These three volumes constitute the first complete English translation of
Felix Klein's seminal series "Elementarmathematik vom höheren
Standpunkte aus". "Complete" has a twofold meaning here: First, there
now exists a translation of volume III into English, while until today
the only translation had been into Chinese. Second, the English versions
of volume I and II had omitted several, even extended parts of the
original, while we now present a complete revised translation into
modern English.
The volumes, first published between 1902 and 1908, are lecture notes of
courses that Klein offered to future mathematics teachers, realizing a
new form of teacher training that remained valid and effective until
today: Klein leads the students to gain a more comprehensive and
methodological point of view on school mathematics. The volumes enable
us to understand Klein's far-reaching conception of elementarisation,
of the "elementary from a higher standpoint", in its implementation for
school mathematics.
This volume I is devoted to what Klein calls the three big "A's"
arithmetic, algebra and analysis. They are presented and discussed
always together with a dimension of geometric interpretation and
visualisation - given his epistemological viewpoint of mathematics being
based in space intuition. A particularly revealing example for
elementarisation is his chapter on the transcendence of e and p, where
he succeeds in giving concise yet well accessible proofs for the
transcendence of these two numbers. It is in this volume that Klein
makes his famous statement about the double discontinuity between
mathematics teaching at schools and at universities - it was his major
aim to overcome this discontinuity.
