That's before a possessive; here the possessive is omitted so anything goes. As I wrote below, you could actually use both the article and the possessive with some nouns (la mia mamma, il mio papà, la mia nonna... I don't think it would work with others) but that's mostly perceived as children's speech.
Yes, you could :) When the possessive agrees with the subject it can be omitted, and in case of close family members it can be switched with the article, so this could have been "nostra nonna". Sometimes the article is also used as endearment, and especially children often say "la mamma", "la nonna", "la zia", even when there is more than one. But it could refer to an old woman too, although that's midway between familiar and rude. And of course it could have been simply "nonna" if the speaker means his own grandmother. A little confusing, perhaps :)
Hi Signor Formica, I am not sure if I have understood you right. So I rephrase some of your points.
"When the possessive agrees with the subject it can be omitted" -- do you mean if I use the word <mia> I don't use <la> like <la mia>? You say <can>. Do you mean I can also include it even if I say <mia> "Invitiamo la mia nonna a cena."?
<children often say "la mamma"> -- Is it common in Italy either to say <mia mamma> or <la mamma>?
<And of course it could have been simply "nonna" if the speaker means his own grandmother> -- do you mean Italians also say ""Invitiamo nonna a cena." if they are talking about their own grandmother?
That's a lot of questions :P
What I meant with "When the possessive agrees with the subject it can be omitted" is that in Italian you can say "Ho parcheggiato la macchina" and if "the car" hasn't been already discussed that would be understood as "I parked my car". So you can say "è andato a trovare la nonna" and that would be understood as "he went to visit his grandma".
"La mia nonna" and "la mamma" would follow into a different category, and that's the affectionate usage of the definite article; a child could say "la mamma mi ha detto di fare i compiti" (mom told me to do my homework) or "la mia nonna mi porta i biscotti" (my grandma brings me cookies), and you could just ignore the article. That sounds childish from an adult though. Not that I haven't heard any say it; it's also common in some dialects like Tuscan.
Using no possessive is fairly common too, especially in a familiar context; of course that's the way to go when calling them to their face, as you wouldn't call your mom "my mom". I actually have a friend who calls his sister "mia sorella" to her face, but I have yet to ask him why.
In some of the family members exercises, I was marked wrong for using il/la etc before mio/mia/mie. eg, in one exercise I used "la mia madre". When I'm on my phone, I don't get the chance to report these things and I can never remember which question it was when I get to my laptop. Old age!
Here's a really helpful link to understand when to use /not use the article in front of a possessive adjective:
In my experience duolingo follow these pretty closely, but I also value f. formica's opinion above because he is offering real experience instead of rules ... I use both as tools to help me understand
Duo accepts both British and American usage. If they have missed something just report it. It may take a while but it will be corrected. This page is full of reminders that "granny" should be accepted but no reference to reporting is seen. Duo does not follow this page for corrections they can only be made through the report option the the exercise page. If this had been done so long ago when the omission was first noticed it might have been corrected by now. See here for some notes etc on Duo and the various guidelines which might come in handy. >https://www.duolingo.com/comment/4821654
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