For a career in the teaching profession, a degree is needed and a must - that is for sure.

Be advised that in the UK there are some university graduates with arts degrees such as BA's and MA's teaching mathematics and physics at LEA schools. They have made a career of teaching those subjects, and can do so because they have passed degree examinations in first year mathematics and first year physics as part of their university degree courses. Though they cannot obtain jobs teaching their majors, they are much sought after as mathematics and physics teachers due to a perennial and acute shortage of teachers in those particular subjects. From a knowledge viewpoint, you can see that they are teaching a subject, mathematics or physics, that they have only studied for one year at university, though they've studied a subject like history for four years there to get their degrees. I doubt very much whether studying subjects such as history can make them better maths or physics teachers!