Part IVElementary Mathematics from a Higher Standpoint—Conception, Realization, and Impact on Teacher Education

Part IV is concerned with Klein’s three volumes on Elementary Mathematics from a Higher Standpoint (Elementarmathematik vom höheren Standpunkte aus). These lecture notes from the early twentieth century were a seminal contribution to mathematics teacher education, presenting a methodological orientation, not just a content-oriented course. They became a model for many later approaches.

From Hand-written Notes to a Bestseller

Klein’s first volume originated from a two-semester lecture course, given in 1907-08-09. It was published as a Nachschrift of notes taken by experienced students and revised by the professor. In Klein’s adaptation of this practice, a great number of lithographic copies could be distributed by the publisher Teubner in Commission: not typeset, but handwritten. Right at the beginning of volume I, one can find the famous quote about double discontinuity (see contribution by Kilpatrick). Volume II, on Geometry, was distributed in 1909 in the same manner. What became later volume III had originally, in 1901, been a separate lecture course on application of differential and integral calculus to geometry.

After the publication of the third, complete and revised edition of the Elementarmathematik from 1924 to 1928, now in regular book format, the German series became a bestseller, was often reprinted, and translated into many languages; the shortcomings, faults, and omissions of the hitherto dominant English translation of 1932 and 1939 are superseded now by the new and complete translation of all the three volumes, published in 2016. Indeed, all translations had so far excluded Volume III. The only complete translation has been the Chinese one, published in the People’s Republic in 1989 and reprinted in the Republic of China in 1996.

Contributions to part IV

Gert Schubring analyzes Klein’s new establishment of the word elementary: not in its everyday meaning of “simple,” but as the outcome of a process of “elementarization” of complex developments in mathematics. Klein does not propose to treat the latest scientific results; rather, he allows proper choices according to criteria of the school system, yielding a certain “hysteresis” behind the recent, not yet elementarized state.

Through examples from Volume III, Precision and Approximation Mathematics, Marta Menghini shows how the relation between applied and pure mathematics was of utmost concern for Klein. Starting from an intuitive or practical approach, Klein develops abstract concepts working in rich “mathematical environments”; e.g., circular inversion is introduced in physics then creating point sets with particular properties.

Klein’s mathematical, historical and didactical perspective is illustrated by Henrike Allmendinger looking at the chapter on logarithmic and exponential functions from Klein’s Volume I, Arithmetic, Algebra, and Analysis. Klein discusses the customary approach to logarithms in school, presenting an alternative way led by the historical development.

Jeremy Kilpatrick shows how, in the three volumes, Klein was able to relate problems in the main branches of mathematics to problems of school mathematics, thus facing a central problem in the preparation of mathematics teachers: a double discontinuity in going from school to university and then back to school, to teach.