Key Points: Normal Distribution –
- Standard Deviation
Key Points: Normal Distribution –
It is not only owed to the Ecole Normale Superieure where the 11 Fields medalists were educated, but the prominent “Math Culture” in French society.
This is similar to the International Math Olympiad (IMO) “craze” in China since 1980s till today, where the parents send their primary school kids to drill in IMO boot- camps, because that is a “direct-entry” gateway to enter top university, bypassing the highly competitive “Gao-kao” 高考 (University Entrance Exams for 500,000 places among 7 million students each year, only 7% successful chance !! vs Singapore 40%).
Quora: read how this middle-age (48) French recounts his “French Math” education since 6 to 15 years old:
The curriculum was designed in the late 60s in part by a group of real mathematicians, the Bourbaki. It was very abstract. I learned about basic set theory when I was 6. I did learn about the basic operators but not before I was able to perform them in arbitrary bases from 2 to 10. I was taught the properties of ordering relationship, equivalence classes and partitions at age 11, vector calculus at age 13. Basic differential calculus was taught at age 15, complex numbers and integral calculus at 17. This was carried over to physics where we were happily solving ODEs for mechanics and electrical circuits at age 16. This was for almost everyone except the most dyed-in-the-wool literary person. I was taught elementary proof techniques at age 13. By age 15 I knew about formal logic, contradictions and recursive proof.
Hadamard estimated that :
About 90% of mathematicians think visually, 10% think formally.
Usually, they think in steps:
Source: [NLB #510.922]
Bill Gates, a top Math student at Harvard entrance exams, recalled his first year Harvard “Math55” Course (Advanced Calculus & Linear Algebra) – the toughest at his time because 4 years of Math coursewares condensed into 1 year (2 semesters) !
Note: Harvard “Math55” is even tougher than the “notorious” French Classe Préparatoire, which is a 3-year Math undergraduate courseware squeezed in 2 years : 1st year (code-name “un-demi” or “1/2”) Mathématiques Supérieures; 2nd year (“trois-demi” or “3/2”) Mathématiques Spéciales.
Though Math 55 bore the official title “Honors Advanced Calculus and Linear Algebra”, advanced topics in complex analysis, point set topology, group theory, and/or differential geometry could be covered in depth at the discretion of the instructor, in addition to single and multivariable real analysis and abstract linear algebra. In 1970, for example, students studied the differential geometry of Banach manifolds in the second semester of Math 55.
Math55 Survivors :
Of those students who could handle the workload, some became math or physics professors, including members of the Harvard Math Department such as Benedict Gross and Joe Harris; also, Harvard physics professor Lisa Randall ’84 and Harvard economics professor Andrei Shleifer ’82.
In addition to these professors, past students of Math 55 include Bill Gates and Richard Stallman.
The Professors teaching Math55: include Siu Yum Tong 萧荫堂(China/HK).
Lycée Louis-Le-Grand, founded since 1563, is the best high school (lycée, 高中 1~3) for Math in France – if not in the world – it produced many world-class mathematicians, among them “The Father of Modern Math” in 19th century the genius Evariste Galois, Charles Hermite, the 20th CE PolyMath Henri Poincaré, (See also: Unknown Math Teacher produced two World’s Math Grand Master Students ), Molière, Romaine Rolland (罗曼.罗兰), Jean-Paul Satre, Victor Hugo, 3 French Presidents, etc.
Its Baccalauréat (A-level) result is outstanding – 100% passed with 77% scoring distinctions. Each year 1/4 of Ecole Polytechnique (*) (France Top Engineering Grande Ecole ) students come from here.
More surprisingly, the “Seconde” (Secondary 4 ~ 中国/法国 “高一”) students learn Chinese Math since 6ème (Primary 6).
Note : Below is the little girl Heloïse (on blackboard in Chinese Math Class) whose admission application letter to the high school :
Translation – I practise Chinese since 6ème (Primary 6), 5 hours a week. I know that your school teaches 1 hour in Chinese Math, which very much interests me because Chinese and Mathematics are actually the 2 subjects I like most.
Interviewer asked Heloïse :
Q: Why do you learn Chinese?
A: It is to prepare (myself) for working in China in the future, to immerse now in the language of environment. Anyway, the Chinese mode of operation is so different from ours.
Note : Louis Le Grand (= Louis 14th). He sent in 1687 AD the Jesuits (天主教的一支: 耶稣会传教士) as the “French King’s Mathematicians”(eg. Bouvet 白晋) to teach the 26-year-old Chinese Emperor (康熙) KangXi in Euclidean Geometry, etc.
Note (*): 5 Singaporeans (out of 300+ French Scholarship students) had entered Ecole Polytechnique through Classes Préparatoires / Concours aux Grandes Ecoles in native French language since 1980 to 2011. It is possible one day some of these elite French boys and girls could enter China top universities via “Gaokao” (高考 ~ “Concours”) in native Chinese language.