I've been listening to the infinity thread, etc. and thinking about some of the discussions and what my response is to some of the points raised.
Okay, first I must admit a bias. I actually enjoy math (several different branches) and deep down want my students to love it enough to become life long mathematical explorers!! Maybe even professional mathematicans.
1. The above isn't going to necessarily happen and that is okay. Afterall there are perhaps better things one could do with their life. : )
2. However, sneaking in a bit of fun (and infinity can be that) won't hurt them, and may even get students to think a bit more. Besides I'll enjoy it. BUT I need to be realistic about my expectations. Enrichment is perhaps the right word.
3. And finally, isn't mathematics about thinking? Aren't some of our goals getting students to think using mathematical tools (not that there aren't other perfectly valid ways of thinking)? And aren't mathematical 'tools' extremely diverse today (I have the number of branches of math somewhere and I think it out of date already)?
Sadly, as I think about it: the fifth graders are having a great time and the eighth graders are bored to tears. What happened?
I found that a fun 'approach' to infinity for eighth graders has been to challenge them to write the largest number they know and then introduce the concept of a mega (which is reasonably easy to define) and then the concept of a moser. I then tell them that they are exempted from the course if they can give me the number of digits in a moser. I've had several phone a relative at some University for an answer. After about two days of pounding the calculators (or computers), I reduce the challenge (and, of course, the reward) to writing a mega. So far, I've gotten no solutions and I 'write' it out for them.
Later, we talk a bit further and, depending on the class, I may use Cantor's approach for counting the rationals.