If you aren't a math person: Some questions about math and reading
I'm going to try to keep this short and sweet today. This is more of a pedagogy post, inspired by a few interesting comments from last week's classes.
Who taught you how to solve word problems? I don't mean, who taught you the steps to take when you see that canonical "two-trains" problem. I mean, a generalized method of looking at any word problem, on any topic, regardless of the style of mathematics involved, and converting the words into numbers, variables and equations such that you can perform whatever kind of operation is necessitated by the structure of the set-up in order to get the numerical information you need.
Think back to that class.
Oh, wait. You probably never took that class, or had that happen for you.
Nobody did. There's no mathematical magic school, where they ship people off to Brakebills and teach the magic trick that solves the hardest of math problems, leaving everyone else to scratch their heads and struggle.
Or... if there is, why didn't I get sent there! Was this, like, a Felicity Huffman thing, and the admissions office was just on the take? 'Cuz, I'd get that, but... damn it!
Anyway, we don't teach anyone "the generalized method of solving all word problems." And in fact, I don't know if it's something we can do. I think that this is a kind of a Turing Test thing.
My little corner of political science-- game theory-- is basically all about word problems, and that can throw a lot of students off-kilter, not because the math is computationally difficult, or uses principles for which you need a math degree, although game theory certainly can be taken to absurd levels of mathematical complexity. No, instead, game theory is all about word problems. To outsiders, it can feel as though some of us got some secret, special training for how to solve word problems.
But... we didn't!
Analogy time.
Who taught you how to get subtext from novels? The generalized process of doing so? I don't mean, who taught you the subtext of any specific novel. I mean, who taught you how to read a novel, and go through the mental processes of figuring out what the subtext is, as a generalized process that will work for any novel from Mary Shelley to William Faulkner to Seth Dickinson? (You don't know him. At some point, I'll probably write some stuff about the Baru Cormorant books, which are awesome.)
Some people have a harder time than others, and it takes practice, and sometimes, it's slow-going, and... does any of this sound familiar?
One novel is, hopefully, different from another, as one word problem is different from another. Otherwise, what's the point? I don't remember anyone giving me an algorithm for deriving the subtext from a novel! I remember being required to do it, and I remember sucking at it as a student, although I don't know how much of that was my disdain for the readings themselves or other things. Some skills just take maturity.
Shut up!
Where was I? Oh, yeah. Maturity. Point being, there's no functioning algorithm, of which I know, to derive the subtext from a novel. If there were, the entire field of literary analysis would be reduced to a computer program, and your lit professor could be replaced by an AI.
So... when's the singularity scheduled to happen?
There's also no algorithm of which I know to convert a word problem into a solvable set of equations, generalized. Sure, there are books aimed at high schoolers, nervous about two-trains problems in high school algebra, and such, but in order to construct any such rule, the problem must be constrained.
The point of the Turing Test is that a machine can fool you for a few exchanges, but after a while, you catch on. Why? A machine can be programed to respond to your words within parameters, but as an interaction goes on, the conversation branches off in so many possible directions that the program cannot accommodate everything you might say, so its responses necessarily become non-sequiturs. You can generate an algorithm for solving two trains-style problems, but the more variation you have in the potential structure of word problems, more difficult it is to generate algorithms that can accommodate everything you might face.
That's why neither I, nor anybody else who can be called "a math person" ever took some secret, special class on how to solve word problems, generalized. Just as there's no secret, special book that tells you the magic trick of figuring out the subtext of any novel, generalized.
I am coming to think, though, that the skill of solving word problems is really the key to getting all things mathematical because it rests on understanding of concept, more than anything else. I've been teaching statistics and game theory for just about 20 years now (which... is strange to type), and empirically, I see a strong association between facility with word problems and facility with other aspects of mathematics. Again, why? They test depth of understanding of the underlying concepts, just as understanding the subtext of a novel is a sort of test of understanding of the novel itself.
And yet, we don't really, to my mind, have a good way to teach word problems, generalized. Of course, we don't have a good way to teach subtext, generalized, either.
There are frequent trends in elementary and high school education, and sometimes we see an ebb-and-flow with word problems. Of course, sometimes an "educator" will just decide that we should re-invent arithmetic. Hilarity ensues.
Where I think a lot of us math-types are stuck, though, is that word problems are, almost by definition, not hard for us. Ask me to teach you calculus? I can do that. Why? I remember how I learned it. Ask me to teach you statistics, game theory, anything like that? Same thing. I can walk you through the steps of proofs, techniques, go back, slow down, reverse, whatever you want. Lather, rinse, repeat. Why? 'Cuz I'm just doing what others did for me. Maybe a little differently, maybe a lot differently if the first time doesn't stick. I have options.
But nobody ever showed me a generalized algorithm for solving word problems. And nobody ever taught me a generalized algorithm for pulling the subtext from a novel.
That doesn't mean the skills can't be learned. I see people improve.
I just don't think we have any coherent way to teach the generalized model of... modeling... models. And that's a problem. Not just in terms of intentionally ugly diction.
A big problem, as far as our educational system is concerned.
