Tall, Dark, and Mysterious

7/20/2005

Prerequisites for college math

File under: Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 4:09 pm.

Are there any high school math teachers - or, better yet, developers of high school math curricula - who read this blog? I’ve decided to take a break from my usual undirected griping about how woefully unprepared students are to do college math, and divert those energies into something more productive - a working list of what students should be learning in high school math classes in order to prepare them for college (or even, for what they are supposed to be learning later on in high school).

Some months ago Rudbeckia Hirta drily observed that contrary to what one might assume, the college prep track in the high school math system does not, in fact, prepare students for college math. As far as I can tell, high school math in general doesn’t prepare them for damned near anything. In fact, I think that students often learn negative amounts of math in high school: in grade school, they learn how to perform basic mathematical operations and such, and by the time they’ve received their diplomas, they’ve been trained to leave those tasks to their calculators. (See: inability to add fractions, talldarkandmysterious.ca, 2004-present.) The claim that foisting heavy-duty calculators upon students frees them to do more complex and creative tasks is wishful thinking that, in my experience, has no grounding in reality.

When I started teaching, I anticipated that students would be weak in much of the material that they were expected to have learned a year or two earlier. The actual state of affairs was far more dire: many couldn’t do the math that they should have learned a full decade earlier. Most weren’t just weak in math; they didn’t even know what math was. They didn’t know what an equation symbolized, or even that it was supposed to symbolize anything at all: to them an equation was just a jumble of symbols. They looked at me blankly when I asked them to think not only about what steps they needed to perform in order to solve a problem, but also about why they needed to perform those steps. They had no experience, nor understanding, of how to reason logically when presented with quantitative problems. Students threw fits when asked to combine simple techniques in…basic ways” (link via Chris Correa) - no one had ever required them to do that before. In his book Innumeracy, John Allen Paulos’ lamented:

Elementary schools by and large do manage to teach the basic algorithms for multiplication and division, addition and subtraction, as well as methods for handling fractions, decimals and percentages. Unfortunately, they don’t do as effective a job in teaching when to add or subtract, when to multiply or divide, or how to convert from fractions to decimals or percentages.

I disagree somewhat with Paulos, who wrote Innumeracy before calculators were ubiquitous: elementary schools no longer do manage to teach the basic algorithms very well. Other than that, he’s correct. Students’ depth of mathematical knowledge is so shallow that they can’t even figure out when perform basic mathematical operations.

The necessary groundwork for doing mathematics at the college level - or even at the high-school-courses-in-college level - is more basic than anything that students supposedly learning in high school in the most superficial and fleeting manner. And they routinely leave high school without it.

Based on my experiences, students graduating from high school should, in order to succeed in even the most basic college math classes:

  1. Be able to add, subtract, multiply, and divide fractions. Moreover, they should understand that the horizontal bar in a fraction denotes division. (Seem obvious? I thought so, too, until I had a student tell me that she couldn’t give me a decimal approximation of (3/5)^8, because “my calculator doesn’t have a fraction button”.)
  2. Have the times tables (single digit numbers) memorized. At minimum, they should understand what the basic operations mean. For instance, know that “times” means “groups of”, which will enable them to multiply, for instance, any number by 1 or 0 without a calculator, and without putting much thought into the matter. This would also enable those students who have not memorized their times tables to figure out what 3 times 8 was if they didn’t know it by heart.
  3. Understand how to solve a linear (or reduces-to-linear) equation in a single variable. Recognize that the goal is to isolate the unknown quantity, and that doing so requires “undoing” the equation by reversing the order of operations. Know that that the equals sign means that both sides of the equation are the same, and that one can’t change the value of one side without changing the value of the other. (Aside: shortcuts such as “cross-multiplication” should be stricken from the high school algebra curriculum entirely - or at least until students understand where they come from. If I had a dollar for every student I ever tutored who was familiar with that phantom operation, and if I had to pay ten bucks for every student who actually got that cross-multiplication was just shorthand for multiplying both sides of an equation by the two denominators - I’d still be in the black.)
  4. Be able to set up an equation, or set of equations, from a few sentences of text. (For instance, students should be able to translate simple geometric statements about perimeter and area into equations. ) Students should understand that (all together now!) an equation is a relationship among quantities, and that the goal in solving a word problem is to find the numerical value for one or more unknown quantities; and that the method for doing so involves analyzing how the given quantities are related. In order to measure whether students understand this, students must be presented, in a test setting, with word problems that differ more than superficially from the ones presented in class or in the textbook; requiring them only to parrot solutions to questions they have encountered exactly before, measures only their memorization skills.

