Also I wonder how would math studies look if we teach people the intuition, a bit of the formalism and lot of how to use numeric and symbolic tools - instead endless rote learning.
I've retrospectively applied some of the ideas on betterexplained to this rather than apply it directly as it's not really suitable for people so young and without some foundation. You can't just pick up that book and apply it without some formal knowledge.
I've been meaning to write the whole thing up as it's about 150 pages of hand written material at the moment. Time rarely allows for such things.
One of the key things I concentrate on is not teaching arithmetic but the relationships between values and the basic rules such as the commutative, distributive and associative laws (which whilst are important in arithmetic aren't understood thoroughly without abstract concepts first). The values themselves are only of consequence once you've worked out the relations. Basically it starts with algebra.