And so it ends.

Monday morning pleasantry: one of my students gave me chocolates today! So I’ll skip the post I had brewing about how it turns out that all of my most petulant, immature, demanding, manipulative, and clueless students happen to be psych majors. Not that there’s much more to say about that, anyway.

Classes end this week. As a last-ditch attempt to get the pre-exam class average in my precalc class above a C-, I gave my students one final assignment: submit test corrections for a chance to earn back some of their lost marks. The corrected tests came pouring in this morning, most of them legible but hardly artfully done. A substantial number of my pupils, though – all young, all female, all failing or coming close – submitted their test corrections in duotangs, headed by title pages, and with one small question per page (and hence a whitespace-to-content ratio of ten or so). Two of them had letters following the title pages, informing me that they really wanted to pass my class; presumably this distinguishes them from the remainder of my students, who really don’t care one way or the other. College math students reading this, take note: I don’t give style marks, and in all likelihood, neither does your instructor. Same goes for pity marks. Other than content, my main consideration is, can I carry this stuff home to grade/fit it in my cramped office? Thank goodness I only have a small number of duotangs, or else the answer would be no.

But I can’t really blame these students for trying. Particularly since, if their high school education was anything like mine, they were taught that presentation is at least as important as content. I still remember the marking schemes for my English essays: “Thesis stated clearly and argued with support from sources: 10 marks. Title page, with name of essay, name of student, and student number listed in that order: 10 marks.”

Book review: Precalculus, 5e, by Barnett/Ziegler/Byleen

Over at Learning Curves, Rudbeckia Hirta had some thoughts about matching a textbook to a course. Our discussion there led me to write a review of the Terrible, Horrible, No Good, Very Bad precalculus text I’m using. (Incidentally, I’ve really warmed up to the finite math text I whined about earlier. It’s not perfect, but it’s clear, well-motivated, and possible to work with. Which is more than I can say about the precalc book:)

* * *
The product description boasts that “[t]he Barnett, Ziegler, Byleen College Algebra/Precalculus series is designed to be user friendly and to maximize student comprehension.” It should have been sent back to the editors before publication, because Precalculus is anything but.

Among the various and sundry flaws with this text:

  1. The fluctuating emphasis on word problems is bizarre: in the very first section (”Linear Equations and Applications”) the book launches into a barrage of word problems (geometry, number theory, quantity-rate-time, distance-rate-time, chemistry, and more), as though students have the basic skills and the experience to grapple with this huge mass of content. From there, it moves on to the more basic skills of solving linear inequalities, and factoring quadratics. There are some word problems, but gone is the all-applications-all-the-time approach in favour of developing underlying concepts. In addition, the book’s claim that “the worked examples are followed by matched problems that reinforce the concept that is being taught” is actually a flaw: students who learn from this text, learn to solve word problems by pattern-matching. Presented with a slightly different problem that draws upon skills they have picked up, they are lost.
  2. The ordering of the content is seemingly random. for example: in Chapter 1, students are presented with a slew of opportunities to express this quantity in terms of that one. Six sections later, in Chapter 2, they learn what a function is (ie, it’s an expression of one quantity in terms of another), and from there they have several opportunities to…express one quantity in terms of another. Some of the exercises in Chapter 2 are virtually identical in content to some of the exercises in Chapter 1 – for instance, students are asked to find break-even points of cost functions in both chapters. Most perplexing choice of ordering: students are taught to find equations of, and graph, circles in Section 2-1, one section before they’re taught to find equations of,and graph, straight lines. One section later we arrive at the “Functions” unit, followed by…”Graphing Functions”.
  3. This book would be around 200 pages shorter if it omitted all of the graphing calculator applications, which it should. There are questions asking students to describe the shape, or find the range, of a function by plotting it on a graphing calulator. As a pedagogical exercise, this is useless. It’s also counterproductive from the perspective of instructors faced with the unenviable task of teaching students to graph functions and find their ranges algebraically. Although the emphasis of a precalculus class should not be overly theoretical, students should come out of it with some notion of what sorts of operations are mathematically and logically sound. Finding the range of a function by plotting it on a calculator is not, while completing the square of a quadratic and analyzing it is. The book presents these two methods on equal footing.
  4. Speaking of graphing functions: the book pays some lip service to graphing functions by applying transformations – for instance, students should know what the graph of y=x^2 looks like, and from there they can apply transformations in order to graph y=-2(x-4)^2+3. But soon it abandons this approach, and graphing functions pointwise becomes the method of choice.
  5. The book description claims an emphasis on computational skills over theory. Indeed, it de-emphasizes theory to the point of presenting content as a series of disjoint rules. The “here’s a word problem, here’s the formula you need for it, now do two just like it” approach is one example, but it’s in the third chapter that the authors’ contempt for mathematical justification really comes through. That section (on polynomials) is awash with a dozen methods of estimating bounds of zeroes of function by using synthetic division. Synthetic division itself is presented without proof, and the applications are all given as formulas without any justification. Later, in chapter 4, we get eight rules of logarithms without a lick of explanation. Contrary to its goal of emphasizing problem solving by opting not to muddy the waters with theory, this approach leaves students with so little grasp of the underlying content that they can’t solve any problems beyond the exact ones presented in the text.

