Unless your characters are somehow omnipresent, then you are forced to use a battle map. As soon as you say, "Character A is there and not over there", you have delineated the physical relationships between the characters. Even if the map is entirely mental, IT'S STILL A MAP.
The idea that, while I can't actually see the guy standing behind me, I also cannot have any knowledge of anything outside of my field of view is ridiculous. If someone steps around me and gets behind me, I don't suddenly forget that he's there. I may not know exactly where he is, but, I am pretty sure he's still there.
Now, if the guy has snuck up on me unawares, that's different. Of course, that's what the surprise round is for. Much easier on OpenRPG where I can place hidden minis that only the DM can see, but, still easily done in a tabletop game. Keep notes of where the badguy is, numbered squares on the battlemap or keep it in your head.
However, once I know that guy is there, I don't studdenly develop amnesia when he is out of my view.
Agreed, however, most people don't drop fireballs in a completely different area than what they wanted to. Why should the DM's misunderstanding mean that I make a mistake? Remember, the mage did the right thing, it's that the DM screwed up. In other words, the DM's mental map differed from the player's mental map.
It wasn't a case that the mage miscommunicated, rather they were not playing the same game.
However, in all cases, THEY STILL HAVE A MAP!