“do you believe you are aware of the inferential connections between your expertise and layperson-level knowledge?”,
since I am not so familiar with the formalism of a “Level 2” understanding.
My uninteresting, simple answer is: yes.
My philosophical answer is that I find the entire question to be very interesting and strange. That is, the relationship between teaching and understanding is quite strange IMO. There are many people who are poor teachers but who excel in their discipline. It seems to be a contradiction because high-level teaching skill seems to be a sufficient, and possibly necessary condition for masterful understanding.
Personally I resolve this contradiction in the following way. I feel like my own limitations make it to where I am forced to learn a subject by progressing at it in very simplistic strokes. By the time I have reached a mastery, I feel very capable of teaching it to others, since I have been forced to understand it myself in the most simplistic way possible.
Other people, who are possibly quite brilliant, are able to master some subjects without having to transmute the information into a simpler level. Consequentially, they are unable to make the sort of connections that you describe as being necessary for teaching.
Personally I feel that the latter category of people must be missing something, but I am unable to make a convincing argument for this point.
A lot of the questions you pose, including the definition of the Level 2 formalism, are addressed in the article I linked (and wrote).
I classify those who can do something well but not explain or understand the connections from the inputs and outputs to the rest of the world, to be at a Level 1 understanding. It’s certainly an accomplishment, but I agree with you that it’s missing something: the ability to recognize where it fits in with the rest of reality (Level 2) and the command of a reliable truth-detecting procedure that can “repair” gaps in knowledge as they arise (Level 3).
“Level 1 savants” are certainly doing something very well, but that something is not a deep understanding. Rather, they are in the position of a computer that can transform inputs into the right outputs, but do nothing more with them. Or a cat, which can fall from great heights without injury, but not know why its method works.
(Yes, this comment seems a bit internally repetitive.)
Ah, OK, I read your article. I think that’s an admirable task to try to classify or identify the levels of understanding. However, I’m not sure I am convinced by your categorization. It seems to me that many of these “Level 1 savants” as you call them are quite capable of fitting their understanding with the rest of reality. Actually it seems like the claim of “Level 1 understanding” basically trivializes that understanding. Yet many of these people who are bad teachers have a very nontrivial understanding—else I don’t think this would be such a common phenomena, for example, in academia. I would argue that these people have some further complications or issues which are not recognized in the 1-2-3 hierarchy.
That being said, you have to start somewhere, and the 0-1-2-3 hierarchy looks like a good place to start. I’d definitely be interested in hearing more about this analysis.
Thanks for reading it and giving me feedback. I’m interested in your claim:
It seems to me that many of these “Level 1 savants” as you call them are quite capable of fitting their understanding with the rest of reality.
Well, they can fit it in the sense that they (over a typical problem set) can match inputs with (what reality deems) the right outputs. But, as I’ve defined the level, they don’t know how those inputs and outputs relate to more distantly-connected aspects of reality.
Yet many of these people who are bad teachers have a very nontrivial understanding—else I don’t think this would be such a common phenomena, for example, in academia.
I had a discussion with others about this point recently. My take is basically: if their understanding is so deep, why exactly is their teaching skill so brittle that no one can follow the inferential paths they trace out? Why can’t they switch to the infinite other paths that a Level 2 understanding enables them to see? If they can’t, that would suggest a lack of depth to their understanding.
And regarding the archetypal “deep understanding, poor teacher” you have in mind, do you envision that they could, say, trace out all the assumptions that could account for an anomalous result, starting with the most tenuous, and continuing outside their subfield? If not, I would call that falling short of Level 2.
My take is basically: if their understanding is so deep, why exactly is their teaching skill so brittle that no one can follow the inferential paths they trace out? Why can’t they switch to the infinite other paths that a Level 2 understanding enables them to see? If they can’t, that would suggest a lack of depth to their understanding.
