This Saturday I'm talking about the Banach-Tarski theorem, and I'm starting from the result, then wroking backwards, deciding what we need to know as we peel it back. I've found that working backwards from a surprising result can create motivation to understand, but sometimes it causes the students to dismiss the whole thing as useless, pointless and irrelevant.
Sorry, I'm rambling. Reply if you're interested, ignore me if not.
I shouldn't be dogmatic about asking for an axiomatic development.
At the same time, it seems like the social attitude towards mathematics has reached the point where it would be useful for schools to ask students to put aside some of their initial attitude towards math.
The best teachers I've had often demanded more than I was initially capable of accomplishing. It's true that such teachers risked losing some of their audience. But if we don't have such teachers we risk even more.
Uh, yeah I've dealt with those students. Notice the last part I add - optional. The thing that kills math interest utterly is those "required" classes which teach nothing to the uninterested.
Modern schooling drags the uninterested through a process of making motions towards understanding - we all know its a waste of time. It really would be better to give up until the students are interested. A motivated student can learn more in a day than the bored learn in the semesters of basic math. That might put you out of a job but those jobs just shouldn't exist. Sorry.