These are just some of the problems with Chapters 1-4; I haven’t taught the second-semester course, which uses Chapters 5-8. The book fails by all standards: it falls far short of its own goals of creating a usable text, it fails by the standards of every instructor I know who has used it, and it fails the students who use it. I have taught a variety of first-year college math classes, and never have I found a textbook so difficult to work from.

Every attempt to instill some theoretical comprehension is undermined by the scattered, calculator-dependent approach of the text. My students agree: I’ve had several tell me that they have never had such an unreadable text. I’d advise math departments to search around for a twenty-year-old college algebra text that’s still in print, and use it.

(If they can’t, they should write to the publisher of such a text for permission to copy it.) Older texts omit the calculator mumbo-jumbo and insist are designed to instill mastery of a rudimentary set of skills, rather than memorization of a handful of formulas – and that’s the best way to prepare future calculus students.

Things my precalculus students said in class yesterday

  • “So we’re just supposed to KNOW that 7x^2 is like 7 TIMES x^2?”
  • “There’s no solution [to the equation x^2=3.52534231], because you can’t take the square root of a decimal.”
  • “Okay, I see how you got 2x=1.46, but how did you get x=0.73 from that?”
  • “Last week’s test was totally unfair – you never gave us a homework problem where we had to graph a rational function where the denominator was like x^2-4x – it was always like x^2-4x plus some NUMBER.”
  • “Can we just plug those numbers into our calculators instead? Because, like, I’m not very good with fractions.”
  • “Miss, I think you made a mistake grading my test. I had (x(x-2)+3x(x+4))/((x-2)(x+4)) , and then I cancelled out the x-2 and the x+4 to get x+3x=4x, but you marked that wrong.”

These are my students who desperately need grades of at least C+ in my class in order to be permitted to take calculus next year. What’s sad is that many of them seem to be under the impression that I am the one thing standing between them and success in university math classes – if only I’d just make my tests easier, they could advance, and surely their calculus prof next year won’t expect them to be able to solve any math problems that they haven’t seen word for word before.

Ideally I’d have taught them well enough that they’d have earned their C+’s and above – but I’m just not a good enough teacher that I can take kids who can’t do grade school math and mold capable university pupils out of them in one semester. As far as I know, there’s only one teacher who was ever able to do that, and they made a movie about him. So, having given up on educating my weakest students, I’ve committed myself to doing the next best thing: ensuring that they don’t get the grades to take calculus next term.

And this is what I’m keeping in mind in designing the final exam. I’m testing them on the material – graphing functions, solving various flavours of equations, factoring polynomials – but I’m specifically designing the questions so as to trip up everyone who hasn’t learned grade 9 math. I could ask them to complete the square of a quadratic with integer coefficients, but I like fractions.

could ask them to solve a problem that reduces to finding the positive and negative square roots of 7 – but I prefer the number 7.5. I could give them a problem in which they’re given the distance in kilometers and the time in hours – but I prefer minutes. I could ask them to sketch the graph of f(x)=2^(x+3), but I rather like the graph of g(x)=-2^x – which many calculators interpret as g(x)=(-2)^x.

