How do you approach your child's math education? My son is one year old, so it is perhaps a bit too early to worry about his mathematical education, but I do. I would like to hear from mathematicians that have older children: What do you wish you'd have known early? What do you think you did particularly well? What do you think would be particularly bad? Is there a book (for children or parents) that you recommend?
(This a community wiki, so please give one advice per answer, as usual.)
Background
I ask here because I believe that the challenges a mathematician faces in educating a child are special. For example, at least some websites and books address the parents' fear of not knowing how to solve homework, which keeps them from becoming involved. On the contrary, I fear I might get too involved and either bore my son or make him think he likes math when in fact his skills are elsewhere.
Christos Papadimitriou said in an interview that, even though his father was teaching math in high-school, they never discussed math. I wonder if that means his father didn't teach him how to count and I wonder if it's a good strategy. (It certainly turned out well in one case.)
Timothy Gowers (in Mathematics, a very short introduction) says that it was inappropriate to explain to his son, who was six, the concept 'zero' using the group axioms. (Or something to this effect, I don't have the book near to check.) That was surprising to me, because I wouldn't have thought that I need to restrain myself from mentioning abstract concepts. (Update. Here's the quote: "[The non-abstract] way of thinking makes it hard to answer questions such as the one asked by my son John (when six): how can nought times nought be nought, since nought times nought means that you have no noughts? A good answer, though not one that was suitable at the time, is that it can be deduced from the [field axioms] as follows. [...]")
There is a somewhat related Mathoverflow question. This one is different, because I'm looking for advice (rather than statistics/anecdotes) and because my goal is to give my son a good math education (rather than to make him a mathematician). I also found an online book that seems to give particularly good generic advice. Here I'm looking more for advice geared towards parents that are mathematicians.
In short, I'm looking for specific advice on how a mathematician should approach his/her child's math education, especially for the 1 to 10 age range.
 A: I have formulated a theorem regarding the involvement of mathematicians in mathematics education.  It says: the personal educational experience of a mathematician is of no value in drawing general conclusions in mathematics education.  I agree that dealing with one's own children is different from "a general conclusion" but even so I'd be careful about sentences that begin "When I was in school..."
A: From my own experience, the best you can hope to do is to entice some sort of (universal) curiosity. Don't try to lecture too much unless it's on the grounds that your kid really wants to know how something works and he's thought abobut it. The worst you can do is try to steer to much of the mental development. Your child may never become a mathematician, and if he does, it might take a good couple of decades to develop the individuality required to go to such lengths successfully. Stay cool about it. Your son won't develop the necessary ability of abstract reasoning for many a year to come either way, so enjoy yourself and try not to think about it. He'll probably be smart enough to handle himself either way.
A: While pushing each of my three sons on a swing when they were wee lads, I would count-by: 1,2,3,4,5,6,7,8,9,10; then 2,4,6,8,10,12,14,16,18,20; then 3,6,9 etc. The swing push would be over when we got to 100. My wife hung count-by sheets in the kitchen. We spoke about counting by eggs as if they were a fraction of a dozen: 1/12, 1/6, 1/4, 1/3, etc. I taught the youngest how to compute squares in his head when he was in 2nd grade. First, he learned one squared through 10 squared, then 10,20,30, etc squared. Then we played a game: what is 20 squared? what is two times 20? what is twenty squared plus  two times twenty? what is twenty squared plus two times 20 plus 1? what is twenty-one squared? These exercises were in the car on a ten minute ride to school. We started to work through computing products as differences of squares. 
Certainly I taught the boys some modular arithmetic, and they all attended the math circle --- even started them a bit too young. 
Also, they were taught how to count to 1023 on 10 fingers. Lots of cute tricks. 
In terms of the mental calculations, even if you can't do the arithmetic quickly, you can teach the child to do so. When the child sees that you struggle with it, then (s)he has someone with whom (s)he can compete.
In addition, I would stress units and developing answers as complete sentences and guiding writing. 
