"Whose sons go to school?"
Translation:Czyi synowie chodzą do szkoły?
Basically, "iść" works in Present Continous - so "they are walking", "they are going".
"Chodzić" is for Present Simple - "they go", "they walk".
But if "they are walking" without any direction and/or purpose, that's "chodzić" as well.
So that's the basic differentiation. As they "go to school" generally, only "chodzić" works.
It's a common mistake, so I am not in the least surprised that a native speaker said that. I can understand that she said "that's how Poles say it". But it is still a mistake.
Even if some mistakes aren't that big and natives make them, we still believe that it's safer to reject them. It would be rather bad if a learner used a sentence that this course has accepted as a correct and someone said to them "You know, this isn't actually correct Polish".
Anyway, "kogo" is the Genitive/Accusative form of "kto", meaning "who". It doesn't mean "whose".
Given the rather free word order in Polish, how could it be explained that "Czyi synowie do szkóły chodzą? " is considered a mistake? I understand that it relocates the emphasis from "Whose(!) sons go to school? " to "Whose sons go to school(!)?" (Consider the (!) as emphasis), but it is nowhere given which part of this sentence is thema and which is rhema.
That's like "Whose sons to school go(!)?". It's just hard to imagine that 'go' would be more important in the sentence than 'to school', putting the verb after the desination is really rather unusual. Perhaps we could defend it and say that "chodzą" is supposed to mean "walk" (rather than take a bus), but to make it clear that this is what we mean it should rather be "chodzą [piechotą/na piechotę/pieszo]".
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Basically, the form of "whose" grammatically works like an adjective, there will be different forms for every grammatical person, as well as grammatical case. Those that you listed are all in Nominative.
"czyja" is feminine singular, "czyi" is masculine personal plural (virile), so it's used for groups with at least one male person - like here. "czyje" is either neuter singular or "not masculine-personal plural" (non-virile), so basically plurals of any words that do not denote male people.