Mathematics In Germany Research Articles

Comparison of the Abitur examination in mathematics in Germany before and after reunification in 1990

In Germany, the Abitur is the highest qualification granted at the end of secondary education after 12 or 13 years of schooling; it provides a general university entrance qualification. Traditionally, written and oral examinations are required to obtain the Abitur. Until 1990, there were mainly decentralized examinations in mathematics in West Germany (Federal Republic of Germany [FRG]), which were taken after 13 school years, and a centralized examination that students took after 12 years of school in East Germany (German Democratic Republic [GDR]). In the unified Germany, examinations are now increasingly set by the 16 individual federal states. This paper has a special focus on changes and permanent features in the written Abitur examination in mathematics in Germany in the context of the social changes caused by the German reunification in 1990. These changes since 1990 are described with regard to the initial situation and framing conditions for the written Abitur examination. Two time periods are considered: (1) the examination situation in the FRG and GDR before 1990 and (2) the changes in the five eastern German federal states (former GDR) under the system change and accession to the FRG after 1990 but before the PISA shock.

Jonas Mašiotas the mathematics teacher

Mathematics teacher Jonas Mašiotas (1897–1953) was the son of Pranas Mašiotas (1863–1940) – the famous Lithuania author of mathematical coursebooks and didactic articles as well as the a creator of books for children. JonasMašiotas studied mathematics in Germany and Switzerland. Prior to the studies, he was engaged as a teacher in Marijampol˙e, after completion of the studies – in Kaunas and during the World War II – in Vilkija. He published 2 articles in didactics of mathematics and 5 coursebooks (including one that consisted of 7 parts), edited and published the coursebook in trigonometry prepared by his father P. Mašiotas. He was one of the enthusiasts for introducing the concept of a function into the school curriculum in mathematics.

“Knowledge gained by experience”: Olaus Henrici—engineer, geometer and maker of mathematical models

The German mathematician Olaus Henrici,1 who was born in Denmark in 1840, studied engineering and mathematics in Germany before making his career in London. Initially, and for only a short time, he worked in an engineering business. He subsequently took on academic positions, first at University College London and then, from 1884, at the newly formed Central Institution (later Central Technical College) where he established a Laboratory of Mechanics. While at University College he became an active promoter of pure geometry and a producer of models of surfaces. In this paper I explore the geometrical side of Henrici's work, setting it into the context of his career and arguing that his interdisciplinary background was a key factor in his success as a creator of models.

Richard von Mises’ work for ZAMM until his emigration in 1933 and glimpses of the later history of ZAMM

Richard von Mises’ work for ZAMM until his emigration in 1933 and glimpses of the later history of ZAMM

“The joy that engineers and mathematicians have come together.”1

“The joy that engineers and mathematicians have come together.”¹

Using Reading Strategy Training to Foster Students’ Mathematical Modelling Competencies: Results of a Quasi-Experimental Control Trial

Ever since the national standards for teaching and learning mathematics in Germany were published, investigation of ways to support students’ acquisition of mathematical competencies has increased. Results of these studies have been of special interest in empirical educational research. In this context, several recent studies have focused on the enhancement of students’ reading comprehension skills as a means of supporting students’ development of subject-specific competencies. Taking into account previous research, the empirical research project FaSaF investigated to what extent students’ mathematical modelling competencies can be fostered using a 15-week training in reading strategy. Treatment effects have been investigated in three conditions: EC A, integrated reading strategy training; EC B, separate reading strategy training; and EC C, no reading strategy training. Data from German secondary school students (N = 380) who were about 13 years old were analyzed. The results indicate that students who have participated in reading strategy training experience an increase in mathematical modelling competencies but that the same increase can also be observed in students who have not participated in reading strategy training. Thus, the issue of fostering the acquisition of modelling competencies using reading strategy training is still open for debate.

