# Intuitive Group

Intuitive Group & Action.

Group Action is an alternative, more powerful than Combinatorics (排列组合) when calculating complex combinations.

If the Math profs had used this intuitive teaching of Group Theory, then Abstract Algebra would be not so scary to all students – even 40 years after “Classe Préparatoire” University Math !

# IMO 2020 (6th Question)

2020 IMO (6th problem) only one Chinese 李 student scored full 7 marks.

This question is similar to Analysis finding epsilon-delta value of N:

(Epsilon-Delta Analysis, aka Advanced Calculus ) definition for “Limit of Series” :
It exists N such that, for all n > N…

[Solution] :

# 3 Math Foundation Skillsets for Engineering / Finance Students

【普通人学数学系列-下集：学到什么程度】

1) Calculus
2) Linear Algebra
3) Probability / Statistics

Prerequisites for the 3 above in
1) O/A level : Trigonometry, Algebra, Geometry
2) Vector Space (with some basic Algebraic Structures : Group, Ring, Field )
3) O/A level : Permutation/Combinatorics

https://m.ixigua.com/video/6967105499342832166/?app=video_article&timestamp=1622200486&utm_source=native_share&utm_medium=android&utm_campaign=client_share

# Math Education Video : The Age of Big Data

Good education video for students on Maths applied in Big Data.

https://v.ixigua.com/ePWfEV6/

# French Baccalaureate Math Paper 2 (Analysis)

【「给我进来做题啦!」法国高考数学题难度如何？居然只有四道题？】

Baccalaureate Scientifique Math 2019 (Analyse).

Yet to see the Baccalaureate (S) Paper 1 “Algèbre” which is tougher on Abstract Algebra (Group, Ring, Field, Vector Space / Linear Algebra,…)

https://m.ixigua.com/video/6867459333400035851/?app=video_article&timestamp=1621316174&utm_source=native_share&utm_medium=android&utm_campaign=client_share

# 吴文俊 Wu WenJun

https://v.ixigua.com/efrQhx8/

# Princeton 《Calculus Lifesaver》

This is a free video lecture series of the Princeton Math Textbook 《Calculus Lifesaver》(link Amazon.com) for university non-math majors.

He explains the “funny” but smart Calculus basics : First & 2nd Fundamental Theorems of Calculus (by Newton’s co-inventor Leibniz), some how never taught in the 1970s GCE A level (now also ? ) bcos not Newton’s Anglo-saxxon invention. Neither the French textbooks teach so (bcos Leibniz was German ?). They only appear in pure American/German Calculus books.

“Funny” bcos by “1st Theorem” you could ‘d/dx’ any intergral to get the anti-derivative . Richard Feymann self-learnt this trick in High school to solve complex integrations.

Also why the “Coset” (+ C) is explained by the 2nd Theorem in this video.

# 2021人口普查的契机：大学普及改革先废高考

2021人口普查数据 ：14.1亿，新生婴儿减少，男多于女，老人（＞60岁）佔20％…

（1）缩短基本 小学-中学教育 ：废除高考，省下2年准备高考的时间，12年减2年。

（2）高中到 大学本科 ：直通车（看高中学校成绩）。

（3）名牌大学（清/北/211/918 等）：不设本科undergrads，直接 研究生 Masters/PhD ， 像法国Grandes Ecoles 制度 (2nd Cycle Tertiary education, 本科设在普通 universities / 工程科预备班 Classes Préparatoires).

（4）所有 普通大学universities 只 教 本科undergrad 学生为主。

10年 (强迫 中/小学) ＋2 or 4年 (专校 or 大学)， 即20岁 / 22 岁 专科/大学本科毕业, 解决“人才荒”的问题。

– 大学改革不可操之过急，要按步就班， 先试点 一些top高中附属大学的 “直通车”（例如：人民大学附属中学）， 这样就省下2年高考准备时间。慢慢推行到全国各省内大学和结盟高中。十年完成直通车，百年树人才。

– 名牌大学集中精力/资金在培养 硕士/博士科研人才，把本科教育的资源分散给各省普通大学，直接提升后者的教育水平。这样也可留住中国人才，防止外流去美国读博深造而不回国。

https://m.toutiaocdn.com/i6960914813585392158/?app=news_article&timestamp=1620752197&use_new_style=1&req_id=20210512005637010212205224381BC609&group_id=6960914813585392158&wxshare_count=1&tt_from=weixin_moments&utm_source=weixin_moments&utm_medium=toutiao_android&utm_campaign=client_share&share_type=original

# Cédric Villani

Mathematician Cédric Villani (Fields Medal) is a MP in President Macron’s young party “En Marche”.

