Math and mental fatigue 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.
 A: I agree with the importance of the points mentioned by Sasha!
But the question is also: what is your process of studying? I found out the hard way that I learn best when I try to write out mathematics to make it as clear as possible to myself, and even as pretty as I can make it. Sometimes this has resulted in rewrites of traditional versions. 
Some well known mathematicians say they learn from conversations with others! 
August, 2014: A few additional points. 
I heard a magician say he practices until the difficult becomes easy, the easy becomes habit, and the habit becomes beautiful. 
I have been helped by having a kind of global question: "What is and what can be higher dimensional group theory?" in which to place many particular problems. The idea is really about the place of multiple groupoids in mathematics, and, hopefully,  physics. This broad programme has allowed lots of flexibility;  it has turned out quite   technically difficult in places, but allowing many pictures, and intuitions. 
I hope that helps. 
A: I was thinking about a specific mathematician, and after Thomas mentioned him I thought I'd make a comment:
Coffee.
As the very productive mathematician Paul Erdős did not say (it was actually Alfréd Rényi, according to wikipedia):

A mathematician is a machine for turning coffee into mathematical theories.

Take a break. It does not really matter how long it is, but it takes at most 5 minutes for an experienced drinker to finish a cheap coffee, mere minutes should be sufficient. I take /very/ regular coffee breaks (every hour?) and can stay active throughout most of the day.
A: I recently read an article on the 40 hour work week and I think it is somewhat related.  The basic idea of it was that in the mid 20th century, they had a 40 hour work week and they had lots of research on it showing that it was optimal in many ways.  That is, if you increased your work week from 40 hours to 60 hours, you wouldn't gain 50% extra productivity.  You would gain 20-30% extra productivity.  But, this is only over the short run.
Once you work 8 weeks of 60 hour work weeks, you end up breaking even.  That is, over that period, you would have gotten the same amount of work done if you had just worked 40 hours every week.  If you do 80 hour weeks, it only takes about 2 or 3 weeks for you to break even and start doing less than if you had just worked 40 hour weeks the whole time.
And, the article mentioned that with jobs that take a lot of mental work, e.g., doing complicated mathematics, in fact you had even less than 40 hours of productive work per week.
So, do some mathematics.  When you get tired and fatigued mentally, go do something else for a while.  Then, come back.  Getting enough exercise and sleep, eating healthy, and having fun activities you do is important.  That is part of the reason the 40 hour work week is good.  Once you start doing too much work, you lose out on all those other important things that help you function normally.
A: This question partially belongs to the sister SE site: productivity.SE
To fight the mental fatigue the following things will help:


*

*doing physical exercises, as they improve oxygen supply to the brain (e.g. walking, working out, etc)

*getting enough sleep

*keeping a healthy diet


Essentially of all the above is to condition the brain to be in the best working order.
A: 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.
A: I know this is an old question, but i still wanted to add my bit to the topic..
There's a studying method, i forgot the name but i'm sure you can find it with google's help.. It goes like this: study 25 mins, then take a 5 min break. Repeat this 4 times, then take a break of a full half hour.
When you take a break, the method also suggests you go out of the room where you're studying and do some light physical exercise, like taking a walk.
It doesn't work that well for me, i must admit. What does work for me when i'm really tired is the following:
Study an hour, take an hour off, and keep repeating this.
During the break i do nothing, preferably i just sit in a chair staring at the wall or relax with my eyes closed. Sometimes i listen to music. If i actually do something my head doesn't seem to rest properly, so i really try to do nothing, just relax and drink some tea.
Taking such long breaks seems ridiculous, but this way i get way more information in my head and it stays there for a longer time. However i must say such long breaks are only necessary when im really tired.
After studying like this for a day or two, i notice i am way more effective and my head gets fresher. I then reduce the breaks to whatever i feel like, normally studying between an hour and two and taking a break of something like 20-30 mins.
Hopefully this still helps.
A: I have the blessing of not being quite sure what type of mathematical insight and clarity each day will bring. Some days, despite little sleep, i may be physically tired, but it allows me to stay at the desk scribbling away and I make some progress (even if it is not directly research related - just coming to terms with "simple" concepts even)! Other days, I wake refreshed and emotionally sound, but my mental clarity isn't there. Either way, I understand that is how I, in particular go about my journey in Mathematics. So I don't get mad if I "desire more clarity" for lack of a decent term.
Sleep is important. I am sure we have had those nights that I call "Feedback Loop Nights" where a Theorem taken for granted in the past just attained radically new salience in much more abstract domain. Thus, the idea continuously morphs back on its previous iteration of conceptual understanding for quite some time until sleep finally arrives. Those are the most unpleasant mathematical experiences because little progress is made. It is just a compulsion.
It is important to be completely absorbed by at least one other, non-analytical hobby. For me, it is music composition (although I often "entertain" analyses of the algebra of harmony and counterpoint). It helps the brain to break out of the compulsion-mode.
As for coffee, I like it in moderation. I find that it does little to improve my cognitive abilities, but it makes conversations more engaging, for sure. Alcohol reacts negatively with me (I get a massive hangover from as little as half a beer) so I don't drink, but I am sure that any psychoactive compound in moderation would aid in broadening one's scope of the World, provided you don't have a history of substance dependence. 
A: I lighted upon this sterling answer by virtue of user Hepth at http://www.physicsforums.com/showthread.php?t=677222:

