Reading the Comics, 16 May 2013

It’s a good time for another round of comic strip reading, particularly I haven’t had the time to think in detail about all the news in number theory that’s come out this past week, and that I’m not sure whether I should go into explaining arc lengths after I trapped at least one friend into trying to work out the circumference of an ellipse (you can’t do it either, but there are a lot of curves you could). I also notice I’m approaching that precious 10,000th blog hit here, so I can get back to work verifying that law about random data starting with the digit 1.

Berkeley Breathed’s Bloom County (May 2, rerun) throws up a bunch of mathematical symbols with the intention of producing a baffling result, so that Milo can make a clean getaway from Freida. The splendid thing to me, though, is that Milo’s answer — “log 10 times 10 to the derivative of 10,000” — actually does parse, if you read it a bit charitably. The “log 10” bit we can safely suppose to mean the logarithm base 10, because the strip originally ran in 1981 or so when there was still some use for the common logarithm. These days, we have calculators, and “log” is moving over to be the “natural logarithm”, base e, what was formerly denoted as “ln”.

How Long Is A Bad Ellipse Question?

Something came to mind while thinking about that failed grading scheme for multivariable calculus. I’d taught it two summers, and the first time around — when I didn’t try the alternate grading scheme — I made what everyone assured me was a common mistake.

One of the techniques taught in multivariable calculus is how to compute the length of a curve. There are a couple of ways of doing this, but you can think of them as variations on the same idea: imagine the curve as a track, and imagine that there’s a dot which moves along that track over some stretch of time. Then, if you know how quickly the dot is moving at each moment in time, you can figure out how long the track is, in much the same way you’d know that your parents’ place is 35 miles away if it takes you 35 minutes of travelling at 60 miles per hour to get there. There are details to be filled in here, which is why this is fit in an advanced calculus course.

Anyway, the introduction of this, and the homeworks, start out with pretty simple curves — straight lines, for example, or circles — because they’re easy to understand, and the student can tell offhand if the answer she got was right, and the calculus involved is easy. You can focus energy on learning the concept instead of integrating bizarre or unpleasant functions. But this also makes it harder to come up with a fresh problem for the exams: the student knowing how to find the length of a parabola segment or the circumference of a circle might reflect mastering the idea, or just that they remembered it from class.

So for the exam I assigned a simple variant, something we hadn’t done in class but was surely close enough that I didn’t need to work the problem out before printing up and handing out the exams. I’m sure it will shock you that an instructor might give out on an exam a problem he hasn’t actually solved already, but, I promise you, sometimes even teachers who aren’t grad students taking summer courses will do this. Usually it’s all right. Here’s where it wasn’t.

Complex Experiments with Grading Mathematics

While I’ve never managed to attempt an experimental grading system as the one I enjoyed in Real Analysis, I have tried a few more modest experiments. The one chance I’ve had to really go wild and do something I’d never seen before, sadly, failed, but let me resurrect it enough to leave someone else, I hope, better-informed.

The setting was a summer course, which the department routinely gave to graduate students as a way of keeping them in the luxurious lifestyle to which grad students become accustomed. For five weeks and a couple days I’d spend several hours explaining the elements of vector calculus to students who either didn’t get it the first time around or who wanted to not have to deal with it during the normal term. (It’s the expansion of calculus to deal with integrals and differentials along curves, and across surfaces, and through solid bodies, and remarkably is not as impossibly complicated as this sounds. It’s probably easier to learn, once you know normal calculus, than it is to learn calculus to start. It’s essential, among other things, for working out physics problems in space, since it gives you the mathematical background to handle things like electric fields or the flow of fluids.)

What I thought was: the goal of the class is to get students to be proficient in a variety of techniques — that they could recognize what they were supposed to do, set up a problem to use whatever technique was needed, and could carry out the technique successfully. So why not divide the course up into all the things that I thought were different techniques, and challenge students to demonstrate proficiency in each of them? With experience behind me I understand at least one major objection to this, but if the forthcoming objection were to be dealt with, I’d still have blown it in the implementation.