The students who belong in university math classes will do fine; the exam is easier than my tests. Those who memorized the problems I did in class without bothering to learn any grade school math will do miserably.

When it’s bad it’s bad, and even when it’s good it’s bad

All hundred-odd of my students wrote tests last week, and so I spent the entire weekend grading them.

My precalculus classes, as expected, did poorly – though I was surprised by just how poorly they did. I knew that my students didn’t know how to factor, but I was surprised by the number (6) of students who thought that the graph of -2^x was a jagged line. There’s also the fact that despite my having spent three (3) weeks on the graphing arts, the preferred method seems to involve graphing functions pointwise and connecting the dots, asymptotes be damned.

(One angry student told me that if I wanted a better graph, I should let him use his graphing calculator.) If I were to test my students instead on, for instance, Fermat’s Last Theorem, I think the results would be similar: pages of gibberish, followed by a dozen students whining that I never showed them how to do that question. In any case, I feel like I’ve utterly botched precalculus. Half of me wants to try it again so that I can get it right, but the other ninety percent of me never wants to teach a precalculus class again.

My discrete classes, though, surprised me: class average in the mid-seventies on what I thought was a moderately challenging test. I was happy about this, until I got into the class today, and found that a good half of them were shocked by their good marks: they’d thought they’d failed. (This didn’t stop many of them from complaining regardless that I shouldn’t have asked this question or that one.) This is making me wonder if I know anything whatsoever about setting tests. At the very least, perhaps I should rethink my generous part-marks policy.

Little Miss Math Teacher

The following conversation wouldn’t be noteworthy, were it not for the fact that I’ve had it a good half a dozen times in the last two months:

Female Stranger: Are you a student at the college?

Me: Actually, I teach there.

FS: (eyes bulge) You do? What do you teach?

Me: I teach math.

FS: (eyes bulge, voice becomes high and squeaky) Really? GOOD FOR YOU!

I’ve gotten this from a bank teller, from the woman who sold me my cell phone, and from a good number of people on the bus. Certain details, such as the timing of the bulging eyes and the squeakiness of the voice, vary from person to person, but the “GOOD FOR YOU!” is constant. Not “I’m impressed” or “oh, interesting, I’ve never known a college math teacher,” or even the standard “I could never do math”, but “Good for you!” as though teaching college math is a milestone on par with making all gone with my brussels sprouts or graduating to big girl underpants.

Sometimes, they clarify that it’s because I’m a girl that it’s so good for me that I’m teaching math. Having stricken “I grade my students’ test papers in menstrual blood, too!” from my list of possible replies, I tend to remain silent or say something noncommittal. I think that my conversational partners are offended that I don’t beam in pride, as our chats tend to end right there. Perhaps they haven’t figured out yet that not only can girls teach math just as well as boys, we can also be just as averse to condescension and paternalism as boys. Any twentysomething male college math instructors or researchersd here had, one a regular basis, complete strangers express pride at their choice of employment? Hell, any young men here ever get verbally patted on the head for their life choices?

(My parents and older relatives are exempt from my indignation here, as they were actually there when I made all gone with my brussels sprouts and graduated to big-girl underpants and are allowed to be proud of how far I’ve come. All others, take heed.)

I know that these strangers mean well, and that they can’t help being idiots, but every single time someone focuses on the fact that I’m a female working in a male-traditional field, I find myself half wishing that I had chosen a job in which my presence were seen as a contribution to my field and not as a political statement. If I had the temperament for feminist activism, I’d get involved with that, but I don’t, which is part of why I inhabit instead the politically bland world of graphs and equations.

