Tall, Dark, and Mysterious


Two quick stories about my classes

File under: Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 11:08 am.

1. Yesterday, a handful of students from my late morning class arrived in class bedecked in holiday garb - goblins, witches, and ghosts were the most popular. “Miss,” one student said, “are you dressing up as anything for Hallowe’en?”

“Yeah,” I replied, “I’m dressing up as a math teacher.”

Twenty-five students rolled their eyes at me.

“Come on,” I challenged, “what are you more scared of - goblins, or next week’s test?”

They concurred.

2. My university has an odd way of scheduling classes, and as a result, many of my students have only my class on Fridays. Consequently, many of my students have recently fallen ill to the Friday Flu, an illness that afflicts approximately five times as many students on Fridays as it does any on other day of the week.

Eight days ago, ten students - out of twenty-eight - showed up to my late afternoon precalculus class. Nearly all had been present for their test two days earlier, and over twenty were there the following Monday. I wagged my finger at the Monday crew, informing them that I’d given a full fifty-minute lesson the previous Friday, and that I had trouble believing that all of them had perfectly legitimate reasons for being away that day. They were appropriately sheepish, but as any math teacher knows, lessons tend to sink in better when they’re presented in more than one way.

We’re covering functions these days, and I presented them with some graphs of increasing and decreasing functions. Anticipating the frequent, “what does this have to do with real life?” query, I gave an example: “For instance, we can look at a graph that gives Friday attendance as a function of time.” I had the data from the past seven weeks: 34, 30, 27, 22, 20, 14, 10. “The larger the value for t,” I said, “the smaller the number of students at time t.”

Twenty-three students showed up yesterday.


One man, n votes

File under: Character Writ Large, Queen of Sciences. Posted by Moebius Stripper at 1:20 pm.

Buoyed by a surge of student queries of the form, “what applications does this have in real life?” I was inspired to dust off an old copy - the library’s - of John Allen Paulos’ A Mathematician Reads the Newspaper. By the time I finished rereading the brief aside on measuring shareholder and voter power, I had abandoned my original goal of making precalculus relevant to my pupils, and was wondering if there were any analyses online about the amounts of power held by the various states in the electoral college that went beyond the standard “wooo, gotta worry about Florida”-type punditry. Naturally, there were, and since this is about the only aspect of the US election that I can think about without wanting to claw my eyes out, I thought I’d post some of them.

To the best of my knowledge, the Banzhaf index is the standard means of measuring power of groups in block voting systems, such as the electoral college, in which each state’s vote is weighted. The Banzhaf power index for Florida, for example, is computed by considering all the state-by-state possible outcomes in the election - one outcome being the possibility that California’s 54 electoral votes go to Kerry, New York’s 33 go to Kerry, Texas’ 32 go to Bush… - and then counting the numbers of those outcomes that are swung by Florida. Here is a state-by-state list of the Banzhaf power indices for the 50 [thanks, Chris] states and DC. (The power indices in the other columns are also defined.) Florida, the largest swing state, has a power index of 0.193864, meaning that in 19.4% of possible outcomes, neither Bush nor Kerry will have enough electoral votes to win the presidency before Florida is counted. Compare this figure to the relatively small ~4.6% of total electoral votes allocated to Florida. (California, meanwhile, is critical in nearly half of all possible outcomes.) The runtime of the programs doing these computations is already pretty high (O(2^n )), but I wonder if there are any probabilistic variations on this index as applied to the electoral college. In the standard computation, for instance, an outcome that gives California’s 54 votes to Bush and Texas’ 32 to Kerry is weighted the same as the far more likely alternative. A friend of mine from Mathcamp has written some Maple routines evaluating different power indices; someone who keeps up with US politics better than I do could probably make the modification pretty easily.

Going a bit further: I haven’t yet read this detailed article about the Banzhaf power index, but it also contains an analysis of how much power each individual voter has - taking into consideration the population of the states as well as their voice in the electoral college. Despite California’s large population, its voters have the most say - each is 3.34 times as powerful as a single Montana voter. (This, I presume, makes certain assumptions - for instance, that the percentage of registered voters who actually show up is constant from state to state.)

Paulos gives a simple and dramatic example of the relative usefulness of the Banzhaf index versus more standard measurements: consider a company with three shareholders, who respectively own 49%, 35%, and 16% of the company. Although the first’s share is more than triple the third’s, all have equal voting power: in a yes/no vote, whichever side attracts at least two of the voters, carries. Consequently, all shareholders have the same Banzhaf power index - in this case, 1/2. On the other hand, if they held 51%, 35%, and 18% respectively, the first shareholder’s vote is clearly the only one that matters. His power index is 1, and the others’ are each 0.


Data retention

File under: Those Who Can't. Posted by Moebius Stripper at 4:56 pm.

One of these days, I should post about how much I like my students. I really do. I have a handful - as in, I can count them on the fingers of one hand - that I wish weren’t in my classes, but the overwhelming majority of the ones I’ve actually spoken more than a few words to are mature and eminently reasonable people; of those, a substantial portion are also putting in the effort required to master the material I teach. Swear to God; I don’t know what I did to deserve this.

