The Legacy of Felix Klein
Abstract
This open access book provides an overview of Felix Klein’s ideas, highlighting developments in university teaching and school mathematics related to Klein’s thoughts, stemming from the last century. It discusses the meaning, importance and the legacy of Klein’s ideas today and in the future, within an international, global context. Presenting extended versions of the talks at the Thematic Afternoon at ICME-13, the book shows that many of Klein’s ideas can be reinterpreted in the context of the current situation, and offers tips and advice for dealing with current problems in teacher education and teaching mathematics in secondary schools. It proves that old ideas are timeless, but that it takes competent, committed and assertive individuals to bring these ideas to life.
Throughout his professional life, Felix Klein emphasised the importance of reflecting upon mathematics teaching and learning from both a mathematical and a psychological or educational point of view. He also strongly promoted the modernisation of mathematics in the classroom, and developed ideas on university lectures for student teachers, which he later consolidated at the beginning of the last century in the three books on elementary mathematics from a higher standpoint.
... In the discussion, we share thoughts about curricular changessomewhat in line with those proposed over a century ago by Felix Klein (see Weigand et al. 2019)that we believe should occur, at least at the secondary school level, to offer students a learning pathway that takes advantage of digital resources such as those discussed. In such a pathway, we see advantages both in terms of addressing and overcoming common student difficulties with certain meanings of function, and in terms of mathematical learning in general, since research has shown that dynamic interactive applets and textbooks can provide an environment for students to communicate in new, meaningful, and inclusive ways. ...
... The German mathematician Felix Klein (1849Klein ( -1925 had a profound impact on the mathematical preparation of prospective teachers for teaching secondary mathematics, particularly in Germany (see Weigand et al., 2019). He presented his ideas for university mathematics lectures for prospective teachers in three books entitled 'Elementary Mathematics from a Higher Standpoint' (Klein, 2016a(Klein, , 2016b(Klein, , 2016c. ...
This paper examines how different approaches in mathematics education conceptualise the relationship between school mathematics and university mathematics. The approaches considered here include: (a) Klein’s elementary mathematics from a higher standpoint; (b) Shulman’s transformation of disciplinary subject matter into subject matter for teaching; and (c) Chevallard’s didactic transposition of scholarly knowledge into knowledge to be taught. Similarities and contrasts between these three approaches are discussed in terms of how they frame the relationship between the academic discipline and the school subject, and to what extent they problematise the reliance and bias towards the academic discipline. The institutional position implicit in the three approaches is then examined in order to open up new ways of thinking about the relationship between school mathematics and university mathematics.
... Esta creación fue, de hecho, el resultado de una propuesta de David Eugene Smith, historiador de las matemáticas y profesor de la universidad de Columbia, documentada en 1905 en la nueva revista l'Enseignement Mathématique, fundada en 1899 por henri Fehr y Charles-Ange laisant, que se convertiría en el órgano oficial de la CIEM (Coray, Furinghetti, Gispert, hodgson y Schubring, 2003). la misión encomendada a la CIEM, cuya presidencia fue confiada al matemático alemán Felix Klein, él mismo muy implicado en cuestiones de educación y formación de profesores (véase Weigand, McCallum, Menghini, Neubrand y Schubring, 2019), hizo un estudio global del progreso de la Educación Matemática en las diferentes naciones. En respuesta a ello, se establecieron comités nacionales en los 18 países miembros y 15 países asociados, entre los que figuraban, para América latina, Argentina, Brasil, Chile, México y Perú. ...
Este artículo trata del desarrollo internacional de la investigación didáctica en matemáticas. Después de recordar la relación especial que existe entre las matemáticas y su enseñanza, evoca en primer lugar la emergencia de la didáctica de las matemáticas como un campo de investigación específico, en un contexto marcado por la reforma de las matemáticas modernas y la influencia de la epistemología piagetiana. A continuación, se examina la evolución de este campo de investigación, centrándose en algunas tendencias globales que trascienden su diversidad inherente, antes de abordar más específicamente la cuestión de las relaciones entre centros y periferias, y destacar la lenta pero real evolución hacia una didáctica de las matemáticas más auténticamente internacional.
