This is continuing an occasional series about my (now six-year-old) son's mathematical development. In an earlier post I mentioned that he still did not find commutativity of addition obvious. The proof of that was that he found questions of the form large number + small number much easier than questions of the form small number + large number. The reason for this was, I'm almost certain, that he thought of x + y as "count on y from x".
I've just tried a few questions of this kind on him and he still seems to find large + small easier, but the difference now is very slight, and I'm not even sure it's there.
However, the main reason for this post is to relay a conversation I had with him earlier today. Out of the blue he asked me what ten threes make. I happen to know that he finds questions of the form "What is n times ten?" easy (because you are taking n tens and adding them together), so I had an ideal chance to test his grasp of commutativity of multiplication. That turned out to be nil: he laboriously worked through multiples of 3 until he reached 30. Interestingly, he then said, "Ah, that's the same as three tens."
I don't know whether that was the moment that he grasped (in the sense of internalizing, rather than being able to understand the reason for) commutativity of multiplication. I suspect not, but I think it was a small step along that road. For some reason, I have a very strong instinct against telling him about it (hey, did you know that you can make the question easier by switching from ten threes to three tens?). I'm not just against telling him the fact of commutativity as though it is a piece of magic, though I'm certainly against that. I'm also against telling him and proving it (by taking an array of dots and saying that you can count it by rows or columns, or some such argument). I feel it will be healthier for his mathematical development if by some mysterious process, which I hope to observe, he comes to find it obvious before anyone ever tells him the general rule. Whether that's by inducting from small instances or somehow understanding the reason for it I don't much mind. I feel that if he does it for himself, he will ultimately develop a deeper personal relationship with numbers, and that that will be extremely valuable to him.