Reading the Comics, April 28, 2013

The flow of mathematics-themed comic strips almost dried up in April. I’m going to assume this reflects the kids of the cartoonists being on Spring Break, and teachers not placing exams immediately after the exam, in early to mid-March, and that we were just seeing the lag from that. I’m joking a little bit, but surely there’s some explanation for the rash of did-you-get-your-taxes-done comics appearing two weeks after April 15, and I’m fairly sure it isn’t the high regard United States newspaper syndicates have for their Canadian readership.

Dave Whamond’s Reality Check (April 8) uses the “infinity” symbol and tossed pizza dough together. The ∞ symbol, I understand, is credited to the English mathematician John Wallis, who introduced it in the Treatise on the Conic Sections, a text that made clearer how conic sections could be described algebraically. Wikipedia claims that Wallis had the idea that negative numbers were, rather than less than zero, actually greater than infinity, which we’d regard as a quirky interpretation, but (if I can verify it) it makes for an interesting point in figuring out how long people took to understand negative numbers like we believe we do today.

Jonathan Lemon’s Rabbits Against Magic (April 9) does a playing-the-odds joke, in this case in the efficiency of alligator repellent. The joke in this sort of thing comes to the assumption of independence of events — whether the chance that a thing works this time is the same as the chance of it working last time — and a bit of the idea that you find the probability of something working by trying it many times and counting the successes. Trusting in the Law of Large Numbers (and the independence of the events), this empirically-generated probability can be expected to match the actual probability, once you spend enough time thinking about what you mean by a term like “the actual probability”.

Kenneth Appel and Colored Maps

Word’s come through mathematics circles about the death of Kenneth Ira Appel, who along with Wolgang Haken did one of those things every mathematically-inclined person really wishes to do: solve one of the long-running unsolved problems of mathematics. Even better, he solved one of those accessible problems. There are a lot of great unsolved problems that take a couple paragraphs just to set up for the lay audience (who then will wonder what use the problem is, as if that were the measure of interesting); Appel and Haken’s was the Four Color Theorem, which people can understand once they’ve used crayons and coloring books (even if they wonder whether it’s useful for anyone besides Hammond).

It was, by everything I’ve read, a controversial proof at the time, although by the time I was an undergraduate the controversy had faded the way controversial stuff doesn’t seem that exciting decades on. The proximate controversy was that much of the proof was worked out by computer, which is the sort of thing that naturally alarms people whose jobs are to hand-carve proofs using coffee and scraps of lumber. The worry about that seems to have faded as more people get to use computers and find they’re not putting the proof-carvers out of work to any great extent, and as proof-checking software gets up to the task of doing what we would hope.

Still, the proof, right as it probably is, probably offers philosophers of mathematics a great example for figuring out just what is meant by a “proof”. The word implies that a proof is an argument which convinces a person of some proposition. But the Four Color Theorem proof is … well, according to Appel and Haken, 50 pages of text and diagrams, with 85 pages containing an additional 2,500 diagrams, and 400 microfiche pages with additional diagrams of verifications of claims made in the main text. I’ll never read all that, much less understand all that; it’s probably fair to say very few people ever will.

So I couldn’t, honestly, say it was proved to me. But that’s hardly the standard for saying whether something is proved. If it were, then every calculus class would produce the discovery that just about none of calculus has been proved, and that this whole “infinite series” thing sounds like it’s all doubletalk made up on the spot. And yet, we could imagine — at least, I could imagine — a day when none of the people who wrote the proof, or verified it for publication, or have verified it since then, are still alive. At that point, would the theorem still be proved?

(Well, yes: the original proof has been improved a bit, although it’s still a monstrously large one. And Neil Robertson, Daniel P Sanders, Paul Seymour, and Robin Thomas published a proof, similar in spirit but rather smaller, and have been distributing the tools needed to check their work; I can’t imagine there being nobody alive who hasn’t done, or at least has the ability to do, the checking work.)

