Looks like it has to do with the combination of the verb fare + (def. art.) + (job), which translates out as "works as a (job)". So, while "la" is usually the def. art., "io faccio il capitano" would be "I work as a captain" or "I am a captain".
On the other hand "io sono il capitano" would be "I am THE captain", and "lei è la dottoressa" would be "she is THE doctor".
It's specifically the usage of the verb "fare" that kicks translation to English into the indef. art.
I think the Italian can be translated as that, although not sure. Sounds a bit strange in English; you'd only say 'my mother works as the secretary' if you were talking about a specific company, you know like 'I work as a manager in this company, my mother is the secretary'. But if you were just talking generally, you'd say 'my mother works as a secretary'.
The sentence "Mia madre fa la segretaria." doesn't indicate whether it is speaking generally or specifically. It's not as though every sentence in duolingo speaks in a general sense. I still don't see how translating the article as definite (instead of indefinite) would get marked as wrong..
"Fare" is a verb with several senses. One that may be useful to know is "to act (like/as) something." It suggests that the object (the "something") is a role that is played, and it is used exactly this way in the theater ("He plays/acts as the King = Lui fa le re").
English also has a similar non-theatrical construction: "To play the fool" (to act like a fool) = Italian "fare il stupido." Italian generalizes this sense to mean "work as" or "is," but the job is still considered a "role" to be acted: "Mia madre fa la segretaria" or "lui fa il medico" (He is/works as a doctor)
EDIT: In my first version, I pulled up the French pronoun "il" instad of the Italian "lui." Didn't look right, but it took me a while to see what was wrong. Sorry. I don't want to add to anyone's confusion.
Wow. I think we all run into the same mistake. La is "the", una is "a". This translation doesn't make any sense whatsoever. Perhaps colloquially, but not logically or structurally. The error is in the language itself. Well, every language has crazy rules I guess. I know both Spanish and English do.
The is a problem for me with the translation of someone "fa + the job" to someone "is a" and his profession.
DL translation "My mother is a secretary." kann be retranslated in " Mia madre è una segretaria." - without using "fa"
And cannot the sentence "My mother works as a secretary."[ Mia madre lavora come segretaria.] - even without "fa" - be translated to "Mia madre fa la segretaria." ?
It seems possible. But thanks for a correction.
Literally, yes, colloquially, no. What I mean is, this is a colloquial set phrase, so the rules of grammar have become twisted somewhat. Usually you'd be right, 'una' means 'a', but in this specific case 'a' can be translated as 'la'. Just go with it.
Also, there have been so very many identical questions on this thread already, have a look through them (on the computer, don't think you can on phone).