Thanks to the increasingly hive-like mind of 43 folders, I landed on this essay about how to think about math on Kalil Azad's excellent BetterExplained. (As we are now in the midst of the era of humees and wudzups, domain names composed of good ol' real words are such a breath of fresh air.)
I honestly wonder how I never came across this guy before. I really like his style, keeping each point short enough to swallow and emboldening the high phrases.
In the above-linked essay he conveys the tricky business of mathematical meaning and how it simultaneously lives in and transcends real-world examples. One skill that makes good mathematicians good at mathematics is the ability to jump between the specific and the general, the concrete and abstract, with tremendous agility. It allows one to see the pathological edge cases without drowning in multiplicity.