| To answer your question, the youngest age at which I have seen a pupil propose a proof by contradiction to me was age eight. The girl had read some Life of Fred http://www.lifeoffredmath.com/ books for children about mathematics, and had newly joined my local mathematics class. On only the second or third week of class, she came up to me after class and said, "I've discovered a proof by contradiction for the parallel postulate." As you can imagine, I found this quite amazing. (I knew her mother, and thus knew the daughter a little before she joined my class, but I would say that's rather precocious behavior even in the social circle I keep.) Her "proof," of course, was really Saccheri's flawed proof http://www.jimloy.com/geometry/saccheri.htm http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean... that assumed the postulate to show the "impossibility" of any quadrilateral that didn't fit Euclidean geometry. But most of us have minds that begin study of mathematics with a stubbornly Euclidean set of presuppositions, so that was all right. The girl eventually advanced from my mathematics class to my colleague's more advanced class, and then did a summer at Epsilon Camp http://www.epsiloncamp.org/ in that program's first year of existence. |