From what I've experienced, it's better to study hard but take breaks in between. By breaks, I mean about 10 minutes to grab something healthy, like fruit, to eat and relax before going back to studying. But it's different for everyone, I suppose.

I naturally go in cycles. For maybe a few days I will do a lot of math, and generally learn a lot. Then, I will suddenly burn out, and not have the patience to do any tough problems that I can't solve in less than 5 minutes. So I kind of move to something else for awhile, such as physics or computer science. Then I will burn out of that... and eventually get back to math. I don't know how long the full cycle lasts. 1-2 weeks maybe?

As someone who coaches cross country running and teaches math, I find this interesting. For running and other sports, after a hard effort, the body needs time to recover and to adapt to whatever stress your training provided. Otherwise, you break down instead of improving.

For math, I don't see that you need time to recover. But I think you do need time to absorb, time for the brain to make sense of what you did before moving on to something else that requires an understanding of that first idea.

Our 7th and 8th grade math classes are on an alternating day block schedule. So our students get 75 minutes of math every other day, all year long, 2 or 3 days each week.

I like having the 75 minutes, since I can do a good deep lesson and give students lots of time in class to work, so they can get help right away if they need it. But I don't like having only half as many classes during the school year, since I either have to skip topics or double up.

So sometimes I feel like I wind up dumping so much on students that they cannot absorb it all before the next class. I would prefer having maybe 60 minutes every day, or perhaps 75 minutes, 3 days a week, every week, so I had more days.

So what's the difference between "absorbing" and "recovering"? I feel like they're, for our purposes, the same.

Of course, there are clear differences between a physical workout and a mental one. You are working different body parts which have different physical characteristics. When you lift weights or run, you create slight tears in your muscles. Thus why recovery is so important: that's where you actually gain muscle, not destroy it.

As for the brain? Well, I don't honestly know the exact mechanisms but I strongly hypothesize that some similar happens when you engage in strenuous brain activities. Perhaps having your neurons fire so much in a short period of time requires you to take some time to recover. Or maybe it has to do with accessing your memory. Whatever the case, from personal experience, you can only work for so long.

One method of working that Tal Ben-Shahar, the teacher of positive psychology at Harvard (well, in the past), taught was instead of just working straight for a long period of time, you should alternate shorter periods of "sprinting" with short rest. Thus, work very hard for 90 minutes, take a 5-10 minute complete break and repeat. This keeps your efficiency up. I like the quote from JP Morgan he gave in class: "I can do a year's work in 9 months but not 12."

Wired had an article in April on Supermemo that touched upon the idea of optimal timing for memorizing info. The basic idea follows from classic research into forgetting curves: reinforcing memory at a certain point in the curve should improve (i.e. decrease) the rate of forgetting with a minimal number of rememorization sessions.

As far as math training goes, the issue is more complex, as both the rate and level of abstraction are context-dependent. For instance, a Supermemo-like approach to memorizing the times table should work fine, but learning algebra for the first time is less like memorizing and more like a paradigm shift. Unfortunately the research into this sort of skill transfer is not as clearcut as the research on pure memory.

From a phsyiological perspective, neuronal fatigue is an unlikely explanation (since the time-scale is on the order of a fraction of a second). Certainly sleep is helpful: a number of studies show that REM plays some sort of role in memory consolidation. However, it's not as simple as taking a nap (even if it is long enough for REM to occur), since memory consolidation (into long-term memory) usually takes on the order of weeks.

This suggests that academic courses should really focus on cumulative tests every week with material covering the past few weeks:
e.g. a test for week X may consist of
50% problems from week X
30% problems from week X-1
15% problems from week X-2
5% problems from week X-3

It's also known that the best method for studying is to take practice tests (as opposed to reading the textbook over and over).

Very interesting -- when I was in college, I would get the class syllabus and work way ahead. If they would give me all the assignments at the beginning of the year, I would do them all by week 5 or 6. By the end of the course, the material seemed very easy to me (even if I didn't really understand it that first time through). It sounds like I was in some way replicating that optimal distribution of exposure to ideas in the course, as each time the course would be hitting material, it would be reviewing it while spending most of my time doing homework on later material. This made college very easy for me.

I just have to ask: what are the "good" substances released when you are training legs that are not released when you are training the upper body?

Certain amino acids. Here's a reference.

How are you starting the ride home and posting this at the same time?
are you doing it on a cell phone?