TITLE: Math and mental fatigue
QUESTION [149 upvotes]: Just a soft-question that has been bugging me for a long time:
How does one deal with mental fatigue when studying math?
I am interested in Mathematics, but when studying say Galois Theory and Analysis intensely after around one and a half hours, my brain starts to get foggy and my mental condition drops to suboptimal levels.
I would wish to continue studying, but these circumstances force me to take a break. It is truly a case of "the spirit is willing but the brain is weak"?
How do people maintain concentration over longer periods of time? Is this ability trainable or genetic? (Other than taking illegal drugs like Erdős.)
I know this is a really soft question, but I guess asking mathematicians is the best choice since the subject of Mathematics requires the most mental concentration compared to other subjects.

REPLY [4 votes]: Often just a change in the way you are thinking about the subject is enough to wake you up and continue learning so let me pass on my own trick to keeping fresh when I get burned out reading through a book.  Take a break for 10 minutes and then switch over to learning how to use the software associated with what you were just studying for a bit.  There's a ton of good software out there that implements what you are learning about and it's all fascinating stuff.  
Some examples:

GAP (for group theory)
Singular or Macaulay2 (for algebraic geometry)
PARI (for number theory)
MIT-Scheme and scmutils (for mechanics using SCIM and differential geometry using functional differential geometry)
Geogebra (for straight up geometry)

General Tools that are useful for general mathematics:

GNU Scientific Library 
Sage

When you take what you are reading about and write a program that uses it or try to understand how such things are implemented in a piece of software then you will lock in the details.  The abstract becomes concrete and your excitement level with the material goes up.  After you've put in some time with the software, take another break, and then turn back to the books. 
It's hard to get bored or fatigued when you're super into what you are studying.  If you are a computer person then this might help.  Anything to make it more exciting is worth considering in my book.