It would be interesting if kindergarten teachers went with you all the way up to senior year of high school. They wouldn't just be a teacher for a grade level, but for the lifetime of a kid till they go off to college. 

Yep, I can see plenty of problems with this and plenty of work arounds to those problems, but it makes for a fun thought experiment. 

It would make it so teachers really "own" the development of their kids. Of course their would be more specialized teachers for a kid as they got older to teach them specific subject matters, but a kid would have one core teacher that would follow along with them.