Someone smarter than I am should work on that. Or... nah. Let's just ask a bunch more two-trains problems and wish this one away.
Who taught you how to solve word problems? I don't mean, who taught you the steps to take when you see that canonical "two-trains" problem. I mean, a generalized method of looking at any word problem, on any topic, regardless of the style of mathematics involved, and converting the words into numbers, variables and equations such that you can perform whatever kind of operation is necessitated by the structure of the set-up in order to get the numerical information you need.
Think back to that class.
Oh, wait. You probably never took that class, or had that happen for you.
Nobody did. There's no mathematical magic school, where they ship people off to Brakebills and teach the magic trick that solves the hardest of math problems, leaving everyone else to scratch their heads and struggle.
Or... if there is, why didn't I get sent there! Was this, like, a Felicity Huffman thing, and the admissions office was just on the take? 'Cuz, I'd get that, but... damn it!
Anyway, we don't teach anyone "the generalized method of solving all word problems." And in fact, I don't know if it's something we can do. I think that this is a kind of a Turing Test thing.
My little corner of political science-- game theory-- is basically all about word problems, and that can throw a lot of students off-kilter, not because the math is computationally difficult, or uses principles for which you need a math degree, although game theory certainly can be taken to absurd levels of mathematical complexity. No, instead, game theory is all about word problems. To outsiders, it can feel as though some of us got some secret, special training for how to solve word problems.
But... we didn't!
Analogy time.
Who taught you how to get subtext from novels? The generalized process of doing so? I don't mean, who taught you the subtext of any specific novel. I mean, who taught you how to read a novel, and go through the mental processes of figuring out what the subtext is, as a generalized process that will work for any novel from Mary Shelley to William Faulkner to Seth Dickinson? (You don't know him. At some point, I'll probably write some stuff about the Baru Cormorant books, which are awesome.)
Some people have a harder time than others, and it takes practice, and sometimes, it's slow-going, and... does any of this sound familiar?
One novel is, hopefully, different from another, as one word problem is different from another. Otherwise, what's the point? I don't remember anyone giving me an algorithm for deriving the subtext from a novel! I remember being required to do it, and I remember sucking at it as a student, although I don't know how much of that was my disdain for the readings themselves or other things. Some skills just take maturity.
Shut up!
Where was I? Oh, yeah. Maturity. Point being, there's no functioning algorithm, of which I know, to derive the subtext from a novel. If there were, the entire field of literary analysis would be reduced to a computer program, and your lit professor could be replaced by an AI.
So... when's the singularity scheduled to happen?
There's also no algorithm of which I know to convert a word problem into a solvable set of equations, generalized. Sure, there are books aimed at high schoolers, nervous about two-trains problems in high school algebra, and such, but in order to construct any such rule, the problem must be constrained.
The point of the Turing Test is that a machine can fool you for a few exchanges, but after a while, you catch on. Why? A machine can be programed to respond to your words within parameters, but as an interaction goes on, the conversation branches off in so many possible directions that the program cannot accommodate everything you might say, so its responses necessarily become non-sequiturs. You can generate an algorithm for solving two trains-style problems, but the more variation you have in the potential structure of word problems, more difficult it is to generate algorithms that can accommodate everything you might face.
That's why neither I, nor anybody else who can be called "a math person" ever took some secret, special class on how to solve word problems, generalized. Just as there's no secret, special book that tells you the magic trick of figuring out the subtext of any novel, generalized.
I am coming to think, though, that the skill of solving word problems is really the key to getting all things mathematical because it rests on understanding of concept, more than anything else. I've been teaching statistics and game theory for just about 20 years now (which... is strange to type), and empirically, I see a strong association between facility with word problems and facility with other aspects of mathematics. Again, why? They test depth of understanding of the underlying concepts, just as understanding the subtext of a novel is a sort of test of understanding of the novel itself.
And yet, we don't really, to my mind, have a good way to teach word problems, generalized. Of course, we don't have a good way to teach subtext, generalized, either.
There are frequent trends in elementary and high school education, and sometimes we see an ebb-and-flow with word problems. Of course, sometimes an "educator" will just decide that we should re-invent arithmetic. Hilarity ensues.
Where I think a lot of us math-types are stuck, though, is that word problems are, almost by definition, not hard for us. Ask me to teach you calculus? I can do that. Why? I remember how I learned it. Ask me to teach you statistics, game theory, anything like that? Same thing. I can walk you through the steps of proofs, techniques, go back, slow down, reverse, whatever you want. Lather, rinse, repeat. Why? 'Cuz I'm just doing what others did for me. Maybe a little differently, maybe a lot differently if the first time doesn't stick. I have options.
But nobody ever showed me a generalized algorithm for solving word problems. And nobody ever taught me a generalized algorithm for pulling the subtext from a novel.
That doesn't mean the skills can't be learned. I see people improve.
I just don't think we have any coherent way to teach the generalized model of... modeling... models. And that's a problem. Not just in terms of intentionally ugly diction.
A big problem, as far as our educational system is concerned.
Someone smarter than I am should work on that. Or... nah. Let's just ask a bunch more two-trains problems and wish this one away.
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