teaching - Curriculum reform success stories at an "average" research university

Here at University of Michigan, we have a completely separate introductory sequence for highly motivated freshmen. It's modeled after the ones at Harvard and Chicago, from what I've been told (although having not taken them, I can't honestly say for sure). We happened to use Spivak, but that's really the only book that is really 'phonebook-sized'.

Essentially, within the first semester, we're fairly familiar with point-set topology (to the point that taking the introductory point-set topology course was a waste of time since I'd already seen it all), the basics of group theory, and a decent amount of analysis (last year, we finished the first semester with the proofs of taylor's theorem and the different error estimate formulas for series expansions.)

The second semester was finishing up covering some things like uniform and pointwise convergence that were left out of the first semester for time constraints, then linear algebra for pretty much the rest of the semester. We covered topics in a similar order to Hoffman and Kunze (i.e. abstract vector spaces before real/complex inner product spaces.). Then we returned at the end of the year to calculus on real and complex inner product spaces (frechet derivative, inverse function theorem, et cetera). During this semester, we were assigned mostly point-set topology and algebra problems unrelated to linear algebra. These problems usually led to the final step on a later homework being a proof of something like the fundamental theorem of Algebra or Sylow's first theorem (which are two I can remember explicitly.) (No textbook necessary for this course, although the official textbook was Hoffman and Kunze.)

The third semester was spent pretty much entirely spent on measure theory and integration, although a little bit of time was spent on complex analysis, but only enough to prove the statements about the fourier transform. (No textbook was assigned for this course. Anyone who actually purchased the course text indicated on the internet was urged to return the book.)

From what I've been told, next semester, we're doing differential geometry, but as of yet, I don't know how our professor intends to teach it (I asked him if he'd talk about the subject from a category-theoretic point of view (i.e., http://ncatlab.org/nlab/show/nPOV, but he laughed as though I were joking).

The upshot on this sequence is that by the end of the first three semesters, we also receive 'credit (toward completion of a math major, not actual credits)' for linear algeba and real analysis in addition to completion of the intro sequence.

Were I forced to take one of the other introductory sequences, I probably would not have become a math major. The rest of the math curriculum is that awful.