Scott Aaronson has a great article explaining how quantum mechanics should be taught. The basis of his idea is to describe quantum mechancis as being the mathematical structure you get when you attempt to generalize probability theory to include negative numbers. Once you do this, the L^2 norm replaces the L^1 norm, an inner product can be derived, and quantum theory falls out. Quite elegant.

It's also a vastly better way to learn than the historically-based approach through which I was taught [1].

Go read it.

[1] Another flaw in the historical approach is that it overemphasizes the photoelectric effect - specifically, it claims that the photoelectric effect proves light is quantized. But that's incorrect - you can derive the photoelectric effect directly from a Schrodinger equation with a classical electric field.