Archive for December, 2008

Amateur Night

\bulletR. Talbot’s “Re-envisioning Calculus”; also a links list (these are very good things).

Just Passing Through

\bullet“Blaming the mathematician”, by I. Laba.
\bulletWeightlifting for the brain by guest cartoonist Mat Moore in TCM Technology Blog.
\bulletPart 3 of JD‘s math war skirmish. Published tomorrow.

Jeremiah 6:14

So it’s complicated. Most of the most prominent of my party in the Math Wars, for example, are of the enemy party in the Class Wars: evidently a determination to cling fiercely to the authority of Mathematics correlates with an eagerness to submit to a Fearless Leader (and revel in said leader’s ability to push people around).

Too bad for my side, ain’t it. But that’s if I had a side.
It’s lonely as hell being the designated mourner for Algebra-as-I-understand-it; the Math Guys all mostly seeming to’ve effected some uneasy truce with the enemies of clarity (“just please let us say what we mean in Calc-and-‘above’ and you other guys just go ahead and tell whatever crazy lies you need to tell to hide the truth from the ‘remedial’ classes; those students don’t want to see us and we don’t want to see them …”)and the publishers just grabbing everything in sight like Visigoths (“let’s make the book bigger and put in more user-proof crudware and be even less precise . . . they’ll order from us no matter what we do!”) before the whole bloated Empire falls.

I speak here of (lower-division) college work, let me hasten to remark. The Math Wars have of course mostly been in the K-12 arena. So that’s where I soon learned (a long time back, in my Newsgroup Era) that I’d often be taken for some kind of Republican because I didn’t think math students should be rated (by their math teachers) on anything but their demonstrated skills in mathematics. The Democrat position evidently being that reasonable people should never stand on principle; math should be just as easy as any other subject for a talented BS artist to qualify in. Well, a plague on both their houses, obviously.

The wingers are always trying to blame everything on The
Union (or Tenure), evidently believing that if Management were only free to push teachers around even more that somehow students would get what they’ve needed all along. Or maybe if we were even more afraid of losing our jobs … or something.
Anyway, they want everybody to be as miserable as possible.
Prisons and wars and all that kind of thing. All so heinous it makes the Math Wars look trivial, as I’ll be first to admit even though the Math Wars obsess me (and I’m not sure I don’t mean this literally).

But then you’ve got the Softies who’re usually right precisely
because they’re willing to be wrong: deciding things as a group
usually is the best way to handle social issues; come let us reason together and all that. Just, dammit, at long last, get away from me with that attitude in the Temple of Mathematics.
It doesn’t fly here. Math tests work. Making it an article of faith that “standardised tests” can’t measure academic achievement makes your program a laughingstock. And, finally, politics be damned: a lie to the face is an insult and the right answer to contempt is more contempt. There’s a whole lot more lies and half-truths and nonsense from this side than the other and I just can’t have ’em as allies.

There’s plenty to go around on both sides of course; if a solid point ever were to be made in one of those old newsgroups, you could pretty well count on it to be replaced with some well-worn strawman real quick … once you’ve decided to talk past each other, it’s a pretty easy trick to quit listening … I imagine I’ll have given the impression that I’ve quit listening …
but it’s the Lefties that are pushing math-without-math and it breaks my heart.

What The Hell Is A Trick Goat?

Don’t Try This At Home

I’m actually playing some videos today since the office is otherwise unoccupied. Here is the famous “Klein Four Group” rendition of M. Salomene’s “Finite Simple Group (of Order Two)”, posted last week by a philosopher whose blog seems like it’ll bear watching. Yep. Mighty entertaining.

In other late-to-the-party news, I should’ve linked to the “Appeal of Mathematics” thread in Ars Math. by now already.

This online equation editor “let’s you write and download equations in seconds!” (sic). Spotted at mathfest.

Why Live

It so happens that x^4 + x^2 + 1 = (x^2 + x + 1)(x^2 - x + 1). And this, all by itself, far from being a cause for despair, is kind of a cool thing to know. What depresses me more than I can safely say is that “factoring trinomials” is a very big deal around here (I blog out of the office even during breaks because I don’t have the social skills needed to establish an internet connection at home) and that this particular equation very clearly has no place in our curriculum.

So. What then are we to do? Well, first of all, obviously, verify it (take nothing on authority in mathematics). Maybe you’ll notice that the RHS (“right hand side”, natch) can be parsed as ((x^2+1) + x)((x^2+1) - x) ; for me this was the easiest way to check (because the Special Product Formulas of our 103 class are second nature to me; I can do the expansion to (x^2 + 1)^2 - x^2 [a “difference of squares”] mentally and mentally use the “square of a binomial” formula to expand further; one x^2 “cancel”s; done). Or not; it’s fun to watch six terms of the longer expansion wipe themselves out. Then what? Well, then ask, “how could I have found this out if I didn’t know it already?”.