I’m not an ambassador for womankind. I stand in front of a math classroom with the same skills and for roughly the same reasons anyone else stands in front of a math classroom, and ignoring those reasons in favour of pointing out that women are such a rarity in their field harms the cause of having women taken seriously in a myriad of fields – it doesn’t advance it. Doubly so if, in the process, you treat the woman in question in the same way you’d treat a six year old. Good for you!

Part of the problem, I think, is that it’s been my experience that the set of people interested in technical subjects (as something to study, not just as something that it’s cool that other people are studying) and the set of people interested in social activism (as something to do, rather than just as something that’s cool for other people to do) are nearly disjoint, particularly among females.

Consequently, those of us , such as myself, who ally themselves with the former camp are interesting but otherwise strange and mysterious to the latter – objects of a psychological experiment conducted behind glass. The latter know all about women, but know squat about physics or math or engineering, and they talk about what they know. Okay, you’re enumerating curves on Hirzebruch surfaces via lattices of dual subdivisions, and I’m sure that’s very nice, but OMG YOU’RE A WOMAN AND YOU’RE DOING MATH and that must be like SO WEIRD, let’s discuss that.

No, I’d rather discuss enumerating curves via lattices of dual subdivisions, thankyouverymuch, but I appreciate your concern. Now go away and leave my profession to the people who are interested in it.

Or maybe they just studied really hard.

My early afternoon precaclulus class has always been a shade weaker than my late afternoon precalculus class; class averages tend to differ by around 5% from one class to the next.

Until this week. Class average on the early afternoon test: 58%. I found this bewildering, as I thought that this test was considerably easier than the previous two. My late afternoon class agreed with me: their class average was 78%.

I don’t like to assume the worst of my students, but no mathematician can look at the data I’m looking at, without incorporatinng their knowledge of standard deviations and variance and accuracy within such and such percent so many times out of twenty and concluding that in all likelihood, something is awry. Since my two classes are one right after the other, with only the ten minute break to get between buildings, I’d naively assumed that it would be okay to give the same test to both sections.

Curiously, my early afternoon class – the ones who bombed the test – tended to leave the classroom early. Most were gone an hour into the sixty eighty (thanks, rohan!) minute test, which left them with half an hour to talk to the late afternoon students. By contrast, I had to pry the test out of half of my late afternoon students’ fingers.

Next week, I’ll write two tests. This shouldn’t be much extra work, as many of my students are so weak that “find the maximum possible product of two numbers that sum to 50″ is a completely different question from “find the maximum possible product of two numbers that sum to 60″. But this still leaves the question of what to do this time. I hope that at the very least, the early afternoon informants were being paid well for their sacrifice.

Standardization, freedom to teach, and the university marketplace File under: Righteo

I have a question for the university instructors, particularly the math instructors, who read this:

How does your university reconcile a commitment to high standards with a desire to create a good work environment for its instructors who wish to teach freely? For that matter, how does it reconcile both with the fact that so many students are overwhelmingly ill-prepared for university?

(I’m assuming a simplified model in which universities actually care about such matters, and have devoted some thought to the issues thereto appertaining. This is not naïvete on my part; it’s a just a mathematician’s smoothing of data in order to create a solvable problem.)

(If you have an answer, feel free to skip the rest and just post in the comments. This entry’s really all over the place.)

My old university, where I did my Master’s degree, took the position that we have high standards, dammit, and so if half of our students fail their first year math courses, then so be it. It was something of an open secret that the first year calc courses were designed with the express purpose of culling the commerce herd: the head of the commerce department had explicitly asked that we fail a third of our students, because his department didn’t have room for all of its applicants.

And so we did, appealing more than once to the defence that “the department made us do it.” Our curriculum was designed accordingly; follow the textbook, cover all of the examples, and don’t digress. We were not permitted to see the final exam; that way, we would not be tempted to teach to it. The instructors were just textbook appendages, and useless ones at that; the deparment head pretty much said as much back when the TAs went on strike. To wit: we were not to adjust the final exam despite the fact that we’d been away for a third of the term – after all, the students could just read the text and learn the material on their own.

