I had the most incredible morning. Today was my first day co-leading a tutoring project in Harlem through New York Cares - it's a morning Saturday school we're running for underprivileged middle school kids, where we tutor them half the time on math and then do a group activity and work on writing skills. I was born to teach - I haven't felt this good in a while.
Something quite incredible happened with a student named Michelle. She took out her pre-SAT study book. First we did a verbal section. I was impressed with how well she retained new words I taught her and prefixes and word roots - she had gotten about 90% of the section wrong originally, and after the half hour of going over it she took it again and got 90% right. It's short-term memory, but I was still impressed. We'll see if she remembers them in two weeks (next week I'm leading a different team on New York Cares Day, where we'll be fixing up an elementary school in Queens). The key to learning the words seemed to be that we spent a minute discussing each word - how it was related to other words, why it meant what it did (if I knew), explaining uses of it she might have heard. I paused a minute on 'cleave', not sure if I should use the example that leapt so readily to mind; instead I explained how a meat cleaver chops down and separates the two halves of meat. And one of the words was completely ridiculous - 'hawser'. Means cable. I guess it's a word, but I'm now among the six people in history that have ever used it.
The incredible thing, though, was when we got to the math section. The first problem was a right triangle with the measurements of two of the sides given, and an x given as the length of the hypotenuse. I asked if she knew how to figure out what x was. She said no. I wrote it down as the Pythagorean theorm and asked if she knew how to solve that. She said no. I soon learned that she had never seen an x in an equation before, had zero knowledge of algebra, and didn't know what the square or square root of a number were. And since she's studying from the pre-SAT book she's probably taking the test at some point this school year. Problem.
But the human mind is an amazing thing. I started explaining what the square of a number is, giving examples, and what the square root is, giving examples. She knew her arithmetic, so she was able to find squares and square roots just fine, but she kept confusing the terms square and square root, like when I said What's the square root of 9, she didn't know which one I was asking for, the big one or the little one. Makes sense, she had no reason no know what they meant. So I drew a square and made one side 3. I said "The side of this square is 3. What is the area of the square?" She knew it was 9, so I didn't have to visually derive the area of a square for her (though explaining that wouldn't be hard either, and in a way would actually be introducing her to calculus. Next time). So I said "Right. All we need to know about a square to figure out everything else about it is how long one side is, because all the sides are the same, and the area is just the side times itself. So we call the length of one side the square's root, where everything else about it comes from. And when we take 3 and turn it into a square, or 'square' it, the square has a size of 9. So if I say What's the square of 3, or What's 3-squared, it means what size square would 3 make. So 3 'squared' is 9. And when I say what's the square root of 9, I mean if I had a square of size 9, what would be its root? The root of the square, or 'square root' of 9, is 3.
Done - she no longer confused the terms, was able to give me the squares and square roots of various numbers. We then tackled X. We worked through a couple simple algebraic equations. I explained that because the two sides of it are equal, they always have to stay equal, so anything we do to one side we must do the the other even if it looks weird or they won't be equal anymore. I introduced the Pythagorean theorem, just saying it's a formula that a genius came up with and is just something you need to memorize. I regretted not being able to explain it, but the geometric diagram proving that one is a doozy and I didn't remember it. It involves drawing a square sticking out from each of the triangle's sides, and then the three squares are related by lots of trigonometry and eventually you get Pythagoras' equation. I'll bet it could be explained from just the diagram because similar angles would just look similar and so on, so I'm going to look it up and see if it's feasible to explain. Memorizing formulas is completely unsatisfying and I imagine why a lot of kids hate math - no one likes being told that something is the way it is just because someone said so. Everything must be derived from something we can see or we just won't get it, I think because our brain just isn't convinced - seeing is believing - and so doesn't give it much importance. We do end up believing formulas, and therefore remembering them, because we use them for a while and see with our own eyes that it produces accurate results, but getting there without an understanding of them is painful. Math like that is boring and tedious.
But back to Michelle. I soon witnessed magic - within 45 minutes she went from not knowing what x was or what a square root was, to understanding how to solve for the hypotenuse of that triangle. I don't know what they're teaching in school these days, but we did that in 45 minutes, so there's no way in hell I'm letting this girl walk into that pre-SAT exam without knowing exactly what she's doing.
The other leaders then wanted a group activity to do, something to discuss, possibly to do with writing which is the other focus of the program, so I suggested discussing the difference between fact and opinion, and what makes a valid argument and what makes a weak argument. So we did - we asked them to come up with arguments for not discontinuing their science class if the administration decided to cut it for budget reasons. We debated whether each argument was a statement of fact or opinion, what made it a fact or opinion, and why an argument must have nothing but facts holding it up if it's going to convince anybody. It was a good discussion, and one I'd like to continue with them - it's sort of the fundament of everything they need to learn - in science, math, writing, any subject really. That to know something, or to prove something, or to convince someone of anything, it must logically derive from a chain of statements that begin with provable, observable facts. In a couple weeks I want to split them into two groups and have a mock debate, letting each team evaluate the other's arguments for validity. I'm really going to enjoy running this thing.
