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# Morendil comments on Understanding your understanding - Less Wrong

69 22 March 2010 10:33PM

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Comment author: 24 March 2010 05:00:18PM *  11 points [-]

Just remembered something which either is a generally useful test for Level 3, or deserves a distinct level of its own: having the ability to guide someone through constructing the knowledge in the first place.

I've just spent the afternoon tutoring a friend of mine's kid, eighth-grader age, who's math-averse. This is a particularly good test, I find, of how well you understand a given bit of material. (And sadly, it seems that his teacher has no particular knack for explaining.)

To someone who already groks math a bit, something like factoring a sum of integers raised to some power is "intuitively obvious". There is no need to spell out the component intuitions of algebra, so you can just say "find the common factor, divide each term in the original sum by that, and you're done".

When you find yourself explaining that to a math-challenged kid you realize that "find the common factor" isn't an obvious, one-step operation, you have to slow way down and break it up: there are usually many different common factors but when the teachers says "the" what they mean is the greatest one, for instance, and finding a common factor might involve decomposing what is (in exercise form) initially expressed in a more compact form.

When you can't rely on the kid's mathematical intuition, you essentially have to give them an explicit algorithm for factoring expressions, much as if you were programming a computer to do it. (This is one of the reasons why I'm probably going to go ahead with my "programming as a useful rationalist skill" post at some point: the gist of it is "teaching a computer what you know about something is an excellent test of whether that knowledge is truly a part of you".)

It's worse than programming, since that kid also has negative feelings about math, plus some unnecessary baggage he got from his teacher that makes it more of a muddle than it should have been in the first place, all of which I must first clear out before we can build something correct.

Comment author: 24 March 2010 06:47:27PM 1 point [-]

Well put. I think that is a good test for Level 3, since it shows how well you can do when deprived of an arbitrary "tool".

I've been on both ends of the situation you've described: in teaching, I've had to break down procedures into ever smaller substeps and tell students what to do if e.g. they don't have their multiplication table or are dividing something that exceeds its bounds (I tutor 4th graders).

Oppositely, when being taught, I've been in situations where the instructor incorrectly assumes certain common knowledge and then can't fill the gap because they have forgotten what it's like to be with out it (not all have this failing, of course). It leaves me suspecting they never seriously thought about its grounding before.