This weekend I prepared a lecture on probabilistic embeddings of metrics by tree metrics (section 8.5 of the Williamson-Shmoys book). I streamlined the presentation a bit to get to a point where I may have the shortest, clearest proof (although still too complicated for a blog post), near-optimal from an NP perspective: the students will hopefully be able to follow easily.
However, it is not satisfactory. Why not? Because the students cannot lead. They have to follow me.
What would I want instead? In the P perspective, I would like the students to discover the algorithm and proof by themselves, perhaps by successive approximations, designing progressively more sophisticated algorithms, where each algorithm encounters an obstacle in the form of some hard input, and the hard input is dealt with by refining the algorithm. When the process is broken down in manageable pieces, the students can do their own design, like someone cooking from basic ingredients.
Unfortunately, in the present case, to teach in that way I am lacking in lecture time, preparation time, and depth of understanding, so the students will have to make do with a ready-made, pre-packaged proof that I will merely microwave during class.