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Having been a full professor at the University of Erlangen, the Technical University in Munich, and the University of Leipzig, Klein joined the University of Göttingen in 1886. He had gained international recognition with his significant achievements in the fields of geometry, algebra, and the theory of functions. On this basis, he was able to create a center for mathematical and scientific research in Göttingen. This brief biographical note will demonstrate that Felix Klein was far ahead of his time in supporting all avenues of mathematics, its applications, and instruction. It will be showed that the establishment of new lectures, professorships, institutes, and curricula went hand in hand with the creation of new examination requirements for prospective secondary school teachers. Felix Klein’s reform of mathematical instruction included all educational institutions from kindergarten onward. He became the first president of the International Commission on Mathematical Instruction in 1908 at the Fourth International Congress of Mathematicians in Rome.

Felix Klein was an outstanding mathematician with an international reputation. He promoted many aspects of mathematics, e.g. practical applications and the relation between mathematics and natural sciences but also the theory of relativity, modern algebra, and didactics of mathematics. In this article about the we firstly refer to his ideas in university teaching of mathematics teacher students and the three books “Elementary Mathematics from a higher (advanced) standpoint” from the beginning of the last century. Secondly we refer to his interests in school mathematics and his influence to the “Merano Resolution” (1905) where he pleaded for basing mathematics education on the concept of function, an increased emphasis on analytic geometry and an introduction of calculus in secondary schools. And thirdly we especially discuss the meaning and the importance of Klein’s ideas nowadays and in the future in an international, worldwide context.

Establishing the habit of functional thinking in higher maths education was one of the major goals of the Prussian reform movement at the beginning of the 20th century. It had a great impact on the German school system. Using examples taken from contemporary schoolbooks and publications, this paper illustrates that functional thinking did not mean teaching the concept of function as we understand it today. Rather, it focusses on a specific kinematic mental capability that can be described by investigating change, variability, and movement.

Teachers’ thinking about the concept of function is well researched. However, most research focused on their understanding of function definitions and properties. This paper addresses a more nuanced examination of teachers’ meanings and ways of thinking that are affiliated with what might come to mind as teachers deal with functions in day-to-day interactions with students, such as “What does f mean in f(x)?”. We report results from using the Mathematical Meanings for Teaching secondary mathematics (MMTsm) instrument (Thompson in Handbook of international research in mathematics education. Taylor & Francis, New York, pp. 435–461, ) with 366 South Korean middle and high school teachers and 253 U.S. high school mathematics teachers. South Korean middle and high school teachers consistently performed at a higher level than U.S. high school teachers, including U.S. teachers who taught calculus.

Klein emphasized geometry and intuition, and made the concept of function central to mathematics education. In fact, number and operations form the backbone of the school mathematics curriculum. A high school graduate should comfortably and capably meet an expression like, “Let y = f(x) be a function of a real variable x,” implying that the student has a robust sense of the real number continuum, the home of x. This understanding is a central objective of the school mathematics curriculum, taken as a whole. Yet there are reasons to doubt whether typical (U.S.) high school graduates fully achieve this understanding. Why? And what can be done about this? I argue that there are obstacles already at the very foundations of number in the first grades. The of the number line, characteristic of the prevailing curriculum, starts with cardinal counting and whole numbers and then the real number line through successive enlargements of the number systems studied. An alternative, based on ideas advanced by V. Davydov, the , begins with - ideas of quantity and measurement, from which the (number) line, as the environment of linear measure, can be made present from the beginning, and wherein new numbers progressively take up residence. I will compare these two approaches, including their cognitive premises, and suggest some advantages of the occupation narrative.

We define notions of mathematical coherence and mathematical fidelity and apply them to a study of the function concept in school mathematics, as represented by the results of image searches on the word “function” in various languages. The coherence and fidelity of the search results vary with the language. We study this variation from a mathematical viewpoint and distill dimensions which characterize that variation, and which can provide insights into the characteristics of professional communities of school educators and into principles for selection and evaluation of curriculum resources.

Aimed at modernizing the teaching of mathematics at German secondary schools around 1900, The “Kleinian Reform Movement” was characterized by Felix Klein’s two key demands: “strengthening spatial intuition” and “training the habit of functional reasoning”. This paper presents a number of examples demonstrating the importance of the concept of intuition () for Klein and explains the role he assigned to intuition in mathematics instruction at school and university.

This paper deals with the questions why and how to introduce into teacher education the history of teaching practices and educational reforms. In particular, we are interested in the developments of curricular school geometry during the 19th century and the reforms at the beginning of the last century in Germany. The life and work of Peter Treutlein—a contemporary of Felix Klein—and a conceptual reformist of geometry instruction, schoolbook author, committed teacher and school principal with educational experience of many years opens to us many opportunities to link present teaching practices in Geometry to its traditions, some of which we will discuss.

Japan caught up with the Klein movement at time it occurred and translated the movement into Japanese to be shared immediately. However, incidents such as the huge earthquake in 1923 caused stagnations. Fruitful classroom experiments were done over the years, and mathematics subjects up to calculus were integrated into the mathematics Clusters I and II secondary school textbooks in 1943–44. The textbooks included in relation to mechanical instruments. This paper shows, compared to some impact from the US, the clear influence of the von Sanden’s on Japan. It also explains how the mechanics and kinematics approaches, known since the era of van Schooten (De Organica Conicarum Sectionum In Plano Descriptione, Tractatus. Geometris, Opticis; Præsertim verò Gnomonicis & Mechanicis Utilis. Cui subnexa est Appendix, de Cubicarum Æquationum resolutione. Elzevier, Lugdunum Batavorum, ), served as missing link for integrating geometry and algebra into the concept of function in teaching in Japan during World War II.

Felix Klein’s vision for enhancing the teaching and learning of mathematics follows four main ideas: the interplay between abstraction and visualisation, discovering the nature of objects with the help of small changes, functional thinking, and the characterization of geometries. These ideas were particularly emphasised in Klein’s concept of mathematical collections. Starting with hands-on examples from mathematics classrooms and from seminars in teacher education, Klein’s visions are discussed in the context of technologies for visualisations and 3D models: the interplay between abstraction and visualisation, discovering the nature of objects with the help of small changes, functional thinking, and the characterization of geometries.