In case you haven’t noticed the absence of posting, this blog is on hiatus for a couple more weeks. I’m in the midst of moving across the country.
In the meantime, my favorite webcomic has a logic and math puzzles forum. Go play.
I have some thoughts I’d like to share on why it might be that so many people experience math as so difficult, even though at bottom it’s just nothing but logic and rule-following. I can identify two factors that are present in those pieces of math that I experience as more difficult than other pieces, and that cause me to slow down. I don’t know that this extends beyond my own experience, but I’ve consistently found that more of my life is generalizable than I’d previously believed, so I’ll run with it for now. (It’s not as if I’m claiming to be scientific here.)
1. The first is symbology. I notice that it takes me noticably longer and takes me several more mental steps to understand a piece of information that is presented in math symbols than one that is presented in words. This is true not only for complicated stuff, but for simple stuff too. It takes me less time and effort to understand “a is greater than b” than to understand “a > b” — seriously, even for examples that simple. I have to pause for a brief moment and consciously remember the old elementary school mnemonic “ok, it eats the bigger one. That means a is bigger than b.” I’m not fluent enough to use the symbols fluidly. I have to mentally translate them to the language in which I am fluent, English. For symbols that I didn’t actually learn in elementary school, it’s harder. For example, I sometimes have to go back and look up whether it’s the brackets that mean a closed interval and the parentheses that mean an open interval, or the other way around. I just have trouble remembering that.
I know this isn’t just me, because my mother reports a very similar experience, not with math, but with directions. For some reason, even though she is a very intelligent woman (she did produce me, you know) she has an extra mental step to remember “left” and “right.” If you’re giving her directions on the fly in the car, and she needs to turn left right now (as opposed to, say, in a block), you’d better say “turn your way” rather than “turn left,” or the split second it takes her to associate the signifier “left” with the signified direction will cause the turn to be missed. Same with me and math symbols.
I’m not sure from whence this problem comes. Obviously, self-interest and self-image forbid me from attributing it to general intelligence, so I’ll go with practice. It seems likely to me that people who do math every day for an extended period (like math majors, for example) will eventually develop a natural fluency in the language in the same way that people who speak a normal language do. Immersion. Has anyone done a comparative study of language acquisition and mathematical acquisition? If there are any psychologists or linguists reading this thing and doesn’t think it’s been done, talk to me, we’ll collaborate or something.
2. I think the second issue is complexity, expressed in terms of the number of memory registers a given discrete object of learning occupies. On those occasions when I struggle with an item and eventually figure it out, I often find that the thing that kept me from figuring it out in the first place is that I failed to take into account the effect of a piece of information that was presented right up at the start, but that I forgot in the process of working through the rest of the details.
Here’s an example. I recently had a little trouble with some theorem about integrating symmetrical functions depending on whether they’re even or odd. The proof operated by dividing the function into two seperate sides, representing the function for the part of the curve on the left side of the origin and the function for the right side. The one for the left side used the variable u, which was defined as -x, and the one for the right side used the variable x. The very last step of the proof combined the integral of the left side and the integral of the right side into 2 times the integral of the right side. That drove me crazy, because of the u being defined as -x. I thought it required one to take the variable u, which one had previously defined as -x, and substitute it for x. Clearly, -x isn’t ordinarily substitutible for x, except where x is zero, and it was driving me up the wall.
It finally hit me as I was driving to work one day. “Wait a minute. These are symmetrical functions. The integral of the function with u represents the area under the curve on the left side of the origin, and the integral of the function with x represents the area under the curve on the right side of the origin. Of course they are equal, by definition! (Note to textbook publishers: for heaven’s sake, would you please put a margin note in places like this?) That’s why the theorem is limited to symmetrical functions. In my struggles with the proof, I’d simply failed to remember and apply the basic piece of information that defined the class of functions to which it applied.
Similarly, I’ve noticed that the more variables and different techniques work their way into a proof or into a method, the more difficult it is. Not because each can’t be easily applied in isolation, but, I suspect, because it simply occupies more memory registers to apply them all at once. It requires more concentration. (In a related note, I’ve noticed that I can’t listen to music while working on the hardest bits.)
