Things vs Coordinates-of-Things

[Stephen Tashi]

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.

 

[SteamKing]

It's not clear what 'coordinates of angles' means.

 

[Stephen Tashi]

It's not clear what 'coordinates of angles' means.

Coordinate systems are permitted to be redundant. In some coordinate systems, the same thing can be represented by different coordinates. The numbers 0 and 360 are obviously different numbers. If you want to talk about them representing "the same angle" in a logically consistent manner then you have to do it without contradicting the fact that 0 and 360 are different numbers. Considering values in degrees to be a method of assigning coordinates to an angle would be one way of doing this. There might be others.