Latest update on the intellectual development of my two young children. As I've said before, with the older one (who has just turned seven) I try to encourage his enjoyment of mathematics, but not to the point where I interfere with his discovering things for himself. From time to time this policy bears fruit. The most recent example was when he announced that to multiply a number by five you could multiply it by ten and take half the result. I suggested that he try it on a couple of examples such as 7 and 9, which he did successfully. I then asked him why his method worked. His answer was not wholly clear but appeared to be not much more than the observation that 15 is half of 30. What pleases me about it is that I think that to be good at mathematics it is a huge help to be friends with numbers. For example, if somebody applying to Cambridge didn't know that 512 was a power of 2, I would be worried about that person. This observation of my son's was good evidence that he is developing that kind of friendship.
It's an interesting question whether his justification of his rule constitutes a proof. At first one would want to say that it obviously doesn't: it's just checking one instance. But my son has no real concept of a general mathematical argument, so it may be, and I think probably is, the case that he somehow intuited that this observation was more than just a coincidence, but lacked the means of articulating that intuition. An interestingly similar phenomenon happened in ancient mathematics, where we see evidence that people could solve quadratic equations and find Pythagorean triples, but no evidence that they had a language in which to express their methods in abstract form. If I understand correctly, historians of mathematics argue that it is wrong to claim that they did not have a good general understanding of what is going on, whether or not they had anything that resembles a modern proof.
Another thing I might mention is that a transition has occurred in my son that seems to occur in everybody but for no clear reason: he now knows, and seems to find it obvious, that multiplication is commutative. I haven't asked him why, and am not sure I can without interfering in the way that I don't want to, though I am curious to know whether he has anything like an explanation in mind or whether this is another case of observing several instances and inducing the rule.
Meanwhile, also within the last few weeks, something seemed to click in my daughter, who will be four soon, and now she completely gets simple three-letter words. There are several things that have changed.
1. She is much more reliable (but still not 100%) if you ask her what something like P-O-T spells.
2. She can now do the reverse process: that is, she can spell words like "pot".
3. She likes inventing her own questions of the "What does P-O-T spell?" variety.
4. She has made short words using magnetic letters on our fridge.
The plan now is to let all that consolidate for a little while, and also try to iron out a few kinks in her letter recognition, before moving to the next stage and teaching her about letter combinations such as CH, EE, OO, AI, etc. After that it will be on to common short words. From past experience, I know that these stages may take months each.