’t learn it properly and wouldn’t ever use it again even if they did. Nevertheless, calculus is a natural choice for students who have not learned to think mathematically: for all the terror it strikes in students’ hearts, it’s one of the easier branches of mathematics to reduce to mindless algorithms in a low-level course. Need to maximize some quantity? Set a derivative to zero, and solve. What were we trying to do again?

Calculus Without Calculus is a collection of methods of solving typical calculus problems without taking a single derivative. The two main methods involve inequalities, or exploiting geometric properties of figures, particularly symmetry. The old “maximize area with given cost of rectangular fence” problem? You can complete the square, take a derivative - or you can apply the AM-GM inequality. The question about getting the best view of the statue on a pedestal, that appears in the chapter on inverse trig functions in every single calculus book? Solvable using elementary circle geometry that could be found in every grade ten math text before it was decreed that geometry should no longer be taught. There are tons of these. I know of three calculus-free methods of finding tangents to ellipses: one using transformations of circles, another using the Cauchy-Schwartz inequality, and one using projective geometry. A few months ago, I was reading some fiction on the Vietnam War - One To Count Cadence, by James Crumley - and one of the characters mentions in passing that he was able to solve the ladder-around-a-corner problem (you know the one) without calculus. I struggled with that for a long time before a camper provided the key insight. A quick application of Holder’s Inequality, and the result falls right out. (Except that now I’m trying to recreate it. Damn; this is going to keep me up again.)

I’m sure there’s a lot more to say on the topic, and I’d like to write a (short) book on the topic. What I’m looking for: book recs. In particular, recommendations for good books on problem-solving, which tend to spend a lot of time on funky inequalities. I’m especially interested in geometric and otherwise intuitive proofs for the old standbys (AM-GM, Holder’s, C-S…) , and lots of examples - both of single variable problems that one would see in calculus texts, as well as multivariable ones that are a lot easier to solve without calculus. One of my favourite books of this sort is Problem Solving Through Problems, by Loren Larson, the talented mathematician and educator who first introduced me to this stuff. If you’ve got any recs, or anything even slightly pertinent, I’d love to hear them - post them below so I’ll have something to check out when I return to this big island.

My other reason for posting this, of course, is that now I’ll feel horribly guilty if I don’t actually have anything to show for this in a few weeks.

And, if [above] isn’t your thing, perhaps the Phallic Logo Awards can keep you busy for the next week. (What’s this you want, a better segue? Here: Galiano Island.)

6/3/2005

‘You have the right to remain silent. So SHUT THE FUCK UP.’

File under: Miscellany, Meta-Meta. Posted by Moebius Stripper at 10:43 am.

There’s no shortage of blogs by cranky educators out there, but to the best of my knowledge, there aren’t many by public defenders. Which bodes well for my traffic, because the occasional “If you’re going to try to convince the department head that you know the material, perhaps you shouldn’t tell your instructor that you haven’t done any work in three weeks” doesn’t hold a candle “If you have some miscellaneous drug charge, think twice about clothing with a marijuana leaf on it or a t-shirt with the ‘UniBonger’ on it.”

[via mythago]

6/2/2005

Is that a bridge I see going up in smoke?

File under: Righteous Indignation, Sound And Fury. Posted by Moebius Stripper at 5:09 pm.

“Hey, Moebius Stripper, any updates on the job front?”

Funny you should ask! Let’s take a look: so far, we have a school that doesn’t screen applications before the job is scheduled to begin; a school that advertizes for nonexistent positions; and a school whose employees are all on vacation around screening time. Anything else? Why, yes: we also have a school whose employment contact email was rendered nonfunctional sometime between the time I applied back in March and yesterday morning. Now there’s a clever way to avoid being pestered. Also, an outright rejection from a college looking for an instructor to do, for the next eight months, the exact same thing that I did (skillfully! conscientiously! with a smile on my face!) for the past eight months. Apparently “after careful consideration”, they decided that I fell so far short of their lofty academic standards that I wasn’t even worth interviewing. The day’s not over yet, but as of this writing, I haven’t responded that excuse me, what in the goddamned fuck are they talking about, they couldn’t even spell my name correctly in their three-line reply, and they’re snotting about standards?

Suspicion that I can’t back up, but that makes me feel better about myself: my incompetent former colleague, the Nice Teacher, whom I know for a fact was at some point employed by Standards U (and who might be employed by them now), was asked for feedback about me by that school’s math department head, who’d considered getting me to come into town for an interview. Either that, or he provided it voluntarily. In any case, I suspect that he approached this task with a level of skill and enthusiasm such that, had either been present when he was actually working with me, he’d certainly have gotten his contract extended. But neither had been, and so midway into the semester I found myself sitting in Department Head’s office, explaining that my colleague, as a consequence of his particular brand of ineptitude, was undermining my work. Thanks for telling me! replied Department Head. We certainly can’t have that! I’ll have a talk with him right away. And no way will he be back here next term. So I guess we’ll have work for YOU.

“Moebius Stripper?” (I’m guessing) said Nice Teacher. “I remember her.” The damned bitch! I’d still have that job at Island U if it weren’t for her! “Oh,” (I’m guessing) continued Nice Teacher, “She’s certainly a nice person. But, she didn’t teach appropriately. She misjudged her students. And I got better evaluations than she did.” Because I basically let my students see their tests in the day before they wrote them, but who’s keeping track?