Although rarely appreciated, the collaboration that brought Felix Klein and Sophus Lie together in 1869 had mainly to do with their common interests in the new field of line geometry. As mathematicians, Klein and Lie identified with the latest currents in geometry. Not long before, Klein’s mentor Julius Plücker launched the study of first- and second-degree line complexes, which provided much inspiration for Klein and Lie, though both were busy exploring a broad range of problems and theories. Klein used invariant theory and other algebraic methods to study the properties of line complexes, whereas Lie set his eyes on those aspects related to analysis and differential equations. Much later, historians and mathematicians came to treat the collaboration between Klein and Lie as a famous early chapter in the history of transformation groups, a development often identified with Klein’s “Erlangen Program” from 1872. The present detailed account of their joint work and mutual interests provides a very different picture of their early research, which had relatively little to do with group theory. This essay shows how the geometrical interests of Klein and Lie reflected contemporary trends by focusing on the central importance of quartic surfaces in line geometry.
Stephen Timoshenko was the author of numerous textbooks on various aspects of applied mechanics, and through them contributed to its rapid development in the world. As always, the community is also interested in the personal side of the scientists. This paper is devoted to a single statement made by Stephen Timoshenko in his autobiographical book. Specifically, it concerns Timoshenko's at least partial “explanation” of the Holocaust that took place during WWII. Whereas Timoshenko's statement constituted, to the present writer, open antisemitism, it does not seem to disturb many researchers who continue to honor him in various ways. “Nonsense is nonsense, but the history of nonsense is scholarship,” according to Saul Lieberman. Hence the present article presents an investigation of Timoshenko's nonsensical and overly antisemitic statement. This author corresponded with a number of historians of science all repudiating Timoshenko's assertions. Most importantly, the correspondents shed additional vivid light on the Holocaust as experienced by Jewish scientists.
The aim of the article is to discuss how university mathematics courses can become more relevant for pre-service teachers (PSTs). We therefore introduce a concept we call mathematical orientation and develop an analytical framework based on Felix Klein’s pervasive approach to elementary mathematics from a higher standpoint and the work of the practical philosopher Stegmaier on orientation. This framework is used to analyze the mathematical orientation of PSTs on two mathematics courses in Norway and Switzerland. In a qualitative empirical approach, we analyzed 85 reflections written by PSTs on the topic of decimal expansions and reconstructed aspects of their individual mathematical orientation. Our results show that mathematical orientation varies across PSTs to a certain extent. In particular, different points of reference, perspectives, standpoints, and types of links between the mathematical content and teaching situations could be identified. We therefore propose that mathematical orientation can be considered a key variable in addressing the mathematical needs of PSTs in university courses.
For secondary mathematics teachers, it is important that their mathematical coursework helps deepen their understanding of the school mathematics they will teach. That is, making connections between advanced and secondary mathematics is vital for practicing and prospective teachers (PPTs). However, forming these connections poses significant mathematical hurdles. In this chapter, I explore the mathematical challenges that arise when PPTs are asked to make connections by recognizing ideas in advanced mathematics as being an instance of an idea studied in secondary mathematics. In particular, I look at the mathematical challenges faced by two PPTs as they tried to reconcile the definition of a binary operation in abstract algebra (i.e., ∗ : A × A → A) in terms of it being a function – something studied in secondary school. In this example, mathematical challenge is evident through the conceptual stages and shifts these two PPTs went through as they came to understand a binary operation as a function itself. I use this example to ground the discussion of mathematical challenges faced, more broadly, as PPTs develop connections from their advanced mathematical coursework. I also elaborate on the purposes such connections might serve, and why, for PPTs, these connections merit the mathematical challenges encountered to develop them.KeywordsMathematical challengeConnectionsSecondary teacher educationFunctionsBinary operationsAdvanced mathematics
One of the challenges of university mathematics courses in secondary teacher preparation is incorporating pedagogical discussions. The focus in a mathematics course is—and should be—on mathematics. But research also suggests that without addressing pedagogical implications these content courses are not meaningful to secondary teachers’ future classroom practice. The thrust of this paper is exploring ideas for how to leverage mathematical practice in university mathematics courses—and, in particular, what have been described as Pedagogical Mathematical Practices (PMPs). The paper reports on a study of (n = 10) pre- and in-service mathematics teachers that explored the viability of the PMP construct, with the intent of specifying particular PMPs. Drawing on interviews with teacher participants who had recent experiences in an inquiry-oriented discrete mathematics course, the study reports on the ways in which they identified a set of mathematical practices as being productive pedagogically. The study contributes a teacher-perspective on the construct of PMPs, including the identification of four PMPs from the study data: explicit visualization; multiple approaches; concrete exemplification; and informal justification. Implications for their potential use in university mathematics courses with regard to teacher education are discussed.
