This morning -- January 1 2009 -- Jake and I went for a walk around the park and greeted various joggers and bike-riders and walkers with hellos, often resulting in replies of "Happy New Year!".
On the way back Jake -- who is almost five -- said to me, "Daddy, do you want to hear something that I know that you don't know?"
"Sure", I said.
"There are two fours in eight, and four twos in eight!"
Actually I did know that, but didn't say it, and instead felt very proud, and asked Jake how many twos there are in six, and how many threes. I also asked him how he got to his first result: He "figured it out". So I told him that he was noticing patterns in numbers, and that that is a good thing to do.
Just for the record, without much prompting from me, basically just encouraging him to count forwards and backwards, counting on fingers, tallying, and recognizing written numbers, my boy spontaneously noticed a pattern illustrating not only a feel for division, but also for symmetry, or more specifically the commutativity of multiplication (a*b = b*a always) albeit in this one case. And this before noticing commutativity of addition (a+b = b+a always), and even the trick of "counting on".
I'll take this as evidence that (at least for Jake) encouraging an awareness of numbers and patterns is not a bad way to go. This contrasts with my experience of doing pages of simple sums and subtractions (4+3= ...) at age four, which apparently I enjoyed!
Not a bad new year's present for a dad!