--------------------------
Learning beyond your level
--------------------------

If you are in school but interested in learning beyond your level,  
you'd proceed in parallel with three things:


1. Make sure that you understand (and practice to do reliably) 
everything COMPLETELY what you need in class.

If you don't get the best grades although you understand everything in 
class, it simply means that you lack practice. So you'd read whatever 
your curiosity leads you to, but you'd practice diligence in applying 
what you know about the stuff taught in class. Check while or after 
you write something whether it indeed conforms to your knowledge. 

Diligence is _essential_ in using math - so this discipline is not a 
waste of time, although it may cost you quite some effort now.


2. Learn what is beyond - whatever interests you, in whatever speed or 
order, but always keeping note of the stuff that you only partially 
understand while you explore the rims of your knowledge.

You need to better learn or practice the 
tools used in the partially understood context - so you need to find 
out how to get this missing knowledge or practice. Knowledge can be 
read in many places (many things are in wikipedia, which is mostly 
reliable on standard curriculum topics), but practice can be gotten 
only by actually doing it - solving exercises, trying to prove things 
yourself, etc.. 


3. Learn how to recognize at which level you understand something, 
and how to move from one level to a deeper one.

Understanding comes on many levels:
- having the name of a concept heard or read often enough so that it 
  has some meaning
- having understood the definition of a concept
- being able to follow an argument containing a concept
- being able to solve exercises containing a concept
- being able to solve exercises not containing the concept but whose 
  solution requires it
- being able to answer standard questions about the use of a concept
- being able to explain a concept to someone else
- being able to ask meaningful questions that you have about the use 
  of a concept
- being able to understand why something is presented the way it is
- being able to improve someone's presentation of something
- being able to teach something
- being able to do research on something

Different but related things may well be understood at different 
levels. Moreover, understanding on each level can be 
- basic (have done it once, just managed), 
- moderate (can do it without mistake, if I concentrate on it and 
  check things)
- advanced (can do it routinely, but some things are still difficult, 
  error prone, or surprising)
- perfect (hardly need to think to do it correctly)