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.
Archive for October, 2004
One man, n votes
Oct 26
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
Oct 22
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.
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
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?