Tall, Dark, and Mysterious

6/8/2006

The Modern-Day Math Test, or, How I Became a Curmudgeon

File under: Righteous Indignation, Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 10:03 pm.

(Or, Wherein the Author Learns Who Subscribes to the RSS Feed.)

The other day, against my better judgement, I took on some contract textbook writing work with a company based in India. The main reason, I confess, is so that I can tell my friends that India is outsourcing work to me, something that I’ll also be sure to mention on my résumé when I inevitably overstay my welcome at my current place of employment. The bad news is that the textbook company produces books for use in the United States, which means that I get paid in American dollars, not rupees. Alas. But here’s the general procedure: someone from an American school board sends specs to the Indian company, which hires an American to write an outline for a text, which then gets sent to a Canadian (me) to turn into multiple choice questions. India the goes over the questions, and sends them to someone else (an American?) to review and revise. This seems to me like an awful lot of trouble to go to to produce (what appears to me to be) a text that is essentially indistinguishable from every other text in use in the public school system, but I don’t make the rules: I just get paid to follow them.

Fortunately, the Indian textbook company is not in any way affiliated with the body that produces and markets fucking graphing calculators. However, no company that mass-produces textbooks for use American public schools can remain solvent without permitting the use of some sort of scientific calculator at the high school level, which makes questions such as

5. Which of the following is equivalent to sqrt(50)-sqrt(8)?
a. …
b. …
c. …
d. …

several orders of magnitude stupider than their non-multiple choice equivalents.

You see the problem.

Calculator use aside: in coming up with the incorrect multiple-choice options, I am finding myself borrowing liberally from some of my earlier work in the field. Needless to say, I’m milking the “everything is linear” fallacy, which features prominently in every single option set I’ve written so far, for all it’s worth. So far, India seems happy with my work, which can only mean that India is confident that, for example, hoardes of students will continue incorrectly expanding polynomials in exactly the same way that teachers have warned students against incorrectly expanding polynomials since time immemorial. Good instruction has minimal effect on the frequency of the most common types of algebra errors, I’ve found, so I don’t have to worry that there will be a classroom of students who overlook Option D en masse because duh, everyone knows that (x+y)^2 does not equal x^2+y^2.

No, I’ve done good work with this text, so far. My employers assure me that my questions will be sufficiently confounding to high school students, of whom negligibly few are even adept enough at plugging expressions into their calculators to compare multiple choice options to an unsimplified expression that it’s well worth asking several questions that can be solved that way; and of whom hardly any understand enough algebra to avoid half of the wrong choices I’ve provided.

A skilled teacher is one who acts as a positive influence on her students, and manages to inspire them. Lofty goals, those, and ones that are seldom attained. It’s far easier - and potentially more lucrative - to be a skilled multiple-choice math question writer, whose success is commensurate with the ability to correctly predict her subjects’ deficiencies, which are never in short supply.

4/1/2006

Drinking (tea) with mathematicians

File under: I Made It Out Of Clay, Queen of Sciences. Posted by Moebius Stripper at 9:02 pm.

This morning, I made myself some tea in one of the matching mugs from my dinner set, waited an inordinately long time for it to start to cool down, and then downed the rest of it in a few gulps because it was quickly becoming too cold to drink. Not the most satisfying tea-drinking experience, but that’s what I get (I think) for throwing narrow cylinders.

So I started to wonder what shape of mug, made out of clay of constant thickness, would be ideal for drinking hot beverages. Specifically: I would like to pour myself some tea, wait a fixed time t for the surface of the tea to cool down to a drinkable temperature before beginning to sip, and then sip at a fixed (and preferably constant) rate until the tea is gone, with the tea remaining at a constant temperature. It seems like this should be doable; at the very least, I don’t think it’s not a ridiculous thing to ask for: the surface of the tea cools much faster than the tea below, and so there should be a way to sip tea from the appropriate vessel in such a way that each sip exposes another layer of tea that cools down to the temperature of the previous layer, just as I’m drinking it. It also seems clear that the top should be wider than the bottom, as the tea at the bottom will not be completely insulated by the tea above. Also, the heat transfer through the mug cannot be ignored: the specific heat capacity of clay is around a third of that of water, and only a third more than that of air.

…that’s as far as I got: I don’t have the physics background to set up this problem, though I spoke with someone who did, and he came up with some rather grisly equations that I don’t have the calculus background to solve. I also don’t know what simplifications could reasonably be made. But if anyone does, and can come up with a reasonable shape of mug that is conducive to tea-drinking, I’ll make you such a mug.

