Missing Things

20 December 2012

Lockhart's Lament

I don't know if you've noticed that I'm really bad at this 'updating the blog on a regular schedule' thing? There's not even really much of an excuse for it; it's not as though I'm working on more important things (well, I am, but I'm spending a lot of poorly-prioritised time on less important things, too). I'll have to try to get back into the habit.

Anyway - I try not to talk about myself too much on this blog because it's really not about me so much as just me thinking about things. (In case that motif wasn't already obvious on the blog background!) But I'm making an exception, because there's something I absolutely HAVE to gloat about at the moment. So, personal anecdote first; if you get lost or bored with the mathematics, skip to the bold text below where things get less anecdotal again!

For a little background - my current college transcript contains references from four different schools already and I'm only halfway done. The problem this has led to is that I took basic classes like Calc I and II at one school, then transferred to another school that accepted them as a prerequisite for Calc III, then transferred again to a school that counted Calc III towards my degree but not I or II, compelling me to have to retake them. The good news is this is advantageous to my GPA; the bad news is... well, guess who decided to use Laplace transformations to solve petty little first-order differential equations? Me. Because BOOOOOOOORIIIIIIIING. It's an open secret in Calc II that I have paid scarcely any attention in class at all, and still set the grade scale for the rest of the class.

Here's the part where I gloat: on the last exam of the year, I got 111%. I scored higher than the grade scale permits. ("I don't always get pass tests, but when I do...")

Here's how. The grade scale in Calc II worked by first scoring all the exams by points-per-problem out of points possible. All the scores are then scaled based on the person who scored highest - if the highest score is 93%, it becomes 100% and everyone else gets (100-93)% = 7% added to their score. Then the extra credit points are factored in; every test has up to 10% worth of extra credit, so the highest theoretical score is 110%. (Did I mention this was an easy class? It was a really easy class.)

There was only one question on the last exam I missed. (NOTE: If you don't follow the rest of this paragraph until you reach the next note, that's okay.) The instructions were to integrate 1/(1+x2) from -1/2 to 1/2 (the antiderivative of 1/(1+x2) is arctanh(x) + C, AKA the inverse hyperbolic tangent, in case you haven't gotten to hyperbolic trig...). Yes, it was literally the formula exactly, so it seemed easy until I realised the calculator doesn't have an inverse hyperbolic tangent button. After mulling this over for a while, I decided I could define y = arctanh(x), solve for x, use the identity tanh(y) = (ey-e-y)/(ey+e-y), and then solve for y to learn the logarithmic equivalent of arctanh(x). (NOTE: Okay, we're back!) Good news: it worked. Bad news: I completely lost the teacher somewhere along the way, who was expecting us to solve it using Taylor series. Oops. So I lost points and got a 93%, the scale was set by someone else who got a 99%, and so after accounting for extra credit I ended up with 104%.

When I got the test back and asked about the big red question mark on that problem, teacher explained his confusion, I explained my process, teacher said OH! That makes sense, and gave me back the seven points I had lost. And, since he wasn't about to go recalculate everyone else's grades in the class, that left me with a score that was 1% beyond what was theoretically possible! :D

Okay, anecdote ends here. Those of you who ended up skipping to this point get something to read as well. Most of you are probably part of the broad portion of society who not only hate mathematics, but are openly proud of it, which is a weirdly socially-acceptable display of ignorance. Ever wonder why that is? Paul Lockhart says it's because of the way that mathematics is (erroneously) taught as a science rather than an art, and you probably do appreciate mathematics when you encounter it - you just don't recognise it when you do, because you've been told all of your life that mathematics is something totally different! You can read it here. It is a bit long, but entertaining, illuminatory, and totally worth it. Post your thoughts when you're done.

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