Here is a sentence I got: "Wir spielen hinter dem Haus. "
I understand that hinter is both an accusative and dative preposition. I saw a thread talking about the motion in the sentence but I'm still having a little trouble wrapping my head around it. Is the motion thing the only way to tell whether the proposition leads to dative or accusative?
It's not motion per se. It's whether there is some change relative to the noun that follows. All of the playing takes place behind the house, so there is no change relative to the house. If there's no change, use dative. If there is a change, use accusative. Of course this only applies to two-way prepositions like 'hinter'.
'Ich renne hinter das Haus' (accusative) I start running somewhere else and end up behind the house.
'Ich renne hinter dem Haus' (dative) All my running takes place behind the house.
It's only the two-way prepositions that can be followed by either an accusative or dative object -- that small group of prepositions are the ones where motion = accusative and location/destination = dative. "All other prepositions take one case only regardless of movement" (pg. 66). So it's just a matter of memorizing the different groups.
Source: English Grammar for Students of German 6th ed.
"Motion or Change of State".
So if you are staying in the back garden (hinter dem Haus), even if you are running around in motion, but you are still hanging out in the back, then you are more-or-less stationary and thus Dativ.
If you were in the front of the house, and are now moving into the back to play there, then you are really in motion and thus Akkusativ.
It's perhaps easier to think of it this way: the Akkusativ answers the question "wohin?" and the Dativ answers the question "wo?". But of course like everything in the German language there are lots of exceptions to the rules, such as certain prepositions which demand one of the cases be used without any clear logic behind it. You just have to remember those by heart in the beginning, and over time you will automatically choose the correct case without thinking about it.
It's not about motion really, which is a common mistake many people make. For instance: Ich renne auf der Straße. There is certainly motion there, but "die Straße" is in the Dativ case nonetheless. Ich renne in die Straße. There is motion there, but now we use the Akkusativ case because the subject is running "to" the street and not "on" the street.
German is fun isn't it?!
True, which matches what I was saying about running around in the back garden. If you are running in a street, you are not leaving the street and you are not entering the street. You are -- in effect -- staying within the "container" that is a street. So although you are yourself in motion, your location relative to the designated place (container) is the same, ... until you run off into some other street perhaps.
German is fun. ... Like listening to a beginner playing the violin at night.
I think the easiest way to explain this issue is by looking at examples and using the preposition in both cases.
"Wir spielen hinter dem Haus" - You are playing where (location = wo)? Behind the house. You are not moving from say in front of the house to the back. You were behind the house and you still are. Thus DATIV.
"Wir gehen hinter das Haus" - Where are you going to (motion = wohin)? Behind the house. You are moving from say in front of the house to the back. You were in front of the house and now you are going to behind the house. Thus AKKUSATIV.
"Ich bin in dem Schlafzimmer" - Where are you (location = wo)? In my room. I am not moving out or into the room. I am just in the room. That is my location. THUS DATIV.
"Ich gehe in das Schlafzimmer" - Where are you going to (motion = wohin)? Into my bedroom. I was outside of the room and I am now going into the room. That is the place I'm going to. THUS AKKUSATIV.
It is sometimes very easy to determine in which case the preposition will be used using the verb (if it is variable and not fixed eg. zu = always Dativ).
Think about the verb "spielen". "Spielen" is playing. There is technically motion involved, but there is no change of state eg. from A to B. You were playing (in state A) and you still are playing (still in state A). "Ich spiele hinter dem Haus". You can't play into a direction. You just play. I can't think of any case where a variable preposition + "spielen" would be Akkusativ. Thus, the verb inherently sort of hints at being DATIV.
Think about the verb "gehen". "Gehen" is going. You are experiencing a state of change eg. going A to B. You were in front of the house (in state A) and now you are going to behind the house (moving to state B). "Ich gehe hinter das Haus". You can't go somewhere without actually having a change of state (unless you are say lost in the forest and you can't get out etc.) In most cases, a variable preposition + "gehen" would be Akkusativ.
NB: "Laufen" is similar to "gehen" in that is almost always akkusativ. !!However!! You can use "laufen" in a DATIV sense. Take for instance: "Ich laufe in dem Wald" = I am walking in the forest. You are walking, so there is technically movement, but you are not moving from A to B. You are still in state A (in the forest) and you are not moving out (still in the forest). The question would be: Where are you walking (Wo?) and not whereto (Wohin?) even when motion is involved. Thus DATIV.
I hope this makes sense and helps a bit.