This is an excellent couple of points. A math teacher doesn't just have to be able to know how to do the problem himself; he also has to be able to figure out w

This is an excellent couple of points. A math teacher doesn't just have to be able to know how to do the problem himself; he also has to be able to figure out where the student's misunderstanding is and how to correct it. Not only that, he needs to be able to do this for dozens of different students, all of whom have their own difficulties and misunderstandings. I'm sure that there are people with only high school math understanding who can do this... but it just seems to me that the kind of personality traits that would make someone learn how to teach math like this would also make that person extremely passionate about learning it to the best of his ability and potential. I can't imagine a passionate history teacher who isn't also passionate about learning more history. Same thing with an English teacher who only reads John Grisham books (not that there's anything wrong with John Grisham).