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Gunnar_Zarncke comments on How my math skills improved dramatically - Less Wrong
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Piagets formal operational stage overly simplifies things. It doesn't go the same way for all people. The basic capability for formal operations sets in much earlier. But using it or recognizing the applicability of specific instances is something else. Some people never get algebra, but that doesn't mean they can't do formal operations. I think what is missing is the intuition behind the formal operations. Just doing the formal operations without intuitively understanding why kills motivation. That is the reason DragonBox works so well. You need to train both. I once draw an ascii art about this: http://c2.com/cgi/wiki?FuzzyAndSymbolicLearning
Yeah, agreed. I think a lot of Piaget's work is considered pretty outdated anyway.
I think it depends. The wikipedia page says that the onset is between 11 and 20 years or so.
My aptitude in mathematics was a bit above average when I was 11 years old. Maybe I had already met the criterion for the formal operation stage, despite not doing well in math the first couple years of high school. But something significant happened when I was 17, and it seemed to be a qualitative change in the way I understood mathematics. I also seemed to be developed the ability to excel in Algebra (with motivated effort) later than my peers. Perhaps it wasn't a specific stage identified by Piaget, but it felt physical/neurological.
I do think Piaget is considered outdated. He might have gotten some of the details wrong or its not the whole story. (For example, I'm skeptical that babies ever lack object permanence.) Nevertheless, Piaget is likely correct that certain concepts develop in stages that are timed with physical development.