I would LOVE to agree with this statement, as it justifies my criticism of poor teachers who IMO are (not usually maliciously) putting their students through hell. However, I don’t think it’s obvious, or I think maybe you just have to take it as an axiom of your system. It seems there is some notion of individualism or personal difference which is missing from the system. If someone is just terrible at learning, can you really expect to succeed in explaining, for example? Realistically I think it’s probably impossible to classify the massive concept of understanding by merely three levels, and these problems are just a symptom of that fact.
As another example, in order to understand something, it’s clearly necessary to be able to explain it to yourself. In your system, you are additionally requiring that your understanding means you must be able to explain things to other people. In order to explain things to others, you have to understand them, as has been discussed. Therefore you have to be able to explain other people to yourself. Why should an explanation of other individuals behavior be necessary for understanding some random area of expertise, say, mathematics? It’s not clear to me.
And regarding the archetypal “deep understanding, poor teacher” you have in mind, do you envision that they could, say, trace out all the assumptions that could account for an anomalous result, starting with the most tenuous, and continuing outside their subfield?
It certainly seems like someone with a deep understanding of their subject should be able to identify the validity or uncertainty in their assumptions about the subject. If they are a poor teacher, I think I would still believe this to be true.
I’ve thought about this some, and I think I see your point now. I would phrase it this way: It’s possible for a “Level 3 savant” to exist. A Level 3 savant, let’s posit, has a very deeply connected model of reality, and their excellent truth-detecting procedure allows them to internally repair loss of knowledge (perhaps below the level of their conscious awareness).
Like an expert (under the popular definition), and like a Level 1 savant, they perform well within their field. But this person differs in that they can also peform well in tracing out where its grounding assumptions go wrong—except that they “just have all the answers” but can’t explain, and don’t know, where the answers came from.
So here’s what it would look like: Any problem you pose in the field (like an anomalous result), they immediately say, “look at factor X”, and it’s usually correct. They even tell you to check critical aspects of sensors, or identify circularity in the literature that grounds the field (i.e. sources which generate false knowledge by excessively citing each other), even though most in the field might not even think about or know how all those sensors work.
All they can tell you is, “I don’t know, you told me X, and I immediately figured it had to be a problem with Y misinterpreting Z. I don’t know how Z relates to W, or if W directly relates to X, I just know that Y and Z were the problem.”
I would agree that there’s no contradiction in the existence of such a person. I would just say that in order to get this level of skill you have to accomplish so many subgoals that it’s very unlikely, just as it’s hard to make something act and look like a human without also making it conscious. (Obvious disclaimer: I don’t think my case is as solid as the one against P-zombies.)
I will reply to this in the sense of
since I am not so familiar with the formalism of a “Level 2” understanding.
My uninteresting, simple answer is: yes.
My philosophical answer is that I find the entire question to be very interesting and strange. That is, the relationship between teaching and understanding is quite strange IMO. There are many people who are poor teachers but who excel in their discipline. It seems to be a contradiction because high-level teaching skill seems to be a sufficient, and possibly necessary condition for masterful understanding.
Personally I resolve this contradiction in the following way. I feel like my own limitations make it to where I am forced to learn a subject by progressing at it in very simplistic strokes. By the time I have reached a mastery, I feel very capable of teaching it to others, since I have been forced to understand it myself in the most simplistic way possible.
Other people, who are possibly quite brilliant, are able to master some subjects without having to transmute the information into a simpler level. Consequentially, they are unable to make the sort of connections that you describe as being necessary for teaching.
Personally I feel that the latter category of people must be missing something, but I am unable to make a convincing argument for this point.
A lot of the questions you pose, including the definition of the Level 2 formalism, are addressed in the article I linked (and wrote).
I classify those who can do something well but not explain or understand the connections from the inputs and outputs to the rest of the world, to be at a Level 1 understanding. It’s certainly an accomplishment, but I agree with you that it’s missing something: the ability to recognize where it fits in with the rest of reality (Level 2) and the command of a reliable truth-detecting procedure that can “repair” gaps in knowledge as they arise (Level 3).
“Level 1 savants” are certainly doing something very well, but that something is not a deep understanding. Rather, they are in the position of a computer that can transform inputs into the right outputs, but do nothing more with them. Or a cat, which can fall from great heights without injury, but not know why its method works.