In Section 1.4, I summarized the situation before 1914 like this: “In the last three decades before World War I, attention to national distinctions and feelings of national pride or imperial supremacy were extremely common, but by and large they peacefully coexisted—if I may put it like this—with increasing contact and collaboration among scientists from different empires or countries.”
The 1930s may well be the most difficult decade of the twentieth century to come to terms with. Given what humanity was about to be led into, i.e. World War II, one may be tempted to consider it a “morbid age”.
Todays International Mathematical Union (IMU) derives its greatest visibility among mathematicians world wide from the International Congresses. Its very foundation was an integral part of the mounting of the first postwar ICM at Harvard in 1950. It is via the experience of the quadrennial ICMs and the published traces they leave behind that an image of mathematics continues to be framed and projected for the mathematical community at large, and for the whole world to see. In this final chapter we present a data-based study of how the most exquisite layer of this image has evolved over the past seventy years.
Turning now to a period which coincides with my own life, I cannot deliver a historical account; the minimum distance to many of the events studied is missing. This is one of the reasons why the style of this third and final part of the book will become increasingly different from that of the preceding chapters.
Today the International Commission for Mathematical Instruction (ICMI) is by far the most prolific of the three commissions of the IMU. But treating ICMI just as a commission of the current IMU misses its peculiar historic significance.
The US-based activities that we briefly recall in this chapter are an important reminder, within the crystalline sphere of scientific endeavors, of the truly global transformation that World War I had wrought.
In each of the three parts of this book we present a characteristic feature that highlights the state of mathematics in the corresponding period.
The foundation of the IAA was discussed in Section 1.3.2. Its structural peculiarities were highlighted by remarks made by one of its architects, the physicist Arthur Schuster, Fellow of the Royal Society. The upshot was that, as an international science consortium, the IAA was somehow suspended in mid-air between nations and disciplines because the member Academies each had their own international network of corresponding fellows, independently of their being part of the IAA, and each one of them tried to represent all disciplines as well as possible. And yet, taken together, these Academies did not count among their individual members all the relevant researchers, even for some of the most prestigious international projects.
This book is about Science International , and what it meant for mathematics over the past two centuries. The emergence of the particular kind of Science International at work in the twentieth century hinges on the concept of Nation States, i.e., states which claim to be political and cultural units at the same time. Science International thus originated from European, especially continental European developments in the nineteenth century.
The delicate dualism of nationalist and internationalist orientations that characterized the pre-war period, as described in Section 1.4 above, came to an abrupt end once war was declared, at the beginning of August 1914. The pride and interest of the nation left no room for ‘internationalist jokes’, as Debussy put it. We will first, in Sections 3.1 and 3.2, document this new state of mind with respect to science, and mathematics in particular. World War I was not only marked by this surge of exclusive nationalism; it was also a modern, technical war which mobilized scientists, including mathematicians; this is the subject of Section 3.3. A short final section considers the (im)possibility of organizing international congresses during the war.
In the nineteenth and twentieth centuries many mathematicians referred to intuition as the indispensable research tool for obtaining new results. In this essay we will analyse a group of mathematicians (Felix Klein, Henri Poincaré, Ludwig Bieberbach, Arend Heyting) who interacted with Luitzen Egbertus Jan Brouwer (the father of the intuitionist foundational school) in order to compare their conceptions of intuition. We will see how to the same word “intuition” (in German Anschauung ) very different meanings corresponded: they varied from geometrical vision, to a unitary view of a demonstration, to the perception of time, to the faculty (shared by everybody) of considering concepts that habitually occur in our thinking separately. Furthermore, we will discover that these different meanings had a philosophical, very relevant counterside: they passed from a racial characterization of mathematics to a pluralistic view of it.
This paper is meant to give some kind of manual instruction for the use of audience response systems for performing peer instruction in class. The different aspects one has to take into account when trying to implement this method are presented. In all of the sections, additional links and references are provided. The author reports on and reflects on his individual decisions when conducting peer teaching in an analysis course at the university. Finally, the results of the corresponding evaluation are presented and discussed.