He introduced “Singapore Modeling Math” to French primary schools Arithmetic in 2017.

https://v.ixigua.com/eUNaWHf/

# 一个数学家的叹息

This American Math educator proposes teaching Primary/Sec math in new ways :

1. HISTORY : how human discovered the math ideas
2. Do not separately teach Geometry & arithmetics: eg.
1+3+5+7 =16=4^2
3. Have fun of Math thru discovery by students, not by remember formula.
4. Intuition 直觉 : use more

https://v.ixigua.com/er2cStN/

# 新数启蒙 (1-5) – 给16岁以上的中学生

This “Modern Math” Introductory Course is based on the French Baccalaureate “Modern Math” for 16+ years old, simplified & customized for local students in Chinese.

https://v.ixigua.com/e6mRFnP/

（1）集合： 等价关系 Equivalence Relation

https://v.ixigua.com/e6mDFhw/

（2）集合与集合的关系：映射 Mapping

https://v.ixigua.com/e6mfC4C/

(3) 新数 历史： Bourbaki 学派

https://v.ixigua.com/e6mjedR/

（4）由外向内看代数结构 : 域Field，环 Ring，群 Group，向量空间 Vector Space

https://v.ixigua.com/e6mj5sd/

(5) 域 Field : 足球场

https://v.ixigua.com/e6mkau7/

# 中国的中学数学教育出错 在一刀切 ？

https://v.ixigua.com/e6aTkww/

# Modern Math Education : Time for Reform

Modern Math Foundation : Set + Logic (Equivalence Relation).

A level (H2 level) / International Baccalauréat Math should study the big red square box (Number System, Set Theory, Logic).

All STEM university Year 1 & 2 study these 3 Math foundations : Algebra, Analysis & Vector Space.

The Math major university Year 3&4 study more : real /complex analysis, Topology, Galois Theory, graph theory, etc…

# 数学与物理的关系

（小学生与诺奖得主杨振宁的对话）

Math was a tool for Physics (Newtonian Physics)

Math & Physics were independently developed ( Physics Gauge Field = Math Fibre Bundle )

Physics is a tool for discovering Math (Quantum Physics : String Theory)

https://m.toutiaoimg.cn/a6936190545714741798/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# Feymann Calculus Trick

Feymann (Nobel Physicist) has many funny speedy Math tricks for Calculus eg. Differentiate an Integral (Applied Fundamental Theorem of Calculus) , and this one below.

# 林群：先通过案例学道理，后才证明

eg. Learn “Group Theory”
Case / Motivation： Symmetry

https://m.toutiaoimg.cn/a6783678419574981120/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# 丘成桐 的雄心计划： 培养800个数学天才苗子

https://m.toutiaoimg.cn/i6928283118424031751/?app=news_article&is_hit_share_recommend=0&wxshare_count=1&tt_from=weixin_moments&utm_source=weixin_moments&utm_medium=toutiao_android&utm_campaign=client_share&share_type=original

# 《数学与人类文明》Mathematics & Civilizations

《数学与人类文明》数学与现代文明：

1. 哲学：希腊 Euclidean Geometry

2. 艺术 : Arabic 文艺复兴， 达芬奇, Golden Ratio

3. 工业革命：Descartes Analytic Geometry, Newton Calculus

4. 抽象: Gauss “Non-Euclidean Geometry” , Paradox in Set Theory, Godel “Incomplete Theorem” .

https://m.toutiaoimg.cn/group/6919012894134764046/?app=news_article&timestamp=1611057637&group_id=6919012894134764046&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# 吴文俊：中学数学教育

1. 文化大革命 / 数学教育革命 灾难

2. 东方数学 ： 小学 – 中学/高中数学 （机械化）eg. 四则运算

3. 西方数学 ：大学”新”数学 （非机械化）

https://m.toutiaocdn.com/i6917519470810366472/?app=news_article&timestamp=1610825278&use_new_style=1&req_id=202101170327580100080431050833ACBD&group_id=6917519470810366472&wxshare_count=1&tt_from=weixin_moments&utm_source=weixin_moments&utm_medium=toutiao_android&utm_campaign=client_share&share_type=original

1） 教少一点 Euclidean Geometry, 用 Analytic Geometry取代， 才可以 computerise （机械化）.

2) 中国古代是有 Geometry, 但 用天元(x), 地元(y) 代数(Algebraic) 化。

https://m.toutiaocdn.com/i6917528352391807499/?app=news_article&timestamp=1610902867&use_new_style=1&req_id=20210118010107010204051088185887A1&group_id=6917528352391807499&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# IMO Geometry Techniques 几何理论基础：分角定理、张角定理，推理证明

IMO Math usually contains 1 or 2 Geometry questions.

