Learning (and Teaching)

I realized something today as I was reading an article about the underlying reasons vegetarians switch over. It went through various reasons, step by step, refuting each one, and as I read I became excited by the elegant words. I could have probably refuted the arguments just as well factually (if not so elegantly), but I didn't pause to try, and read on. Then I knew the conclusion was coming, the real underlying (and perhaps subconscious reason), and I became very excited, as if reaching the dramatic conclusion of a good book. It was at this point that I realized what I'm now writing about, so I quickly typed a few points so I wouldn't forget later. And I read the conclusion, and my joy increased for understanding it along with the steps to get to it. It made sense, the steps to reach it made sense, I comprehended this that someone less intelligent might not comprehend. It made me feel all warm and fuzzy.

But then I went to talk to a vegetarian about their irrationality (to test my new arguments), and I found that I struggled recalling the arguments and the reasoning and the steps to reach and support the conclusion. For fear of saying something wrong, I went back to the article, and even typed things verbatim, using the author's elegance instead of my own. His elegance was such that I struggled remembering much other than it was elegant, and I knew if I tried to rephrase it then it would likely be less elegant. I do this more often than I like to admit, and it's really annoying when I catch myself at it. Quoting someone (it doesn't matter if you source it or not) takes away the responsibility of thinking for yourself. I like quotes, and I try to think about them, but if a situation comes up in an argument where that quote would elegantly explain my point, I am very tempted to use it. I'm now trying to only use quotes as introductory material. "That which can be destroyed by truth, should be." A great opener for a post on why it's good to be rational.

I believe to know the problem now. The information I was presented was done in an elegant, linear way, with little deviation and no "whoops, backspace that bit, it's wrong" mess-ups. The article gently led me through each step of the process, carefully and elegantly arguing, appealing to my intelligence for not being talked down to, and then it presented the elegant conclusion, with a few final words to back it up. This makes for a very exciting read, and in any mystery books that is the formula the author takes for a successful book. As you read the reasoning, you mistakenly believe you yourself have also reasoned it. You get excited. Then the final bad guy is unmasked, and you proclaim "I knew it!" Well, not really, you were spoon-fed the whole way. You may have predicted it back in the beginning, but were not very sure, or if you were then it's a poor mystery book.

The really good mystery books can be reread and still fully enjoyed. I read the Redwall book when I was younger, and I believe it has a decent enough mystery subplot about it to keep it from the group of "poor mystery", at least to a youngster, but even after reading it I doubt I could have told you the exact steps, the reasoning, and so forth the mice performed to find Martin's stuff. Next time you read anything with a mystery element that kept you guessing until the end, see if you can recall all the steps after reading.

So you see, while this process is great for literature, it's not so great as a teaching device. And yet how do math professors teach you where the quadratic formula comes from? You might not even remember the quadratic formula at the time. But the professor starts with the standard form of a quadratic equation, and proceeds to derive it by completing the square and so forth, until voila! the quadratic formula.

I can't remember the exact process because it's shown in a linear manner without much thought, and so you try and memorize the entire series of steps rather than understanding the algebraic reasoning behind the steps and recognizing patterns. I can't even really remember completing the square. But I bet if I thought about it for enough time I'd derive it; but few students wants to spend an hour on one math derivation, or derive it every time, so they cram the full solution into their brain and forget the steps, and all that's left (if they're lucky) in the end is the conclusion. This is why math is so hard for people, I think. They put in one part of their brain this old solution they saw once:

2x = 6; x = 3

into a separate compartment from another solution they also saw once:

2x - 4 = 0; 2x = 4; x = 2

instead of the quite simpler memorization of a pattern. "Get x on one side, then reduce until it's only 1x using basic arithmetic skills." I can remember those two steps very easily, and in multiple step scenarios I might deduce a missing one if I remember others; it's significantly harder to remember the entire string of one of the above solutions which only works for one solution. They're treating learning math like learning new vocabulary, with sheer memorization of static symbols. But math is about variables, and things are never always the same or expressible in one way.

