Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis Hardcover – November 1, 2009

All that I expected. An historical reference for all serious students of how to think differently than what one learned in fourth grade.

I read Felix Klein's [1849-1925] 3-volume Math book [publised in German, 1924] in Chinese translation bought recently in Beijing bookshop. The chinese translators are themselves mathematicians, so they corrected some translation mistakes made by the 1939 English translators from whom they translate into Chinese.

Klein was a great mathematician, being the Father of Modern Geometry based on Group Theory [Erlangen Program]. He was also a great Math educationist, who built the Gottingen University into the World Center of Mathematics before WW II, attracted the Germany's and the World's best mathematicians to teach and learn there. eg. David Hilbert, Noether, Laskar, etc.

This 3-volume Math books were compiled by his assistants from the lectures he gave over 20+ years in breaching the gap of Secondary School Elementary Math and University Modern Math, hence the title "Elementary Math from an advanced standpoint". Although written 80 years ago, his concerns and teaching are still valid today. Except for France where they teach modern math concepts in Lycee (secondary school from 3eme to Terminale), other countries' secondary schools don't introduce these modern concepts (Group, Ring, Field...), some with little Set theory at best. For Singapore and the English commonwealth countries, secondary and high school math are still using the same syllabus 100 years ago: 100% computational-bias (calculus, trigo, classical algebra).

Klein spotted this gap 80 years ago, as indicated in the volume 1, which he labelled as Syllabus A, Syllabus B and Syllabus C.

In Syllabus A (which is still used worldwide): all math subjects (algebra, geometry, trigo...) are taught as separate independent modules. The secondary school students don't see them as inter-connected.

In Syllabus B, Klein proposed to base whole math teaching on Functions, which is intuitive as functions can be visualised in Graphs, and link to all branches of math.
eg. Function exponential e: e^x = 1 + x + 1/2! x^2 + 1/3! x^3 +....
also e^ix = Cos x +i.Sin x
where the function exponential has linkage to trigo (cosine and sine), and algebra's complex number i. Same can be applied to calculus: integration and differentiation link to functions like Log, exponential, trigo, etc.

Although he didn't elaborate on Syllabus C (out of his life / book context), but I would presume he had predicted the Bourbaki school of Math based on Set Theory as math structures. This approach of teaching was very prominent in French Mathematics Education. It has proved with little success to students because of being too abstract, distancing the Math from real-life (pure math vs applied math). Klein, although a pure math par excellence, didn't encourage this approach which was advocated after his death by the French Bourbaki mathematicians.

I admire Klein's foresight in Syllabus B intuitive pedagogy which would introduce secondary school students gradually to the University modern math concepts.

After 80 years, his dream is still not fully appreciated by math educationists. It is hard to change the mindset in the math education bureaucrats worldwide.