The use of prepositions has to be learnt. It's tedious, but there's no way around it.
When I learned English, I was given long lists "to think of, to concentrate on, to insist on..." In German, "to think of" is "denken an".
I was thinking that i was on my way to mastering German after figuring out the cases... Until I read this article and realized just how complicated cases with prepositions and adjectives can become
Not entirely sure whether it is an accepted variant and please correct me if I'm wrong, but isn't "we think on" an accepted variant in Scottish English?
As an example, from Tam O' Shanter, by Robert Burns
"We think na on the lang Scots miles, The mosses, waters, slaps and stiles, "
Regardless of whether's it's correct or not it's still cool to see similiarities between English and German in older works!
I don't understand why it is accusative here. If it is because of the "denken", then why is there another example in this lesson with "Ich denke an meinen Trainer", where it is dative?
'Ich denke an meinen Trainer' is not dative. It is accusative. Think for example 'Ich esse einEN Apfel'.
I wrote: ''we are thinking about our families''. Can anybody tell me why is it wrong? (Or shall I report it?)
Can I put "an" at the end of the sentence and still be correct? What would be the difference between the two sentences?
No, the "an" is a preposition and therefore must come immediately in front of the things (or persons) referred to. It doesn't make sense at the end of the sentence with nothing following.
This is a different situation than e.g. in "ich schalte das Licht an", where the "an" is a part of the soplittable verb "anschalten" and not a preposition.