Read "Alice in Wonderland" and "A Wrinkle in Time" to the child at about 1st or 2nd grade. Emphasize the connections between math and human development.  
A: The truth is, that almost just by being around an educated parent, children grow up to be smart/successful. Pushing them in certain directions, or trying to teach them may be effective, or it may backfire. Being hands off, likewise, may be a good or bad decision. Either way, they won't be failing 4th grade arithmetic.
If you want all your kids to grow up to be multimillionaires and senators, then you're probably going to have to push, and push hard. If you're fine with them being content but unspectacular (with the option to go for spectacular if they're inclined), then take a more relaxed stance.
Of course, I've never had kids, only been one. So take this with a grain of salt.
A: 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.
A: I have been trying to make sure that my daughter hears me say with some frequency "I don't know (but I'm curious)" and "I'm wrong (but I'd like to know what's right)," since that's the best way for her to learn to say these things when appropriate.  Looking in the big picture well beyond math, the world would be a better place if people would say those things more frequently.  But math is one of the best places to come to terms with these sentiments (which is one of the many reasons some people dislike it).
Children are sometimes OK with not knowing, but hate being wrong.  I don't want my daughter to have the impression that I know everything, so I'm really happy when she asks science questions I don't have good answers for.  That's not going to happen in math for a while, but I will sometimes intentionally lead her astray and push along until she catches me - that has a big element of fun to it (she gives me these looks when she starts to suspect me) and catching me being wrong is a step, I think, in her developing the ability to catch her own mistakes.
A: get familiar with Piaget and Seymour Papert's works
A: Teach him to use language precisely. The idea that words can have hard, exact meanings is fundamental to mathematical thinking, and will also serve him well in most other subjects.
And I mean all language, not just mathematical language. Being clear and precise in one's language fosters a mathematical attitude by developing one's ability to make and reason about fine distinctions.
A: I hope this answer is not too general but here it goes:
How Not to Talk to Your Kids.
A: I would recommend a great book by Alexandre Zvonkine, "Math for little ones", but it is only available in Russian (here); however, two articles by Zvonkine which were published earlier are translated into English, see here. (You might also want to check other materials linked on Andrei Toom's webpage.)
And, as a tiny bit of more general advice, you know about Piaget's works, right? They are highly relevant when trying to teach kids anything at all, I think.
A: Here are some activities that my son (almost 3 years) has enjoyed.  They are all motivated by the idea:  make mathematics visceral (especially for the young ones).
As Kevin says, count, count, count things.  Count backwards.  Count by twos.  Do it while you're moving.
Rather than show them the symbols $3 \times 2 = 6$, take 6 bottle caps and arrange them into a rectangle.  Can you do the same with 7 bottle caps?
Let them play with a nice length of rope.  Show them the "trick" of a slip-knot.  Do it repeatedly (you've taught them to crochet!)
Take off your T-shirt while keeping your sweater on.  Or put on a shirt that is upside-down and inside out so that it comes out right.  
Draw big shapes with chalk on the sidewalk.  A perennial request from my son:  "Draw it bigger!"
When you do get to the stage of learning the strange code called "alphabet," keep it tactile.  Cut out big letters with scissors.  Recognition of symmetry seems to be a pretty natural phenomenon when you can hold the object in your hands.  "What happens when you turn M upside down, flip over the b?"
Most importantly, don't push it.  If their interest wanders elsewhere, then let it go.
A: The story I heard from a senior colleague when I was at this stage was:
"Twenty years ago I had no children and five theories on how to bring up children.
Now I have four grown-up children and no theories."
A: Make sure your son knows how to identify patterns and associate meaning to them (when the patterns and meaning are actually there...you'll have to sort out why numerology is bunk).
This is all that mathematics is. For instance, demonstrate the idea of multiplication by arranging arrays of objects. Make the connection with geometrical area in the same way. 
Finally, make it clear that patterns can be manipulated. Make this explicit by exploiting their meanings. To continue the example: given a rectangular shape, you can introduce division by working in reverse. 