  5. Be able to interpret graphs, and to make transitions between algebraic and geometric presentations of data. For instance, students should know what an x- [y-]intercept means both geometrically (”the place where the graph crosses the x- [y-]axis”) and algebraically (”the value of x [y] when y [x] is set to zero in the function”).
  6. Understand basic logic, such as the meaning of the “if…then” syllogism. They should know that if given a definition or rule of the form “if A, then B”, they need to check that the conditions of A are satisfied before they apply B. (Sound like a no-brainer? It should be. This is one of those things I completely took for granted when I started teaching at the college level. My illusions were shattered when I found that a simple statement such as “if A and B are disjoint sets, then the number of elements in (A union B) equals the number of elements in A plus the number of elements in B” caused confusion of epic proportions among a majority of my students. Many wouldn’t even check if A and B were disjoint before finding the cardinality of their union; others seemed to understand that they needed to see if A and B were disjoint, and they needed to find their cardinality - but they didn’t know how those things fit together. (They’d see that A and B were not disjoint, claim as much, and then apply the formula anyway.) It is a testament to the ridiculous extent to which mathematics is divorced from reality in students’ minds that three year olds can understand the implications of “If it’s raining, then you need an umbrella”, but that students graduating from high school are bewildered when the most elementary of mathematical concepts are juxtaposed in such a manner.)
  7. More generally: students should know the basics of what it means to justify something mathematically. They should know that it is not enough to plug in a few values for x; you need to show that an identity, for instance, is true for all x. Conversely, they should understand that a single counterexample suffices to show that a claim is false. (Despite the affinity on the part of the high school text I am working for true/false questions, the students I am working with do not understand this.) Among the educational devices to be expunged from the classroom: textbooks that suggest that eyeballing the output of a graphing calculator is a legitimate method of showing, for instance, that a function has three zeroes or two asymptotes or what have you.

What else? I invite comments from everyone with a dog in this fight - college students who’ve taken math classes, college math instructors, curriculum developers, textbook authors. What unwritten (or written!) prerequisites are students missing? What should every student know in order to succeed in the class you took, taught, or wrote the text for?

7/17/2005

With apologies to those readers of mine whose lives have been improved by propecia

File under: Sound And Fury, Meta-Meta. Posted by Moebius Stripper at 3:37 pm.

Or levitra, or fioricet, or viagra. Or, for that matter, online craps, roulette, or strip blackjack. Spam round these parts has been nigh unbearable as of late (200 spam comments so far today alone), so from now on anyone mentioning drugs by brand name is going to be assumed to be a spammer, and their comments will disappear into the abyss without me even having to see them. Ditto for those trying to lure get my readers hooked on various forms of gambling. The other vices are still fair game, though, so go nuts.

First thing I’m going to do when I get my computer back: install better anti-spam plugins, even though they’re just temporary stop-gaps. Does anyone know if the old rename-your-comments-file trick still works with Wordpress 1.5?

7/15/2005

Dispatches from the library

File under: Sound And Fury, This One Time, At Mathcamp, Those Who Can't. Posted by Moebius Stripper at 5:03 pm.
  1. Apparently my computer has some hard-to-diagnose computer ailment. In a sequence of events that bears striking parallels to those often experienced in my country’s health care system for humans, this has resulted in the blasted machine being discharged - thrice - from the hospital for sick computers before it was fully cured, only to be sent back for more tests and treatment. Fortunately, my computer’s health care has another important feature in common with mine: it is free. Unfortunately, I don’t know how long the waiting list is before it can see the appropriate specialist. Blissfully, this whole thing has been handled by my father, who has advocated tirelessly on its behalf. And now that I’ve praised my father, I won’t feel as guilty when I post this hilarious story involving him making a really stupid bet with my brother back when we were kids. Stay tuned!
  2. I had a lovely, lovely week at Mathcamp. Briefly:

    • I taught two classes - a squishy, very visual version of projective geometry, and Calculus Without Calculus - to the best audience of ten that anyone has ever taught. Rather than having students bitch and moan about how I gave them homework that, like, was totally unfair because it made them think and shit, I had students request that I skip the routine calculations because there were only ten minutes left in class and they wanted to get to the cool stuff. I also had those same students construct a model of the real projective plane using carrots and toothpicks.
    • On the whole, our campers kick all kinds of ass in all kinds of ways. However, every now and again we encounter some unpleasant behaviours that need to be addressed. For instance, we often have a small but vocal contingent of campers who boast, at length, about their mathematical prowess. Last year, some of the staff had the idea of addressing these sorts of things by presenting humourous skits that parodied the unpleasant behaviour.

      Fellow Mathcamp staff member A had the idea of writing a skit in which one character, played by me, would list all of the insanely difficult (”for a beginner, I suppose”) math classes she was taking. In researching for the part, we decided we needed to come up with extremely technical-sounding class names - ones that appeared to be about math, but were actually nonsense. We spent some time trying to come up with such technical sounding gibberish, until A had an idea: “Go to the ArXiv,” he suggested, “and look up quantum algebra!”

      A simple permutation of preprint titles resulted in our fictitious braggart boasting about her exploits in hypercategory theory, m-difference representations, quantum affine algebroids, cohomology of semiregular twistor spectra, and quasi-coherent sheaves on Calabi-Yau manifolds, Moore Method.

      I think our message got through. And if any of those topics do exist, I’m not sure I want to know.

  3. And now I’m back at not-Mathcamp, still waiting the remaining two weeks for the hardworking bureaucrats at the Employment Insurance headquarters to finish transfering my file from one address to the next. Fortunately, as long as the good people at Texas Instruments have their claws in high school curricula, there will be work for unemployed math instructors willing to tutor high school math.
  4. The kid I’m working with is in grade n, but I can’t for the life of me figure out what possessed any of his math teachers in grades five through n-1 to promote him. He’s got a really good attitude about learning, and is on the whole quite pleasant to work with, though, so I shall lay off the snark. I’ll say only that if there is a Hell, I’d like to put in a suggestion to management that assign everyone who played a role in introducing calculators to the classroom to spend eternity watching eighteen year olds extract same from their backpacks, turn them on, and key in a sequence of commands in order to figure out what two times one half is.

7/9/2005

My country, timid and unsure

Meanwhile, back in the homeland, we have apparently learned the finer points of how to convict loathesome pieces of shit for hate speech by studying the materials used to prepare debutantes for matriculation from the nation’s top finishing schools. Really, we’re just that polite: witness [part of] Judge Marty Irwin’s explanation for convicting David Ahenakew of hate speech and stripping him of his membership in the Order of Canada:

[Irwin] noted that, rather than being “timid, unsure or rattled,” Ahenakew’s demeanour “bordered on self-confidence to the point of arrogance.”

Sure, Mr. Ahenakew, you openly and publicly declared that Hitler was just helping rid his neighbourhood of the “Jewish disease” when he “fried six million of those guys”, and you appear to have appointed yourself the official spokesperson for the updated edition of The Protocols of the Elders of Zion, and for that we’ll take back your medal and fine you a thousand bucks, but - self-confidence? Arrogance? Failure to be rattled by criticism? Heavens above, that’s just unCanadian!

7/7/2005

Not from the solutions manual of Calculus With Analytic Geometry

File under: This One Time, At Mathcamp, Queen of Sciences. Posted by Moebius Stripper at 1:46 pm.
  1. You’re in the middle of the desert, 2 km north of an east-west river. Your destination is 1 km north and 5 km east of where you stand. You need to stop at the river for a drink at some point in your journey, but it doesn’t matter when. What is the shortest path you can take?

    Solution. Let the river be represented by the x-axis, your starting point be P(0,2), and your destination be at Q(5,3). The shortest path between P and Q that touches the x-axis is equivalent to the shortest path from P to Q’(5,-3). This path is clearly a straight line; properties of similar triangles show that you arrive at the river at (2,0).

  2. A statue is 4 m tall and mounted on a 2 m pedestal. Your eye level is 1.5 m above ground. Where should you stand in order to get the view of the statue deemed “best” by the writers of calculus textbooks - that is, the view such that the angle subtended by the statue at your eye is a maximum?