I’m treading into the philosophy of mathematics, and I realize my naivete about questions like what constitutes a proof are painful to anyone who really studies the field. I apologize for inflicting that pain.

Real Experiments with Grading Mathematics

[ On an unrelated note I see someone's been going through and grading my essays. I thank you, whoever you are; I'll take any stars I can get. And I'm also delighted to be near to my 9,500th page view; I'll try to find something neat to do for either 9,999 or 10,000, whichever feels like the better number. ]

As a math major I staggered through a yearlong course in Real Analysis. My impression is this is the reaction most math majors have to it, as it’s the course in which you study why it is that Calculus works, so it’s everything that’s baffling about Calculus only moreso. I’d be interested to know what courses math majors consider their most crushingly difficult; I’d think only Abstract Algebra could rival Real Analysis for the position.

While I didn’t fail, I did have to re-take Real Analysis in graduate school, since you can’t go on to many other important courses without mastering it. Remarkably, courses that sound like they should be harder — Complex Analysis, Functional Analysis and their like — often feel easier. Possibly this is because the most important tricks to studying these fields are all introduced in Real Analysis so that by the fourth semester around the techniques are comfortably familiar. Or Functional Analysis really is easier than Real Analysis.

The second time around went quite well, possibly because a class really is easier the second time around (I don’t have the experience in re-taking classes to compare it to) or possibly because I clicked better with the professor, Dr Harry McLaughlin at Rensselaer Polytechnic Institute. Besides giving what I think might be the best homework assignment I ever received, he also used a grading scheme that I really responded to well, and that I’m sorry I haven’t been able to effectively employ when I’ve taught courses.

His concept — I believe he used it for all his classes, but certainly he put it to use in Real Analysis — came from as I remember it his being bored with the routine of grading weekly homeworks and monthly exams and a big final. Instead, students could put together a portfolio, showing their mastery of different parts of the course’s topics. The grade for the course was what he judged your mastery of the subject was, based on the breadth and depth of your portfolio work.

Any slightly different way of running class is a source of anxiety, and he did some steps to keep it from being too terrifying a departure. First is that you could turn in a portfolio for a review as you liked mid-course and he’d say what he felt was missing or inadequate or which needed reworking. I believe his official policy was that you could turn it in as often as you liked for review, though I wonder what he would do for the most grade-grabby students, the ones who wrestle obsessively for every half-point on every assignment, and who might turn in portfolio revisions on an hourly basis. Maybe he had a rule about doing at most one review a week per student or something like that.

The other is that he still gave out homework assignments and offered exams, and if you wanted you could have them graded as in a normal course, with the portfolio grade being what the traditional course grade would be. So if you were just too afraid to try this portfolio scheme you could just pretend the whole thing was one of those odd jokes professors will offer and not worry.

I really liked this system and was sorry I didn’t have the chance to take more courses from him. The course work felt easier, no doubt partly because there was no particular need to do homework at the last minute or cram for an exam, and if you just couldn’t get around to one assignment you didn’t need to fear a specific and immediate grade penalty. Or at least the penalty as you estimated it was something you could make up by thinking about the material and working on a similar breadth of work to the assignments and exams offered.

I regret that I haven’t had the courage to try this system on a course I was teaching, although I have tried a couple of non-traditional grading schemes. I’m always interested in hearing of more, though, in case I do get back into teaching and feel secure enough to try something odd.

Looking to Euler

I haven’t forgotten about writing original material here — actually I’ve been trying to think of why something I’ve not thought about a long while is true, which is embarrassing and hard to do — but in the meanwhile I’d like to remember Leonhard Euler’s 306th birthday and point to Richard Elwes’s essay here about Euler’s totient function. “Totient” is, as best I can determine, a word that exists only for this mathematical concept — it’s the count of how many numbers are relatively prime to a given number — but even if the word comes only from the mildly esoteric world of prime number studies, it’s still one of my favorite mathematical terms. It feels like a word that ought to be more successful. Someday I’ll probably get in a nasty argument with other people playing Boggle about it.