3/12/2006

Driving with mathematicians

File under: Queen of Sciences. Posted by Moebius Stripper at 6:49 pm.

“How many gas stations do you think we’ll pass on the way home?”

“You mean on the side of the road, or at exits?”

“Whichever - places where we’d consider buying gas.”

“I don’t know, six, eight…why?”

“Well, if we pass the first 1/e gas stations and then stop at the first place that offers cheaper gas, then we’ll maximize the chance that we’ll end up buying the cheapest gas. I mean, assuming that gas prices are random and don’t follow any trend with regards to location, which obviously isn’t the case.”

“Oh. Ha! But that’s not the algorithm we want - that one maximizes the probability of getting the cheapest gas; it doesn’t minimize the expected price of the gas we buy, which is what we want.”

“Oh, yeah, you’re right: if we apply that first algorithm, then in the likely event that we don’t get the very cheapest gas, there’s still a reasonable chance that we’ll end up with pretty expensive gas.”

“Right.”

“So what’s the strategy for minimizing the expected price of gas?”

“I don’t know.”

So, is there a probabilist in the house?

3/2/2006

Math class: now with more social justice (and less math)

File under: Character Writ Large, Righteous Indignation, Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 8:54 pm.

The adage that addresses the issue of judging books by their covers counsels unambiguously against, but I’ve always rejected it on the grounds that it assumes, generally incorrectly, that authors have no editorial control over the presentation of their work. I unabashedly judge books by their covers in the figurative sense; but at the moment I’m being quite literal. Specifically, I am judging Rethinking Mathematics by its cover:

The authors of this tome aim to “provide examples of how to weave social justice issues throughout the mathematics curriculum and how to integrate mathematics into other curricular areas”, and if the cover is one such example, then we can safely conclude that the integral of SOCIAL JUSTICE ISSUES + MATHEMATICS CURRICULUM equals GIBBERISH. It’s an equation! No, it’s an inequality! No, wait…it’s bullshit! Seriously -what are the units of MULTICULTURALISM * POVERTY / INEQUALITY?

I haven’t read the book. My local library doesn’t carry it, and if I were to pay the $16 cover price to purchase it from pair of white guys who write about economic racism, well, I fear that that would make me part of the problem.

Nevertheless, even those of us who haven’t read the book can find plenty to criticize (cf “critical thinking”) in the introduction alone, which decries the “unfortunate scarcity of social justice connections” that are to be found in conventional high school mathematics curricula. Sadly, some old fuddy-duddies think that math classes should deal with stuff like trigonometry, as opposed to, say, the War on Iraq Boondocks Cartoon. Those naysayers, however, are just party poopers who totally don’t get it, but they can easily be convinced that a a social justice approach is consistent with your (yes, your) state’s mathematics standards:

Occasionally, a teacher needs to defend this kind of curriculum to supervisors, colleagues, or parents. One approach is to survey your state’s math standards (or the national standards) and to find references to “critical thinking” or “problem solving” and use those to explain your curriculum. Also, the NCTM clearly states that “mathematical connections” between curriculum and students’ lives are important.

There’s a valuable lesson to be learned here, actually, and it is this: if the description of your curriculum is impenetrably vague and long-winded, then people can use it to justify anything. For instance, the paid-by-the-word folks behind the Illinois math curriclum talk about a goal, within the math class, for students to “express and interpret information and ideas”, from which one could argue that it follows that we should be teaching interpretive dance in lieu of geometry.

This book pisses me off. It pisses me off a lot. Not because I think that math and life should be disjoint - quite the contrary, as I’ve frequently argued in this space. Hell - during the first ten minutes of the statistics-for-social-scientists course I taught last year, I stated, in so many words, that no one can accurately claim to be a fully-functioning member of a democratic society if they can’t interpret quantitative data. (Nor am I the first to make this argument; John Allen Paulos, author of the marvellous - and bestselling - book Innumeracy, says as much himself. The fact that the authors of Rethinking Mathematics are so unfamiliar with the literature that exists on the topic of mathematical literacy that they claim that theirs is the only resource of its kind, does not speak well to their expertise on the subject.) I also don’t think that the social-justice-based math curriculum is the dominant force behind students’ appalling inability to work with quantities. Sure, it’s not helping, but to claim that students think that 2/3+5/7=7/10 because their teachers are ideologues who have rethought mathematics is to diminish the roles played by innumerate elementary school teachers, innumerate curriculum developers, absent fathers, working mothers, fucking graphing calculators, sugary breakfast cereals, sex on TV, and shock rock in bringing about that sorry state of affairs. Hell, a good many of the topics in this book look quite worthwhile: the section on how unemployment figures are reported, for instance, seems like a nice topic for the “how you present data impacts what people think” section that appears in every single statistics chapter in every single high school math text, not just this “first of its kind” resource, but anyway! No, I am not opposed to mathematical literacy, and I wish that folks who are more politically-inclined than I would invoke it more often.

