Tall, Dark, and Mysterious

1/31/2005

Infinitely many kilometers per liter

File under: Know Thyself, Hubris, I Like To Ride My Bicycle. Posted by Moebius Stripper at 2:53 pm.

At the beginning of the term, I was unhappy with my teaching schedule: sixteen hours per week, thirteen of which are spread over two days, and the other three of which are spread over another two days. I didn’t anticipate one huge benefit of that until the term began: my light teaching days don’t leave me ready to collapse at the end of the day, and that makes them good days to bike to school.

I put 100 km on my bicycle last week, and when I get back home today, it’ll have travelled another 20 km. I used to hate exercising, because I have an unusual mix of preferences and abilities: I’m woefully uncoordinated (which rules out team sports and dancing), I have bad knees (running would be the death of them), I hate being indoors (nix aerobics and the gym track), and I like being able to keep track of my progress (making things like hiking less than ideal). A bicycle equipped with a speedometer/odometer is tailor-made for people like me.

One of these days I should write about the dozen more conversations I’ve had that began with, say - is that an electric bike? It’s made me the most popular kid on the Island.

1/29/2005

Two thoughts on the statistics quiz

File under: Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 6:13 pm.

Last week, Rudbeckia Hirta at Learning Curves posted a handful of definitions of the Pigeonhole Principle, as given by her students. Go read them if you haven’t already: they’re not only spectacularly incorrect, they’re also bizarre, incoherent, and could rival the output of The Random Sentence Generator. I was jealous. My students’ incorrect answers just make me cry, not laugh. I said as much in the comments, and RH replied, You would get equally comic answers if you were able to ask this sort of question. Maybe you could try, “What is a function?”

I have no qualms about designing quizzes with the singular purpose of acquiring fodder for entertaining blog posts, so The opportunity presented itself this week, when I decided to check if my students really knew what the standard deviation measured, and not just how to evaluate it. “What characteristic of a data set does the standard deviation measure? Explain in a sentence or two,” I instructed, and tingled with anticipation as I collected the papers.

I sat down to grade them later that day, and…

I got nothin’ for you, folks. Apparently my students actually know what characteristic of a data set the standard deviation measures.

Hunh. I guess that’s okay, too.

* * *

One aspect of my job that never fails to surprise me: the strange and many ways in which my students don’t understand the subject. I’m still caught off guard by these, even after tutoring and teaching math for years. And I’m not referring to the sort of dear God, why can’t they add fractions already frustration that I experience on occasion and have chronicled in detail, but rather the very frequent oh, they’re confused by THAT. I didn’t even realize that that could be confusing realizations.

For instance: on the stats quiz, I asked students to write down the formula for the standard deviation, and also to specify what each of the variables represented. The question was, for the most part, quite well done: most got the formula right or close to right, and most were able to tell me that x-bar was the mean, the xi’s were the data values, and n was the number of values.

…and then, half a dozen or so of them also went on to explain that the capital sigma meant to add stuff up, and that the symbol that looked like a checkmark with a horizontal tail meant to take the square root.

They don’t know the difference between variables and functions. I had no idea.

1/28/2005

Why invest in the children when you can invest in Texas Instruments?

File under: Sound And Fury, Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 8:57 pm.

Last week, I gave my stats students an assignment: to figure out how to use the statistical functions on their calculators. The distribution of times at which twenty or so of them came to my office for assistance with [above] was heavily skewed to the left, with five or so of my pupils asking for help before the quiz day, and around fifteen coming in between forty five minutes and three hours before they would be called upon to find the standard deviation of a month’s worth of daily snowfalls in Edmonton.

I am not a calculator expert, and took pride in my nonownership of one from the time I ditched my physics minor until the beginning of this term. But I have a decent intuition for these things - most of the time. Half of my students, God bless ‘em, had taken my advice and invested in $15 calculators; I was able, with little effort, to figure out how to finagle standard deviation computations from a variety of those. The other half carted in their TI-83 Pluses, which they’d been required to use in their high school math classes. Most of these students had left their manuals at home, though one girl brought in a slightly older version of the machine, along with the accompanying guidebook. Clearly I’m behind the times; back in my day, we used wee little scientific calculators with ten page instruction booklets and you never heard us complain; but a manual the size of the TI-83 Plus’s could fit, with room to spare, all of the directions that Moses himself would have needed to free his people, lead them through the desert for forty years, and compute the standard deviation of a month’s worth of daily snowfalls in Edmonton. In case you’re interested, it’s available online - all 3.7 MB of it.

