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Elithrion comments on Rationality Quotes April 2012 - Less Wrong
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It means nothing, although Greg Egan is quite impressed by it. Sad but true: Someone with an IQ of, say, 90 can be trained to operate a Turing machine, but will in all probability never understand matrix calculus. The belief that Turing-complete = understanding-complete is false. It just isn't stupid.
I would imagine that he can be taught matrix calculus, given sufficient desire (on his and the teachers' parts), teaching skill, and time. I'm not sure if in practice it is possible to muster enough desire or time to do it, but I do think that understanding is something that can theoretically be taught to anyone who can perform the mechanical calculations.
I fear you're committing the typical mind fallacy. The dyscalculic could simulate a Turing machine, but all of mathematics, including basic arithmetic, is whaargarbl to them. They're often highly intelligent (though of course the diagnosis is "intelligent elsewhere, unintelligent at maths"), good at words and social things, but literally unable to calculate 17+17 more accurately than "somewhere in the twenties or thirties" or "I have no idea" without machine assistance. I didn't believe it either until I saw it.
Do you find this harder to believe than, say, aphasia? I've never seen it, but I have no difficulty believing it.
Well, I certainly don't disbelieve in it now. I first saw it at eighteen, in first-year psychology, in the bit where they tried to beat basic statistics into our heads.
Have you ever tried to teach math to anyone who is not good at math? In my youth I once tutored a woman who was poor, but motivated enough to pay $40/session. A major obstacle was teaching her how to calculate (a^b)^c and getting her to reliably notice that minus times minus equals plus. Despite my attempts at creative physical demonstrations of the notion of a balanced scale, I couldn't get her to really understand the notion of doing the same things to both sides of a mathematical equation. I don't think she would ever understand what was going on in matrix calculus, period, barring "teaching methods" that involve neural reprogramming or gain of additional hardware.
Your claim is too large for the evidence you present in support of it.
Teaching someone math who is not good at math is hard, but "will in all probability never understand matrix calculus"!? I don't think you're using the Try Harder.
Assume teaching is hard (list of weak evidence: it's a three year undergraduate degree; humanity has hardly allowed itself to run any proper experiments in the field, and those that have been run seem usually to be generally ignored by professional practitioners; it's massively subject to the typical mind fallacy and most practitioners don't know that fallacy exists). That you, "in your youth" (without having studied teaching), "once" tutored a woman who you couldn't teach very well… doesn't support any very strong conclusion.
It seems very likely to me that Omega could teach matrix calculus to someone with IQ 90 given reasonable time and motivation from the student. One of the things I'm willing to devote significant resources to in the coming years is making education into a proper science. Given the tools of that proper science I humbly submit that you could teach your former student a lot. Track the progress of the Khan Academy for some promising developments in the field.
What are the experiments that are generally ignored?
Some of it is weak evidence for the hardness claim (3 years degree), some against (all the rest). Does that match what you meant?
I'd intended a different meaning of "hard". On reflection your interpretation seems a very reasonable inference from what I wrote.
What I meant: Teaching is hard enough that you shouldn't expect to find it easy without having spent any time studying it. Even as a well educated westerner, the bits of teaching you can reasonably expect to pick up won't take you far down the path to mastery.
(Thank you for you comment - it got me thinking.)
No, I haven't, and reading your explanation I now believe that there is a fair chance you are correct. However, one problem I have with it is that you're describing a few points of frustration, some of which I assume you ended up overcoming. I am not entirely convinced that had she spent, say one hundred hours studying each skill that someone with adequate talent could fully understand in one, she would not eventually fully understand it.
In cases of extreme trouble, I can imagine her spending forty hours working through a thousand examples, until mechanically she can recognise every example reasonably well, and find the solution correctly, then another twenty working through applications, then another forty hours analysing applications in the real world until the process of seeing the application, formulating the correct problem, and solving it becomes internalised. Certainly, just because I can imagine it doesn't make it true, but I'm not sure on what grounds I should prefer the "impossibility" hypothesis to the "very very slow learning" hypothesis.
I can't imagine how hard it would be to learn math without the concept of referential transparency.
Not all that hard if that's the only sticking point. I acquired it quite late myself.
What was your impression of her intelligence otherwise?
Suzette Haden Elgin (a science fiction author and linguist who was quite intelligent with and about words) described herself as intractably bad at math.
This anecdote gives very little information on its own. Can you describe your experience teaching math to other people - the audience, the investment, the methods, the outcome? Do you have any idea whether that one woman eventually succeeded in learning some of what you couldn't teach her, and if so, how?
(ETA: I do agree with the general argument about people who are not good at math. I'm only saying this particular story doesn't tell us much about that particular woman, because we don't know how good you are at teaching, etc.)
I can't imagine how hard it is to learn to program if you don't instinctively know how. Yet I know it is that hard for many people. Some succeed in learning, some don't. Those who do still have big differences in ability, and ability at a young age seems to be a pretty good predictor of lifetime ability.
I realize I must have learned the basics at some point, although I don't remember it. And I remember learning many more advanced concepts during the many years since. But for both the basics and the advanced subjects, I never experienced anything I can compare to what I'd call "learning" in other subjects I studied.
When programming, if I see/read something new, I may need some time (seconds or hours) to understand it, then once I do, I can use it. It is cognitively very similar to seeing a new room for the first time. It's novel, but I understand it intuitively and in most cases quickly.
When I studied e.g. biology or math at university, I had to deliberately memorize, to solve exercises before understanding the "real thing", to accept that some things I could describe I couldn't duplicate by building them from scratch no matter how much time I had and what materials and tools. This never happened to me in programming. I may not fully understand the domain problem that the program is manipulating. But I always understand the program itself.
And yet I've seen people struggle to understand the most elementary concepts of programming, like, say, distinguishing between names and values. I've had to work with some pretty poor programmers, and had the official job of on-the-job mentoring newbies on two occasions. I know it can be very difficult to teach effectively, it can be very difficult to learn.
Given that I encountered a heavily preselected set of people, who were trying to make programming their main profession, it's easy for me to believe that - at the extreme - for many people elementary programming is impossible to learn, period. And the same should apply to math and any other "abstract" subject for which biologically normal people don't have dedicated thinking modules in their brains.