I assumed that there was more specialization in high/middle school teachers, whereas elementary ed is just elementary ed. So you could find a teacher who liked math to teach 3rd grade math all day, but that teacher wouldn't necessarily be qualified differently. Once I started thinking about it, I realized that I don't really know how this works in the US. I assume to teach foreign language you have something more than just an ed degree?? Similarly, is the teacher doing AP statistics trained more in math?
Yes there is more specialization in anyone with a single authorization. I'm in California so I can't speak for every state but we do it like most other states. I am authorized to teach social studies in any K-12 classroom or adult vocational class. But I'm not authorized to teach "elementary" school because I do not have that authorization. As long as I wasn't in a situation where I had to teach math, like in a class with the same students all day, there is no reason I couldn't teach a bunch of different social studies classes.
To get an authorization, you need to either have a bachelor's degree in your subject, a master's degree or pass an extensive examination PLUS take your 5th year teaching certification.
However, to teach elementary school, in most cases you need a "liberal studies" degree or an education degree and that's it. Some programs wrap in the 5th year certification into the liberal studies degree, some don't. This is why you find elementary teachers that can't teach math very well. They don't have upper level math classes themselves, in most cases- they only have a class in "how to teach elementary school math." I haven't had math since high school. My teaching credential included a class called "Computers for the Educator" which was my math class.....
We digress, sorry!
