My question is about the continuum between school math and research math.
This started as a discussion at the Mittag-Leffler institute where I am visiting, and then I thought to post it here, to see if anyone has something wise to contribute.
The school curriculum we have in most of the world was proposed by Felix Klein and laid out with colleagues as the Merano Syllabus in 1905. It essentially establishes that we teach towards calculus, and this plan, which focuses on the study of functions and limits, was adopted in most of the world (France had a similar solution, developed by Poincare and Borel, which was much more rigorous but without limits). Since calculus is technically "easy" but conceptually "Hard", we get a situation in schools where we end up teaching the techniques of calculus without necessarily bringing out the beauty of the subject.
What I am hoping to find are examples from calculus/analysis that get at why the subject is beautiful.
- One example could be Cantor's diagonalization argument. It can raise issues about properties of the real line which otherwise don't get discussed at the high school level.
- Another example could be Riemann surfaces. Even without understanding much about them, the visual representation draws you in.
