I've been inspired by Dr. Adequate's series on Geology (an amazing thread), so I would just like to propose a similar introduction to quantum mechanics.
The only mathematics involved will be matrices and vectors.
EDIT: Vectors and matrices form the subject known as linear algebra, the amount of linear algebra necessary for the course will be explained in a separate pdf/post. To learn linear algebra you only need to know what a number is!
So the idea is that the topics would be something like:
1. Derive quantum mechanics from looking at the Stern-Gerlach experiment. Basically that all the features of Quantum Mechanics (e.g. Complex numbers, probabilities) naturally follow from taking a serious look at how the spin of electrons is affected by a magnetic field.
First Set Of Notes
2. Then, look into more depth at the framework we have derived. Deriving the uncertainty principle and entanglement for example.
3. Next, focus on the probabilistic aspects of Quantum Mechanics and their meaning. Proof of the Kochen-Specker theorem and other such results, which serve to prove how probability in quantum mechanics differs from normal statistics and probability.
4. Interpretations and Decoherence.
5. Quantum computing, as an application of the preceding four sections.