The really crucial notes still have not been added by them! At least we have an extremely buggy and not that useful clubs feature which you don't really need to complete the course! Its like they are just assuming what mobile users want instead of actually demanding for...
I find this sentence somewhat unnatural. I am inclined to write 'his mother' which I did the first time around but was punished for doing so.
Ironically, there's another exercise where you are not allowed to write 'while brother reads', I work' ( Пока брат читает, я работаю) because Duolingo will tell you that " in English the sentence sounds unnatural without a pronoun before the word 'brother'."
Warning: I speak British English! "mum" is a a form of address: "Hi, mum!" i.e. it can be used like a name, because for a small child, the 'name' of that parent is 'Mummy', but "brother" is not used like a name in that sense. I would never say "Hi, brother!", I would use his name. (I know that in some dialects it can be used that way, which complicates things.) I think this is why 'mum' can be used without a pronoun when 'brother' cannot - it is being treated like a name.
Он думает что мама дома is better. Как can mean how, what, like or as, but not "that." There's an exception that had a structure similar to your sentence, though. Check out http://russian.cornell.edu/grammar/html/le63_70_b.htm
It's a speech error. I think that sometimes you can hear something like this from native speakers but only from those who are not very educated or at least hates reading and missed classes of Russian language at school :) However you can say "Он думает о том, что мама дома" but it will be something like "He is thinking about his mother being at home".
I disagree. The word "mum" is always accepted by Duolingo, so "mom" vs "mum" is not the issue. And the omission of the word "that" in noun clauses (which was my initial point) is neither non-grammatical, nor incomprehensible in American English. So I would maintain this is an error.
It seems to me that the implied context is one of a sibling speaking to another sibling, about a third sibling, in effect meaning that "He" (i.e., our brother) thinks that (our) mom is at home. This is the only context that would make sense to me as otherwise a possessive pronoun would be needed. In cases like this I capitalize "Mom" to indicate that it's a name as well as a relationship, and it would have been helpful if DL had as well.
I put in "He is thinking that Mom is at home" and it was considered incorrect for it when in reality, it should be correct. Because он думает could mean "He thinks", "He is thinking", and "He thought." So my answer should be correct because there is only 1 way to say the phrase despite the past tenses and my answer should be considered acceptable since there are 3 different correct translations to this sentence that involve the он думает part.
You can sit around and think about want you're going to do next year. Einstein can sit around and think about blackholes and not believe they exist. "He thinks that mom is at home." Thinking without believing in this case doesn't make any sense. Of course he believes that mom is at home. It is my assertion that think and believe might be synonyms.
For all those interested "what" used as "that" in English. https://www.google.com/#q=lyrics+to+bellboy+the+who