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simplicio comments on Understanding your understanding - Less Wrong

69 Post author: SilasBarta 22 March 2010 10:33PM

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Comment author: simplicio 23 March 2010 04:11:08PM *  18 points [-]

Excellent post, I like the breakdown. So let's take a simple example for clarification:

"Why do race cars sound high-pitched when they are coming toward you and low pitched when they're heading away?"

Level 0: "That's because of the Doppler effect."

Level 1: "The frequency is f = (v + vr)/(v+vs) * f0 . Make sure you get the signs right."

Level 2: "Sound waves have a wavelength, but the wavelength is shortened when the car is coming toward you, because the car 'catches up' with its own waves, and lengthened when it's heading away. Since the waves travel at a constant speed in the medium, a short wavelength implies a high frequency, and frequency is what we hear as pitch. Yes, I know the equation. The phenomenon also happens for light, although the equation is different."

Level 3: "No, I don't remember the equation. Here, gimme a minute and I'll derive it."

Comment author: SilasBarta 23 March 2010 04:42:43PM *  10 points [-]

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?"

Comment author: Lightwave 24 March 2010 09:42:29AM 8 points [-]

Because that's what the equations say!

It seems to me that this is usually the answer to questions about quantum mechanics. Does that mean most people (including physicists) understand it at Level 1?

Comment author: gwillen 23 March 2010 06:31:01PM 6 points [-]

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".

I had a friend express frustration with me once, because we would have conversations about some subjects she felt she understood, in which I would say things like "I don't understand X", and then proceed to demonstrate what she felt was a better understanding than she had. I think it felt to her like I was "sandbagging" when claiming I didn't understand things, whereas I was merely expressing that I was somewhere between levels 1 and 2 and was unsatisfied with it.

Comment author: prase 23 March 2010 05:31:47PM *  1 point [-]

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?

Do you remember what insight helped you to overcome these questions? My experience with Level 1 -> Level 2 transitions is that I somehow mysteriously got used to the phenomenon, without knowing exactly how that happened. Also, I am not sure how could I explain the Doppler effect to somebody at Level 1, or answer questions such as the ones above. It seems that explanations reliably work up to Level 1 only.

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?"

I suspect the rationalist would be already confused at Level 1, if he got the signs right.

Comment author: SilasBarta 23 March 2010 05:49:46PM *  6 points [-]

Do you remember what insight helped you to overcome these questions? My experience with Level 1 -> Level 2 transitions is that I somehow mysteriously got used to the phenomenon, without knowing exactly how that happened. Also, I am not sure how could I explain the Doppler effect to somebody at Level 1, or answer questions such as the ones above. It seems that explanations reliably work up to Level 1 only.

I don't remember when I finally got a "Level 2" answer, but to move someone else in that direction, I would explain it this way: First, make sure they understand what's actually happening in a compression wave in air. Help visualize it with a slinky if necessary. Then say,

"The sensation of sound comes from when your ears recognize a quick sequence of compressed-air, less-compressed-air, compressed-air, less-compressed-air, etc. And the rate at which this sequence cycles determines the pitch you hear, with quicker cycling meaning a higher pitch.

"If the source of the sound isn't just standing still, but moving toward you, then each compression it makes of the air happens at a point where it is closer to you that it was before, so that bit of compressed air hits you sooner than otherwise. So as the compressed/less-compressed groups reach you, they cycle through faster, which you experience as a higher pitch, for the same reason you'd experience anything as a higher pitch."

(Someone let me know if I'm seriously off; it's been a while.)

Comment author: srjskam 15 June 2010 09:17:16PM 3 points [-]

My first instinct would have been something like that, but on second thought, I'd start with a example of a boat moving in water and the waves it makes, maybe drawing a picture and ask them to visualize it. This is admittedly very crude and inaccurate, but gives a good overview of the phenomenon. After that I'd elaborate on the differences of surface waves vs. pressure waves, wavelength & frequency and anatomy of hearing etc.

Generally speaking (and not directed against anything anyone has said): give the explainee an intuitive framework to hang details on, don't pour a litany of seemingly unconnected facts. Just make sure he doesn't confuse the crude framework for the actual phenomenon.

(...And more generally: of course, the best would be to explain in a mode that is natural for the individual... for me (and, I assume, quite a few others) it's visuality & real-world analogies.)

(And hello, everyone. First post.)