    Solution. Because I’m lazy and don’t feel like creating pictures, let’s say that the base of the pedestal is at O(0,0), the base of the statue is at B(0,2), and the top of the statue is at T(0,6). Then your eye will be somewhere on the line y=1.5. Construct the circle tangent to y=1.5 that has the segment BT as a chord. Then the point of tangency, P, is such that the angle BPT lies in the circle, while BQT lies outside the circle for all other Q on the line y=1.5. Hence BPT>BQT for all such Q, so you should stand at the P. The Pythagorean Theorem quickly shows that you should stand 1.5 m from the base of the statue.

  3. When widgets are priced at $50 apiece, 100 people buy them. For every $5 increase in price, 2 fewer people buy widgets. How much should a widget cost in order to maximize the widget company’s revenue?

    Solution. Letting x be the number of $5 price increases, the company’s revenue, which we wish to maximize, is R=(50+5x)(100-2x). Equivalently, we may maximize 5/2*R=(50+5x)(250-5x). Applying the AM-GM inequality to this quantity, we have sqrt(5/2*R) < = 1/2*((50+5x)+(250-5x))=150. The revenue is thus maximized when 50+5x=250-5x, ie, x=20. So, twenty $5 price increases, for a price of $150 per widget, gives the maxmimum revenue.

  4. Find the largest possible perimeter of a rectangle whose sides are parallel to the axes and that can be inscribed in an ellipse with equation x^2+2y^2=1.

    Solution. Letting (x,y) by a point on the ellipse such that the rectangle’s vertices are at (x,y), (-x,y), (x,-y), and (-x,-y), we have a perimeter of P=4x+4y to maximize. Apply Cauchy-Schwartz to the vectors (x, sqrt(2)y) and (1,1/sqrt(2)) to obtain P/4=x+y< =sqrt(x^2+2y^2)sqrt(1+1/2)=sqrt(3/2). The perimeter is a maximum when x=2y. Substituting into the equation of the ellipse, x=2sqrt(6), y=sqrt(6), and so the largest perimeter possible is 12sqrt(6).

  5. Two hallways meet one another at right angles. One hallway is 27 inches wide, and the other is 64 inches wide. What is the length of the longest ladder that can be carried horizontally around the corner?

    Solution. The ladder’s length, L, is that of the shortest line segment that passes through the point (64, -27) and is bounded by the axes. The lines passing through (64,27) have equations of the form y=mx-64m-27. The intercepts of such a line are at (0,(-64m-27)) and ((64m+27)/m,0), so we need to minimize L^2=(64m+27)^2+((64m+27)^2)/m^2=(1/m^2+1)(64m+27)^2. Apply Holder’s Inequality with three 3-norms to the vectors (1/m^(2/3),1), (4m^(1/3), 3) and (4m^(1/3), 3). This gives L^(2/3) = ((1/m^2+1)(64m+27)^2)^(1/3) >= (64)^(2/3)+27^(2/3), that is, L>=25^(3/2)=125. So the longest ladder that can be carried horizontally around the corner is 125 inches long.

7/3/2005

Transition

File under: Home And Native Land, What I Did On My Summer Vacation. Posted by Moebius Stripper at 1:23 pm.

Greetings from the happiest place on earth - Mathcamp - where I just realized that I’ll be spending ten consecutive days in the same place, the longest stretch of time I’ll be in one place in two months.

Meanwhile: all of my material possessions (except for the goddamned computer, goddammit) now reside in a storage locker in Vancouver.

Yes, I'm stealing bandwidth. This is what happens when you're at an internet cafe and can't transfer your files. When you live, by yourself, on an island (*) and don’t own a car, such a maneuver requires a lot of careful planning. Rent van; make itinerary of visits to lockers to check them out before committing to any; schedule trips to and from the mainland so as to allow sufficient time to scout out downtown lockers before rush hour and then to transfer materials to chosen locker; return before the last ferry of the day so as to avoid staying overnight in hotel. All of these are subject to last-minute inconveniences: car rental places may be understaffed and deliver your car late, traffic in downtown Vancouver may suck even harder than it did last time, a ferry may run aground and take out twenty-odd boats and result in the cancellation of all service to and from the Island for the rest of the business day. Granted, it’s nice when your ferry schedule makes national news and you don’t have to call 1-800-BC-FERRY to be put on hold for five minutes and be given outdated information anyway (**), but still. No one was hurt, but several hundred people were kept on board for seven (7) hours after the crash, and but for a series of coincidences, I’d have been one of them. Unhurt, but stuck on a ferry with a van full of heavy boxes, kept aboard until after the storage lockers closed for the day, on June 30, a day before Canada Day, when all the storage lockers in Vancouver are closed for the day, and don’t open until after I’ve boarded a train to the US.

