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pwno comments on Understanding your understanding - Less Wrong
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Well done. Coincidentally, I think my first frustration regarding (different meanings of) understanding was about that exact issue, back in high school. When my friends and I had learned about the Doppler effect, we were all at Level 1, maybe with minor inroads into Level 2. The problem? I classified that as "not understanding", while my friends called it "complete understanding". So we had conversations like this:
me: Okay, I don't really get this Doppler effect. I mean, why should coming closer make the pitch higher?
them: Because the equation says...
me: Well, I know the equation, but I don't get it at the gut level. What is it about moving closer that shortens the waves like that?
them: Because it like, compresses the waves.
me: But why should the sound wave act like some stiff rod connecting me to the car? Why do you get to apply that kind of reasoning to sound waves?
them: Because that's what the equations say!
me: Argh!
Btw, another important thing about Level 3 is that you could change the question to "... low-pitched when they are coming toward you ..." and vice versa, and the Level 3 rationalist would still derive the same result, except to add that, "wait, I'm confused -- are you sure you're reporting that right?"
Some thoughts regarding the difference between level 2 and 3:
Seems like a level 3 understanding necessitates an insight-producing ability (i.e. ability to improve existing models) -- otherwise your models wouldn't regenerate if destroyed. The question is why your insights with a level 2 understanding aren't evidence of a level 3 understanding. Or whether it's even possible to have insights with a level 2 understanding.
If we're able to regenerate a model, we obviously have model-making abilities. But isn't the same happening when you draw connections between your models? The moment you realize two or more models are connected, you've added to your model of reality. Neither model predicted their relationship with the other, your insight connected the two, improving your older model.
How's level 3 different from level 2?
In short, if you are simply informed about the connections between the fields, you are at level 2, but if you could discover the links yourself with no hints, you are at level 3. For example, if you know how the parameter "speed of light", c, has implications for both general relativity and quantum phenomena, you have a level 2 understanding (to the extent that these fields are involved), but if you couldn't discover the need for a "speed of light" parameter, how to find it, and how it affects the disparate fields, you haven't reached level 3.