In advanced mathematics, one must eventually learn the distinction between a mathematical object and some coordinates of that object. For example, eventually students are supposed to understand that a "vector" is not an n-tuple of numbers. I wonder why this distinction is not introduced in elementary mathematics when students are taught about angles. Perhaps it wouldn't be simple! Teaching students that numbers like 0 degrees and 360 degrees are coordinates of angles instead of being angles would relieve the teacher of having to double-talk about them being "different, but really the same" angle. Yet there are situations when zero degrees and 360 degrees denote different things. For example, a moving object making a "turn of 360 degrees" is different than its making a "turn of zero degrees". Perhaps teaching angles in a way that made sense would involve teaching both equivalance relations and coordinate systems.