Tall, Dark, and Mysterious

7/23/2005

Is there a cognitive psychologist in the house?

File under: Those Who Can't. Posted by Moebius Stripper at 4:36 pm.

The comments to my post on math prerequisites that aren’t being met are the reason that, despite all of the comment spam, I leave these pages world-writable. Some excellent stuff there, and I hope that some curriculum developers in my province are reading what my readers have to say. I have more to say in the prerequisites thread, but first I wanted to address some comments from high school math teachers who assure me that their students aren’t leaving their classrooms without knowing what an equation was, how to add fractions, and so on.

I believe them. To look at the BC high school curriculum, it’s hard to find anything specific that is explicitly wrong with the course. (Aside, that is, from the “This book is brought to you by the letters T and I” thing, which I suppose is a big one.) By the time British Columbian teenagers leave grade twelve, they’ve been exposed to fractions, exponents, quadratic equations, graphs, and logarithms - the prerequisites for college math. They’ve even been exposed to questions that are a lot more open-ended than I expected - questions that demand a modicum of creativity.

The problem is that many students - perhaps most - never properly learned this stuff. Others knew it to some degree at some point, and promptly forgot it.

I’ve been tutoring a student in that second category - he can’t remember how to solve a linear equation (when do you divide both sides by the same number? When do you add stuff to both sides?), he can’t remember how to evaluate powers, he can’t remember…much of anything he learned. And nothing makes me feel like a failure as a teacher than hearing that the mathematics I’ve been trying to teach has come across as something merely to remember.

Can anyone recommend any reading material, accessible to a layperson, on how knowledge is stored? Why, and how, some information gets stored in short-term memory, some finds its way into long-term, and other types of data - a first language, for instance - is not something that people think about as knowing and not as remembering? Because I don’t think it’s something intrinsic about mathematics that it seems to get stored into the most unreliable section of my students’ brains.

As I was thinking about this post, reader Susan made an insightful connection between math (something students memorize) and language (something students learn):

It’s possible for a non-expert to determine whether a child is literate by asking him or her to read something unfamiliar (ideally both outloud and silently) and to explain what they’ve read in their own words.

We need to define and expect something similar as far as being numerate.

To which I add: not expecting anything similar for math gives students no reason to process and learn math instead of just remember it.

I’m only now remembering that I actually did do something similar last term, although I didn’t draw the connection to reading that Susan did. Last term, after one particularly miserably-done test, I gave my students an opportunity to submit (for credit) corrections to the word problems. I gave them a template to follow: they had to describe, in words, what information they were given in the problem, and what they were missing. They had to describe, in words, how those things were related, and then, based on their previous work they had to provide the equations they needed. A few students actually came up to me and said that that exercise really helped them understand what they were doing with word problems. I was shocked - this was the horrible precalculus class - and encouraged. A month later, however, I found that those same students drew a blank when faced with the word problems on the test.

We’ve got a long way to go.

One could argue that I went into it with a negative attitude

File under: Know Thyself, What I Did On My Summer Vacation. Posted by Moebius Stripper at 3:57 pm.

Because I don’t think I want to spend the rest of my life teaching precalculus to students who can’t compute 5*0 without a calculator, and because the local colleges don’t seem intent on letting me do that anyway, and because a temporary lapse of self-awareness made me completely forget the “independent to a fault, don’t need nothin’ from nobody” aspect of my character, I decided to pay a visit to a career counsellor.

Summary: big, fat waste of time. Upon reflection, I think the biggest problem is that I’d assumed, incorrectly, that an employment counsellor was an expert on jobs. It turns out that an employment counsellor’s expertise is actually a step removed from, and hence a step less useful than, that: mine clearly specialized in the “job-seeking process”. Which meant that when I came in with a list of my skills, interests, and possible employers that I’d like to research (this last one is non-trivial, as there are confidentiality issues involved with the employers I’m interested in; I’m not going to go into details), he couldn’t help me with that. He could, however, refer me to a “career exploration program”, where I’d be able to “explore my strengths and weaknesses”, “discover my interests” and other somesuch; this would be explained in greater detail in the pamphlet he gave me. Leafing through it, I noticed that the first day of the three-week program would be devoted to discovering my Myers-Briggs personality type.

