Tall, Dark, and Mysterious

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.

1/3/2006

The Gentle Art of Driving Your Employees Insane

File under: Sound And Fury, Welcome To The Occupation. Posted by Moebius Stripper at 7:12 pm.

A few weeks ago, a friend of mine sent me this marvelous guide to writing unmaintainable code. Though long, it’s an easy read, and it’s too good to excerpt, so go. But impressed as I am by the deliberate use of accented letters, comments masquerading as code, and variable names that sound like keywords, none of this holds a candle to my own employer’s practice of maintaining uncodability. And they do it all without even trying! You can’t nurture this kind of talent; some are born with it, and others can but gaze upon it from afar and wonder, “indexing a record in four distinct ways, storing those indices in eight different tables in seven directories, and using each type of index for a different purpose? That would never have even occurred to me.” It’s beautiful in its own way, really, kind of like…you know that Far Side comic where a fidgity passenger’s movement could cause the wings of a plane to fall off? Sort of like that.

As the resident accidental database developer, though, I get to deal with this genius more intimately than most of my coworkers, save one, the IT guy who’s got his own database going. I asked him if he had any advice for me. He did.

“You can’t fight City Hall,” he said, staring vacantly at his own monitor. “So don’t waste your time trying.”

Anything else?

“City Hall’s a bitch.”

Noted. But I plodded ahead anyway, creating an interface that was useful and more or less maintainable. The database was centred around a form that contained data that I use regularly - one page for each warehouse I deal with, and a button on each page that brings up the monthly warehouse report.

The paths and filenames for the monthly warehouse reports were easy enough to generate: each month had its own directory, and each filname was of the form followed the template [Warehouse code][Item code].doc. The monthly reports themselves were not all produced at the same time, so I’d have to poke around in the code every now and again to make sure that I was pointing to the right directory, but other than that, the thing ran itself.

Until the December reports came in, and suddenly I was getting “File Not Found” errors.

I poked around in the December directory, and lo, all of a sudden the filenames were following a completely different convention. Instead of [Warehouse code][Item code].doc, we had [Supplier name][random eight-digit string].doc.

I spent a few minutes searching for a table that decoded the eight-digit strings before realizing that there was no point trying to search for those things. I figured the IT guy would know.

“Eight-digit string? Yeah,” he said, “that’s the item number. Sometimes the filenames follow that convention.”

“Item number?” I said. “How is that different from the item code?”

“The item code,” explained the IT guy, peering over the tops of his glasses, “has six alphabetic characters. The item number has eight digits.”

“I can see that, but -”

Right. He had a file relating item numbers to item codes, and I imported it into my database, tweaked my code, and -

FILE NOT FOUND. Again.

Back to the directory, where I discovered that the eight-digit codes in the IT guy’s table didn’t match the eight-digit codes in the folder of December warehouse reports. Some of them different in one place. Others differed in three or four. Still others were completely different.

“These numbers aren’t the item numbers,” I regretfully informed the IT guy. “Do you have any idea what else they could be?”

Of course he did; he’s been with the company eleven years. “Probably the product numbers.”

The product numbers? And these are different from the item numbers how?

“Well,” explained the IT guy, “If you look at the table of product numbers and item numbers you’ll see that they’re clearly different numbers.”

Oh, so that table existed? I could get one relating the product numbers to the item numbers?

“Yeah, of course we have that table,” said the IT guy. “I mean, product is just another word for item.”

I didn’t ask. I just returned to my desk, imported the relevant data, checked the eight-digit strings against the ones I had, changed a few field names, and ran the program, confident that I wouldn’t get any errors.

The first warehouse report came up without any problems. So did the second. And the third. And the fourth. And the fifth. Programmers who work with sane datasets might be satisfied with those sorts of results, but I am not a programmer who works with sane datasets. Indeed, I got an error on the sixth report.

Peeking into the directory of December reports, I saw that Product #6 had a completely different product number than the one generated by my program. I approached the IT guy again.

“What did I tell you about City Hall?” he asked.

“Big bitch?”

He nodded. “What you’ve got there is the other version of the report for Product #6. Some of the warehouses have the first version. Others have the second. Others have both.”

“Is there any way to tell whi-”

“No.”

Because I am sucker for punishment, I poked around the December directory to find some products for which there were two version of the report. I found one, and opened the two files, and compared them.

They were identical. Word. for. word. identical.

I asked the IT guy if there was something I was missing, because I couldn’t see why we’d have two versions when there isn’t a difference and was there any way to tell which version I’d get so as to avoid errors, because I couldn’t see -

“City Hall,” he said.

I gave up on the program. The way I see it, come time for the January reports, we’ll be back to the old naming conventions anyway.

11/20/2005

Moebius Stripper’s Guide to Public Speaking

File under: Sound And Fury, Welcome To The Occupation. Posted by Moebius Stripper at 11:02 am.

Actually, there’s only one point, and it is this: An essay and a speech are different media. If you’re going to drag several dozen people to a different city to hear your talk, rather than just giving them essays to read, you need to justify the time and expense involved. When your presentation consists of you just reading an essay you wrote, your message is not best delivered as a speech. It is not even best delivered as a speech if:

  • It is accompanied by a PowerPoint presentation that consists entirely of excerpts from the very paper you are reading.