France, UK, Singapore, and some countries which reduce Secondary school syllabus in Euclidien Geometry, are disadvantaged in scoring Gold.

https://m.toutiaoimg.cn/a6910816390312100360/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# Dr. Eugenia Cheng: “How to Bake Pi”

Key Points of the Talk:

1) Math is interesting but is only so after undergraduate school. Before that, Math is taught as computation subject from Elementary to High school.

2) Braid : Bach music, Juggling 3 balls

3) Platonic Icosahedral (20面体) Structure discovered by ancient Greek Plato 2000 years ago, but can’t find a real world Icosahedral object until in Viruses found by Louis Pasteur in 20th century – also now in Sars, Covid19.

4) Group Theory : Battenberg Cake, Bed Mattress Rotate/Flop/ Flipping

5) Mobius Strip & Donnut Cutting.

6) Fermat’s Last Theorem : Andrew Wiles in 1994 proved in 7 years still with a “hole”, but fixed a year later by himself & his student.

7) MacLane Pentagon : Higher-Dimension Categories (PhD Math)

# 为啥法国数学这么强，平民的算数这么烂 ？

{?€ 6.5 是这样算的 ：313.5+0.5 [凑整]＝314.0, 然后 314＋6＝320，得[找] 0.566.5}

https://m.toutiaocdn.com/i6896389485832389134/?app=news_article&timestamp=1607277607&use_new_style=1&req_id=202012070200070100140540151610B97E&group_id=6896389485832389134&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# Chinese Remainder Theorem 《韩信点兵 》& Ideal

Chinese Remainder Theorem

By 1930, even Bourbaki founder Dieudoné didn’t know “Ideal” when he read 《Algebra》from Noether’s lecture-note compiled by her student Van der Waeden

https://m.toutiaoimg.cn/a6559856147912720909/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# 日本战后数学教育家 远山启

https://m.toutiaocdn.com/i6895034925657883144/?app=news_article&timestamp=1605514504&use_new_style=1&req_id=2020111616150301020405504612025E05&group_id=6895034925657883144&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# 听杨振宁讲物理课

[7.1] 好书推荐

1. E. T. Bell: 《Men of Mathematics” 》 – ［Comments:］ Irish Hamilton & Quartenions – Bell said Halmiton later’s life was a tragedy in trying to make quartenions universally applied in Physics but failed. Prof Yang : Quartenions will be “universally useful” in future.

Note : Quartenions (1,i,j,k) & Special Relativity – Does it ring a bell to you the UNIVERSAL 4 FORCES (Gravity, Electric, Magnetic, Particles Weak force) :

2. E. Segrè (吴健雄的博士导师) & the story of the missing Periodic Table Element ’43’ (Tc) discovered by accident from unwanted trash.

[7.2] 重力势能和弹性势能

https://m.toutiaoimg.cn/group/6608369844179960327/?app=news_article&timestamp=1605036314&group_id=6608369844179960327&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

https://m.toutiaoimg.cn/a6606089315711713796/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# 中国数学家之乡 – 浙江 温州

https://m.toutiaocdn.com/i6686319487479710220/?app=news_article&timestamp=1604069365&use_new_style=1&req_id=202010302249240102020851974C100485&group_id=6686319487479710220&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# 丘成桐：我研究数学的经验

‘路漫漫其修远兮，吾将上下而求索‘ – 屈原 《离骚》(340BC)

1. 在香港中文大学遇到美国人年轻数学讲师，引导他进＂数学之大门＂，并推荐他去美国拜他的教授 陈省身 ( 微分几何 的 “Gauss II” 宗师) 读PhD。
2. 14岁前父亲给他打下古文学/哲学的基础，”真/善/美” 艺术的直觉 Intuition， 对高深数学理解有大帮助。
3. “Read directly from Masters” (Abel 说的) . 读经典原著。
4. 多听别人的讲座，多看 journal papers. “抛砖引玉”，“他山之石可以攻玉”。Get inspiration from others.
5. 高中/ undergrad 基础功夫要打好，Algebra, Analysis, Geometry, Physics ，都要 “懂” （只有多作习题才知道）。

https://m.toutiaocdn.com/i6888616174393754123/?app=news_article&timestamp=1603895425&use_new_style=1&req_id=20201028223025010202055140190F8C1C&group_id=6888616174393754123&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

https://m.toutiaocdn.com/i6888616174393754123/?app=news_article&timestamp=1603895425&use_new_style=1&req_id=20201028223025010202055140190F8C1C&group_id=6888616174393754123&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share