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.
I think what you say [especially about symbology] reveals one of the larger problems: that mathematics education in primary and secondary schools, if not post-secondary instutions as well, is too caught up in practicality.
What I mean by that is that we have forgotten how to approach mathematics creatively. Instead we are introducted to it algorithmically: high school teachers spend years [yes, YEARS!] explaining how to algorithmically manipulate algebraic equations to produce roots. And so when students are presented with a word problem with algebra at its core, the students have no idea how to proceed: there is too little understanding of the point of mathematics. We teach rules and procedures, not mathematics …
I think your comments on symbology kind of demonstate this — that we’re too concerned with the concrete and haven’t been taught to think as abstractly as we should. Inequalities are a fine example: it’s been my experience that a vast amount of students have a really hard time dealing with algebraic [especially quadratic] inequalities because they don’t really think about what the symbology means; they just want an algorithm to know how to solve the problem.
Now, granted, there’s nothing wrong with algorithms. But I think it’s the wrong way to *teach* mathematics. I often agree with those students who ask, “Why in the world do we have to learn this?” — because very few people, on average, need to know the quadratic formula. We need to start approaching mathematics creatively and abstractly again, so that there really is a point in teaching it — namely, the cultivation of general intelligence, of general techniques of problem-solving.
As to your thoughts on complexity — well I don’t have something as cogent to say. Complexity is a difficult problem, especially, for example, in dealing with concrete cases in linear algebra: it’s just plain hard to think about four by four matrices. In my experience, it takes a very long time to build up proficiency in thinking mathematically. But it’s also a struggle in learning to think creatively, abstractly, metaphorically –
Comment by Matt — August 12, 2006 @ 5:10 pm
I think you’re definitely right here. In addition to making it difficult to solve actual problems, and failing to cultivate general intelligence, I also think the problems you mention kill the attraction of math. And there is one. I’m sure there’s one for you, but there is even one for non-math people like me. Some of this stuff is just really cool!
Like limits as x approaches zero. I have a really enjoyable fascination with this concept, because, holy shit, this is a trick to let you divide by zero! You can’t divide by zero, but then you play around with this limit business, and you can make sense of a fraction with a zero in the denominator! This is sneaky and it appeals to me aesthetically as well as just intellectually. It’s fun! And even stuff I don’t fully understand yet — like Euler’s identity. One of these days, I need to find and learn the proof for that, because it’s just so bloody beautiful. I put it on a bumper sticker, for pete’s sake. And this isn’t even considering the really wonderful applications, and the advanced stuff. Probablity theory. Game theory. Set theory. Number theory. Crazy stuff like infinitesimals. Complexity theory. THIS!! Good lord, I don’t even know anything about these things except the lay outlines, but that’s enough to make me fall in love with them. Everything I hear thirdhand about those things is beyond fascinating and I can’t wait for a chance to learn them.
And yet the people teaching math don’t teach this stuff! When I was actually taking math classes (it’s been a dozen years now — a dozen years!), I was always extremely good at it, always got the highest grades in the accelerated classes that I took even before the general acceleration that we won’t talk about here for identity-preservation purposes (you know what I’m talking about). But they were so boring, so damned boring! So when I got through the one math class I had to take in undergrad (with a truly horrible professor), I quit there. It’s a shame. I wish I’d actually taken a lot more back then. (Oh well, I’m making it up from now on.)
I think a lot of these problems would be solved if students were actually shown the coolness and the beauty in all of this earlier on. And I do mean earlier on. The notion that it takes high school teachers years to teach the quadratic equation just horrifies me, even apart from the fact that it’s really bloody easy (I recall learning it in the 7th grade). They could be pushing forward in all that time. They could be challenging their students. They could be developing an interest, a fascination in this stuff in their students!
I think you nailed it with your point about it being a strugle to think mathematically but also to think creatively, abstractly, metaphorically, etc. I also nailed it with my analogy to languages. And the way we both nailed it in both cases was that all of these things are best developed early. Children develop creative, abstract, and metaphorical thought early, and they should develop languages early (that’s another rant entirely). But are they helped to develop serious mathematical thinking early? Nope. Nope. Instead they’re locked in institutions having their brains turned into stunted robotic mush.
I’m of the school of thought that holds that the vast majority of children can learn a lot more and a lot faster than they have been taught. Maybe this is a biased perspective, but we don’t bloody well know, do we? Because we don’t try except for the most fortunate students. It’s an outrage.
Comment by Johannes Climacus — August 12, 2006 @ 9:27 pm
I agree completely, particularly with what Herr Climacus said about mathematics and language. There are two things which schools (or, at least, schools in America) do not do: give appreciation for maths/language or teach their methodologies.
The quadratic equation is just exploded onto them like a flood of afterbirth (which is, after all, what it most resembles). I’m not saying that a rigorous proof is necessary, but SOMETHING.
Compare that with these two stories I’ve heard from people who went to British schools.
First, a whole lesson was spent discussing a Mrs Heloise Johnson who lived at 12 Spens Avenue and left her house, with £10 in her purse, at 10.01am on the 1st of September 1981 to walk 2.4 miles to the market whereat she bought some apples and then 20 cloves for each apple. Each apple cost 15p and each clove cost a single penny. She spent a total of £1.75. She then went to the hairdressers and returned home one hour and thirty-eight minutes later. How many apples did she buy?
The students are forced to write down the ENTIRE STORY in their copy-book, and then to work out the answer any way they like. Then they’re asked how they could’ve done it faster. They work out pretty quickly what is relevant and what is irrelevant. Every time someone says “We don’t need to know her address” or similar, everyone has to write down the WHOLE thing, sans whatever was deemed unnecessary. Eventually, natural laziness lead someone to say “We can write ‘a’ instead of ‘apples’” and the rest is algebraic history.
Another similar example is giving them a curve on a piece of graph paper and asking them to measure as accurately as they can the area underneath that curve. You’d be surprised at how much more sense an integral makes to a child who’s worked out that some really narrow columns underneath a curve do the trick.
The same in grammar. I was appalled when I discovered that the “rule” children were being taught for choosing between ‘who’ and ‘whom’ was ‘replace it with “he/him” and see which one “sounds right”‘. No mention of subject/object — no EXPLANATION of why these words are different. We aren’t giving our children concepts, were giving them words and rules, which are unmemorable and unpleasant.
It’s my firm belief that this has come to pass because of sheer pedagogical laziness. Children are naturally good at grammar and naturally good at math — to a point. At some stage, their natural intuition will give out. The experience of “hitting a wall” in mathematics is almost universal amongst those who do not consider themselves mathematical. “Oh, I was alwayd really good at math until fractions/algebra/geometry/trig/whatever.” But that’s not the real sin. The real sin is that this has been going on so long that those who TEACH mathematics are precisely those whose natural intuition never gave out (or, at least, didn’t give out at the point they’re trying to teach). So when students don’t “get it” they are at a loss to understand why, let alone explain anything to them.
And so the greatest sin a teacher can ever perpetrate is committed on a daily basis: they say to the poor, bewildered would-be mathematician, “Can’t you just SEE that you should multiply by the LCD/divide through by x squared/construct a line between P and Q/use some trig identity/whatever?” No, they can’t “just SEE”, and neither should the teacher: the teacher should know WHY and be able to communicate that.
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