Hello,
IN my opinion, the "box" is a good tool to visualize the vital space of the enemy King.
I have written a blog post about this checkmate that details the whole method:
http://chesstrainerapp.blogspot.fr/2013/12/how-to-checkmatewith-king-and-rook.html
When I started playing chess - many years ago - I learnt to checkmate with a K&R by forcing back the K rank by rank (or file by file) - getting my king to face opponent's king and then giving check with a rook.
I now see that that the 'Box' method is recommended - it is far quicker but not quite as easy to explain. Also, students think that the only way to checkmate is with the opposing K in a corner - thus missing possible checkmates in the late middle game when the opposing king is on the edge of the board but not in the corner.
What do people think is the best method to teach?