Somehow my old university managed to micromanage us while giving us just enough freedom to completely screw our students, which is pretty remarkable, when I think back on it. If we wanted to prepare our students for the exam and for the next course, we had to do everything by the book, example by example, question by question; on the other hand, we designed our own midterms, so if we didn’t give a crap about our students, we could just babble about the easiest aspects of the course, and leave our pupils to flounder on the final exam. But don’t worry: the students in the latter type of classes would just fail the final, and term marks would be adjusted in order to be in line with exam grades, so at least the students would have their university careers ruined.

My current university takes the opposite position, handing us textbooks, telling us to cover chapters n through m, and then letting us go play. I loved the freedom, and entered the classroom with the optimism of a first-time full-timer: I would not just teach my students how to factor quadratics and graph polynomials and solve systems of equations; I would teach them to think! They would learn how to check their work! To estimate ballpark figures for the values they were computing! By God, they would LEARN!

I anticipated that that radical philosophy would encounter some opposition (see also, “you make us do DIFFERENT problems on the test!”) , and I chalked much of that up to the fact that some students fresh out of high school take a while to realize that, well, we’re not in Kansas anymore. But recently, I’ve begun to seriously question – and somewhat regret – my approach.

My students are doing fine – average in the mid 60’s, which is where it ought to be – but some of them are failing despite what I believe are their best efforts. I went in to this course deciding that I wouldn’t teach them to memorize formulas; I’d teach them to think mathematically. But so many of them simply aren’t at that level. As a result, not only are they not learning to think mathematically, they’re not even learning to memorize formulas. I now know what the secretary meant.

And my students are angry. And they’re not just angry because I make them do problems they haven’t seen before on tests, whereas their high school teachers didn’t; they’re angry because the Nice Teacher who teaches section 104 doesn’t make his students solve new problems on tests, either.

Nice Teacher is another temporary instructor here, one who has complained at length to me and to others about what lazy-asses his students are. Some of my students are lazy, too, but I wouldn’t describe them as lazy en masse; and if they were, I’d remedy their laziness by making them work. Given the option between being lazy and failing, or working hard and passing, prudent students – even lazy ones – tend toward the latter.

Nice Teacher chose a different approach. If the students are lazy, then the path of least resistance is to challenge them so little that even their lazy asses can pull B’s without effort. Nice Teacher spends half a class or so on applications – two or three of them per section – and then assures his students that they won’t see any applications that deviate even slightly from the ones he did in class.

His tests are shorter than mine, and the front page of them contains a list of all of the formulas that any first-year math student could possibly want. As though that weren’t enough, the test problems – which are pretty easy, as far as the subject goes – are annotated with hints. An example: “Use Formula #3.”

For the benefit of those students who are so weak and so lazy that even that’s not enough, Nice Teacher gives what he calls “pretests”, which are word-for-word identical to the tests themselves, with only a few numbers changed.

I know this because I supervised Nice Teacher’s students during a test last week, and saw most them leave halfway into the allocated time they had to write, and fielded questions such as “It says to use Formula #3 – so do I plug in a=10, p=5, or the other way around?” I replied that I couldn’t answer that question, and the student pouted: “Nice Teacher would have told me.”

Anyway, Nice Teacher’s class full of lazy students has an average of 80%. Some of my students are acquainted with some of his, and have asked me why I can’t give pretests, why I can’t give formulas on the test, and why I can’t overall not require them to do any work in order to get B’s. I’m hard-pressed to give an answer that isn’t of the form “Because I’m actually teaching you how to do math, not how to be a slacker.”

Most of my students would rather learn nothing and get A’s than learn something and get B’s (or even C’s), and so there’s been a migration of my students into the other sections. The department head tells me that he often gets students complaining that they paid for this course, so they deserved to pass. Department Head is of the dry British persuasion, and treats these claims with the irreverence they deserve.

Once I had a student complain about same to me, and I remarked that a university education was a poor investment for them, as a diploma mill could provide them with higher marks for less money. But my students have an even better option now: transfer sections for the high marks, and get the same university degree. If you’re unhappy with a product, switch brands. This is what happens in an environment with few standards.

