"My mother is also a teacher."
It should be 我的妈妈.
的 makes I/me into my, as in my mother. You are grammatically incorrect but still understandable.
In priciple, you are right, but 我妈妈 and 我的妈妈 both are grammatically correct. The particle 的 indicating possession is optional with close relationships (close relative, boyfriend/girlfriend, but also the school you attend or the company you work for). Actually, it's more natural to omit the 的 in these cases.
Is it important if it is 'also is' or 'is also' (in Chinese, not English), because Duolingo only seems to accept 'also is' here...
Are you asking what is the difference between 我的妈妈也是老师 and 我的妈妈是也老师? Actually the 2nd sentence is a wrong one.
我的妈妈也是老师 can mean
- (same as another person) my mother is also a teacher .
- (in addition to her other identities) my mother is also a teacher
We need other context words to know the exact meaning.
Actually, it does not matter which order you place "also" in the sentence. In fact, the problem is that the solution is manually input by developers instead of the "machine" itself that can naturally detect language.
How could I figure out whether there is another person who is a teacher or the mother has another profession?
In a sentence like this wouldn't it be 教师? 老师 is usually a form of address, but the profession is 教师。
老师 can also be used as a profession. However, we also refer to anyone teaching anything or be respectable as 老师 (and now because of TV talent shows it is almost abused); 教师 is not used in that way.
It is all about the context. If it is already understood that the subject is "my" mother then 我的 can be omitted.
You are speaking to your sibling (you have the same mother).
You are speaking to your teacher (the identities being clear, vs. when you are speaking to a classmate and she has her own mother and confusion becomes possible).
"My" mother has already been the subject of discussion earlier.