Generally, to teach HS and below, the only degree allowable is an education degree. A PHD in math will not be allowed to teach algebra, and a Nobel prive winning physicist will not be allowed to teach physics, unless of course they additionally have a BA in education.
Where do you live? My wife is a high school English teacher, and has a degree in English. I have a friend who teaches elementary school who double-majored in Psychology and Italian. Here in California, teachers who teach specialized subjects (English, math, foreign languages, etc) generally have a degree in the subject they teach, while teachers who don't specialize (elementary school teachers, for example) might have a degree in education. There is a lot of flexibility, though.
All teachers here, except in rare circumstances, have to have teaching credentials, though. These are usually acquired after the bachelor's degree by completing graduate-level coursework that can be finished in a year, as well as taking an exam and getting some real classroom experience (usually through student teaching). The credential is not a degree, but it does require continuing education, and many teachers end up eventually getting a Master's in education due to all the additional education classes they need to take.
My wife did have a friend at her former school who ran into a problem similar to what you're describing. He was a 7th and 8th grade biology teacher, but had a doctorate in some branch of biology from Oxford University. Aside from having a solid background in the science, he was an excellent teacher, having been selected as Teacher of the Year for the district. However, he received notice that he wasn't "highly qualified" for his position and would have to take additional classes to keep his job. Apparently due to some bureaucratic snafu, the powers-that-be didn't recognize his Oxford degree because it wasn't a PhD, but a DPhil, DSc, or something similar that we never see in America. They couldn't be reasoned with, though, so he ended wasting a bunch of time taking classes that he had *taught* when he was a grad student.