And the awful truth there is: Google it. This online rootfinder was pretty easy to find and gave back our result more-or-less correctly (I’m quibbling with some notations and suchlike conventions here; nonexperts needn’t be troubled by my weaselwordage). “Google it” is the right answer to a lot of questions!

No, really. Stick with me here. Lord Satan appears to you and says “Factor x^4+x^2+1 over the integers and I will spare your tribe; fail and I will annihilate you all without appeal.”. What do you do? Well, if you’re good in algebra, you solve it (and check it very carefully); if not, well, if you’ve got any sense, you get up on the internets and find a page that’ll do it for you. The last thing you’d do (if you’ve got any sense) is take our 103 course or anything like it: not only won’t they tell you how to find this out, they’ll probably leave you with the impression that you know for sure that x^4+x^2+1 is prime or something; alas, your very teacher will probably think it’s prime.

So what are we spending so much time and money for, trying to teach these poor devils to do badly what an easily-accessed robot slave can do well? And, assuming for the moment that there are good reasons for doing so, is it at least possible to be honest about it?

Because, check it out: if we rewrite the question as “How could I have found this out without a computer?”, we’ll encounter some interesting math, along with some insights into the Remediation Congame that has replaced Mathematics Education rightly-so-called with its putrefying corpse.

First of all, notice that Lord Satan said we had to factor our polynomial over the integers. But if he’d said “over C” (i.e., over \Bbb C; this typesetting interface is free but otherwise far from perfect), one would (do I get to say “of course” here?) have a different answer: x^4 + x^2 + 1 =(x-\zeta)(x-\zeta^2)(x-\zeta^4)(x-\zeta^5), where \zeta = e^{{\pi i}\over 3}. My point here, for now, is simply that the domain over which we are to factor a given polynomial matters: it’s part of the problem and needs to be part of the problem statement (again, assuming we’re interested in trying to be honest). I’m not proposing here that 103 students need to learn about the complex number field before learning about factoring polynomials (and neither do I say that that it’s necessarily a bad idea for them to do so); one could point out that x^2 – 2 is prime over the rationals but x^2 -2 = (x+\sqrt2)(x-\sqrt2) over the reals (and, indeed, seeing that our texts and courses seem determined not to make this point, one is led to wonder how—if at all—they expect to be taken seriously in making such a fuss about the distinction between the reals and the rationals in an earlier part of the course … every chance to actually use the distinction having apparently been rigorously suppressed).

OK. Once we do allow ourselves to mention C, we can make a great deal of progress. It’s probably not wildly optimistic to hope that the median-level students of a 104 class could use the quadratic formula to write down the four complex solutions to x^4 + x^2 + 1 = 0, namely x = \pm \sqrt{{-1\pm\sqrt{-3}}\over2}. It probably is wildly optimistic to hope that they could then write down the four roots separately. I’m not kidding. A few “A” students might be able to produce the equation
x^4+x^2+1 =
(x +\sqrt{{-1+\sqrt{-3}}\over2})(x-\sqrt{{-1+\sqrt{-3}}\over2})(x+\sqrt{{-1-\sqrt{-3}}\over2})(x-\sqrt{{-1-\sqrt{-3}}\over2}) (which is of course equivalent to the C-factorization I gave earlier). A student that could produce this equation with no other instruction from me than “Factor x^4+x^2+1 over C” would gladden my heart more than I can say even though I don’t approve of square-roots-of-negatives written without the “plus-minus” sign. Bitter experience convinces me that most 104 students, even given the “x=” solution I provided above, couldn’t produce the factored-form equation with me right beside ’em telling ’em how to do it. It’s like they feel themselves physically incapable of writing down anything so messily algebraic. Not that I blame ’em (much); you’ve gotta walk before you run.

OK. But a 148 student could probably do it. Still couldn’t multiply out the result and check it, though: for that we’ll need a better notation. So let \zeta = {{1+i\sqrt3}\over2} and take it from there: \zeta^6 = 1, \zeta + \zeta^5 = 1, and so on (inscribing a hexagon in the unit circle is helpful here). But we sure as heck don’t do this in 148 around here; it’d take a week. And nobody cares. Not the students, not the publishers, not the department, nobody. Just me.

What am I even supposed to be doing here? Does everybody else involved actually want me to get up in front of the room and say, “Look, in real life you’re going to let the computer take over whenever anything gets at all complicated, so never mind trying to understand the big picture: that’s for the experts anyway, and the creators of this course don’t want you to learn to think like an expert …”? Because I don’t need this god-damn job that bad. I can beg on the street.

Walter Kovacs

\bulletDan M’s Rorschach test.

\bulletM.A. Chandler kicks off a thread on “imaginary” numbers.

Why Be Poet

\bulletMark CC’s manifesto: “Why Math?”.
\bulletLaTeX to MathType at TCM Technology.

\bulletNew TIMSS results noticed at KTM.
\bulletCPM prez Tom Salee’s math ed presentation praised at dy/dan. And what the heck, Dan’s.