Now, to point form, since things are crazy here and I’m being called away every few minutes:

  1. There’s a $400 surcharge for dropping off a van rented on the Island, on the mainland. Since travelling between the Island and the mainland with a car costs $45 each way and takes half a day, and since I can’t imagine employees at a car rental place make much more than $100 for that amount of time it takes to transfer the van back, and since the car rental place has locations in both cities I was travelling between anyway and has a million vans and what’s the difference anyway if one of them stays in Vancouver, this is a very sweet deal indeed for the folks at Budget Rent-a-Car. Consequently, my plan to take the Queen of Oak Bay at 8:30 am on June 30 to Horseshoe Bay had to be changed: to Vancouver on the 29th, back to the Island later that evening, and then back to Vancouver by foot the next day.
  2. Over at the storage locker: a massive truck pulled up along the driver’s side my rented Dodge Caravan, leaving me unable to open the driver’s door. I did what anyone who found herself in such unremarkable circumstances would do: I unlocked the passenger door, with a key, slid over, and stuck the key in the ignition. A siren sounded, and the engine didn’t take. Confused, I called the guy at the rental place. After a false diganosis (leading me to recruit a cabbie from the taxi company next door for a jump, which didn’t work, and being told by said cabbie tht he was being “very generous” by charging me only $10 for the five minutes he was at my service), the rental dude figured out that the car alarm sounds whenever you unlock the passenger-side door first. Because this is how cars are stolen these days: some cunning thief unlocks the passenger-side door, and starts the engine with a key. Except that when the thief’s target is a Dodge Caravan, he’s been thwarted. Thanks, Dodge!

    Related - other, related inane “security” measures I’ve run across lately:

    • I haven’t yet seen a dime of my employment insurance. I made the mistake of using my parents’ permanent address instead of my temporary one, which resulted in my file being flagged. The good folks at EI, while unfamiliar with that downwardly-mobile demographic of which I am a part - twentysomething transients with more education than job prospects, who almost-yearly move to wherever there is work or school - are apparently well-acquainted with that group of unemployed folk who try to defraud the system by filing a single application, under their actual names and social insurance numbers, but listing a different province than the one they’re filing the application from. I may be unemployed, but I’m not stupid, and if I were trying to bleed the system I’d like think I’d be a bit more imaginative, but whatever. So: my file is being transferred from Ontario to BC, which takes four (4) weeks. Feel free to speculate, in the comments, what in the act of transferring a file could possibly take four weeks. Is the EI office hiring? Because I can totally streamline this process for them.

    • If you don’t live in the US, but wish to travel in the US by Greyhound, you’re out of luck. They won’t take your credit card. You can get an American friend of yours to purchase the ticket on their credit card, but then you will have to pay a $15 “gift charge”. Sources of various degrees of reliability tell me that this crackdown on Canadian residents is a consequence of the PATRIOT Act. This seems completely stupid to me, but it’s less stupid than voluntarily turning down business from Canadians, so it’s probably true.
  3. After dropping off my belongings in the storage locker, I made my way to the ferry, forgetting that it’s faster to walk on one’s hands than it is to drive through downtown Vancouver during rush hour. Because of that, combined with the delay caused by the car-alarm fiasco, I missed the second-to-last ferry back to the Island. I didn’t get home until midnight. Exhausted, I slept in. Past seven. Missing the June 30, 8:30 am ferry to Horseshoe Bay.
  4. Which ran aground at 10:10, leaving passengers stranded on board for seven hours.

And that was how a series of little inconveniences allowed me to avoid one big inconvenience, or something. I’m busy and tired and haven’t proofread this, so maybe this post is about something entirely different.

(*) As in, no one shares my apartment, not that I own a private island or anything.

(**) In all fairness, it should be noted that ferry service to and from Vancouver Island is, in general, spectacularly efficient.

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