Intuition for regression to the norm So for regression to the norm it says if someone has a high score on a test (relative to the average) then they are likely to score lower and lower on each following test? This seems very counter intuitive.
As with all probability problems the setup is very important, perhaps you heard of the problem of 3 coin flips, if we know 2 of them are heads what is the probability the third is heads? Which has probability 1/4 This is because the third coin to fall heads is not fixed. 
Likewise consider if you were to pick a high score from the class. For any given person they are much more likely to be picked if they did relatively well for themselves on the test rather than relatively poorly (with respect to their own average). Another way of thinking about this is consider a class with 10001 people 10000 average 70 on tests which we call group B and 1 that averages 90 which we call group A. You pick a paper with grade 80 is it more likely it came from group A or B?. Of course B is the natural answer which means that this person is likely to score lower than 80 on the next test.
They would score lower and lower, and then higher and higher, and then lower and lower, etc. such that the mean is realized over large $n$. If a higher score is indicative of some real improvement in the student's average ability (as you suggest) you would need to alter the mean to reflect this.