With respect to GRE math, most graduate programs in mathematics require the GRE subject test in math, which is much more challenging and covers all of the material one would go through for a BS in math (calculus, topology, real analysis, set theory...). I'm not sure what other subject exams are given, but it might be that the GRE general exam is used to show competence, and that the subject exam is to differentiate ability.
Sadly, no. The average score on the mathematics subject test is in the low- to mid-600s, with a standard deviation in the low- to mid-100s. So
this year (2011), the average was 650 and the SD was 134. A score of 790 would be about the 84th percentile. For non-US people: the scores range from 200-800.
Some subject tests are even worse: the average score on the chemistry subject test is 700, with an SD of 115. In physics, the average was 692 with an SD of 157 (!). So no score on either of these tests is above average. But you can score -4 SD on chemistry and -3 SD on physics!
It seems wrong to me that you can't be above average on a test, but your poor performance can be in the range 1:10,000. Worse, the scores look like this every year. If they wanted to change things, they would have done so in, say, 1995.
I'm also a bit dismayed that the education students score so low and are assessing children on abilities in these areas...
So am I. But it explains a lot.