This sheer memorization technique is poor for learning anything, really, including vocabulary. I used to "learn" lots of big words for fun, and consciously use them in my vocabulary, but the desire to do that is gone and those words weren't in common usage anyway, so I now have forgotten many of them. Repetition is better than sheer memorization, but some cases of "repetition" are really sheer memorization in disguise. The example is looking at a vocabulary list multiple times; it's not repetition, it's sheer memorization iterated. Repetition would be more common in program language syntax, where some complex problems just happen to require a foreach() loop, so you use it, and the syntax is learned by using it over and over, not by looking at some examples over and over.

While this is the topic of a different post, I think it's worth mentioning. When I want to really learn something, I make a program to do that something for me. This really helps in math, where your program has to do some logic instead of just substituting values into an equation. Physics problems easily lend themselves to such needs of logic.

What are some remedies to this? I know some who might advocate forcing everyone to reinvent the wheel all the time (which sometimes works well), but this doesn't lend itself to progress. When professors insist that we can't use calculators, I explain that it's insulting to the mathematicians of the old days to think that they didn't need to use all of their brain power to do calculations without computers, and that if we are forced to do the same then we'll never get anywhere. (Human brains are inefficient with math compared to computers; let's merge with them instead of sweeping them under the rug.) Personally I like two similar approaches instead of reinventing the wheel.

The first: the teacher presents the conclusion first, the answer, and if necessary the first step toward getting that answer. Let the learner go from there. Give them lots and lots of time, and make sure they're focused on the task instead of just wasting time thinking about their hot date that evening. With this approach, the student knows when they have reached the correct answer, and they will also come out knowing the steps of the patterns better because, as in programming, they just happened to be required to take the log of both sides to bring down an exponent.

The second is more guided than the first. The first step is named, and then the student is left to explore from there, but if they reach a dead-end or start straying they are given the proper next steps. Let them say when they're done, and tell them if it's true. Let them make mistakes, mention your own mistakes in papers. I remember seeing how Paul Graham wrote one of his essays. Entire paragraphs were deleted from the end result, he didn't do it in a straight linear fashion; it was enlightening.

What might be classified as a third approach is simply stating the conclusion, because the reasoning is unimportant. This works for many math problems once the student has a certain knowledge; it's insulting when teachers make you show work step-by-step for patterns you really know how to solve. I like programming the least when I'm programming things I already know how to do, like a login system. But this approach isn't really what my point is trying to address; few things are instantly understood from just the conclusion.

The only breakdowns from these approaches are when the student genuinely doesn't know how to proceed to the next step. A recurring case of this with myself is trigonometric identities. I'll get some ways into a derivation, and then I find I can't seem to go any further. I get guidance on the next step, and wow, I had forgotten that I can factor out this thing here and turn (sin(x)^2 + cos(x)^2) into 1, then finish up. There are some identities (like that one) that I remember, and others that I don't. But from that one I can come up with another two or three that other people unnecessarily memorize, making life harder for themselves. So if a student genuinely doesn't know (or has forgotten) the next step, they need more problems that involve that step by chance, not intensive (if new) or further (if forgotten) study of that step. I don't usually need the trigonometric identities, so I forget them, but that doesn't mean I should again study how they came about in more detail.

I'm going to try using those approaches when I educate myself, even if I would get more excitement from the mystery-book approach. And not just on math, either, I believe it is more conducive to learning anything if you avoid the mystery-book approach. I'd love to see schools try this; they're all about linear teaching of things and expecting lightning speed of response. Learning things linearly does have its place, and I will expect lightning speeds of response in the future when we have faster nanobot-neurons which allow us to think a week's worth in a second, but until then, schools and teachers really need to rethink their strategies. Mnemonic devices such as songs help for memorization, but actually understanding the processes, actually reasoning through the steps as much by yourself as possible, helps more.

Posted on 2009-06-12 by Jach

Tags: philosophy, school

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