For a mere $150, students who think that 5/2+5/2=10/4 can (in theory, anyway) multiply matrices, graph functions (in Cartesian, parametric, or polar form; no need to do any conversions by hand), evaluate antiderivatives of algebraic and transcendental functions (with all of the intermediate steps helpfully displayed) program directly in assembly (according to rohan), and find the covariance of two sets of data. They can also hook up their calculators to the internet and download games that they can play in class. (I’d hate to be the math teacher instructing her students to put that calculator away, right now, or leave the class.) When one girl brought hers to my office, I was tempted to check the menu to see if it could also vacuum my living room carpet and clean my bathtub.

The writer of the otherwise valuable stats text I use (Essentials of Statistics, 2e, by Mario Triola) refers to this beast in every chapter; ditto for the Precalculus book, while my calculus text only “recommends” that its users own graphing calculators, “such as the TI-83 Plus”. I have seen no evidence that the “all calculators, all the time” approach to elementary and high school mathematics education has been anything short of an unmitigated disaster, and I know of no math teachers who disagree with me on that front. I tend, in general, not to assume that my ideological opponents are motivated entirely by money, but I can’t think of any other explanation for this unholy alliance between a wide variety of math text books and this one brand of calculator. One prof at my grad school once railed against the oft-cited explanation that students should use graphing calculators as a means of making their math classes “relevant to real life”: “When on earth does anyone have to use a graphing calculator in real life?” he demanded. “I certainly never do.” Nor am I convinced for even an instant that calculators “[liberate students] from ‘computational algorithms’” so that they may “pursue higher-order activities, like inventing personal methods of long division.” To the contrary: never having become familiar with computational algorithms, many of my students cannot make connections between the nuts and bolts of math and the pictures and equations produced by their calculators. I have dealt with students - plural - who do not know that 3/5 means “3 divided by 5″. Their calculators, you see, have “fraction” buttons, not to be confused with the “division” buttons.

The market for such a powerful calculator should be limited to applied mathematics and physics graduate students and a handful of professionals - far from every single high school math student in British Columbia, and I can’t fathom a charitable motivation for requiring every high school student in British Columbia to possess one. Requiring those students’ parents to shell out such a huge sum of money in order to secure their children decent math marks (not to be confused with decent mathematics educations) makes me ill.

In the calculus and precalculus textbooks I’ve used, the graphing problems have been accompanied by suggestions that students check their work on their graphing calculators. “If you have a graphing calculator,” I tell my students, “it may be worthwhile to use it for this purpose. But don’t rush out to buy a graphing calculator for this class; it’s not worth it. And if you do use a graphing calculator,” I continued, “make sure that you know how to check your calculator’s work.”

“What do you mean?” said one guy in the back row. “We never learned how to do that.”

Sure enough, the ample manual omits those instructions.

1/27/2005

Trying to make precalculus work

File under: Those Who Can't. Posted by Moebius Stripper at 9:23 pm.

My precalculus students wrote a quiz today. On the top of the front page I’d given the instuctions Solve the following equations for x. Show your work. If I’d instead typed Hey, kids! I’m under the impression that you know how to solve quadratic equations, but if you convince me otherwise, I’ll give you ten dollars! I would have received the exact same thirty papers.

This is the class with the crappy attendance. On Tuesdays, two thirds of my students show up. On Thursdays - quiz days - I get a full house, and then close to half of them jump ship after the break. A few weeks ago, I was willing to believe that a third of my students knew this material well enough that they didn’t need to come to class. Apparently I’m just that naive.

