* Pool-lings Unite! *

You've put your finger on an important point there, Cambridge make a huge deal about the potential a candidate has to improve, which is why the grades matter less than what the admissions tutor discovers at interview with regard to a candidate's capacity to progress. You may be the better mathematician now, but they are more interested in potential than current status. Interviews test potential not what you already know.

But isn't being a better mathematician the same thing as having more potential? How else can one define a good mathematician? The whole premise behind an elite university is that it takes people who are naturally talented and gives them the resources they need to develop their talent. Mathematics is one of the subjects in which natural talent is most important (another is music). What is meant, anyway, by "capacity to progress"? In one sense of the phrase, the people who have the most capacity to progress are the people who have had no mathematical education at all. In some sense, education can do more for the people with no natural talent at all than for those with a great deal. But that is not the sort of education that Cambridge seeks to offer.