"Das Kind macht Übungen."
Does this mean the child is playing etudes on an instrument, or is in P.E. class?
It can be both physical exercise or exercising on an instrument, and also doing exercises for school, etc. Given the context you would perhaps specify the kind of exercise, e.g.:
Das Kind macht Klavierübungen. (exercising on the piano)
Das Kind macht Turnübungen. (gymnastic exercises)
Das Kind macht Englischübungen. (English language exercises)
Hmm, I think I have to disagree. For me, 'Übungen machen' refers primarily to physical exercises. (Although it might also be used in other contexts). I would never say 'Das Kind macht Klavierübungen' but rather 'Das Kind übt Klavier'. I'd also prefer 'Das Kind übt Englisch' but I can at least imagine saying 'Das Kind macht seine Englischübungen' (It still sounds a bit strange to my ears). But I'm not sure whether this is just my personal aesthetics or a general usage pattern. @Germandy: Do you really say 'Klavierübungen machen?'. Perhaps we can collect some further opinions of native speakers from different parts of Germany here.
I agree that I'd prefer saying "Das Kind übt ...", but when I read the given sentence it can still have all three meanings to me. To me, "Klavierübungen" seems common. When you buy a set of notes it may say at the front "Klavierübungen für Anfänger" or a mother might say to their child "Du musst noch deine Klavierübungen machen".
To be honest, I generally don't use the word "Übungen" very often, even for physical exercise. Maybe "Dehnübungen" for 'stretching'. When I exercise, I never tell anyone "Ich mache Übungen". I'd rather say "Ich trainiere" or something like that. Of course, maybe that's just me. I also say either "to work out" or "to practice" instead of "to exercise".
My problem is not with the word 'Klavierübung' as such but with the collocation 'Klavierübungen machen'. I'd never say that. When hearing this sentence I have the picture of someone doing piano-related physical exercises in mind, e.g. stretching your fingers. Of course, I'd understand the intended meaning but still... it's akward ;-) However, I'm absolutely not sure whether this is just me or the common usage.
I am certainly not a German speaker, native or otherwise and should probably not be commenting on this topic, but when I was a teenager the exercise book was called "Übungsbuch". It was not for physical exercise but for doing written tests provided by the teacher of grammar, composition and poetry.
@Cephalium: Thanks, there is no issue with "Übungsbuch" or in general to use the word "Übungen" for piano exercises. Both are perfectly fine, very common and sound natural. My problem is with the collocation (i.e. combination of words) "Übungen machen" used for piano exercises which sounds awkward to me. I'd never say that. The natural way of putting it is "Klavier üben".