Won't put me out of a job - you've jumped to an incorrect conclusion. I'm not a teacher. I run two companies, and I go out to schools to give talks on why math is fun, interesting, useful, and occasionally exciting. Most of the students I deal with are motivated and interested, but even then, some don't like puzzles, and don't like starting from the ground up axiomatically.
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.
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.