  • You periodically glance up at your audience for a tenth of a second at a time, to make it look like you’re giving a speech rather than reading from a paper.

  • You write faux spontaneity directly into your essay. This includes, but it not limited to:

    • Jokes of any type.
    • On the spot “observations” such as “You all look very excited to be here.” When you can’t even look up at your audience when you say that line, it loses a shred of credibility.
    • The line “This award comes as a complete surprise to me, so I didn’t prepare a speech.” Dude, you’re not fooling anyone: you walked up to the podium with a goddamn folder, for crying out loud, which you proceeded to open and then read from FOR TEN MINUTES. You even read that line from the folder.
  • Its title is Effective Teaching: Nurturing Active Learners. Tell me, do you think that you are underlining or undermining your message when you deliver your findings by monotonously reading a paper at your audience for two straight hours without interacting with them? Me, I’m gonna have to go with undermining.

(In other, unrelated news, the business conference was superfantastic and not painful at all! But it’s good to be back at home so that I can blog about things that have nothing to do with the conference and were not in any way inspired by it.)

10/26/2005

‘A journalist doesn’t think she should know statistics. Bloody hell.’

File under: Sound And Fury, Queen of Sciences, I Read The News Today, Oh Boy. Posted by Moebius Stripper at 9:38 pm.

Thus commented Geoff the other day. Timely remark, that one, for tonight a whopping eighty-five percent of Canadian adults waited eagerly to see if they took home the Lotto 6/49’s record-breaking $40 million jackpot, and we all know what that means: it means that it’s time for journalists nationwide to present us with a panel of mathematical geniuses who regret to inform you, Joe Ticketbuyer, that you’re probably not going to win:

The results of the biggest lottery jackpot in Canadian history will be announced tonight, but experts are warning people not to get their hopes up.

For the uninitiated: you play the Lotto 6/49 by selecting 6 numbers out of 49. You win (part of) the jackpot if all six of your numbers all match the six numbers drawn.

According to experts, this event - whose probability, by the way, I had my never-were-any-good-at-math-and-always-hated-the-subject psych majors compute in my discrete math class last year, and most actually managed to do so correctly - is unlikely.

What are the chances that your ticket will hit the $40 million Lotto 6/49 jackpot?

Not good, according to Simon Fraser University Professor (well, senior lecturer, but who’s keeping track? - Ed.) Malgorzata Dubiel. She has calculated that the odds are just short of one in 14 million.

All the while muttering to herself: “For this I got a Ph.D.?”

Semantic quibble: the odds are slightly better than one in 14 million; they’re one in 13 983 816. So I take issue with the use of the phrase “just short”, which implies “a bit less than”, no?

Anyway, is followed by three paragraphs about the would-be philanthropist who said he’d donate half his winnings to charity if he won the jackpot (he didn’t), and then this:

[Dubiel] also debunked the myth that a person can “crack the code” of lotteries.

“Everything we know about mathematics says no, it can’t be done.”

This makes it sound as though the sum total of the mathematical canon to date, from Archimedes to Zariski, was brought to bear on the age-old question of “Stochastic Processes: Totally Stochastic, Or Just Kind Of?”. And, at last, produced the long-awaited conclusion that as a matter of fact, God occasionally does play dice with the universe, at least when He’s choosing the Lotto numbers. On top of that, I wince at the “everything we know” wording, which is reminiscent of Underwood Dudley’s dealings with aspiring angle trisectors. Many of those sorry folks explained their obsession with the problem with something along the lines of “mathematicians say that trisecting an angle using compass and straightedge alone is impossible, but they’re just not trying hard enough.”

But I digress: whether or not the lotto code is crackable isn’t a mathematical question, dammit. If the code is crackable, it’s because the random number generator selecting the numbers is somehow not completely random, and seriously? Take it to a computer scientist, dude. (Except that…lotto numbers are still selected by that spinny thing with the balls, no? And we’re asking a mathematician if it gets all spinny on the balls in a crackable way? Is this what people start thinking when they watch shows like Numbthreers?)

Dubiel admitted that there have been cases of people winning multiple times, but put it down to luck.

Note the use of the transitional term ‘but’, which is typically used to contrast one idea with an ostensibly opposed one. As in, there’s an apparent contradiction between the existence of multiple lottery winners, and the absence of mathemagical gnomes that select them deterministically.

“People simply put too much faith in something that is just coincidence,” she said.

They’re also too easily wowed when mathematicians remind them of the stuff they saw in Chapter 8 in their grade twelve high school math text, but that’s neither here nor there.

10/20/2005

Show me the data.

File under: Righteous Indignation, Sound And Fury, Those Who Can't. Posted by Moebius Stripper at 10:56 pm.

What’s that you say? You’re sick of all those long-winded education rants that never go anywhere? Me too! However, I can’t pass up this opportunity to commend the California Commission on Teacher Credentialing for so succinctly summarizing everything that’s wrong with elementary-school-level mathematics education today. And in an exam that every prospective teacher in the state is required to write, to boot - now that’s efficient delivery!