I too get mental fatigue if I'm working too hard. Usually my problem is if I work on research (read papers/do math/program mathematica) for 8+ hours it takes another 4+ hours for my brain to slow down and I can relax. This leads to problems as if I work late, maybe to 9pm before going home, and then get in bed at 11pm, there is 0% chance of sleeping for another 3 hours due to my mind just running and running. Its horrible, and then the next day I'm even more fatigued and tired and subsequently get less done.
The key to solving this for myself was to :
1) Go to bed early
If you're a student, this is difficult. But stop studying late into the night. Set a curfew where you don't do homework or study after say 8-9pm. Give yourself time to relax.
If you're of working age, spend time with your kids after work (most important), but don't check your emails often/fret about tomorrow's workload/etc after a certain time. Relax, and do something you can enjoy without a ton of mental stimulation (take the wife/SO to a movie, the mall, the modern art museum, etc.)
2) Get up early
If you went to bed early, you should be able to get 8+ hours of sleep and get up at a decent time. The discipline needed to get out of bed quickly and get ready for the day is tough to acquire, but if you nail this down in college, you'll be ready for those 1am, 2am and 3-5 am wakeup calls from your newborn without feeling like your heart is going to explode every time the "alarm" wails.
3) You MUST take breaks during your workday.
Sometimes I feel like I shouldn't. I'll be on a roll, working hard, making progress, skip lunch, then all of a sudden its 6pm. I FEEL like I was getting a lot done, but I really just wound myself up and zoned-in. While I was working for the full 8-10 hours, I didn't get 10 hours of work done.
Instead, if I take a break every hour (or a little less), and go walk and get some water, get some fresh air, I find that during this break my mind will reorganize the priorities of what I'm actually trying to accomplish, and I'll nail down a single task as soon as I sit back in my chair. This also helps avoid the brain-burning overload of studying/working for long periods of time.
4) Stop browsing the internet when trying to study/work.
 Remove facebook/etc from your bookmarks, don't save it so it stays logged in/etc. Make it difficult for yourself to access those sites. While you might think browsing the internet is harmless and basically a "break" from working, its not. You're still thinking about what you're reading, and its a non-stop flow of new, but worthless, input.
As for the whole "sick of learning new ideas" problem, it sounds like you're in the part of your studies where you're working on a bunch of difficult material that you have no interest in. If you were interested in it (like I was in physics) you'll have no problem studying it and learning it quickly. But if you think its worthless for you to learn and just hate it (Organic Chem for me, do I really NEED to know how to properly identify and name 1 cis-3 methylcyclopentane ?) then you just need to "soldier on" and try your hardest to be interested in it.
It's the interest in a subject that makes learning easy; nothing is actually too difficult to learn.
As an aside:
Organic Chemistry was my lowest grade in undergrad (I think like 77%). I was "placed" into it due to my entrance exam in my first semester of college. I was taking 23 credit hours of courses and way overloaded. I struggled to get that 77%, hate the class, and it took up about 80% of my study/homework time as we had to complete homework assignments in the chem computer lab every other night.
The last semester before I graduate I'm told that Organic Chem was NOT required for my degree, that the school should not have made me take it, and the pre-requisite (Principles of Chemistry) IS required, and though I tested ahead of it I didn't get credit, and I'd have to go back and take it. I ended up doing well, but I realized it's errors like this that are the reason so many people drop out in their last semester.
A: I'm not a mathematician. I'm actually too young.
However, I wholeheartedly do not recommend coffee. Coffee leads to emotional instability (in my experience and perhaps others) as well increased stress levels. There is also a huge negative effect on the quality of your sleep.
I used to spend a lot of nights working on mathematics with the aid of coffee and various forms of caffeine. This was fun; I can't deny that. However, I don't think it's productive in the long run. If you do this, you deny yourself the opportunities outside of mathematics and ways to improve yourself as a person. As you probably know, it also destroys your sleeping patterns. 
Did I mention that it's hard to maintain a romantic relationship when you're not on the same sleep schedule as she or he is and that you're tired whenever they're awake?
I now spend my days trying to live on a fairly regular sleep schedule and without caffeine. (I still haven't got adjusted to waking up at 6 a.m. on weekends, but I will eventually.) I feel that there are more than enough hours in the day and you do not need to strain yourself by staying up late at night and working diligently on a problem. The best idea, in my mind, is to work from 6 p.m. to whenever you are comfortable. This allows you to dedicate yourself to a problem if need be; that is, if you become obsessed with it, you have enough time to work on it for hours. Obsession, I feel, is inevitable in mathematics.
With all that said, you should take a step back from time to time and feel proud of what you've accomplished. I am sure that you have done things that are atypical and worth feeling proud for. Revel in those accomplishments from time to time if you're down on yourself. It reminds you that you're not nearly as terrible as your mind wants you to think. Also, I highly recommend having an emotionally positive romantic relationship. Personally, it causes me to have a much more stable self-image.
Good luck on your mathematics!
A: I used to think that the best way and the only correct way was to sit down for hours get really concentrated. Staying up late at night. Never thinking about anything but math. It was my opinion that only when you get in this zone can you really produce some good thoughts, when you really start to live the problem.
But then I realized that while this sometimes works just fine (when, for example, one just can't stop going), often it didn't. Pursuing this method often led me to get frustrated. I just couldn't do it. I am sure than for some (like Erdős?) this is how it works, but for mere mortals, not.
So as suggested above, one way to deal with mental fatigue, if simply by taking breaks. But I would suggest that you actually schedule those breaks. So you could for example work intensely for 50 min and then take a 10 min, or you could for example work for 2 hours and take a 30 min break. I am guessing that it depends on the person how exactly one schedules breaks. It is my experience that if one doesn't schedule breaks, then one either forgets to take breaks, or the breaks end up becoming much larger.
Also (as alluded to in other answers) I think that it is important to keep the mind working on other things than math. Even though you want to "mainly" be thinking about the problem ahead, I think that it can be beneficial to have other projects to work on. Often this might just be another math problem than the one you are currently working one, but it could also be something completely different. Having some other project to work on, will also help with the feeling of hopeless when no progress is made.