Comment author: SilasBarta 15 June 2010 09:22:28PM 1 point [-]

Welcome to Less Wrong! Feel free to introduce yourself on that thread. Here's the rest of RobinZ's newcomer welcome package.

And thanks for the reply to my article and comment. I hope to have an article about how to explain up soon, which will expand on the ideas here (this thread and the article).

Comment author: pwno 22 January 2011 06:47:06AM 0 points [-]

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?

Comment author: SilasBarta 23 January 2011 02:47:01PM *  1 point [-]

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.

Comment author: pjeby 23 March 2010 05:18:56PM 4 points [-]

Actually, this is a good example of why I don't think this is really a linear hierarchy.

My understanding is like your description in level 2, but I don't know the equation. I could probably derive it if I had enough time, knew the speed of sound in air, had a way to check that I could properly relate wavelength to frequency, and I could use a computer to check my results by programming it to simulate the sound of a constant-pitch siren as an ambulance passes by.

Is that level 2, or level 3? 2.8? I find the whole idea of levels rather confusing here.

Comment author: RobinZ 09 June 2010 08:52:14PM *  1 point [-]

This would suggest a typology of understanding: type N* understanding knows the passwords, type I understanding knows the equations (edit: or can otherwise find the answer - thanks, SilasBarta), type II understanding knows the connections to other fields of knowledge, and type III understanding generates type I and II knowledge within the domain and connections to other fields of knowledge (type II knowledge in other domains).

* Zero in roman numerals. Which is appropriate, of course, because the passwords are just names and can be wrong in a number of ways.

Comment author: SilasBarta 09 June 2010 08:57:48PM *  0 points [-]

Very good, but I just want to emphasize that Level 1, as I've defined it, doesn't necessary involve equations; it just means that you get the right answers somehow, as long as it's not cheating. (So I think the "calculate" part in its name is a bit misleading and I should probably pick a different one.)

To put it another way: for purposes of determining whether you have type I understanding, ad hoc is okay, but post hoc is not.

Comment author: RobinZ 09 June 2010 09:07:44PM 0 points [-]

I see - yes, that's a good point. I'll edit in a note.

(By the way, I switched from Arabic to Roman numerals to distinguish the typology from the hierarchy - it's level 1 and type I because the two are related, not perfectly identical.)

Comment author: SilasBarta 09 June 2010 09:33:31PM 0 points [-]

Oops, I had thought to correct my first reference (which should have been "Level 1"), but only corrected it halfway! Fixed now.

Comment author: SilasBarta 23 March 2010 05:44:05PM 1 point [-]

The original draft of this made sure to note that a) the levels aren't really discrete in that you can be e.g. partway/ halfway/ mostly toward completing Level 2; and that b) it's conceivably possible to complete them out of order, but that should be extremely unusual. I decided that explaining all of that would be a distracting "caveat overload"

Also, I don't think your situation counts as bypassing levels. Level 1, by design, doesn't say that you must have the standard equation in your model, or that you could provide it right this second. It just requires that you have a model that works. So if you know enough to generate a predictive, successful model, then you're at least at Level 1, even if you need some time to flesh out the specific predictions.

Comment author: simplicio 23 March 2010 05:37:25PM 0 points [-]

So maybe there should be level 2a: conceptual understanding, and level 2b: quantitative + conceptual understanding.

The fact that nobody quite falls perfectly into a discrete level doesn't mean it isn't a useful heuristic though. Even in your case you could say that you're "on your way" to level 2.

Comment author: pjeby 23 March 2010 05:48:10PM *  1 point [-]

Even in your case you could say that you're "on your way" to level 2.

Right - but that's what I think is wrong with the definition of levels 2 and 3. Since I could get to the equation, if I had to, shouldn't that also be a valid description of level 3? Requiring me to know the equation in order to be level 2, yet not requiring it at level 3, kind of makes the point that this is not really a linear progression.

[Edit to add: I don't necessarily mean that understanding itself isn't linear, just that this particular set of definitions does not seem to be.]

Comment author: [deleted] 23 March 2010 04:28:23PM 1 point [-]

So what do you call it when you know the level-2 explanation but don't know the equation and can't derive it?

Deriving the frequency equation. I should do that some time.

Comment author: Nanani 24 March 2010 12:34:46AM 1 point [-]

I'd call it "time to dust off the math books". Incidentally, I've got to do just that.