Here are two questions (both from Plato’s Euthydemus dialogue) that can provoke much thought:
1. Which sort of human beings are those who learn, the wise or the ignorant?
2. Do learners learn what they know or what they don’t know?
I will leave you to ponder those for a day before commenting myself. Try to think about what those questions open up concerning the nature of learning.
shared ignorance 
1. I think learning is a skill one develops; the more educated a person becomes the better they are at acquiring new knowledge. On the other hand, ignorant people are such because they haven’t properly developed the skill.
There is also a big difference in their assessment of the field of knowledge; ignorant people don’t feel that there is much to learn, whereas learned people are keenly aware of how much there is to know and how much they don’t.
2. Do we learn what we know or what we don’t? I know what Plato would say. But I think of learning as of putting a puzzle together; we may not have a piece in place yet, but we have the surrounding pieces, we have some idea of the shape of the unknown piece, and we find it, recognize it, and incorporate it by the way it fits into our current body of knowledge.
Good comment, Quay!
Let me try to summarize your points and ask some follow up questions. Correct me if I got got you wrong:
1. If learning is a skill that the wise possess, then it is the wise who learn. And yet, the wise are self-consciously ignorant whereas the truly ignorant are not. If we put these two points together, we would have to say that the either/or form is deceptive, that the answer must have a both/and quality, yes? But if it has a both/and quality then doesn’t this threaten to undo the difference between the wise and ignorant, to introduce confusion in place of necessary distinctness?
2. We know what we don’t know by a felt absence in what we do know. I am interested in what you are referring to by “some idea of the shape of the unknown piece” — where does this idea come from? Is it just from the other pieces or from something else as well? By one part, we know another part only if we know the whole out of which each is a part. But where does this knowledge of the whole come from?
Note that I am not trying to argue against what you wrote in a sophistical way; I am just trying to draw out more clearly what you mean.
Very interesting comment! I really enjoy the comments on #2. I wish to clarify, for my own curiosity, what knowledge is obtained? What is knowledge? – a series of objective truths, an answer to the universe which we slowly piece together like in the puzzle analogy above? Or perhaps knowledge and its learning is completely subjective. Perhaps the style in which we learn or what information we acquire and all of the choices we make is a form of personal identity. Maybe as we we place each puzzle piece down we do lock into an image of a vague picture (our identity) still incomplete. However, I believe an ignorant person might try to define an image of them self too early in the puzzle and reinforce what he already knows – “learning what they know”. A wise person may accept that this image is incomplete allowing them to, at least, have a better chance of learning what he did not already know.
Tough question! Probably beyond the scope of what I can do in a comment, but let me just gesture in the direction I think is most fruitful. I will give a heuristic definition of knowledge, not by itself adequate but helping you search for it:
Knowledge is clarity concerning the criterion that makes a defect appear as a defect.
In Quay’s example of the missing piece of a puzzle, what must be known are the generic norms of how puzzles work: chiefly that a completed puzzle results in a identifiably whole picture. If a puzzle doesn’t obey that norm, then one cannot anticipate the character of a missing piece even if one is given all the other pieces. See if you (or anyone else) can complete the analogy using that example toward an adequate understanding of knowledge as such.
1. So, this is hard to put in clear terms, but the wise are “truly” wise yet self-consciously ignorant, whereas the ignorant are “truly” ignorant yet self-consciously wise (I put “truly” in quotation marks to show that I don’t yet know what makes someone truly wise or not).
Socrates is our prime example of the wise yet self-described ignorant person. One of his distinct skills is the ability to separate knowledge from opinion. That must be an essential difference between the wise and the ignorant.
I think you’re right that we’ve muddled the distinction, but I think we now have 2 distinct trademarks of a wise person: (1) facility with learning and (2) meticulousness in separating fact from opinion. Conversely, these are skills that the ignorant person clearly lacks.
2. By “some idea of the shape of the unknown piece” I mean this: when we pose a problem to ourselves, the question itself dictates a certain kind of answer (rather than any answer whatsoever). The question, by its very nature, gives us clues to the answer by (a) delimiting the field of possible answers and (b) giving us hints as to what characteristics the answer is going to possess.
Also, if we’re attentive enough, we know when we’re missing something. I think this is close to what you mean by a “felt absence.”
With the puzzle metaphor, I mean that new knowledge must fit into our existing body of knowledge. When we find something that contradicts our beliefs, for example, we’re forced to rethink or reconfigure the new pieces or our existing pieces (or throw them out entirely) to make them fit, and this is why thinking through paradoxes can be so stimulating.
I don’t think we need to even picture the whole of any field of knowledge to make headway in it. It’s usually good for us to not assume we know what shape the whole is going to take before we get there, if we ever do. In fact, because new knowledge itself is always productive of new questions, I think it’s a fallacy to think we’re ever going to solve anything once and for all.
Quay, I’m not sure the both/and necessarily muddles an important distinctions. We agree that there is still a big difference between the ignorance that seems to itself to be wise and the wisdom that seems to itself to be ignorant.
And you are right that we don’t have to have omniscient knowledge of everything to know even one thing. Read my interchange with Celephais above. Perhaps in addition to knowledge of the whole, we can have a practical wisdom of dwelling within the encompassing whole of being itself. This gets back to your notion of knowing as a skill. In the puzzle analogy, I don’t have to know what the final picture looks like to locate the missing piece, but I do have anticipate something like a whole as just one of the norms of puzzle construction.
By the way, thanks to my two commenters. I am learning a lot thinking through this with you both!
I just added a new post, “Ignorance as absence and presence,” which is a continuation of this ongoing conversation.