Tatiana Alexeyevna Afanassjewa (from 1904 Tatiana Ehrenfest-Afanassjewa) was a mathematician, a physicist, and a teacher. All three of these vocations come together in her philosophy of geometry, which bases a novel approach to the teaching of geometry on her understanding of the proper roles of intuition and logical reasoning in geometry, grounded in our experience of concrete objects occupying physical space. Having been a student at Göttingen during the time of its greatest flowering as a centre of mathematical research, and a member of the physics community during the revolutionary period from Einstein’s annus mirabilis of 1905, she was close to the centre of some of the most exciting developments in science of her time. Since early on she was also deeply invested in teaching, and in developing new and better ways to communicate her subject to her students. Afanassjewa’s reflections on the teaching of geometry are thus those of a mathematician and a theoretical physicist who was passionate about scientific discussion and teaching: her ideas originate in her own experience as a student, researcher, and teacher, and in the debates with her scientific contemporaries—debates in which she played an active and important role.
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.KeywordsFunctional thinking
Meraner Lehrplan
Principle of movementMathematical mental representationsFundamental ideas
Recent research on the professional competencies of mathematics teachers, which has been carried out during the last decade, is characterized by different theoretical approaches on the conceptualization and evaluation of teachers’ professional competencies, namely cognitive versus situated approaches. Building on the international IEA Teacher Education and Development Study in Mathematics (TEDS-M) and its follow-up study, TEDS-FU, the paper compares cognitive and situated approaches on professional competencies of teachers. In TEDS-FU, the cognitive oriented framework of TEDS-M has been enriched by a situated orientation including the novice-expert framework and the noticing concept as theoretical approaches on the analyses of classroom situations. Correspondingly, the evaluation instruments were extended by using video vignettes for assessing teachers’ perception, interpretation, and decision-making competencies in addition to cognitive oriented knowledge tests. The paper discusses the different kinds of theoretical frameworks and the consequences for the evaluation methods, the strengths, and weaknesses of both approaches. Furthermore, connecting the results of TEDS-FU with TEDS-M allows comprehensive insight into the structure and development of the professional competencies of mathematics teachers, the complex interplay between the different facets of teachers’ competencies, and the high relevance of teaching practice for the development of these competencies. The analyses show on the one hand that both approaches—cognitive and situated—are needed for a comprehensive description of teachers’ professional competencies. On the other hand, it is shown that both approaches can be integrated in a productive way. The paper closes with prospects on further studies coming even closer to the real classroom situation.
How Humans Learn to Think Mathematically describes the development of mathematical thinking from the young child to the sophisticated adult. Professor David Tall reveals the reasons why mathematical concepts that make sense in one context may become problematic in another. For example, a child's experience of whole number arithmetic successively affects subsequent understanding of fractions, negative numbers, algebra, and the introduction of definitions and proof. Tall's explanations for these developments are accessible to a general audience while encouraging specialists to relate their areas of expertise to the full range of mathematical thinking. The book offers a comprehensive framework for understanding mathematical growth, from practical beginnings through theoretical developments, to the continuing evolution of mathematical thinking at the highest level.
Intuition is often regarded as essential in the learning of geometry, but questions remain about how we might effectively develop students’ such skills. This paper provides some results from analyses of innovative geometry teaching in the early part of the 20th century, a time when significant efforts were being made to improve the teaching and learning of geometry. As examples, we examine the tasks for students that can be found in Treutlein’s “Geometrical Intuitive Instruction" (Germany) and Godfrey’s geometry textbook (England). The analyses suggest that educators at that time attempted to develop students’ intuitive skills through various practical tasks such as drawing, measurement, and imagining and manipulating figures, which could be useful for current geometry teaching. We also identify different approaches taken to the development mathematics teaching in Germany and England.
