Precise Definitions of Mathematical Maturity | On Teaching and Learning Mathematics

Five Skills which define “Math Matutity”:

For High School Math, the following deficiencies in math maturity :
1. Fluent procedural skills to differentiate & integrate (skill #2)
But unable to …
2. Understand and strategically apply the “Fundamental Theorem of Calculus” in “Differentiate under an integral” (Skill #1 & #3).

For Undergraduate (Abstract) Math study, “Math Maturity” lacking is quite obvious among Math Majors, even though they score well in math exams (which test ONLY on Skill #2) , but never master the true meaning of some concrete concepts behind the abstract definitions, hence they never become a true mathematician (much less a good math teacher) :

Examples 1: Vector Space, Vectors, Algebraic Topology. (Skill #1, 4, 5).

Example 2 (Monkeys & Coconuts “2nd Solution” ) : True meaning of “Eigen-vectors and eigen-values. (Skills #1, #3).

Example 3: What is the true meaning of an” Ideal” 理想, its relation to The Chinese Remainder Theorem. (Skill #1).

Example 4: What is the true meaning of a Group’s Kernel (Skill #1).

Example 5: Abstract Nonsense makes sense. (Skill #1, 4, 5).

Example 6: Algebraic Topology (Homotopy) (Skill #1, 4, 5).

Example 7: Rigourous Reasoning (Skill #1)





Epsilon-Delta Analysis Concept :

How to prove the “Negation” :


Cedric Villani: The Birth of (Abstract) Theorem always comes from some concrete object(s) .

China “Gaokao” Math Syllabus

This is a summary of Gaokao Math syllabus from a top high school 河北 衡水中学 :

Gaokao 高考 : Chinese “Concours” (“科举”) for University Entrance Exams equivalent to International / French Baccalaureate & Cambridge GCE A level.

Besides the hardcore traditional high school math (Algebra, Calculus, 2D/3D Geometry, Trigonometry Inequality, Probability, Series & Sequence), Gaokao Math has 2 topics covered quite deep compared to A-level :

1. Set Theory (集合)

2. Functions (函数)

This paves a solid Math foundation for Chinese high school students to excel later in local or overseas universities.

大数学家会解 IMO /高考 / Concours吗 ?


答案是: 不能!

19 CE Evariste Galois 是 ”抽象代数” 群论之父 (Abstract Algebra – Group Theory), 大学入学考试 (Concours = 法国科举) 连续2年不及格 – 因为他准备不充足,不适应考题的技巧。

反之,中国IMO 奥数2届满分金牌的一位学生,北大数学系不能毕业

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小算盘 大乾坤 Abacus

浙江临海市 国华珠算博物管主人 木匠 雷国华 收藏算盘20年,凭个人力量,保存中国算盘发源地的文化遗产,为此散尽钱财借债,免费开放给公众参观。说到家人因支持他而受苦,感慨激动不已。

中国的算盘上面2珠 (2×5),下面5珠(5), 共15,加1 = 16 进位。古代是16位 (Hexadecimal) 制, 半斤 八两 (= 1/2 斤 x 16 两 = 8 两)。

日本改良成10位制: 上面1珠 (1×5),下面4珠(4), 共9,加1 = 10 进位。

明朝万历年间的 “平民王子”朱载堉 (1536 AD – 1610 AD) 发明现代音乐的 十二平均律 (12-tone Equal Temperament) , 用81档的大算盘算出:先开立方根,后开平方根 2 次

\boxed { \displaystyle \sqrt[12] {2} = 2^{\frac {1}{12} }= 2^{{\frac {1}{3}}.{\frac {1}{2}}.{\frac {1}{2}}} = \sqrt {\sqrt {\sqrt[3]{2}}}}