Anyway, my precalculus students wrote a test the other day. I tend to make my tests “semicumulative” - 90% of the test is drawn from new material, and the remaining 10% is culled from topics they were exposed to before the previous test. On this test, their second, I put two questions that could have been on Test #1; of these, my students were to choose one, and I’d count the better one. One of my top students came to me after the test and mentioned that she’d done almost perfectly on the new material (she had), but couldn’t get more than half marks on either of the “old” questions. She wasn’t complaining; she was commenting that she’s clearly able to learn and understand the material, but that she has trouble retaining it. She asked me if I had any advice.

I didn’t offhand, but I told her that hers was a valid and worthwhile question, and that I’d think about it over the weekend and see if I could come up with any useful advice. I know that there are a number of experienced high school and university math instructors reading this blog, and I’d love to hear if you have any insights into how math students (and students in general) could better retain the material they learn - as well as how math teachers could teach for better retention. (Please ping this post if you think your readers might have any ideas!) Every year in grade school and high school, math teachers spend several weeks reviewing the previous year’s material, so this is clearly a pretty widespread problem.


Home and native land by the dawn’s early light

File under: Character Writ Large, Home And Native Land, I Read The News Today, Oh Boy. Posted by Moebius Stripper at 4:43 pm.

I wanted to comment on Diana Moon’s evisceration of a particularly egregious FrontPageMag.com article, because her characterization of US/British sentiments could be applied almost just as well to US/Canada ones:

I took one trip to Britain, in the summer of 1989. George Senior was a-comin’ fer a visit and the hot topic on every BBC show was “Is the special relationship still special?” Now, I knew what this special relationship thing was, but to see the anxiousness displayed on the British media was, well, surprising. Of course, I couldn’t help but note that when a British PM comes to the US, there is no such reciprocal anxiety…which might account for a certain resentment on the part of the British towards the overwhelming power imbalance…which I did not encounter.

Yeah, that about covers it, except that while the British are (from what I can tell) obsequious (in that formal British backhandedly contemptuous way) toward the US, Canadians are more passive-aggressive. My Canadian readers will recall Prime Minister Martin’s campaign this past summer, which featured the Liberal leader assuring us, in turns, that 1) we wanted so very much foster a close relationship with our good friends the Americans and by God we would under a Liberal government; and that 2) we’re not American, with their privatized health care and their Iraq war and their votes on abortion, okay, we’re Canadian, we’re different, THANK THE LORD GOD WE’RE NOT AMERICAN. Witnessing the US presidential campaigns from my vantage point north of the 49th, I found myself thinking more than once - the Republicans are calling Kerry a flip-flopper? They have no idea. But anyway, similar dynamic - wee Canada sitting up north, chewing its nails and assuring itself that if it behaves properly (and unlike those brutes the Americans, it always behaves properly, it’s Canadian for heaven’s sake!) then the US will love it. Meanwhile, south of the border, the US making its decisions and formulating its views pretty independently of anything Canada thinks. (Independently, that is, when they’re not trying to piss us right off.)

I was reminded of this last night, watching an unintentionally hilarious CBC piece about Machias Seal Island, one of four (!) disputed islands that lie on blurry segments of the Canada/US border. According to the US, the island is part of Maine; Canada maintains that it’s property of New Brunswick. The latter doesn’t sit terribly well with one John Norton of Kennebunkport, Maine*, who leads puffin tours of the island. His family has been in Maine for hundreds of years, apparently, and the last five generations of his family have fought for the US government to take Maine’s property claims more seriously. He defends the island with a righteousness that makes me kind of selfishly glad that the US troops are all tied up in Iraq for the time being. Meanwhile, Canada has had a lighthouse on the island since before Confederation, which seems like a pretty strong case for Canadian ownership; but at the same time, our federal government actively allows its fishermen to fish off the Island coast before the Canadian fishing season has started…as long as American fishing season is underway. Cakes and consumption, Canada.

In any event, Norton is adamant: “Eye lead puffin tours of Machias Seal Island, YOU. ESS. AY,” he enunciated from his boat, with all the passion and clarity of a stage actor. He then continued, “Canadians are very smooth…they can stick the knife in your back and twist it around” - cue crude arm gesture - “and say ‘thank you’.” Norton’s family has run the puffin tours for generations, but he still feels threatened by New Brunswick (that’s threatened by New Brunswick, for those of you keeping track), and he won’t back down: “Canada,” he announced, “will have to shoot me in the head” - this he demonstrated visually by pointing his finger at his head in the classic “shooting self in head” position - “to keep me off this island.”

From there we met Paul Cranford, a Cape Breton fiddler and writer who’s worked as a lighthousekeeper on Machias for the last decade. “It’s Canadian territory,” he told the camera calmly, and chuckled, “though I know John Norton thinks otherwise. Sure,” he continued, “the island is disputed…but there’s no real animosity.”

And then the camera cut to Cranford walking through the overgrown Maritime fields against a backdrop of Celtic music, which it then showed him playing on his fiddle.

