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In 1905, at the meeting of the Deutschen Mathematiker-Vereinigung in Meran, the “Meraner Lehrplan,” a mathematics syllabus, was proposed. This document, which contains many of Felix Klein’s ideas on teaching geometry in school, proposed approaching geometry via intuitive geometry, which is the ability to see in space, in order to provide elements for both interpreting the real world and developing logical skills (see Treutlein in Der geometrische anschauungsunterricht als unterstufe eines zweistufigen geometrischen unterrichtes an unseren höheren schulen. Teubner, Leipzig/Berlin ). Klein’s ideas still hold today: An intuitive approach to geometry can be facilitated using information technology. Some activities related to a space geometry approach based on the analogy among figures and on the use of a dynamic geometry system will be presented in this chapter.

This chapter studies Klein’s conception of elementarisation; it is first put into the context of other approaches for mathematics teacher education in Germany. Then, approaches in mathematics education and in history of education to conceive of the relation between academic knowledge and school disciplines are discussed. The wrong translation of Klein’s German term “höher” in the long time prevailing American translation is commented on, in preparation for the analysis of the concept of “element” in the history of science. Klein’s practice and his introduction of the term “hysteresis” to emphasise the independence of school mathematics are discussed. The last section reflects the consequences of the hysteresis notion for integrating recent scientific advances into school curricula.

The relationship between applied and pure mathematics is of utmost concern for Klein. Examples from Volume III of his “Elementarmathematik” illustrate how, starting from an intuitive and sometimes practical approach, Klein develops abstract concepts working in rich “mathematical environments”. The examples concern the concept of empirical function and its comparison with an idealised curve, point sets obtained through circular inversion that lead to compare rational numbers and real numbers, and the “continuous” transformation of curves with the help of a point moving in space.

In the first volume of Elementary Mathematics from a Higher Standpoint: Arithmetic, Algebra and Analysis, Klein closely adheres to several principles which contribute a great deal to the understanding of Klein’s higher standpoint—such as the principle of mathematical interconnectedness, the principle of intuition, the principle of application-orientation, and the genetic method of teaching. In addition, Klein conveys not only a mathematical but also a historical and a didactical perspective, all of which broaden this standpoint. This versatile approach to the mathematical content will be illustrated in this article by taking a closer look at the chapter on logarithmic and exponential functions.

Felix Klein was the first to identify a central problem in the preparation of mathematics teachers: a double discontinuity encountered in going from school to university and then back to school to teach. In his series of books for prospective teachers, Klein attempted to show how problems in the main branches of mathematics are connected and how they are related to the problems of school mathematics. He took three approaches: The first volume built on the unity of arithmetic, algebra, and analysis; the second volume attempted a comprehensive overview of geometry; and the third volume showed how mathematics arises from observation. Klein’s courses for teachers were part of his efforts to improve secondary mathematics by improving teacher preparation. Despite the many setbacks he encountered, no mathematician has had a more profound influence on mathematics education as a field of scholarship and practice.