I stayed at St. John's one summer and I remember they had a "final" where they had to solve geometric problems by actually drawing the proofs on a chalk board, shapes and all. I watched that girl go through the whole process while studying and I was fascinated.
That's what you do literally every day in math - take turns going to the board and working through proofs.
Ptolemy is Waterloo for anyone who has trouble drawing big circles.
