Now you can use both forms, according to the new spelling rules. (Does not matter if it is a father or a mother of three children). TŘÍ / TŘECH is the same, both is OK, but Třech is better to use in spoken Czech only. Here in DUO they probably want to show you both forms. A couple years ago it was only possible to use the form TŘÍ.
It looks grammatically correct but sounds strange to me. Probably because you're not talking about any specific 3 children, so there's no point to emphasise that you are the mother. But "I'm the mother of THESE three children." is perfect, and in that case you can't use "a" instead of "the" (except if the children have more than one mother).
I hope this makes sense :)
It actually IS grammatically correct, and it sounds perfectly normal to me (native US English speaker).
A few minutes ago, "THE father of XXX children" was accepted in another exercise. So I've reported this one, in case DL wants to accept "the" here, too... or maybe to "un-accept" it in the other exercise. :-)
It's true, though, that there was no "the" equivalent in either Czech sentence, which might make both answers with "the" instead of "a" incorrect.
I just got marked wrong for typing "the mother" instead of "a mother" on one of these, even though the former is by far the more common way of saying that to me (native UK English speaker). I'd be more comfortable with "a mother to X children", but if it's "of X children" then I'd say "the mother" every time.
I've changed the main translation to use "the" for this exercise, the exercise you mentioned in your response, and the "father of" exercise referred to in other comments above. As I come across other similar ones, I'll update the default there as well. (Variations using "a" are also accepted.)
It's not possible in English to skip both a/an and the (or some variation this/that/my etc). My understanding: in Czech, nouns default to "unspecified" unless you put "ten" or one of the other variations infront. English doesn't have this default. The only exception I can think of is people's names, where you (almost) never have an article e.g. "I see Joe" (I guess it's obvious that it's a specific Joe) but "I see an/the apple".
I'm not sure I understand your question, but you cannot say "I am mother of three children" in English. A demonstrative (an article) -- "a" or "the" here -- is required. Demonstratives which would be required in English are often not used in Czech, when context makes it clear what is meant; Czech has no articles. (I am native AmE.)
If you say so, I need to believe. My thoughts were like: If I as a mother wanted to say that I am (a) mother of 3 unspecific children, you could just skip the article to underline the unspecificness of the sentence. When I was wrong there, my apologies to you for stealing time.