Initially, this was supposed to be a comment about point 3) in Peter Tingley's answer, about a math teacher being able to "convey the beauty of the subject", but it got too long so I turned it into an answer.

While I generally agree with it, I think it is also a tad too idealistic. Dr. Johnson, in one of his invectives against the Scots, barked that "much may be made of a Scot if he be caught young". Contrapositively, if a student arrives at university without any appreciation for Mathematics, there is only the slimmest of chances that he will gain it there. And teenagers? Mathematics has to compete in a teenager's mind with Lady Gaga and porn sites; with mtv rap and online gaming; with entertainments that range from the irrational to the inane. Call me an elitist cynic, but the plain matter of fact is that the majority of students will be completely unmoved by the beauty of mathematics, much as great literature can only be understood and enjoyed by a relative minority within each generation. I guess what I am saying is that a quality that a good teacher must have is a thick-skin and somehow, by whatever means, keep the flame alive inside his own heart so that when that special, receptive student does come along, he will be able to kindle the fire and infect him with the rapturous love for Mathematics. Or in the words of Samuel Beckett, "Try Again. Fail again. Fail better."