SPACE PHILOSOPHY

INSPIRED by a dynamist Naturphilosophie and looking for a mathematics of the natura naturans, the founders of modern mathematics in Germany made some lasting contributions in the attempt to go beyond perceptible space (“Anschauungsraum”). Hermann Grassmann’s extension theory, Johann Benedict Listing’s topology, Bernhard Riemann’s non-Euclidean manifold theory, Carl Gustav Jacob Jacobi’s approach to non-mechanistic theory and last but not least Georg Cantor’s transfinite set theory were all influenced by the tradition of Naturphilosophie. One central motivation for the new mathematics was to decipher the so-called “inner side of nature” which was thought to be invisible but nonetheless physical. As a result, mathematics was understood not only in terms of quantity but of morphogenetic quality and as a tool to recognize the intrinsic possibilities of structuring and organizing nature. This essay focuses on the philosophical viewpoints of Christian Samuel Weiss, Justus and Hermann Grassmann, Bernhard Riemann and their reception of Schelling’s philosophy. Riemann deconstructed empiricist theories of space so as to glean the fundamental constitution of space. One result was that a given, fixed and static metric is not necessary for the understanding of space. The underlying topological substratum (Hans Freudenthal’s term) has no specific metric but is capable of exhibiting any possible metric. Riemann’s claims are compared with Schelling’s theory of space in his “Darstellung des Naturprocesses” (1843–44). My thesis is that Schelling’s philosophy of the emergence of space can be read in Riemannian terms and that it is an interesting model in the context of modern questions.

Preface Issue 2-2015

Friedrich Hirzebruch, who died in 2012 at the age of 84, was one of the most important German mathematicians of the second half of the twentieth century. This applies not only to his mathematical achievements –mainly in topology, geometry and number theory– and his huge school of at least 52 PhD students. He also made enormous efforts and had great success in reshaping mathematics in Germany after the Second World War. The Sonderforschungsbereich Theoretische Mathematik, the Arbeitstagung and the Max-Planck-Institut fur Mathematik played an important role in making Germany once again a place which mathematicians from all over the world would like to visit. The Deutsche Mathematiker-Vereinigung (DMV) owes a particularly deep gratitude to Friedrich Hirzebruch who served twice as its chairman, in 1962 and 1990. In 1962, one year after the wall was built, the Gesellschaft fur Mathematik der DDR was founded and urged to branch off from the DMV. One purpose of his second election in 1990 was that he should chair the reunification of both societies and to celebrate at the same time the 100th anniversary of the DMV. In the 28 years in between he ensured that the ties to the East were not severed. One of his best known former PhD-students, Don Zagier, gives a fairly detailed picture of all these and many more facets of Friedrich Hirzebruch’s life within mathematics. Often, pictures are available only in a corrupted form and one has to try to reconstruct the original ones. One may think for instance of a real valued function describing the gray-value of the picture at the actual position. If one assumes that the corruption mechanism (the corruption mapping) is known up to some “noise” one has to seek a preimage of the corrupted picture under this mapping. In principle one could apply the corruption mapping to all possible pictures and search for those whose images minimise a suitable distance functional from the given corrupted picture. Due to the “bad” properties of the corruption mapping this problem is in this form not well posed. In order to resolve this issue, one adds a regularisation term containing first and second order derivatives of the unknown preimage to the distance functional. Gabriele Steidl explains in her survey article on “Combined first and second order variational approaches for image processing” the recent development of such techniques, and illustrates them by two rather different applications. Functions of bounded variation and related function spaces are fundamental for Gabriele Steidl’s survey article and are also the topic of a recent book by Jurgen Appell, Jozef Banas, and Nelson Jose Merentes Diaz, which is reviewed by Michel Willem. The second review concerns a book by Luis Barreira and Yakov Pesin on smooth ergodic theory and is provided by Gerhard Knieper.

Mathematics teaching before and after the Meiji Restoration

This paper outlines mathematical education before the Meiji Restoration, and how it changed as a result. The Meiji Restoration in 1868 completely changed the social structure of Japan. In the Edo period (1600–1868) Japan was divided into domains (han) governed by local lords (daimyo). Tokugawa Shogunate supervised local lords and governed Japan indirectly. In the Edo period there were no wars for more than two centuries and many people participated in cultural activities. Japanese mathematics developed in its own way under the influence of old Chinese mathematics. Japan also had a good education system so that the literacy rate was quite high. Each domain had its own school for samurai but mainly education was provided privately. Private schools for elementary education were called terakoya, in which mainly reading and writing and often arithmetic by the soroban (Japanese abacus) were taught. In the Edo period the soroban (abacus) was the only tool for computation and Arabic numerals were not used. The Meiji government was eager to establish a modern centralized state in which education played a key role. In 1872 the Ministry of Education declared the Education Order, whereby in elementary schools only western mathematics should be taught and the soroban should not be used. But almost all teachers only knew Japanese traditional mathematics “wasan” so they insisted on using the soroban. This was the starting point of a long dispute on the soroban in elementary education in Japan. Two Japanese mathematicians, KIKUCHI Dairoku and FUJISAWA Rikitaro, played a central role in the modernization of mathematical education in Japan. KIKUCHI studied mathematics in England and brought back English synthetic geometry to Japan. FUJISAWA was a student of KIKUCHI at the Imperial University and studied mathematics in Germany. He was the first Japanese mathematician to make a contribution to original research in the modern sense. He published a book on mathematical education in elementary school, which built the foundation of mathematical education in Japan.