“Thank you for your input,” (I’m guessing) said the department chair at Standards U. “We won’t waste time with an interview then.”

So, if this is how things went down, I suppose this could be construed as poetic justice of some sort: I screwed Nice Teacher out of a job, and then he screwed me out of a job. Except that it isn’t, because the difference was that I was right, and the Nice Teacher was wrong. No good deed, yadda. I could also get all self-righteous and say that it’s good this happened, because I don’t want to work for an institution that would not only hire my school’s sloppy seconds, but listen to him, anyway, so there; except that I still do want to work for them.

Even if that’s not how it went, feh. Fortunately, I have some technical writing work lined up for the coming months, and I’ve decided I want to visit all of the major Gulf Islands. (Five down, eight to go.) I’ve also decided to work on a math book I’ve been thinking about for a long time; more about that in a bit.

In the meantime, I guess I should be prepared to be in this for the long haul.

5/31/2005

On teaching college students what they should already know

File under: Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 6:41 pm.

Rudbeckia Hirta succinctly explains that if you can’t do algebra, then you can’t take calculus:

Due to reasons beyond my understanding, high school math and college math are completely unaligned. The K-12 system sends us students whose knowledge is a mile wide and an inch deep: we get students who are shaky at algebra, frightened of fractions, and unsure of how to find the areas of basic plane figures (and completely unable to accept the idea that it is a reasonable request to ask them to solve non-standard problems where the method of solution is not immediately obvious), but they have been exposed to matrix arithmetic, computations from polynomial calculus and other supposedly “advanced” procedures. You would think that the “college prep” track would prepare students for college, but it doesn’t. Recently I read somewhere (maybe in Focus?) that there is more calculus taught in high schools than there is at colleges.

See also: mathematics by pattern matching, word problems, et al. So much of this problem can be traced to the fact that students do not understand that an equation is a relationship among quantities. Each equation they see is a concept unto itself, to be memorized and applied to the word problem on the test that looks like the word problem that I did on the blackboard and that used a similar equation. If the word problem on the test doesn’t look like any of the problems I did in class, then that question is “totally unfair”. Most students will leave the question blank, or solve a completely different problem (one with a ready-made equation) in the space provided.

This New York Times op-ed is about teaching freshman English to illiterate college students, but I’m sure that anyone teaching freshman math to innumerate college students can find plenty to relate to. I’m not sure I’m willing to buy into, wholesale, the author’s belief that content should be ignored in favour of form, but I can’t argue with success:

On the first day of my freshman writing class I give the students this assignment: You will be divided into groups and by the end of the semester each group will be expected to have created its own language, complete with a syntax, a lexicon, a text, rules for translating the text and strategies for teaching your language to fellow students.

…14 weeks later - and this happens every time - each group has produced a language of incredible sophistication and precision.

How is this near miracle accomplished? The short answer is that over the semester the students come to understand a single proposition: A sentence is a structure of logical relationships. In its bare form, this proposition is hardly edifying, which is why I immediately supplement it with a simple exercise. “Here,” I say, “are five words randomly chosen; turn them into a sentence.” (The first time I did this the words were coffee, should, book, garbage and quickly.) In no time at all I am presented with 20 sentences, all perfectly coherent and all quite different. Then comes the hard part. “What is it,” I ask, “that you did? What did it take to turn a random list of words into a sentence?” A lot of fumbling and stumbling and false starts follow, but finally someone says, “I put the words into a relationship with one another.”

An equation is a relationship among quantities. How many times have I tried, and failed, to get this idea across? Imagine getting students to realize it for themselves! Unfortunately, many students lack the intuitive ideas about math needed to know, even subconsciously and with the help of leading questions, that equations are anything other than a jumble of letters, numbers, and symbols. (Even though I spent twenty minutes showing how we could use the definition of a circle and the Pythagorean Theorem to derive the equation of a circle in the Cartesian plane, nearly all of my students were angry that I wouldn’t provide the formula on the test. I mentioned, incorrectly, that they could derive the formula themselves, on the test, if they needed to; of course, I was wrong.) But I would gladly sacrifice 80% of the poorly-learned content in a first-year college math course if I could instead effect a solid understanding of what equations really meant - and how to get one from a sentence or two of information. I wonder if Fish’s lesson could be adapted to the first-year math classroom.

At Critical Mass, where I found the NYT piece, Erin O’Connor isn’t optimistic even about applying it to the English classroom:

[M]ost university composition courses are taught by graduate students who are a) not necessarily good writers themselves, and b) often more interested in using the composition classroom to practice teaching the content they hope to teach as non-composition teaching English professors, and you’ve got a situation in which the Fish vision, regardless of its merits, is pure pipedream.

Ditto. No one teaches precalculus if they can avoid it; at Island U, it got passed from temps to new faculty to the department head, who teaches the courses that no one else will teach. Everyone says that the precalculus course needs to be completely revamped, but given the option, they’d rather take a less thankless teaching assignment than put forth the effort to revamp it. And around and around we go.

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