Apparently, though, Euler didn’t dub this quantity the “totient”, and the word is a neologism coined by James Joseph Sylvester (1814 – 1897). That’s pretty respectable company, though: Sylvester — whose name you probably brush up against if you study mathematical matrices — is widely praised for his skill in naming things, although the only terms I know offhand that he gave us were “totient” and “discriminant”. That term in the quadratic formula which tells you whether a quadratic equation has two real, one real, or two imaginary solutions, was a name (not a concept) given by him, and he named (and extended) the similar concept for cubic equations. I do believe there are more such Sylvester-dubbed terms, just, that we need a Wikipedia category to gather them together.

I’m amused to be reminded that, according to the St Andrews biographies of mathematicians, Sylvester at least one tossed off this version of the Chicken McNuggets problem, possibly after he’d worked out the general solution:

I have a large number of stamps to the value of 5d and 17d only. What is the largest denomination which I cannot make up with a combination of these two different values.

Animated Sieve of Eratosthenes

Reblogged from The Math Less Traveled:

Here's something I made yesterday! (Note, I strongly suggest watching it fullscreen, in HD if you have the bandwidth for it.)

Can you figure out what's going on? The source code for the animation is here; I was inspired by Jason Davies' visualization which was in turn inspired by this.

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The Math Less Traveled has a lovely video here, animating the Sieve of Eratosthenes, one of the classic methods of finding all of the prime numbers one wants. I suppose it won't eliminate writing out and crossing off numbers for extra credit on a math test. I actually remember that being one test I had in, I believe, seventh grade, for reasons that I don't think I ever got. Possibly the teacher wanted to have an easy time grading, or was giving everyone a break from too much computation by shifting to evaluation of our crossing-out abilities.

A quick guide to non-transitive Grime Dice

Reblogged from bayesianbiologist:

A very special package that I am rather excited about arrived in the mail recently. The package contained a set of 6-sided dice. These dice, however, don't have the standard numbers one to six on their faces. Instead, they have assorted numbers between zero and nine. Here's the exact configuration:

Aside from maybe making for a more interesting version of…

Read more… 584 more words

The bayesianbiologist blog here has an entry just about a special set of dice which allow for an intransitive game. Intransitivity is a neat little property, maybe most familiar from the rock-paper-scissors game, and it's a property that sneaks into many practical applications, among the interesting ones voting preferences.

Reading the Comics, April 5, 2013

Before getting to the next round of comic strips that mention mathematics stuff, I’d like to do a bit of self-promotion. Freshly published is the book Oh, Sandy: An Anthology Of Humor For A Serious Purpose, edited by Lynn Beighley, Peter Barlow, Andrea Donio, and A J Fader. This is a collection of humorous bits, written out of a sense of needing to do something useful after the Superstorm. I have an essay in there, based on the strange feelings I had of being remote (and quite safe) while seeing my home state — and particularly the piers at Seaside Heights, New Jersey — being battered by a storm. The book is available also through CreateSpace.

Jenny Campbell’s Flo and Friends (March 23) mentions π, and what’s really a fairly indistinct question for a tutor to ask the student. “Explain pi” is more open-ended than I think could be useful to answer: you could write books trying to describe what it’s used for, never mind the history of studying it. After all, it’s the only transcendental number with enough pop cultural cachet to appear routinely in newspaper comic strips; what constitutes an explanation of it? Alas, the strip just goes for the easiest pi pun to be made.

Scott Hilburn’s The Argyle Sweater (March 25) returns to the gimmick of anthropomorphized numerals. It’s a cute enough joke; it’s also apparently a different pair of 1 and 2 from earlier in the month. I do wonder what, in this panel’s continuity, subtraction might mean. Still, Hilburn is obviously never far from thinking of anthropomorphized numbers, as he came back to the setting on April 3, with another 2 putting in an appearance.