Since starting to really look for math education information, I have come across such a broad range of opinions and methods - it's amazing and overwhelming!
But one piece that has really stuck with me is how some gifted kids can handle the abstract typically way sooner than they have mastered the computation facts. And that many of those who are naturally gifted mathematically, who "think math" I guess you could say, will often realize as they start to do the higher level stuff that having the computations memorized would make things go faster. So then they do like your son, Ania, and memorize it when it is relevant.
Others, like my husband, and perhaps Trinity ;), are skilled at that mental math and don't need to memorize because they are so good at the mental computation. My husband always amazes me by how quickly he can multiply or divide fairly large numbers. But he does it in his head kind of like how Trinity describes it. While I'm still computing the standard algorithm in my head, he's already got the answer! Which seems to be what some of the "reform math" programs are trying to teach. But I would question whether it can be taught. Is that mental math capability a natural talent that can only be developed if the person already thinks that way? Or can you teach a child how to do it, even if they aren't inclined to think way?
I personally was the math computation whiz in grade school, great at those timed tests because I could memorize so easily. But then when I got to high school I didn't have the mental math flexibility to really understand how I was applying it all. Failed miserably in higher level math classes. My husband was the exact opposite. Failed miserably on anything timed - but excels at the abstract stuff! I see DS heading in the same direction! Last night he got all upset because the new version of Math Blaster we just got has a timed game. He was so upset that he started crying about how much he hates timed games! But I'm going to use it to help him understand how effort pays off. He needs to learn that soon - things being too easy all the time end up with such a twisted idea of how things should be all the time. Hopefully he will learn to love a good challenge AND be great at mental math!