I have no data on the subject (Kimberly? Do you?), but I would wager that the most zealous faction of the anti-standardized-testing crew barely overlaps with the set of mathematics instructors. By and large, math instructors would be delighted to have a classroomful of students who have mastered some set of basics, the content of which we could for the most part agree upon.

When a student arrives in university unable to add fractions, unable to deal with negative numbers, unable to set up a simple equation from a paragraph of information – it’s not because that student’s teachers were spending so much time preparing for a test that they couldn’t teach creatively; witness the disaster that was the New Math. Often, in grading a test, I wonder what on earth some of my students were doing in math class for the last decade, because it obviously wasn’t math.

Department Head was unaware of my situation until I brought it to his attention, of necessity telling on Nice Teacher in the process. What should I do about the gulf in standards, I asked? We arrived at a compromise: scale the grades after the final exam, so that no one would be penalized for being in my class. It’s not an ideal situation, as I strongly believe that my students who are getting 40’s in my class do not deserve to pass, but it’s a decent balance between achieving fairness across sections and maintaining my standards.

Nice Teacher’s students get punished by not actually learning anything; alternatively, depending on your perspective, my students get punished by only possibly learning anything and getting bad marks in the process. At my old university, poorly-taught students would be punished by failing the common final exam, and hence the course. In both universities – one superstandardized, the other giving full freedom to the instructors – the only quality assurance was the sort that would (in some form) punish the poorly-taught students. Has your university come up with anything better?

Lest I ever consider teaching elementary school math in Georgia

…and lest I ever make the mistake of thinking that standardization is necessary compatible with high standards:

I’m writing an introductory math text for – swear to God – an Indian company that’s outsourcing work to me, howdoyoulikethemapples. In searching for some inspiration for applications of fractions, I ran across this teacher guide. Quoth the introduction:

The lesson is created for day 145 of the 180 day sequence.

No, really. Don’t go off on any pertinent tangents, teacher; you may fall behind schedule, and your students will never have a chance to

…review the concepts presented in the unit by creating a “how-to” booklet. Students will create an entry for each concept presented. Finally, in the next lesson in the sequence entitled Fractions, Decimals, & Percents Unit Review, Day 2, students will randomly choose one of their entries to share with the class.

This is a grade 8 curriculum, by the way. Math teachers: ever wonder why your students regard math as a series of disjoint facts and tedious procedures, devoid of imagination? I don’t anymore, but if you did, it might have something to do with the fact that eighth grade teachers in Georgia must

[e]xplain to students that to review the concepts in this unit, each student will need to create a how-to manual. Each of the ten concepts addressed in step one will need to be addressed in the manual. The manual should consist of a cover page, table of contents, and 10 chapters or entries. The manual should also be bound together in some way such as stapled, connected with yarn, or in a report cover of some kind. Encourage students to be as creative as possible.

Emphasis mine. Eighth grade is old enough to be tackling logic puzzles and word problems, but whoeever conceived this bastard curriculum apparently couldn’t think of any way to do mathematics in a creative way, hence the yarn. By the way, teachers are told to spend five minutes in the middle of class explaining the format of the how-to manual – this after the ten minute brainstorming session, but before the thirty-five minute period where the children make their books explaining fractions creatively with glue and yarn. Yes, that’s the whole fifty-minute class period. For the benefit of those teachers thinking, “Well, that’s a good start, but what I really want is to be micromanaged,” Step 3 is fleshed out in detail:

Allow time for students to complete the assignment. Monitor students while they are working and assist any students experiencing difficulty.

This guards against the possibility of the teacher choosing instead to release a box of firecrackers into the centre of the room and then tell the kids to go play.

At the bottom of the webpage is a series of links about how to modify this sorry-ass curriculum to better accommodate students with various special needs. There’s less objectionable stuff there, but the guidelines for accommodating gifted students are curiously vague, considering the source. “Encourage creative thinking and expression by allowing students to choose how to approach a problem or assignment.” What on earth does that mean in this case – allow them to pick which colour of yarn they want to use for the how-to book, hmm?

Two quick stories about my classes

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

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.