INTJ,” I told him. “And I know what my interests and strengths and weaknesses are. I’m a highly analytical, independent worker with no patience for small talk and routine. I have some types of careers in mind; I need more specialized direction than this.”

From the look on his face, I gathered that no one had ever made such a request before.

That wasn’t the only problem. The meeting, actually, started going poorly even before I shook hands with the man: I arrived on time, and spent the next thirty-five minutes in the waiting room while the counsellor was “almost done, really, we’re sorry about this.” At the stroke of n-thirty, he finally emerged from his office and presented himself to the (newly-hired) secretary, and proceeded to admonish her gently for booking half hour appointments instead of full-hour appointments. He needed a full hour, he explained, for new clients. Somehow this didn’t translate into my own consultation lasting for more than twenty-three minutes, or involving the counsellor doing things like actually reading the resume I’d been instructed to print out, but I was ready to leave after twenty-three minutes (see above) anyway, so I wasn’t about to object.

Once in the room, he took a minute, literally, to scan the form I’d filled out in the waiting room. Like all government forms I’ve ever filled out, this one contained an optional section in which one can identify oneself as belonging to one or more of various groups; like all government forms I’d ever filled out, I opted to leave this part blank. Noticing that I am severe-featured and dark-skinned, the employment counsellor quickly proceeded to engage me in a variant of Twenty Questions that I swear to God I play every other month:

“So, where are you from?”
“Ontario.”
“You were…born there?”
“Yes.”
“Where were your parents born?”
“Quebec.”
“I see…”

Just his luck, I’m third-generation, so he reluctantly abandoned that line. But seriously, I understand why there’s an ethnicity field on these sorts of forms, but I understand even more so why filling out such a section is optional, and I resent it when people who really should know better try to coax such information out of me. Am I the first person who has ever chosen not to fill in a few optional fields on a government form? Shee.

(Aside: every now and again someone asks me, point-blank, “What is your ethnicity?” I’ve had bad luck answering honestly, as my experience has been that more people fancy themselves experts on the Israeli-Palestinian conflict than is warranted, and a lot of them are just dying to give their input on the matter. Finding out that the bloodline of their interlocutor intersects with that of some folks who live in that region, by the way, apparently constitutes a capital opportunity to do so. So when the Swedish hosts of a B&B on Denman Island posed the question, I responded in my preferred way: by selecting, at random, a country that lies roughly on the line joining Warsaw to Bombay, and claiming ancestry. “I’m half Turkish,” I lied, and the Swedish wife turned to the husband and said something rushed and excited that had the cadence of I told you, didn’t I tell you? and the husband turned to me and smiled weakly, as though to say, No she didn’t, but what can I do?)

Back to the meeting: the best I could say about it is that unlike almost all of the academic types who have counselled me on employment, Employment Counsellor was not of the mind that I’d never get anywhere without a Ph.D. Alas, he opted for the other extreme, and wrote off my education altogether: “Oh, I see you can use a computer,” he remarked approvingly, as he glanced at my resume, skipping over things like the title of my thesis (understandable), my Dean’s List placement at my alma mater (less so), and a description of the ceramic dinner set I’d been commissioned to make (ok, fine). And, yes, I can use a computer to a degree that puts me in direct competition with only three quarters of the youth in my province, rather than all of them, but, my lord.

Like many members of my demographic - gifted kids of professionals, who were directed to seek scholarship, rather than employment, in their studies, and who were never given much guidance with regards to the latter - I am finding myself suspended between two distinct groups that are, for opposite reasons, ill-suited to help me. On the one hand are the intellectuals who can’t fathom a universe outside the academy, and hence cannot help me find my way in that world; on the other, the folks who never studied a subject as abstract and as technical as mathematics beyond the high school level, and consequently can’t provide the specialized direction I need to apply my own abstract and technical interests and skills outside the academy. Frustrating, because I know that there are math folks employed in statistics and in finance and in the military and elsewhere, and they didn’t hatch ready-made inside their cubicles.

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!

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