This is a big part of what I enjoy about teaching. Aside from all the usual stuff about not properly understanding something until you've taught it to someone else, there's something else going on, which (by crude analogy to your analogy!) is a bit like the rsync algorithm: you have a large "bitmap" to sync, and limited bandwidth to sync it over, so you proceed by checksumming the receiver's existing copy and trying to figure out what diffs you need to communicate to bring them up to date.
In practice this means that teaching is a deeply interactive activity which involves first exploring the contours of someone's existing understanding and then trying to move them to a richer understanding in the most direct way possible. Perhaps this is a more general case of what you're talking about: intuition is one thing that you can try to teach (or communicate) to another person, but maybe all you're trying to teach is something mundane and concrete like the difference between a Google account and a Google Apps account, which is less about intuition and more about having a simple but exactly correct mental model of the world in order to not be hopelessly confused by it.
My co-founder and I recently made a decision regarding a partnering with another company in a somewhat new field. We made our decision pretty quickly, and it wasn't until I was later asked why we made the decision we did that I went back and considered what had really led us down the path we took:
A mentor who works at a big hosting company talked about the way they go about choosing open source technology is not by looking at what the best technology at the time is, but at the quality of the community contributing to that technology, and make a bet that that one will "get better faster" than the others. Without thinking about it, that was the primary criteria that led us to make our decision: the company we chose isn't currently the best at what they do (which is why I had to give this some thought when queried about the decision), but they seem to have set up some systems that (we believe) will let them "get better faster" than their competitors.
It wasn't something we considered at the time of making the decision, but in hindsight, that anecdote was the "vector" that gave us the reasoning that probably contributed the most the decision, without us really being conscious of it.
Edit: Thinking more about this more now, these incredibly well-built-up and well-tested transfer functions that experienced entrepreneurs and investors have built up is what makes YC and other mentor programs so incredible. This is how Ron Conway and Jessica Livingson judge entrepreneurs so well from just a simple conversation.
It's nothing that can be taught, but through books like Founders at Work, or simply sitting and listening to an entrepreneur tell their story, and recall the tough decisions they made early on, one can probably build up pieces of this transfer function. It's what makes Mixergy interviews valuable: there's not a much traditional educational value to them, but just hearing the story of what decisions were made, when, and why, helps build these functions for the viewer.
It was exhausting, and probably even harder on him than me (I can be difficult about design stuff), but it was hard to argue about the quality of the results.
I don't like the word "intuition" though because it insinuates that there's no logical reasoning behind the thought process.
Intuition to me is inductive thinking based on impartial information, rather than deductive thinking on complete information, which is often the case in many situations in life and in business.
It's one of the key differences between bridge/poker (logic + "intuition/feel") and chess (complete information + purely deductive reasoning).
I think of intuition more like a data bank you have in your mind, which only gets more accurate in a field that you apply yourself to over time. Accumulate a lot of data through hard-earned experience and your thinking starts functioning like a monte carlo simulation with all these reference points.
It's also why one of my most favorite tweets of all time was by PG about how studying history refines this sort of judgment or "intuition" in your thinking: http://twitter.com/paulg/status/23203210182
However, the description about how we learn language is completely wrong.
Whatever things we experience are stored in our memory. The way to "draw" pixels on our brain is through experiencing( as in observing through our senses). So each of the experience is a pixel according to me.
Trying to find lines and shapes in your vector graphics is basically finding patterns in the 2d pixels. Or what I would call learning in our case. What I think about learning that it is a top down approach. Each experience is very very specific. If you take any visual image from your memory, a huge amount of context is attached to it. The way we learn is we try to find patterns in these experiences and this commonality we save as a concept. Eg, we learn the concept of addition by seeing that when i get "2" "apples"
from someone and if I had "3" "apples" already than I have "5" "apples". I have quoted these because in your childhood you will observer that whatever is in the quotes can very. You can have oranges instead of apples in the quote, or any other number other than 2 and 3. So by observing this you learn that if I get two numbers i can 'add' them. You need to already have an understanding of 'numbers' though. This understanding of 'numbers' or 'addition' is the pattern that you have found, we call it a concept.
So, according to me, 2d pixels are your experiences. and concepts or notions are the shapes in the vector graphics.
IMO Why you cannot convey your intuitions can be attributed to the fact that we have words associated with well known concepts. You learn words by associating them with the concepts you have learned. But many concepts do not have words associated with them. Also The concept might made up of complicated combination of small concepts that you find difficult to break down in simpler language.
(I am also not a neuroscientist, but I have worked on the development of a neurology research s/w for few months.)
I wish I could get better at conveying my intuition. One way I have of doing this is if I find say a framework or library that 'feels' good. If it comes with strong rational explanation of why this is a good design, I can offset that work and just point to that. It's the same way with design patterns and essays or articles.
The best knowledge is knowledge that you already had, clearly laid out.
Sometimes I think it's just me forgetting the details of what I've learnt so I can't explain it to someone properly.