Fields Medalists on School Mathematics
Most people form their impressions of math from their school mathematics courses. The vast majority of school mathematics courses distort the nature of mathematical practice and so have led to widespread misconceptions about the nature of mathematical practice. There's a long history of high caliber mathematicians finding their experiences with school mathematics alienating or irrelevant. I think this should be better known. Here I've collected some relevant quotes.
I'd like to write some Less Wrong articles diffusing common misconceptions about mathematical practice but am not sure how to frame these hypothetical articles. I'd welcome any suggestions.
Acknowledgment - I obtained some of these quotations from a collection of mathematician quotations compiled by my colleague Laurens Gunnarsen.
I have talked with many others who became mathematicians, about the mathematics they learned in school. Most of them were not particularly inspired by it but started reading on their own, outside of school by some accident or other, as I myself did.
In his autobiography Ferdinand Eisenstein wrote about how he found his primary school mathematical education tortuous:
During the first years [of elementary school] I acquired my education in the fundamentals: I still remember the torture of completing endless multiplication examples. From this, you might conclude, erroneously, that I lacked mathematical ability, merely because I showed little inclination for calculating. In fact the mechanical, always repetitive nature of the procedures annoyed me, and indeed, I am still disgusted with calculations lacking any purpose, while if there was something new to discover, requiring thought and reasoning, I would spare no pains.
There is some overlap between Eisenstein's early school experience and the experience that Fields Medalist William Thurston describes in his essay in Mariana Cook's book Mathematicians: An Outer View of the Inner World:
I've loved mathematics all my life, although I often doubted that mathematics would turn out to be my life's focus even when others thought it obvious. I hated much of what was taught as mathematics in my early schooling, and I often received poor grades. I now view many of these early lessons as anti-math: they actively tried to discourage independent thought. One was supposed to follow an established pattern with mechanical precision, put answers inside boxes, and "show your work," that is, reject mental insights and alternative approaches. My attention is more inward than that of most people: it can be resistant to being captured and directed externally. Exercises like these mathematics lessons were excruciatingly boring and painful (whether or not I had "mastered the material").
...for me, one starts to become a mathematician more or less through an act of rebellion. In what sense? In the sense that the future mathematician will start to think about a certain problem, and he will notice that, in fact, what he has read in the literature, what he has read in books, doesn't correspond to his personal vision of the problem. Naturally, this is very often the result of ignorance, but that is not important so long as his arguments are based on personal intuition and, of course, on proof. So it doesn't matter, because in this way he'll learn that in mathematics there is no supreme authority! A twelve-year-old pupil can very well oppose his teacher if he finds a proof of what he argues, and that differentiates mathematics from other disciplines, where the teacher can easily hide behind knowledge that the pupil doesn't have. A child of five can say, "Daddy, there isn't any biggest number" and can be certain of it, not because he read it in a book but because he has found a proof in his mind...
In Récoltes et Semailles Fields Medalist Alexander Grothendieck describes an experience of the type that Alain Connes mentions:
I can still recall the first "mathematics essay", and that the teacher gave it a bad mark. It was to be a proof of "three cases in which triangles were congruent." My proof wasn't the official one in the textbook he followed religiously. All the same, I already knew that my proof was neither more nor less convincing than the one in the book, and that it was in accord with the traditional spirit of "gliding this figure over that one." It was self-evident that this man was unable or unwilling to think for himself in judging the worth of a train of reasoning. He needed to lean on some authority, that of a book which he held in his hand. It must have made quite an impression on me that I can now recall it so clearly.