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.