At the beginning of the 20th century, the industrial development of many European countries led to a necessary reform of education; its purpose was to ease the entrance of the younger generations into the world of work. As a result, schools were required to provide more technical skills, and to also educate the new ruling class. The teaching of secondary school mathematics in schools with a ‘cultural’ curriculum was therefore assigned the additional task of providing modern tools and methods that could be used in mathematical applications. From this perspective, the secondary curriculum had to aim towards a better integration with its counterpart in higher education, and therefore towards a redefinition of the topics studied. The first official reform of school mathematics programs was in France in 1902, with the creation of the modern curriculum; this was closely followed by the German reform in 1905, and then by those of most other European countries. Despite the interest shown by many Italian mathematicians, and also frequently shown by the Institutions, the Italian school system did not register these changes. Our aim is to identify the causes and the consequences of the Italian attitude during this time. With this in mind, we will retrace the development of the teaching of mathematics in Italian schools that allowed access to the University system, from the unification of Italy to the middle of the 20th century. We will describe the evolution of the programs of the Ginnasio-Liceo, and the development of the physics-mathematics section of the Istituto Tecnico, the school which played the role of a ‘cultural’ school of scientific curriculum until the establishment of the Liceo Scientifico in 1923. We will discuss the debates and the proposals that surrounded the teaching of mathematics at the turn of the 20th century, and, finally, the programs of the Liceo Scientifico and their following modifications (see Marchi & Menghini, 2011). As a result, it will be clear that the setup that inspired the first school programs of the newly unified Italy has remained a dominant feature in Italian schools. It was these programs which led to the position of Italy in the international arena at the beginning of the 20th century, the programs of the Liceo Scientifico of 1923, and the little attention given to all the modifications which aimed to establish a modern model of scientific culture in this same school.
IMPACT (Interweaving Mathematics Pedagogy and Content for Teaching) is an exciting new series of texts for teacher education which aims to advance the learning and teaching of mathematics by integrating mathematics content with the broader research and theoretical base of mathematics education. The Learning and Teaching of Geometry in Secondary Schools reviews past and present research on the teaching and learning of geometry in secondary schools and proposes an approach for design research on secondary geometry instruction. Areas covered include: teaching and learning secondary geometry through history; the representations of geometric figures; students' cognition in geometry; teacher knowledge, practice and, beliefs; teaching strategies, instructional improvement, and classroom interventions; research designs and problems for secondary geometry. Drawing on a team of international authors, this new text will be essential reading for experienced teachers of mathematics, graduate students, curriculum developers, researchers, and all those interested in exploring students' study of geometry in secondary schools. © 2017 Patricio Herbst, Taro Fujita, Stefan Halverscheid, and Michael Weiss. All rights reserved.
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 II presents a paradigmatic realisation of Klein’s approach of elementarisation for teacher education. It is shown how the various geometries, elaborated particularly since the beginning of the 19th century, are revealed as becoming unified in a new restructured geometry. As Klein put it: “Projective geometry is all geometry”. Non-Euclidean geometry proves to constitute a part of this unifying process. The teaching of geometry is discussed in a separate chapter, which provides moreover important information on the history of geometry teaching and an international comparison.
About the Author
Felix Klein (1849-1925) was a leading German mathematician whose research interests included group theory, complex analysis, and geometry. His work influenced many areas of mathematics and related subjects, ranging from mathematical physics to mathematical didactics. To this day, Felix Klein is considered one of the most important mathematicians of the 19th century.
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.
In Volume III, Klein explores the relationship between precision and approximation mathematics. He crosses the various fields of mathematics – from functions in one and two variables to practical geometry to space curves and surfaces – underlining the relation between the exactness of the idealised concepts and the approximations to be considered in applications. Logical procedures are confronted with the way in which concepts arise starting from observations. It is a comparison between properties pertaining only to the theoretical field of abstract mathematics and properties that can be grasped by intuition. The final part, which concerns gestalt relations of curves and surfaces, shows Klein to be the master of the art of description of geometric forms.
About the author:<
Felix Klein (1849-1925) was a leading German mathematician whose research interests included group theory, complex analysis, and geometry. His work influenced many areas of mathematics and related subjects, ranging from mathematical physics to mathematical didactics.
To this day, Felix Klein is considered one of the most important mathematicians of the 19th century.
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.
About the Author:
Felix Klein (1849-1925) was a leading German mathematician whose research interests included group theory, complex analysis, and geometry. His work influenced many areas of mathematics and related subjects, ranging from mathematical physics to mathematical didactics.
To this day, Felix Klein is considered one of the most important mathematicians of the 19th century.
