Re: the group axioms. My eldest is 10 and mathematically very able, but I don't think he's ready for group theory. On the other hand he has been using algebra, at some level, for years now: I taught it him by asking: "what is 3 add 2? What is 3 apples add 2 applies? What is 3 million add 2 million? What is 3x add 2x? [it's 5x---but what does x mean?] [Oh---x can be pretty much anything, right?]". But I don't think he's ready for "a group is a set equipped with a map such that blah". Just because you know that they should learn sin(x):=x-x^3/3!+...rather than sin(x):=opposite over hypotenuse doesn't mean that they're ready to do so.

But here's my general answer to your question: I didn't read any books (how can a "general" book tell a specialist mathematician what to do, and a book written for specialist mathematicians wouldn't sell enough because there aren't enough of us. Is that an arrogant thing to say? Not sure. Perhaps it is). All I did (and continue to do) is to make mathematics welcome in our house. It is around a lot in our house now. My 4-year-old knows her numbers much better than her letters and I'm sure that's because I'm forever just counting, counting, counting random things, counting the steps we go down as we go into the subway, counting this and that, randomly firing fun questions at my other kids and, if they don't take the bait, never pushing it (if they're not interested in the question then I have to let go: that's one of the hardest things, especially if I felt that I was just about to say something fascinatingly interesting and they're not interested; you just have to leave it and wait until you have their attention). I seize options to turn the topic of conversation in a mathematical direction, and if it ends up going that way then that's great.

The one thing I never do though, is to try and push my kids ahead in the UK mathematical curriculum. I leave that for the schools. The last thing I want them to be is bored at school because they "know it all". So my 10-year-old has just learnt that the sum of the angles in a triangle is 180 degrees, and I don't think I'd ever told him that, but he knows about goofy things like binary and arithmetic mod N and distributivity of multiplication over addition and that the sum of the first 1000 odd numbers is a square and other random things that came up when we were doing nothing in particular. In particular I don't feel like his "tutor", more like a "book of random maths facts".

I think that in summary, I am just myself in front of my children, and that works fine.