This past year our seven-year-old should have, by the world's standards, been sitting at a desk for large portions of the day, listening to a teacher and doing his schoolwork. Instead, he ate, breathed, and slept robots, and I, his otherwise responsible mother, completely let him. Even encouraged him. He drew pictures of them, made them out of magnet pattern blocks on the dry-erase board, built them out of Legos and other less robot-ish materials like leaves and sticks, watched YouTube videos about them, went to a science festival in DC where he watched Asimo, a humanoid robot, do cool stuff, built some simple robots (popsicle sticks and tiny motors) with his Dad, etc. He did a few non-robot things like a weekly nature study class, an eclectic bunch of other things with another co-op, and a gymnastics class. Together we read some books that were VERY non-robotic, like Little House on the Prairie and No Flying in the House. But in his free time (and he had a lot of it), he dreamt of robots.
I mention this because today he demonstrated to me that all that dreaming did not slow his ability to do logic-based academic things. I've finally gotten around to giving him his first grade test (we have to do one standardized test per year in Virginia), and on the math section he didn't make a single mistake. They had him doing single- and double-digit addition and subtraction, and despite the fact that he's never done math like that, it didn't phase him for a second. Right before the test I wrote down some problems and said, "Now when they write an up-and-down subtraction problem like this, it means that you take the bottom number away from the top number." Just a few little tips on the formalities, and he was off and running.
I watched him look at the more difficult problems, then gaze off into the blackboard inside his brain. He has used Cuisenaire rods for building and measuring things, and I'm sure he has a good mental-image set of them that he was using internally--along with pulling the physical ones out a few times for the very large numbers like 59+23.
Toward the end of the test he was presented with 17 - 8. Borrowing/regrouping is a concept that stumps a lot of young kids, but he didn't think about it for more than a few seconds before he said it was 9. I asked him to tell me how he'd gotten that answer, and he said he did Bounce-off Attraction. I don't know when he made that up--perhaps right on the spot, or perhaps he's been using that for a while, but he described it to me thus:
The 10 from the 17 bounces off from the 7, and then breaks into a 2 and an 8 (we've played lots of "Ten Pair" games, like the ones that come with my Magical Math program). Then the 8 gets taken away and the 2 is attracted to the 7, so it makes a 9.
This is NOT the way I would have thought that he'd have figured this out, so I asked him if Bounce-off Attraction is something the robots do when he's creating them in his head, and he said yes, it was. He's fascinated with magnetism and anything that looks like a Rube Goldberg machine has him instantly enthralled, so this fits right in--if robots break up and stick numbers together, he's all over it.
All of this leads to a logical conclusion, and it's not that our son is superior to other children in any way. He's actually quite average in most things you want a seven-year-old to be able to do, like not be obnoxious. His reading skills are at the Bob Book level, as he likes those, but most of the time he prefers to not bother actually looking at words, and I've been fine with that for first grade (though we're going to focus on that more in second).
No, the logical conclusion of a perfect math score in a child who's never been taught math is that, for some kids at least, math instruction is overrated, and possibly harmful. I say that because I have a child who attended public school for years and is now completely math-phobic. A problem like 17 - 8 will send her searching for a pencil and paper. She wants to be good at math, but is so worried about making mistakes that she handicaps herself with all the tricks and strategies she's learned at school, quite often applying the wrong one to the situation at hand, and coming up with the wrong solution.
Early development of mathematical thinking is less, I believe, about instruction, drills, and practice in writing symbols on paper, and more about getting out of the way and allowing a child to think logically about the world around him. The world is filled with math--you can't stop a child from learning math. But you can hinder it by throwing math instruction at him with explanations full of jargon about problems he's not interested in. And you can do immense damage by then testing him to see if he grasped that concept in the way you wanted, and then comparing him to other children. Both the desire to understand math and the child's concept of himself as a capable person wither away in these situations. Avoiding them at least until the child is older and more resilient is, I feel, wise.