For all the time schools devote to the teaching of mathematics, very little (if any) is spent trying to convey just what the subject is about. Instead, the focus is on learning and applying various procedures to solve math problems. That's a bit like explaining soccer by saying it is executing a series of maneuvers to get the ball into the goal. Both accurately describe various key features, but they miss the what and the why of the big picture.

I certainly do care about measuring educational results. But what is an 'educational result?' The twinkling eyes of my students, together with their heartfelt and beautifully expressed mathematical arguments are all the results I need.

The whole apparatus of the calculus takes on an entirely different form when developed for the complex numbers.

Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience.

The completion of a rigorous course in mathematics - it is not even necessary that the student does well in such a course - appears to be an excellent means of sharpening the mind and developing mental skills that are of general benefit.

Indeed, nowadays no electrical engineer could get along without complex numbers, and neither could anyone working in aerodynamics or fluid dynamics.

Cardinal arithmetic will be quite important for us, so we spend some time on it. Since, however, it tends to be trivial, we shall not need to spend much of this time on proofs.