No, what bothers is this: is anyone familiar with a movement among social studies educators in secondary schools to use math in their courses, or does the movement toward interdisciplinary studies of social justice only go in the other direction? I am aware of none. Why are the educators who are motivated by political issues - and who see numeracy as a means to that end - injecting those issues into the math curriculum, rather than injecting math into social studies classes - which seems more natural to me? If I think that potters would improve their craft by learning some elementary Newtonian mechanics, I’d sooner give impromptu physics lessons at the pottery wheel than drag my physics classmates to the studio.

Is the overall effect to the high school curriculum, a net reduction of mathematical content?

The authors of Rethinking Mathematics are unabashedly politically-driven, and from the table of contents it is apparent that the math they present in their book leads students, none too subtly, to such conclusions as the one that capitalism is a fundamentally damaging economic system. Leaving aside for the moment the validity of this conclusion - I personally dispute it - let’s consider just how very involved a topic economics is. To come to any conclusion about capitalism requires one of two things: 1) a great deal of in-depth studies of economics and related issues, issues that Ph.D. students have written theses about; or 2) some superficial examination of pre-selected data (is this the Global Capitalist Economy Cartoon mentioned in the book’s table of contents?) that leads directly to the desired conclusion. In the context of a high school math class, (1) entails a huge use of the mathematics class’s time to teach and learn economics, while (2) constitutes brainwashing.

Given how ill-prepared the majority of high school students are to either do mathematics or think (let alone “think critically”, and the first person to point out case of that phrase being used by anyone who doesn’t have an ideological axe to grind, gets a cookie), you’ll forgive me if I can’t get on board with either of those two options.

This book, if used more than very sparingly, will give innumerate high school students highly skewed foundations on a wealth of complicated topics, and direct them to predetermined conclusions. Judging from the table of contents, it might prepare students for jobs preparing statistical expositions of positions espoused by lefty think-tanks. And, hell, that’s more than a lot of high school classes prepare students for, so I can’t even find fault with that; the problem is that while grooming students for that path, the social-justice math class will inevitably omit, because of time constraints, some other topics that might prepare students for further study in other areas. Will students whose teachers are motivated by social justice concerns learn enough trigonometry to hold their own in a university engineering course, should they wish to pursue that path? Will they learn enough algebra to succeed in the chemistry courses required by every medical school? The authors of this text talk about using mathematics to “potentially change the world”, which is hardly the exclusive domain of the social justice activists: anyone who thinks that engineers and doctors haven’t used math to change the world, has spent too long at rallies and is brainwashed beyond salvation. Engineers and doctors have changed the world for the better, even if measured in social justice terms. A robust, demanding, contentful high school mathematics curriculum, even one that suffers from an “unfortunate scarcity of social justice connections” (yes, they did use that phrase to describe the standard high school math curriculum, because you know that when I see the “how to use your fucking graphing calculator to plot a straight line” unit, the first thing I think is “but where’s the social justice?”) will leave the door open for students to acquire the tools they will need to use math to change the world - whether or not they later choose to become social justice crusaders. A curriclum designed to “guide students towards a social justice orientation” will cripple them if they choose any path other than that one.

And given how deluded high school students seem to be about the nature of equations , it can’t be a good idea to let them anywhere near that horrible cover.

1/31/2006

Technology: the cause of, and solution to, all of life’s problems

File under: Sound And Fury, Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 8:58 pm.

Reading this article about a series of math workshops directed at students and parents, I am reminded of a famous fifty-year-old psychology experiment:

In Festinger and Carlsmith’s classic 1959 experiment, students were made to perform tedious and meaningless tasks, consisting of turning pegs quarter-turns, then removing them from a board, then putting them back in, and so forth. Subjects rated these tasks very negatively. After a long period of doing this, students were told the experiment was over and they could leave.

However, the experimenter then asked the subject…to try to persuade another subject (who was actually a confederate) that the dull, boring tasks the subject had just completed were actually interesting and engaging. Some subjects were paid $20 [for this], another group was paid $1…

When [later] asked to rate the peg-turning tasks, those in the $1 group showed a much greater propensity to embellish in favor of the experiment when asked to lie about the tasks. Experimenters theorized that when paid only $1, students were forced to internalize the attitude they were induced to express, because they had no other justification. Those in the $20 condition, it is argued, had an obvious external justification for their behavior, which the experimenters claim explains their lesser willingness to lie favoring the tasks in the experiment.