The drawing of the square, and also how I taught Michelle words - by showing how they were related by roots, prefixes and suffixes and by putting each one in a common-usage sentence, made me realize that we most easily remember things that we understand and can explain why it is the way it is. This is true of everything, even a play - when an actor doesn't know his line, it's because he doesn't understand it and exactly why he's is saying it at that moment. When I was doing some directing a few years ago, and some moment in the scene wasn't there, I'd ask "Why did you say that?", and if the actor didn't know (and if the moment wasn't there, they didn't know), we'd figure out why someone would say such a thing at that moment, and often to do that I'd have the actors start at some other moment in the scene that is working - that is, that creates a genuine, true experience, which are theater's version of provable, observable facts (because they're the only things the audience accepts as true), and see if that led to a better understanding of the lines we didn't understand. That's all a play is - a proof by demonstration of why a set of people would say a specific sequence of words and do specific things to each other given a set of circumstances and some inciting incident. If the actor does understand what all his lines mean and why his character was moved to say them at that moment, on a level as deep and detailed as if he had actually lived the character's entire life to date (which is what makes acting so much work), he could move through the play with authority, precision and ease, and the lines would be on the tip of his tongue exactly as he needed them to achieve his character's desires (which is why good acting looks so easy). This doesn't happen nearly as often as it should.
I'm going to see if I can't help Michelle ace the verbal section too. It's no different than the math in a way, just on a much larger scale, but luckily our brains are extremely hardwired to deal with the complexities and scale of language. We're going to learn why words mean what they mean, not memorize them. I have a hunch she'll retain much more that way. I'll certainly learn a thing or too - I'll bring a dictionary. The problem is that there are so many more special cases in vocabulary than in math, so many historical sources of change, or sources of change whose logic were so specific to whoever invented a given word and however it caught on, that you can't boil it down to a few unifying principles as you can with math. But all words have an explanation - the dictionary holds the entire history of the human race. Just encoded.
All this made me think about the artificial intelligence class I'm taking (a grad class at Columbia), and about how the mind remembers things. It's really good at remembering things that are somehow linked to other things it knows, especially things that are already deeply rooted in our memory. This made me think about knowledge representation in an artificially intelligent system, and that facts and ideas are all linked together in some extraordinarily complex graph (though we haven't gotten there yet in my class so I could be wildly wrong). Revealing the nature of that graph as it exists in the human mind is what we do when we express ourselves somehow, and I personally believe we're all expressing ourselves at every moment in some form or another when in the presence of other human beings. Which makes a street full of people a set of human minds all demonstrating the makeup of their internal graphs, and perhaps also exposing a larger composite graph of all the peoples' graphs combined. Then everything gets all zen and groovy.
I also realized that understanding a concept is really just another form of association, the most powerful kind - associating the new information to things we already know and remember. Since we are really good at remembering things that we associate with other things, that's probably how the fundamental wiring of our brain works - knowledge and memory ARE associations, nothing else. Association doesn't just help us know things, it IS how we know things - we intern new information by linking it up with various points in the rest of the network of stored ideas in our head. I think this is what Lacan means when he says that the subconscious is structured like a language. Michelle will probably always remember squares and square roots in terms of an actual visual square because that's how she first linked the idea into the rest of her brain's network of memories. Since the square was a visual link, perhaps Michelle's thoughts about squares and square roots will always start in the visual cortex of her brain. I'm guessing about all of this, but it's my blog so nah nah nah.
The math, the debate, the vocabulary - they're all expressions of causality. Linking things causally is the defining strength of the human brain that let us conquer the world - by predicting what an animal would do before it knew itself. It's easy for us - we're hardwired for it and we enjoy doing it. Reveal the the cause of a fact and we never forget it. We love watching mysteries get solved for the same reason - things happen for a reason and we derive great satisfaction from discovering that reason. Maybe because without seeing the cause we never really believe it. Kids that are given math formulas out of thin air, and then conclude that math is nonsense and a waste of time are understandably rejecting the information as nonsense because they haven't been shown what makes it true. Their brain has no reason to deam the information important enough to focus on without a demonstration of causality. It's not convinced. No wonder they think math is a bunch of arbitrary and tedious busy work - in that form it is. We need to be shown absolutely everything or we don't buy it.
Anyway, it was a good day. Thanks for reading.
