On Teaching Oneself Calculus
Calculus really isn’t all that hard, but it would be a lot easier if they would explain things in the right order.
As many of you know, I am teaching myself calculus. There is a very good explanation for this, but it requires identity-undermining revelations. (Perhaps later.) Suffice it to say that I was on (ahead of, even) the usual math track until about age 14. For complicated reasons, it’s 100% technically accurate to say that I never set foot in a high school math class, and I only took one math class in undergrad, some boring abomination called “college algebra,” which seemed to be mostly about logarithms.
Well, in under two months, I’ll be needing math. Specifically, I’ll be needing calculus to survive the stats requirement in the a hyper-quantitative poli sci phd program. So over the last few months, I’ve been working on this little project.
I started out by trying to take an actual class, via a sneakily low-priced local adult education program. (Yay government subsidies.) Sadly, the instructor wasn’t very good, and, worse, the class was at 6 pm Monday nights about a half hour’s rush our drive away. Most people, even in the real world, could make a 6 pm class 30 minutes reverse-rush-hour drive from their office one night a week. Most people are not lawyers. After missing three classes in a row (including the midterm) I gave up the ghost. Thank you, practice of law, for forcing me to throw away the $400 I spent on tuition for that class. Literally, all I learned for my $400 was f(x+h) – f(x)/h. That’s it. I didn’t even get to the power rule.
Fortunately, I got one good thing out of that class: a not unreasonable book. It took me a while to delve into it. I first had to psychologically satisfy myself by buying three separate easy-tutorial books that I opened roughly twice each, and that I’ll probably donate to a library or something before I leave, since in all cases, they appear to be less clear and/or less useful than an appropriately slow reading of the appropriate section in the actual textbook.
I did find something other than the textbook that was useful, however: the Teaching Company DVD series. Michael Starbird, if you are reading this, you have only to say the word and I will gleefully have your babies. The DVDs aren’t very rigorous, and several of the 24 half-hour classes are fluff (I’m not bothering to listen to the last two), but it provides a broad-brush conceptual overview of the whole thing, so that when in the midst of a page full of functions, one has some notion of what it is that one is trying to flesh out. I really think the “one pass for broad overview, then a second for the gritty details” approach is the best way to learn this sort of thing.
I also read about half of the trigonometry for dummies book just so I’d know what the calculus book was talking about with sines and cosines and all that. I may scandalize the purists here, but I don’t see the point of all the focus on trig stuff in calculus. Sure, the functions are useful for various things (I’m sure electrical engineers love them). But it’s so much boring memorization. My theory is that the trig part of calculus can be reduced to a couple pages of cheat sheets with identities, derivatives, etc. Then, trig for them that cares, I say.
Anyway. Then it was into the calc book. At first, I started reading from page 1, slowing down to do the problems only when I wasn’t 100% clear that I got it, i.e. for confirmation. That was slow going, and mixed the interesting bits (the definition of a limit, getting to cheat and divide by zero with aid of the difference equation) with the boring (trig) bits. By parametric functions, I’d lost interest in the slow going and stopped regularly reading. I brought myself back on track by focusing on the interesting bits and relying on the forthcoming math craziness (whose premise is that “you’ve been exposed to calculus but you don’t remember it, so we’ll jam the stuff you need for the methods requirement straight into your brain via a jackhammer”) to fill in any gaps I inadvertently leave. Thus, I’m skipping all of derivatives after the chain rule and implicit differentiation, except minima and maxima, the mean value theorem, differentials and linearization. Those things I’m deferring so I can add some variety by diving right into integration.
Integration by substitution is sexy. I think one service that I can provide to the community with this blog is to express the symbol-bound techniques like integration by substitution in plain English. That will be the next post.
This really isn’t that hard. None of it is. There are bits that took a few leaps of reasoning for me to understand, but as far as I can tell, that’s the deficiencies of the book rather than anything else. For example, the book doesn’t really explain that Leibniz’s notation for derivatives means something other than “here’s a different way to say f'(x).” Suddenly, implicit differentiation enters the picture and all this dx/dy df/dy dy/dx stuff means stuff other than “take a derivative, but do it in a silly German accent because the idiot continentals couldn’t acknowledge that Newton was much smarter and hotter than them.” (In case you noticed, the Newton vs. Leibniz thing is going to be a running theme of this blog.) It took me a while to figure out why it is that they got a y’ whenever the variable to be mucked with had a y in it, and I still might call in one of my math friends to make sure I’m clear on the concept, even though I seem to be able to solve the problems. But, anyway, if the book had presented that ratio/not really a ratio/oooh, look, it’s a ratio again as meaningful on its own account rather than a fancy way of saying “take a derivative” in the first place, or for that matter bothered to explain in words whose relationships were previously defined what it means to “differentiate both sides of the equation with respect to x, treating y as a differentiable function of x,” I wouldn’t have had to spend a couple of hours searching the internet for an explanation that made sense.
Anyway, onward and upward. This really doesn’t seem so difficult so far. And it’s kind of fun.