Tall, Dark, and Mysterious

3/8/2005

Sometimes I can’t laugh about it

File under: 1000 Words, Sound And Fury, Those Who Can't, Queen of Sciences, Know Thyself. Posted by Moebius Stripper at 8:43 pm.

Not too impressive on the surface, but get this - I’ve only marked the first page of the test papers. (Strike’s been delayed ’till next week.) Eight to go, and I wouldn’t bet against me.

I’ll finish them tomorrow; why give myself nighmares?

Question that I ask without a hint of facetiousness: what do students learn in math class for the first twelve years of their educations? I’m completely serious about this; I have a few dozen students who seem to be so utterly disconnected from the subject; numbers and equations have no meaning to them. I can’t fathom how they could go through twelve years of classes and still be at that level, but apparently it’s possible. I try to address the disconnection, by explaining things that they should know by now, but I can’t give them over a decade of background in the two weeks of leeway I have in a pretty tightly-packed curriculum. (Nor do I necessarily think I should; Jenny D has a good post on watered-down courses, and I expressed my ambivalence in her comments.)

Last week, the equations finally got the better of one of my students, who handed in her test forty-five minutes into the ninety-minute period with all the resignation of a defeated battalion. Before I’d returned to my office, she’d already sent me an email telling me where things stood. She was dropping the course, she said; she’d tried and tried and tried, but she’d always had so much difficulty with math. She’d barely passed grade 11 math, the last class she’d taken, and that was with the almost-daily assistance of a tutor. It wasn’t that she didn’t understand the material, she told me, it was that she couldn’t remember all of those formulas. (There’d been all of two equations I’d required the students to commit to memory for the purposes of the test.) “At least I tried,” she concluded her message, “and I probably did learn some statistics that will be useful to me one day.”

I didn’t reply to her email immediately. My instinct was to invite her to my office to go over strategies for learning math, as she clearly had no idea what was involved. A lot of my students think that I have a phenomenal memory; nothing could be further from the truth. I have trouble remembering math, too, but I understand it, and that’s the difference. But whenever I try to go over strategies for reasoning out the problems, I see four dozen eyes glaze over, and half the time a student will raise his hand and ask me which part of the junk I was saying he can ignore, and what the important formula in the question is. I have no idea how to deal with this. Technically it’s not my responsibility - it was their high school teachers’, or their elementary school teachers’ - to teach them these basics, to teach them what math is - but if I don’t help them, I don’t know who will. If they fail my class, they’ll just end up taking it again with another instructor, and that’s not what they need.

I decided to mark this student’s test, even knowing I’d probably never see her again. I stopped when I got to the second question. In the space below it she’d written her answer of (6/26)^10 - which was incorrect - and then left it in that form, with a sentence explaining, “Calculator doesn’t have a fraction button…what do I do?”

It’s not a good sign that a mere week after vacation, I had to sit on my hands to keep myself from pointing out that she had been one of the students to get help from me with her calculator, and if I recalled correctly, I was pretty sure that it had a fraction button.

Real-life applications

File under: Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 4:29 pm.

If success in teaching a math class can be measured by how few times one’s students ask, “What’s the point of this in real life?”, then Statistics for the Social Sciences at Island U has gone swimmingly. If, on the other hand, it is more accurately measured by the proportion of one’s students who can add fractions and solve linear equations, then, well, not so much, but I’m working on writing a teaching dossier, so it behooves me to be selective about how I measure my abilities and I’m trying to stay in character.

This week we’re talking about what pollsters mean when they say that a poll is accurate within 3%, 19 times out of 20. Fun stuff, and a subject of some curiosity for some of my students, kinda, now that I mention it. Last month, we did various flavours of probability, and I decided to dwell a little longer than I needed to on the conditional probability section. Conditional probability is often the first time that people’s intuition about how numbers work butts heads with the reality. I introduced that section with an example of a population that had a 2.5% rate of a certain type of cancer. A reasonably accurate test was available, but it wasn’t perfect: it gave false positives to 4% of healthy patients, and false negatives to 4% of sick patients. The relevant question: suppose you test positive; what’s the probability you have cancer?

“96%”, a student declared with confidence, so we worked it out. It’s not 96%, in this case; it’s 38%. If you test positive on this test, it’s more likely than not that you’re just fine. Ten or twenty years ago, some statisticians did a survey of doctors. The majority of them also gave the 96% figure, and many would send a positive patient for treatment without further testing. Lawsuits, I told my class, have been filed over this sort of thing.

Another highly relevant example of conditional probability, which I’d forgotten to teach, comes up in the case of racial profiling: suppose that Ethnic Minority X is much more likely than Ethnic Majority Y to commit Heinous Crime A. Suppose Heinous Crime A is committed. What’s the probability that the criminal is a member of Ethnic Minority X? If the minority group comprises a relatively small proportion of the total population, it’s actually more likely than not that the criminal is from a different group.

This type of question is similar-yet-different enough from the cancer test one that I thought it would be suitable to include something like it on the test. I’m not as brave as some people, though, when it comes to these controversial real-life applications. (If you don’t see why [above] is controversial, try replacing “Ethnic Minority X” with an actual ethnic minority group, and “Heinous Crime A” with a heinous crime in which that group is [thought to be] disproportionately highly represented. See what I mean?)

I wasn’t about to dig through actual relevant data, but after having realized what a great topic racial profiling was, I was too invested in the topic to simply abandon the question. So, I needed to come up with a problem that was similar enough in structure to a profiling scenario, and one that I could later explain as being an example of it, but different enough that I wouldn’t require my students to grapple with data about, say, gang activity among Hispanics - and under the pressure of a timed test, at that.

I spent the next little while, on and off, trying to come up with completely neutral “ethnic” groups into which to divide my population, along with a “crime” that wouldn’t upset anyone. After more time than I’d like to admit thinking about this, I finally got it.

From the test:

A study has revealed that 1/3 of flat-footed students leave their cell phones on during class, while only 1/9 of normal-footed students leave theirs on. According to the same study, 10% of students have flat feet, and 90% have normal feet. Suppose a cell phone rings in class. Assuming that students with cell phones are all equally likely to receive calls during class, what is the probability that the offender has flat feet?

They didn’t get the connection to profiling until I explained it, but none of them missed the subtext.