In what I can only infer to be the 2006 version of this experiment, two math experts who believe that students rely too much on calculators, are then sent into schools to…teach students to use calculators.

Sunshine and Speier will show students how to do math problems without having to reach for the calculator.

Sunshine and Speier both said students rely too much on using the calculator to solve math problems.

“Get the pencils and papers into their hands as soon as possible…,” Sunshine said.

Sounds about right. I can’t wait to see where this is going!

Speier will also work with Lego Robotics and show high school students how to use graphing calculators.

Huh? But didn’t you just say…? Oh, never mind:

Speier and Sunshine will help students understand basic math because they said they have seen students struggle with basic math concepts like multiplication.

So have I, and so, I presume, has everyone who has ever taught math on this continent. And I agree with Speier and Sunshine when they talk about how the best way to understand basic math is to put pencils and papers, rather than fucking graphing calculators, into students’ hands as soon as possible.

What, then, accounts for the schedule of these workshops?

Monday, Jan. 30: Providing a good start in math at home: graphing and multiplication.

Tuesday, Jan. 31: What is Algebra all about? A two-hour crash course in the subject.

Wednesday, Feb. 1: Programming and Robotics with the Lego Robotics systems.

Thursday, Feb. 2: Programming the TI-83+ Calculators.

Given Speier and Sunshine’s lack of enthusiasm for the calculator-based curriculum, my guess is that Texas Instruments put them into the $20 group.

1/26/2006

What do you call it when a person deliberately seeks out psychologically unhealthy attachments?

File under: Righteous Indignation, Sound And Fury, Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 6:34 pm.

I often forget just how dysfunctional a relationship many of my weaker students have with mathematics.

It’s been a long time since I’ve been surprised at the extent to which such students harbour an unproductive and damaging belief that mathematics is nothing more than a mishmash of symbols and voodoo procedures. After all, this is understandable, what with students being taught that memorizing templates of questions and plugging memorized formulas into their fucking graphing calculators is homologous with “doing mathematics”.

What surprised me for a long time after - at least the first fifty times I encountered the phenomenon - was how resistent these same students are to seeing mathematics as anything other than a collection of disconnected formulas and calculator algorithms.

The other week, I found myself teaching introductory graphing to a handful of students. Partway through a lesson, one student asked me - how do I graph the line in this question? Do I find two points and join them, or should I just find one point and the slope and then graph it that way?

Giddy with delight at this hint of outside-the-box thinking, I replied: you can do it either way you want! It’s your choice! Both of these options are totally valid methods of graphing the line! Two points, point slope, it’s up to you! In fact, you can graph it one way, and then if you want to check your work, you can graph it the other way, and ISN’T MATHEMATICS SUPER?

Pregnant pause. Hesitation. The barely-perceptible tremours of a worldview beginning to collapse unto itself.

There are two ways to do this question?

Yes! Not one, but two (2) ways to achieve the goal of graphing a straight line! Pick one! It’s entirely up to you!

But which way should WE do it?

EITHER way! The easy way! The quick way! Try ‘em both for practice, and then on your homework you can do the graphing questions whichever way you prefer, unless you’re explicitly instructed otherwise!

Facial expression indictating flicker of hope. Oh, so sometimes you’ll tell us which way to do it?

Well, yes, sometimes, because I want to make sure you understand both methods, but in general I’ll -

But which is the right way THIS time?

Do it both ways, and if you did it right, you’ll get the same line both times!

Shock and awe. Oh, we get the same line? No matter which way we do it?

Yes! That’s the point - these two methods are different ways of answering the same question correctly! Remember last class, when we talked about how we can think of a line as a path, and the two methods of being different ways of giving directions? I can say “start at 8th and Burrard and head north”, which is like a point - 8th and Burrard - and a slope - the direction “north”. Or I can say “start at 8th and Burrard, go to 7th and Burrard, and keep going in the same direction”, which is like two points. These are two ways of describing the exact same route. Just like the two points, or the point-slope method, will give you the exact same line.

Pause that gets pregnant, gives birth, and raises twins to adulthood. But which method is BETTER?

SWEET JESUS GOD, THE POINT-SLOPE METHOD, ARE YOU HAPPY NOW?

Remind me why I bother again? Give them freedom, and they beg for a dictator.

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