(Yes, this comment seems a bit internally repetitive.)
Ah, OK, I read your article. I think that’s an admirable task to try to classify or identify the levels of understanding. However, I’m not sure I am convinced by your categorization. It seems to me that many of these “Level 1 savants” as you call them are quite capable of fitting their understanding with the rest of reality. Actually it seems like the claim of “Level 1 understanding” basically trivializes that understanding. Yet many of these people who are bad teachers have a very nontrivial understanding—else I don’t think this would be such a common phenomena, for example, in academia. I would argue that these people have some further complications or issues which are not recognized in the 1-2-3 hierarchy.
That being said, you have to start somewhere, and the 0-1-2-3 hierarchy looks like a good place to start. I’d definitely be interested in hearing more about this analysis.
Thanks for reading it and giving me feedback. I’m interested in your claim:
Well, they can fit it in the sense that they (over a typical problem set) can match inputs with (what reality deems) the right outputs. But, as I’ve defined the level, they don’t know how those inputs and outputs relate to more distantly-connected aspects of reality.
I had a discussion with others about this point recently. My take is basically: if their understanding is so deep, why exactly is their teaching skill so brittle that no one can follow the inferential paths they trace out? Why can’t they switch to the infinite other paths that a Level 2 understanding enables them to see? If they can’t, that would suggest a lack of depth to their understanding.
And regarding the archetypal “deep understanding, poor teacher” you have in mind, do you envision that they could, say, trace out all the assumptions that could account for an anomalous result, starting with the most tenuous, and continuing outside their subfield? If not, I would call that falling short of Level 2.
I would LOVE to agree with this statement, as it justifies my criticism of poor teachers who IMO are (not usually maliciously) putting their students through hell. However, I don’t think it’s obvious, or I think maybe you just have to take it as an axiom of your system. It seems there is some notion of individualism or personal difference which is missing from the system. If someone is just terrible at learning, can you really expect to succeed in explaining, for example? Realistically I think it’s probably impossible to classify the massive concept of understanding by merely three levels, and these problems are just a symptom of that fact.
As another example, in order to understand something, it’s clearly necessary to be able to explain it to yourself. In your system, you are additionally requiring that your understanding means you must be able to explain things to other people. In order to explain things to others, you have to understand them, as has been discussed. Therefore you have to be able to explain other people to yourself. Why should an explanation of other individuals behavior be necessary for understanding some random area of expertise, say, mathematics? It’s not clear to me.
It certainly seems like someone with a deep understanding of their subject should be able to identify the validity or uncertainty in their assumptions about the subject. If they are a poor teacher, I think I would still believe this to be true.
I’ve thought about this some, and I think I see your point now. I would phrase it this way: It’s possible for a “Level 3 savant” to exist. A Level 3 savant, let’s posit, has a very deeply connected model of reality, and their excellent truth-detecting procedure allows them to internally repair loss of knowledge (perhaps below the level of their conscious awareness).
Like an expert (under the popular definition), and like a Level 1 savant, they perform well within their field. But this person differs in that they can also peform well in tracing out where its grounding assumptions go wrong—except that they “just have all the answers” but can’t explain, and don’t know, where the answers came from.
So here’s what it would look like: Any problem you pose in the field (like an anomalous result), they immediately say, “look at factor X”, and it’s usually correct. They even tell you to check critical aspects of sensors, or identify circularity in the literature that grounds the field (i.e. sources which generate false knowledge by excessively citing each other), even though most in the field might not even think about or know how all those sensors work.
All they can tell you is, “I don’t know, you told me X, and I immediately figured it had to be a problem with Y misinterpreting Z. I don’t know how Z relates to W, or if W directly relates to X, I just know that Y and Z were the problem.”
I would agree that there’s no contradiction in the existence of such a person. I would just say that in order to get this level of skill you have to accomplish so many subgoals that it’s very unlikely, just as it’s hard to make something act and look like a human without also making it conscious. (Obvious disclaimer: I don’t think my case is as solid as the one against P-zombies.)