I teach at a private school in a vibrant math department. Over the past couple of years we have made curricular changes to allow a place for alternative math courses, in both the required and elective courses. We're focusing on student choice as a guiding force, making the math more personal.
In the required sequence students have two options. Differences are usually in approach, but the core content remains the same. In all of the classes, however, students are practicing mathematics - asking questions, writing up "proofs," defining terms, presenting material, conversing formally and informally.
*Algebra 1: Techniques and Applications (pretty standard course) *Algebra 1: History and Aesthetics (follows historical developments)
*Geometry: System Building (euclidean axiomatic course) *Geometry: Investigations (more informal and experiential)
*Algebra 2: Analytic Geometry (pretty standard course) *Algebra 2: Functions and Abstract Algebra (set theory and foundations +)
The last of these is what I have been working hardest on. We cover a pretty standard list of topics, but we start with sets and relations as a foundation, and everything is later put in these terms. We think about functions and maps everywhere. Doing this also gives me the opportunity to introduce Graph Theory and Topology in a semiformal way and ask one or two great problems. We can also discuss the axiomatic method without necessarily tying ourselves to it.
Secondly, we offer semester electives, mostly for juniors and seniors:
*The Complex Plane *Fractals and Chaos *Number Theory *Non-Euclidean Geometry *Modern Algebra *Topics in Topology
Having exciting course options thrills our students. Students are signing up for multiple math courses where previously they had no interest.
HERE'S THE TRICK: 1) Get to the good stuff. Don't worry about locking everything in or ensuring complete mastery when it comes to advanced, modern topics. Bring in your very best, most compelling and gripping problems and let the kids get to work on them.
2) Use lots of informal narrative to motivate what you're doing. It doesn't have to be perfectly sequenced so that everything can be laid perfectly on top of what came before. Human learning doesn't really work like that. Informal precedes formal. This is how humans learn; They just make sense of something, relating it to other things, talking it out, and later putting it all together, cleaning house more and more. Axiomatic process is sometimes just editing.
3) Make it personal. Our students feel that their choice of math class is a personal statement, a reflection of who they are intellectually. By making them ask their own questions and write up and present their work, we force the students to put themselves in the work. On another note, sharing your own thoughts and research is very helpful. I talked about commutative linear functions under composition, and they ate it up.
The trouble has been resources. I have found lots of good books, but nothing really really great. I just found a really solid and admirable book called "Introductory Mathematics" by Seymour Hayden. I recommend it highly. I also have two good books called "Introduction to Algebraic Structures" and "Fundamental Concepts of Algebra," published by Dover.
The truth is, of course, we always need new resources and texts. The people who do this kind of thing need to work tirelessly to share their best stuff - writing books, posting on blogs, tweeting, etc. Speaking of which I better get to work on "Algebra 2: A Modern Course." :)