Many people believe children will never learn mathematics if allowed to use pocket calculators. Having spent countless hours memorizing multiplication tables and doing long-division problems unaided by any mechanical device, many adults cannot conceive of anyone acquiring this knowledge without similar effort and practice. ______________. What many people fail to understand is that mathematics is constantly evolving; it is not a fixed body of facts. Students must still learn basic skills, but they do not need to perform the endlessly repetitive exercises that calculators largely eliminate. Youngsters can better use their time—time they would have spent performing long-division problems—to learn mathematical concepts that will enable them to become better problem solvers.

Which sentence, if inserted into the blank line, would best focus attention on the main idea of the passage?

(A) It is true that mathematics is not the easiest subject in the typical elementary school curriculum.

(B) Many of you have doubtless heard about the bitter classroom experiences of students who learned mathematics this way.

(C) There is much to be said for instilling this kind of discipline in students.

(D) Although it was clearly not fun, students trained in this manner rarely forgot what they had learned.

(E) Such views, however, seem to reflect a resistance to change rather than a rational approach to mathematics instruction.

(F) Contrast this instance of common sense with the hallucination that follows.

Just kidding! (F) isn’t an option. The correct answer is (E).

Yes, yes: fish in a barrel are targets for amateur marksmen, I know. But what we have here is a barrelful of tranquilized guppies that been given free reign over early mathematics education, so let’s have a go at it before any more metaphors get mixed:

What many people fail to understand is that mathematics is constantly evolving; it is not a fixed body of facts.

You know, I skewered this one months ago. I even used the word evolved, for crying out loud. My readers found this -

Yes, if there’s one field that has evolved beyond recognition in the past quarter-century, it’s introductory calculus. Why, back when I was a tot, the area of a rectangle was length plus width, the derivative of sin x was 5, and we only had whole numbers, so whenever we needed to compute the area of a circle we had to use pi=3, AND YOU NEVER HEARD US COMPLAIN.

- to be hysterically funny, because it was so over-the-top. Except that…it wasn’t. It was very much beneath the top. It was smack dab in the middle of what those enlightened educators, unlike “many people”, have succeeded in understanding.

Students must still learn basic skills, but they do not need to perform the endlessly repetitive exercises that calculators largely eliminate.

They don’t? Really? I’m not convinced. Let’s see what a cognitive scientist has to say about this!

It is difficult to overstate the value of practice. For a new skill to become automatic or for new knowledge to become long-lasting, sustained practice, beyond the point of mastery, is necessary.

… By sustained practice I mean regular, ongoing review or use of the target material (e.g., regularly using new calculating skills to solve increasingly more complex math problems, reflecting on recently-learned historical material as one studies a subsequent history unit, taking regular quizzes or tests that draw on material learned earlier in the year). This kind of practice past the point of mastery is necessary to meet any of these three important goals of instruction: acquiring facts and knowledge, learning skills, or becoming an expert.

But what does that guy know, anyway? The times (tables), they are a-changin’! Mathematics is evolving! Get with the program!

I’m curious about something, though: is this anti-repetition view unique to mathematics education? Have the musicians among my readers, for instance, noticed a similar trend in music pedagogy? Music is EVOLVING! It is not a FIXED BODY OF FACTS! We have new-fangled technomology that enables students to bypass all that boring stuff, like learning scales! Or…do music students still practice scales, even though scales really aren’t that much fun to practice? I know that when I teach pottery, I spend a fair bit of time focusing on the basic skill of centering the lump of clay, even though it’s more rewarding to throw teapots.

Onward:

Youngsters can better use their time—time they would have spent performing long-division problems—to learn mathematical concepts that will enable them to become better problem solvers.

Joanne Jacobs speaks for me:

I’ve spent countless minutes (that seemed like hours) waiting for students I’ve tutored to multiply 4 times 5 or 3 times 6. Oddly enough, they weren’t adept at understanding mathematical concepts or solving problems.

The idea that students who don’t waste their time learning the boring basics will have freed up valuable time and brain space to become creative problem-solvers is one I’ve seen cited often. It’s an idea that every single mathematics educator with whom I have ever communicated - and I have communicated with hundreds, across several continents, both in person and through this blog - considers to be utter hogwash. But it’s possible that we’re just outliers. It’s possible that on the whole, students who don’t learn their times tables beyond mastery are brighter, bolder, and more creative problem solvers than their predecessors. And if that’s the case, then there should be plenty of statistics to bear that out. And surely the proponents of calculator-based curricula have plenty of rigorous statistical studies at their fingertips.

Let’s see them. Put up or shut up.

Show me the data.

Show me a single peer-reviewed study that indicates that students who were raised with calculators show greater facility and more creativity in problem-solving than those who were raised without. I want to see test scores. I want to see comparisons of performances in university-level mathematics classes between students who were made to memorize and practice their times tables, and those who weren’t. I want to see evidence that learning the basics and learning advanced mathematics are negatively correlated.

Until then, I’m going to rationally resist change, and teach mathematics in the way that my peers and I actually learned it.

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