And that right there, ladies and gentlemen, is Canada/US relations in a nutshell.

* I’ve been to Kennebunkport, and I would postulate that there is no one else in the entire state of Maine - save perhaps his family - who’s anything like John Norton. Mainers make Canadians look unfriendly and impolite by comparison. Tell an average Mainer that there’s an island disputed between Maine and New Brunswick, and you’ll probably get an answer along the lines of “Oh, is there? Well, you can have it, you know, if you want it. Is there anything else you want? Another island maybe? Really, just let us know.”

Back to basics

File under: Those Who Can't, Queen of Sciences, I Read The News Today, Oh Boy. Posted by Moebius Stripper at 11:35 am.

The other day, I wrote about my university math students not knowing their times tables or how to add fractions. (By the way, you should go read the excellent comments to that one, which I will reply to eventually - where did all of my awesome commenters come from?) A few days earlier, Erin O’Connor, an English-prof-turned-high-school-English-teacher, posted an English teacher’s equivalent: grammatically ignorant English students. There are plenty of parallels:

Most high school students these days are not on the grammar curve at all. The parts of speech are largely mysterious to them; the rules of punctuation and agreement are likewise unfamiliar. Semi-colons, colons, and dashes do not come into play in their writing because they do not know what they are for…

Don’t get me wrong. Kids today are as smart, creative, and sharp as ever. Their grammar deficit is not their fault. They can’t be blamed for what they were never taught. It’s increasingly unfashionable to emphasize grammar and the rules of syntax in school, the reasons ranging from the hang-loose notion that the rules of usage are confining and binding and irrelevant anyway since language is a living, breathing thing, to the feel-good notion that grammar is boring and mind-numbing and kids will be turned off to reading and writing forever if they have to learn it.

The notion that “language is a living, breathing thing” doesn’t apply to math; math is seen as boring, and in need of sexing up (hence the skipping over the basics), and all of the people I know who love math and actually do see it as living and breathing are absolutely militant about making sure that the basics of math are taught. But just as the parts of speech are mysterious to Erin’s students, the whole notion of quantitative data is foreign to mine. Two of my commenters from the previous post confessed that they, too, never learned their times tables, but both are comfortable with math. But I was appalled not at the fact that one of my students didn’t know off the top of her head that 3x8=24; rather, what appalled me was that she lacked the ability to figure out the value of 3x8. She left her calculator in her car, and therefore she would not know how to compute that quantity. She could not count groups of eight (or three) on her fingers. She could not make eight and eight and eight marks on a sheet of paper. Not only did she not know what 3x8 was, she did not know what 3x8 meant. And this is the real issue; students not getting the basic, basic aspects of what mathematics means.

Year after year, I’m reminded that my students, almost to the individual, are nigh incompetent when it comes to word problems. There’s this basic inability to translate quantitative data into useful equations. I’ve taken to reminding my students at the beginning of every section of word problems, “an equation is a relationship among quantities. So we need to figure out what quantities we’re interested in, and find relationships among them.”

I make a point of emphasizing this because, based on my students’ “solutions” to word problems on tests, their thought processes are something akin to the following:

Okay, let’s see, this question says that it takes me twice as long to get 25 km upstream, with a current of 3 km/h, as it does to get downstream. Okay, so I have a 25 in my question. And a 3. So I need an equation with both of those. How about…25+x=3? No, because 25 is bigger than 3. So - 3+x=25. No hold on, there’s something about “twice as long”, so…multiply something by 2? 2(3+x)=25? Yeah, let’s try this, that’ll work.

My students, almost to the individual, have little idea how to relate word problems to mathematical equations. The parts of math, in other words, are largely mysterious to my students, and it’s a crying shame that this is still the case in university.

[Timely update - Joanne Jacobs reports that LA schools are beginning to teach algebra in elementary school. The article is annoyingly contentless (what are these algebra games the kids are playing? Inquiring minds want to know), but dammit, it’s about time; I’ve been suggesting this for years. It’s completely ridiculous that every single first grader knows that two dogs plus three dogs equals five dogs, but then in seventh grade they’re completely perplexed to see that 2d+3d=5d.]


This student is in my class, too.

File under: Sound And Fury, Those Who Can't. Posted by Moebius Stripper at 10:49 am.

I’ve gotten the math teacher version of this complaint:

…all my other professors give me A’s and you didn’t! And I proof-read my work and everything. I don’t understand what you want, so could you write up some personalized guidelines telling me what to do so that I can make sure and follow your instructions? Thanks!

The math teacher version, of course, is “I memorized how to do the problem you did in class, but then on the test you put a DIFFERENT problem, and you never showed us how to do THAT one, and it’s not fair! My method of doing math by memorizing formulas and then blindly applying them to problems that are identical to the ones I’ve seen has gotten me A’s until now, so what gives?”

It’s been several years, and I still haven’t come up with a tactful way to say, “the reason you used to get A’s is because your education has failed you, as you clearly don’t have any concept of how to reason mathematically. My goal is to challenge and teach you, not further your illusion of yourself as being clever enough already.”

It’s been a bit of a rough week; can you tell?

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