Deep understanding 

Rough Draft 
What is?
By "deep" understanding, I mean understanding which is
It is characterized by its breadth, by knowing a things components and their connections, and most importantly, by knowing what is important.
It is not taxonomies or vocabularies. These can be useful preludes to, and frameworks for, understanding, are not in themselves "deep" understanding. It is not trivia collections, nor skills only useful or or safely applied on carefully constructed examples. It is not even knowing all the equations, and thus being able figure things out, unless this latent potential understanding has been explored to cultivate a feel for what the equations mean.
An argument I hope to make elsewhere is that "deep" understanding is very accessible, even in a completely introductory context. You just have the difficult task of finding someone who deeply understands what is going on to write your introduction. A search far far more difficult than finding someone merely competent.
The
"The purpose here [...] is to encourage and develop a sense of the right order of magnitude for some important quantities. Exact numerical relationships are less important than knowing what is reasonable. Thus it is essential to be able to approximate quantitative relationships through mental images, sketches, or simple math. You cannot understand science without understanding some numbers, at least to the extent that you have a sense of what is reasonable. There is no escaping the fact that science is quantitative." [Triplehorn,v42p47] 
Why valuable?
"Deep" knowledge can be played with, facilitating education, and being fun.
I suspect far greater integration of knowledge is possible than seems now common.
Comments encouraged.  Mitchell N Charity <mcharity@lcs.mit.edu> 
History: 1997.Aug.01  First draft, from notes previously here.