4 ms·

Brett Victor writes: When most people speak of Math, what they have in mind is more its mechanism than its essence. This "Math" consists of assigning meaning t

by ezyang 14y ago

Brett Victor writes: When most people speak of Math, what they have in mind is more its mechanism than its essence. This "Math" consists of assigning meaning to a set of symbols, blindly shuffling around these symbols according to arcane rules, and then interpreting a meaning from the shuffled result. The process is not unlike casting lots. This mechanism of math evolved for a reason: it was the most efficient means of modeling quantitative systems given the constraints of pencil and paper. Unfortunately, most people are not comfortable with bundling up meaning into abstract symbols and making them dance. Thus, the power of math beyond arithmetic is generally reserved for a clergy of scientists and engineers (many of whom struggle with symbolic abstractions more than they'll actually admit). I think gamification is a great way to teach symbol manipulation, and I think (contrary to Bret) that symbol manipulation is a prerequisite for deeper STEM ideas. I do also believe that harder mathematical problems can also be gamified, but this process is much less well understood, and you'll probably want a few theorem prover experts around if you attempt a system like that.

dbaupp 14y ago
I 100% agree that having symbol manipulation is required for deeper understanding and learning. In my experience, being able to quickly and (fairly) reliably move symbols around in one's head makes futher learning much easier and faster, as one can follow proofs and explanations intuitively without feeling the need to work through each step laboriously to justify it. As a simple example, being able to see the steps that were used to go between (x - 1)^2 - 2 = 0 x = 1 +- sqrt(2) at a glance gives one a higher "maths-learning" bandwidth, since one spends just the 3 seconds it takes to read those lines, instead of the extra 30 seconds physically doing the working. (And, on a slightly different note, having some basic facts memorised like (for example) "d/dx sin(x) = cos(x)" is a little like the difference between data in L1 cache and that just in RAM. However, this doesn't mean that one should not understand why, or not be able to rederive it in a flash by drawing a diagram or whatever.)
jessriedel 14y ago
> I think (contrary to Bret) that symbol manipulation is a prerequisite for deeper STEM ideas. Can you elaborate on this further? I tend to disagree, but I don't know if I have all that much to say to back it up. I mean, sure, there's just no good way to understand shear forces without being able to manipulate matrices (so I agree with dbaupp's sibling comment), but that doesn't mean you want to learn the rules of matrix algebra as if they were arbitrary rules enforced by the stick and carrots in a game.
- ezyang 14y ago
  In the case of mathematics, there is a fundamental sense in which the symbols and their manipulations are what you're studying. Think of it as if you were an archeologist: the symbols are the artifact of study. It's only half of the picture: you must(!) come up with a different way of intuitively understanding it--but at the end of the day if you really want to precisely say what it is you're talking about, it's symbols.
  - jessriedel 14y ago
    OK, I think I see what you're getting at. But we could devise a AltDragonBox game with a completely different set of arbitrary rules. (Possibly even rules that are inconsistent.) And as far as game play goes, AltDragonBox would work just as well. But there's a reason that the rules of algebra are what they are. And if you can't distinguish between DragonBox and AltDragonBox, I don't think you're learning what you need to learn. Here's another way to look at it: we all know people who could ace their high school math tests because they had memorized the rules of manipulation but weren't good at math--as evidenced by their poor performance in college and failure to succeed in future STEM classes. If "the symbols and their manipulations are what you're studying" (which, I agree, there is some truth to), then what is it exactly that these people were lacking?
    - ezyang 14y ago
      I agree that it is important to recognize the limitations of this approach. As a high schooler (and even as an undergraduate in mathematics), you learn all sorts of formal systems. As a mathematics researcher, one of your charges is inventing interesting mathematical concepts to play around with. Obviously this is a bit much to ask out of DragonBox! And even more modestly, the ability to sit down with a problem and say, "Ah, but what really is going on here?" is a deep and difficult skill to impart. I don't know what these people were lacking, but if they didn't know how to move symbols around, I might have started there... (BTW, an inconsistent set of rules would correspond to a version of DragonBox where there was a cheat code you could enter, and then easily solve every problem. So it would not "work just as well", and this would be pretty clear to a cheating gamer.)

by ezyang 14y ago

dbaupp 14y ago

I 100% agree that having symbol manipulation is required for deeper understanding and learning. In my experience, being able to quickly and (fairly) reliably move symbols around in one's head makes futher learning much easier and faster, as one can follow proofs and explanations intuitively without feeling the need to work through each step laboriously to justify it. As a simple example, being able to see the steps that were used to go between (x - 1)^2 - 2 = 0 x = 1 +- sqrt(2) at a glance gives one a higher "maths-learning" bandwidth, since one spends just the 3 seconds it takes to read those lines, instead of the extra 30 seconds physically doing the working. (And, on a slightly different note, having some basic facts memorised like (for example) "d/dx sin(x) = cos(x)" is a little like the difference between data in L1 cache and that just in RAM. However, this doesn't mean that one should not understand why, or not be able to rederive it in a flash by drawing a diagram or whatever.)

jessriedel 14y ago

> I think (contrary to Bret) that symbol manipulation is a prerequisite for deeper STEM ideas. Can you elaborate on this further? I tend to disagree, but I don't know if I have all that much to say to back it up. I mean, sure, there's just no good way to understand shear forces without being able to manipulate matrices (so I agree with dbaupp's sibling comment), but that doesn't mean you want to learn the rules of matrix algebra as if they were arbitrary rules enforced by the stick and carrots in a game.

ezyang 14y ago
In the case of mathematics, there is a fundamental sense in which the symbols and their manipulations are what you're studying. Think of it as if you were an archeologist: the symbols are the artifact of study. It's only half of the picture: you must(!) come up with a different way of intuitively understanding it--but at the end of the day if you really want to precisely say what it is you're talking about, it's symbols.
- jessriedel 14y ago
  OK, I think I see what you're getting at. But we could devise a AltDragonBox game with a completely different set of arbitrary rules. (Possibly even rules that are inconsistent.) And as far as game play goes, AltDragonBox would work just as well. But there's a reason that the rules of algebra are what they are. And if you can't distinguish between DragonBox and AltDragonBox, I don't think you're learning what you need to learn. Here's another way to look at it: we all know people who could ace their high school math tests because they had memorized the rules of manipulation but weren't good at math--as evidenced by their poor performance in college and failure to succeed in future STEM classes. If "the symbols and their manipulations are what you're studying" (which, I agree, there is some truth to), then what is it exactly that these people were lacking?
  - ezyang 14y ago
    I agree that it is important to recognize the limitations of this approach. As a high schooler (and even as an undergraduate in mathematics), you learn all sorts of formal systems. As a mathematics researcher, one of your charges is inventing interesting mathematical concepts to play around with. Obviously this is a bit much to ask out of DragonBox! And even more modestly, the ability to sit down with a problem and say, "Ah, but what really is going on here?" is a deep and difficult skill to impart. I don't know what these people were lacking, but if they didn't know how to move symbols around, I might have started there... (BTW, an inconsistent set of rules would correspond to a version of DragonBox where there was a cheat code you could enter, and then easily solve every problem. So it would not "work just as well", and this would be pretty clear to a cheating gamer.)