There's a never-ending story that many of us have sunk our teeth into. How do we go about teaching subjects like calculus and analysis "well?" Most universities that I'm familiar with do a significant amount of "service" work where the majority of our students have their focus on other subjects, be it engineering of physics or computer science or economics or whatever. So our curriculum frequently has to strike awkward balances between issues such as:
How we view mathematics.
What the students are ready to learn.
What our students would like to learn.
What other departments expect us to teach our (their) students.
In universities where resources are not limitless (I want to exclude examples like Harvard, MIT, Stanford, Oxford, etc -- not to say they don't have these problems, but the focus is different) this leads to endless compromises and fussing about with strained resources. Sometimes the compromises are extremely far from ideal.
I'm curious to find out what some "joe average" math departments have been doing. Are there some interesting success stories out there? Some novel approaches to teaching things like calculus and/or analysis to a broad audience, on a tight budget?
Have any departments out there got away from the expensive "phone book" style textbooks? Into on-line material? Interactive software? Has anyplace started seriously using things like Wikipedia as a resource for elements of their courses? Are any departments having success using "muscular" calculus books like Hubbard's "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach" ?
I've seen some examples of on-line homework management, like "WebWork". I believe there are a few others similar platforms out there. We use content-management software here, things like "Moodle" and "Blackboard".
How about interesting ways of merging (or separating) highly-motivated math students into/away from the service curriculum? Does your department have honours courses starting in the n-th year where students would learn axioms for the real numbers? Set-theoretic constructions of the real numbers? Do you ease them into foundational issues slowly (axioms for real numbers before a definition, etc?), or do you whip it out right away? Do you avoid the issue completely?
What kind of background do your students have before learning things like basic point-set topology? Modules over rings? Manifold theory? Lie groups? Representations of finite groups? Basic differential geometry? The uniformization theorem for Riemann surfaces? -- if they have chances to learn anything of the sort. ie: what are the "high points" of your curriculum?
This is a massive sprawling question but I'm curious to hear your insights. In case there is any confusion I do want to keep to specifics as much as possible, things like: we tried A, it was a problem because of X, then we tried B and it worked well with Y. What I'd love to see more than anything is a response like: here at the University of Z we just started trying C and it has doubled the enrollment in our analysis classes!! That'd be candy.