An old favourite topic of mine to daydream about on pleasant afternoons is this:
If you could completely redesign the university-level mathematics curriculum from the ground up to be as good as it could be, but without external constraints (e.g. designing a math curriculum for an isolated group of mountain monks with no interest in employment or in going to universities, but wouldn't mind winning Fields Medals), how would you do it?
Constraints are as follows:
- The students do not necessarily have good pre-college background, and are not necessarily of "high math aptitude", whatever that means. So the curriculum should be easier than a regular undergraduate mathematics curriculum.
- There is no need to break things up into courses, and there is no need to grade or to write tests. Thus, nothing ever needs to be taught "because it's easy to write easily-graded exercizes about it".
- There is no back-compatibility requirement. No need to conform to anyone's expectations of what student ought to learn or what will land them nice jobs.
- Students are assumed to be highly motivated and to always pay attention.
- There is no need to restrict to "pure math". A physics or engineering or finance example or technique or insight are entirely admissible.
- The objective is to maximize student's understanding of mathematics and their abilities both to construct theories (Grothendieck) and to solve problems (Erdos).
Relaxing some of these constraints, and in particular assuming students of high mathematical aptitude, an attempt along these lines might include Bourbaki's work or Math55 in Harvard, although these are strongly biased in favour of "pure math" and are hard.
I posted on this topic on Facebook, and received a wonderful array of comments from mathematicians and math educator friends. The ideas were interesting, and collecting more would also be fantastic (but that isn't the question). I also saw this intriguing argument by Alexandre Eremenko that vector calculus ought to be dropped in favour of differential forms.
My actual question follows below:
Question: Has there been an attempt to design a utopian mathematics undergraduate-level curriculum, one of whose goals is to be easier for students than current curricula, with no external constraints (in particular, no need to test students and no compatibility constraints)? Has there been any research on the topic of what such a curriculum would include and how specifically it would differ from existing curricula? Could you point me towards such references?