Category Archives: Background

Fluency, Fluidity, Symbology, Complexity, Difficulty. Speed Bumps.

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.

To start, a request for technical advice.

I expect to be putting quite a few bits of math notation here. Does anyone have any bright ideas on how I can go about doing that? The basic options seem to be MathML (I’m not sure how it works) and just using inline .jpg files. Is there anything better out there? (And does anyone know of any simple, straightforward software that lets you type math symbols and outputs image files directly?)

Introductory Matters

Hello. This is a math blog. I am a math blogger. Yes, there is such a thing. There is such a thing now, at least.

This may be a math blog, but I am not a math person. Quite the contrary, in fact. This is the diary of an adventure through mathland, sort of the way that Dante adventured through — well, whether we’re in the Inferno or the Paradiso is a question yet to be determined.

Who am I?

I am an ex-lawyer (soon-to-be-ex, as of this writing), and nascent political scientist. I won’t give too much of my personal history here, since I’m trying to keep this thing somewhat pseudonymous. I will, however, probably release information in drips and drabs as we move forward on this romp through the Math Dimension.

I am keeping my identity weakly secret. This means that I have told several people who know me in person that this blog exists, and I really don’t care if people figure out who I am. However, I am not going to outright tell people who I am, and I ask that those who do know not go publishing it, either in the comments here or elsewhere. This isn’t a request that I can enforce, but I’m only telling people who I believe are honorable and have respect for my autonomy. I have no particular fear of publicity, and I have no plans to say anything indiscreet here. Nonetheless, I don’t relish the idea of being cyber-stalked based on a math blog. I also reserve the right to decline to disclose various personal details, or to change meaningless ones.
What is this blog about?
In just over a month, I return to grad school from a few years in the practice of left-wing activist law, to pick up a Ph.D. in political science from Unnamed Super-Quant Department. Immediately on beginning this happy adventure, I will be in a two-week intensive math course designed to get people who majored in political science in undergrad (i.e. people who never took math) to a position where they have enough minimal competence with calculus and linear algebra to survive stats. Then, I will jump straight into several semesters of very difficult statistics and other quant-type courses. I will probably take more methodology/math courses, or teach myself methodology/math type things, thereafter. Here is where I blog the experience. I begin, actually, in the midst of teaching myself (sorta) basic calculus, so that I don’t get overwhelmed by the cram-course.
Why this blog?

Girls! Uh. Wait. Maybe not. I have it on good authority that I will never have another date again if I start a math blog. Apparently, my soul will be indelibly stained, so that not even the pseudonym will protect me from social ostracism and leperhood.

So the real reason is to get one of those sweetheart half-million dollar advance book contracts to tell the compelling story of my love affair with lim h ->0 (f(x+h) – f(x))/h. Clearly, the market for this thing is so staggering that merely contemplating it will cause a stroke in the unprepared brain.
Seriously? You want serious reasons? Oh, ok. Primarily, I thought that my trek through math might be interesting or amusing. At worst, it could have the same gruesome appeal as the classic train wreck. At best, it might be some kind of inspiring Disney-movie story of someone who, at last count, hasn’t done a math class in thirteen years and his recovery of the mad math skeellz that existed those bakers dozen years ago — honest! I used to be quite good at math and veryaccelerated in it.
Secondarily, I hope to develop a community of math neophytes and the cool math people who love them. (Hey math girls! I’m single!) There seem to be many people out there with an interest in math who haven’t gotten around to doing it, and perhaps they can read along and learn something. This is particularly the case if I can lure cool math people into reading this for the trainwreck factor, and then seduce them into talking to the non-math people. And, you know, to helping with any questions that I might have, too. Now, you see my nefarious plot! Uh, or something. In case you haven’t realized yet, being a good sport with a sense of humor will help in reading this blog.
Finally, I hope this will be an amusing artifact of this stage of my life.

So what’s with the names?

The title of the blog, Madness of Fluxions, is derived from Newton’s posthumous book about calculus, Method of Fluxions. I am bringing my madness to Newton’s method. Newton, incidentally, was much smarter and hotter than Leibniz.

The name I’m trying to publish this under (assuming I can beat the software into shape), Johannes Climacus, was originally one of Kierkegaard’s better pseudonyms. Kierkegaard, of course, was the most awesome pseudonymous writer in history (closely followed only by Junius and Publius).