One of my all-time favorite books is about math.
It’s Infinity and the Mind, by Rudy Rucker. I was reading adult mathematically-themed books and books about physics by the time I was in the fifth grade.
I understand most mathematical concepts – including some very high-level ones – better than many who have scored in the top echelon of students.
As a for instance, for several years I was friends online with a student at Stanford. He was in their quantum program. It’s a bit more technical than I want to get in here, but our dispute was over determinism and probability in quantum systems. His argument was that determinism was ruled out by quantum systems. I argued (correctly) that determinism was perfectly possible under several different valid at-the-time interpretations and probability models of QM. We argued about it for a long time, and then I told him to go ask his professor.
He later sheepishly told me that he’d talked to his professor, and that I was right.
The details don’t matter, and even if I’d been wrong, they still wouldn’t matter – the point is that I understand the concepts just fine in many areas where even those who practice using those methods don’t really understand their implications or exactly what it is they are doing when they use infinitesimals\limits or many other abstruse mathematical tools.
However, for the life of me, I cannot actually calculate (usually) even the simplest math problems that would be quite easy for a middle-schooler. I also cannot at all play chess (despite being in the chess club and playing hundreds of games over two years), I believe for similar reasons, despite understanding it conceptually just fine. I’d guess these two facts are related, as whatever cognitive deficit I have expresses itself in these two arenas. Also, I cannot solve even the simplest logic problem, no matter how long I spend or how patient I am.
This post originally intended to talk about teaching kids math using soul- and curiosity-destroying techniques, how the “transfer effect” that supposedly makes teaching math help thinking in other arenas is completely false, and the opportunity cost of these failed methodologies – but already this entry is too long, so I will just stop here for the moment.