Again, I suspect this is related to practice. There are a lot of things that one can do unconsciously after long practice so that the technique doesn’t have to be in working memory, as such, when one does it. For example, when you start to drive, you have to consciously think about when one starts to do a left turn. After you’ve been driving for a while, you no longer do so. I suspect the same applies to, e.g., applying the chain rule.
Extrapolating from my experience, I suspect that other people experience these same things as requiring more mental resources. Particularly to the extent these things require mental time to work through, I can see how people get more and more behind. They sit in a math class and hit one of these bumps, and they don’t get enough time in the class to process it before the instructor moves on, for example. Suddenly, they’re behind and they don’t have the self-awareness to realize that they need to spend the time out of class working through the speed bump in order to catch up. So the problem cascades.
Perhaps? Or perhaps I’m just repeating things that math instructors (not to say psychologists) have known for generations? I don’t know. But it’s interesting to me to make sense of my own experience this way, at least.
David Chalmers (insane brilliant philosopher of mind in Austrailia) has apparently written not just one paper on the two-envelope thing, but two papers on the two envelope thing! (Cross-reference: How to make people crazy with probability.)
I have it on good authority from my Mathematician Friend (TM) that the answer is that there’s no probability distribution on the real numbers. I suspect he wanted to add “so shut up about it already!!”
One very good reason to start a math blog is that it motivates you to crack the math book slightly more often, simply to have something to blog about. Because if you don’t blog frequently, people won’t keep coming back, and nobody will add you to their blogrolls, and you will die at age 30, alone, cold, wet, and poor in a back alley in Düsseldorf.
So the other major bit of calculus so far that took me inordinately long to parse in text because of the sheer number of symbols boomeranging about was the chain rule. As before, here is my (no doubt amusingly wrong) attempt to translate it into English. Here be a good online symbol-version. Here’s another. Hopefully one of those will make sense to you. (It mildly squicks me to link the second version, as I seem to have dated one of the people on that project who may have created it. This is, like, math-cest. Ewww. This TMI comes to you courtesy of The Blogosphere, Bringing You Irrelevant Personal Revelations Since 2003(r)).
So, ANYWAY! You have zis function, and you want to take its derivative. Only, it’s a little messy (clearly the root of all evil). Lucky for you, it can be written as one function inside another function. For example, consider the function F(x) = (x2 + πx)4. This can be seen as one function — g(x) = x2 + πx inside a second function f(u) = u4.
(Isn’t it interesting how whenever mathematicians want to notate a second function for substitution purposes — in this case replacing the g function with a variable to show how it fits into the f function — they name the big function f(u)? Is this some kind of latent resentment at the world because physics gets all the grant money?)
So here’s what you do to get the derivative of this sucker. You take the derivative of the outside function — f(u) and then replace the u with the inside function, without taking the derivative of the inside function. In the example, f'(u) = 4u3 by the power rule, and that is expanded to 4(x2+πx)3. Then you multiply that result by the derivative of the inside (g, or u) function. Which, of course, is g'(x) = 2x+π. So the final derivative of the whole big function is F'(x) = 4(x2+πx)3(2x+π). Shazam!
Yes, I just put pi in this one so I could play with HTML math symbols.
EDIT: Ok wordpress, get the line breaks right. I do not want to hand-write the HTML, but I will if you cross me.
SECOND EDIT: I’m serious here. Stop removing my paragraph tags. Stop it. Now. NOW.
There’re a bunch of probability puzzles that seem to drive people crazy even today. Beyond the classic Monty Hall problem, there’s the puzzle about picking between two envelopes, where one has twice the amount of money as the other. The second of those problems gave rise to over 300 debating comments in its recent appearance in the Volokh Conspiracy. It is also broken down here.
If you ever want to really bust up a party full of people who are intelligent and aggressive, but don’t know probability, try that one. It’s guaranteed to turn a group of otherwise normal people into snarling math-maniacs.