RE: A: When and how do schools outside the United States teach geometry?
I just listened to a podcast but a theoretical physicist who suggested that we had under-estimated the learning abilities of children around 5-8 years to be able to accept higher forms of mathematics and physics. That at that age, they are okay with handling the concepts that they would later reject because of their "Classical" education. Interesting idea... as for me, my father had been showing me some pretty cool things around this time... and so I think that is where I learnt and started to enjoy these things (I'm trying to do the same for my daughters....).
On a seperate topic... As a private tutor, I often gripe about how (in general and speaking as a general stereotype...), primary and even high school teachers often don't really understand the topics that they are teaching to a great enough depth and passion to be able to pass on the knowledge effectively... of course, like I said, it is a bit of a generalisation... but sometimes (often...) sadly true.
When I first time successfully explained Newton's laws to a 4-year-old, I stop believing in that "kids are not ready" crap. The point is to make it fun and to adjust your methods, they have no problem with understanding the knowledge as long as you explain it to be interesting to them. I am glad that you are doing such an awesome job with your kids and as for the teachers in schools, yeah, we have the same problem here. They experience a burn-out after some time and become completely uninterested. Not all of them, of course, but unfortunately the vast majority of them. The problem is their paychecks are too low and the joy is becoming harder with each day so I do not blame them, on the contrary. I feel sorry for them and am grateful for all that they are doing.