On a related note: a few weeks ago, I was chatting with Dr. Matt about review sheets for math tests. He was ambivalent about them, mentioning (I’m paraphrasing, and possibly misrepresenting, here) that he gives out the sheets one day, and then he goes over them the next class, and he wonders how much good that does. I was thinking about this, and also about a recent post at Learning Curves, during today’s class, when I was fielding questions about the homework. Last class, I’d assigned three word problems; this class, my students asked me to go over…three word problems.

They were difficult questions (relatively speaking), and my students have limited experience setting up word problems, so I wasn’t surprised, nor was I bothered. What did bother me was that when I prompted the students with leading questions - okay, we are given a perimeter and we need to find an area…what equations should we set up? I was met with silence. I’d realized it before, but it was only then that I fully saw just how disengaged my students were from the material. Math, for them, is a monkey-see-monkey-do affair. I may talk at length about how we need to figure out relationships among variables and translate them into equations, but my students just see how I apply those to this word problem or that one. They can’t apply a method of a problem about a rectangle to a problem about a right-angled triangle.

I started to set up some of them problems, but then realized that I’d tried this before, and it didn’t work; my students had never learned, on a large scale and to my satisfaction, how to set up word problems that differed from the few I’d shown in class. So I decided I was going to try something new. “I’ll go over these next week,” I said, “but I’ll only go over them if I get some feedback about how you tried to set up the problems. It’s okay if you don’t do it correctly; you can learn from false starts. What’s important is that you try.”

I pointed out that I don’t always know how to approach a word problem when I first see it, and that solving word problems isn’t typically something that comes to people all at once. But, I said, it never comes to anyone if they don’t try their hands at word problems first.

I know that other instructors of this course have thrown in the towel, and were satisfied with their work if their students were able to answer the same question about maximizing the area of the fence that they’d seen solved in class the week before. I personally don’t see the point in that; if the only problems my students can solve are the ones I did in class, then they can’t do math in any useful sense. From now on, I’ll do a couple of word problems in class, and then assign some that are similar in concept but slightly different in structure. And I’ll only go over those problems in class if I can see that my students have at least attempted to apply my lessons on finding relationships among unknowns. If they haven’t managed to make at least some sense of that, they’re not intellectually mature enough to learn what I’m teaching them, and there’s not much point spending more time on this material; doing word problems with students who can only mimic them isn’t any better than not doing them at all. We’ll see how this experiment goes.

Something in the air

File under: Those Who Can't. Posted by Moebius Stripper at 5:28 pm.

I give weekly quizzes in all of my classes, so I hear from my ill students more than I otherwise might: missing a class isn’t necessarily worth alerting the instructor, but a quiz worth part of the mark requires a reason, and possibly a doctor’s note.

I just got my fourth or fifth email - three weeks into the term - from my fourth or fifth student claiming that a crippling migraine was preventing them from coming to class.

Except in extreme circumstances, I tend to think that my students are honest. But I don’t think I had a single student with a migraine last term, and suddenly I have several. If these migraines are real, why the sudden prevalence? And if they’re not, why is this a more popular excuse - by far - than it was last time?

Apparently problem solving isn’t my forte

File under: Meta-Meta, Know Thyself. Posted by Moebius Stripper at 5:23 pm.

Last weekend, I took my new laptop to Vancouver, and when the two of us returned to Island Town, the sound system didn’t work. At all: I couldn’t play CDs, I couldn’t use headphones, I couldn’t get any audio on streaming videos. And this is a new computer! After several iterations of the Universal Computer Troubleshooting Procedure (shut down computer; turn it back on), I consulted the control panel for assistance. One of the questions in the help file was, “Have you checked ‘mute’ on the volume control?”

“No, you idiot,” I answered aloud to an empty room, “What do you think I am, stupid?”

Nothing worked, and after three days of computer silence, I got ready to phone the long-distance help line, when I noticed a little dial on the side of the laptop.

It was the volume dial, and it was set to minimum.

This is my excuse for not having posted anything for the last couple of days: it was out of a justified fear that the resulting entries would have been commensurate with the intellectual capacity that left me unable to turn up the volume on my computer for three straight days.

However, I do have a big post (which may turn into two or three smaller ones) about IQ, achievement, and socialization in the wings. With a graph and everything.

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