Math and mental fatigue

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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!