This paper discusses Aspects and “Grundvorstellungen” in the development of the concepts of derivative and integral, which are considered central to the teaching of calculus in senior high school. We focus on perspectives that are relevant when these concepts are first introduced. In the context of a subject-matter didactical debate, the ideas are separated into two classes: firstly, mathematically motivated aspects, such as the limit of difference quotients or local linearization within the concept of derivative, as well as the product sum, antiderivative, and measure aspects of integration; secondly, the “Grundvorstellungen” associated with the concepts of derivative and integral. We regard finding a comprehensive description of Aspects and “Grundvorstellungen” an important objective of subject matter didactics. This description should clarify both the differences and the relationships between these perspectives, including mathematically motivated Aspects and “Grundvorstellungen” that are central to the students’ perspective. The primary objectives of this paper include a specification of the concepts of Aspects and “Grundvorstellungen” within the context of differentiation and integration, and a discussion of the relationships between the Aspects and “Grundvorstellungen” associated with the concepts of derivative and integral. We first present the characteristic properties of Aspects and “Grundvorstellungen”, including an account of related concepts and the current state of research. Aspects and “Grundvorstellungen” are analyzed, based on a subject-matter didactical analysis of the concepts of derivative and integral. We conclude with an account of how these insights can be beneficially exploited for introducing differentiation and integration in real-life environments, within the framework of a theory of concept understanding and subject matter didactics.
In the early 20th century, a demand arose for a course of university studies considering the special needs of future teachers. One of the well-known representatives
of this movement is Felix Klein. Inter alia, he held lectures in Elementary mathematics from a Higher Standpoint. In the work at hand, I will present results of my
Ph.D. theses, in which the lecture’s manuscripts are analyzed concerning the underlying intention and inner structure.
The article focuses on the didactical perspective, that among other characteristics specifies Klein’s understanding of an advanced standpoint. The comparison with
modern concepts of mathematics didactics may give an indication for a provound validation and possible adaption of Klein’s concept.
This chapter discusses the role of Felix Klein in mamthematics. During the past decades, historiography of the sciences has markedly changed: formerly largely monolithic concentrating on the history of ideas, it has become rather multi-faceted. In the aftermath of Thomas Kuhn's Structure of Scientific Revolutions (1962), the focus has shifted— for almost all disciplines—from isolated scientific heroes to the larger communities that produce and communicate knowledge. The historiography of mathematics has also participated in this broadening of perspectives. Symptomatic of this trend has been a growing interest in styles of mathematics peculiar to certain periods, cultures, or scientific schools. The theory provides a frame-work for studying the relations of subsystems to their environment as a characteristic pattern within modern, functionally differentiated societies. The chapter also discusses about the professional careers and disciplinary orientations.
Dass die Integration eines historisch-genetischen Zugangs zur Mathematik insgesamt förderlich für das Lehramtsstudium sein kann, ist mittlerweile unstrittig, wenn auch die praktische Realisierung allzu häufig an mangelnden Ressourcen scheitert. Vor dem Hintergrund der Erfahrungen aus dem Siegener Lehramtsstudium sollen mögliche Funktionen einer solchen mathematikhistorischen Komponente diskutiert werden.
In this study we investigated above-average high school calculus students from Japan and the United States in order to determine any differences in their conceptual understanding of calculus and their ability to use algebra to solve traditional calculus problems. We examined and interviewed 18 Calculus BC students in the United States and 26 Suugaku 3 (calculus) students in Japan. Each student completed two parts of a written examination. The first part (Part I) consisted of problems emphasizing conceptual understanding but requiring little or no algebraic computation. Problems on the second part (Part II) required sound algebraic skills in addition to good conceptual understanding. Following the examination, we interviewed each student in order to assess their mathematical and educational background, their college and career plans, their thinking on the examination problems, their understanding of concepts, and their computational and reasoning skills. We found little difference in the conceptual understanding of calculus between the two groups of students, but the Japanese students demonstrated much stronger algebra skills than their American counterparts.
The aim of the paper is to show on hand of the curricula and the textbooks for the lower gymnasium the idea of intuitive geometry that was growing up in Italy between the end of the 19th century and the beginning of the 20th century, in comparison with different interpretations in other countries. We compare, on some specific points, the first texts on intuitive geometry, those of Veronese and Frattini, and analyze in particular the use of proofs, of experimental checks, of geometric transformations,... The history of the curricula and the textbooks continues, observing how the latter shift towards a rationalization of the geometry for the level corresponding to what is now called middle school, till to the inversion of the trend due to Ugo Arnaldi and Emma Castelnuovo. In the conclusion the different points of view are taken up again and characterizations of intuitive geometry are proposed which are valid also nowadays.