Mini-Workshop: History of Mathematics in Germany, 1920-1960

The Mini-Workshop dealt with the history of mathematics in Germany between 1920 and 1960, with a particular focus on the social history of mathematics. For the period under discussion research in the history of mathematics is faced with some fundamental problems, which cannot be addressed by individuals. Consequently the Mini-Workshop’s aim was to develop future perspectives and methods for research and ways to organise new research projects. Among the topics addressed during the Mini-Workshop were the international relations of mathematicians in Germany before, during and after World War II; the prosopography of mathematicians in Germany from before 1933 into the 1950s; the role of mathematics as a key technology in World War II; and the professional policies from the 1920s to the 1950s.

Documents on the mathematical education of Edmund Külp (1800–1862), the mathematics teacher of Georg Cantor

Despite some studies by the historian Wilhelm Lorey, Edmund Kulp is rather unknown today. His role in the development of mathematics and mathematics teaching in the nineteenth century, however, deserves closer attention. Having been the director of the hohere Gewerbe—und Realschule in Darmstadt, he can be counted among the founders of the Technische Hochschule Darmstadt. Moreover, he had been, still at the Realschule, the mathematics teacher of Georg Cantor. Recently detected documents concerning Kulp’s mathematical formation in Brussels by A. Quetelet permit revealing insights into the evolution of Kulp’s mathematical ideas and of his views on the context of mathematics in Germany. The contribution presents extracts from these documents (in French) and analyses them. Furthermore, the paper discusses possible influences exerted by Kulp on Cantor.

Mathematical world views and achievement in advanced mathematics in Germany: findings from TIMSS population 3

Mathematical world views and achievement in advanced mathematics in Germany: findings from TIMSS population 3

Episodes in the Berlin-Göttingen Rivalry, 1870-1930

One of the more striking features in the development of higher mathematics at the German universities during the nineteenth century was the prominent role played by various rival centers. Among these, Berlin and Gottingen stood out as the two leading institutions for the study of research-level mathematics. By the 1870s they were attracting an impressive array of aspiring talent not only from within the German states but also from numerous other countries as well. The rivalry between these two dynamos has long been legendary, yet little has been written about the sources of the conflicts that arose or the substantive issues behind them. Here I hope to shed some light on this theme by recalling some episodes that tell us a good deal about the competing forces that animated these two centers. Most of the information I will draw on concerns events from the last three decades of the nineteenth century. But to understand these it will be helpful to begin with a few remarks about the overall development of mathematics in Germany, so I will proceed from the general to the specific. In fact, we can gain an overview of several of the more famous names in German mathematics simply by listing some of the better-known figures who held academic positions in Gottingen or Berlin. As an added bonus, this leads to a very useful tripartite periodization:

Hilberts Ruf nach Bern

Attempts to find a successor for the late Berne professor of mathematics Johann Heinrich Graf in winter and spring 1919 proved fruitless when, unexpectedly, the renowned Gottingen mathematician David Hilbert intimated that he himself would, under certain conditions, accept a call to Berne. Subsequent negotiations between Hilbert and the Bernese and Prussian authorities mainly concerned the question of salary, with offers of unusually high compensation both in Berne and in Gottingen. Although Hilbert eventually declined for financial reasons, it is argued that his consi-derations may well have been rather complex and reflect both his own situation in Gottingen and the general state of affairs of university mathe-matics in Germany and Switzerland after the First World War.

Mathematician or pedagogue? On the education of teachers of mathematics in Germany

(1992). Mathematician or pedagogue? On the education of teachers of mathematics in Germany. The Curriculum Journal: Vol. 3, No. 3, pp. 291-309.

Anmerkungen zur Rechendidaktik von 1945 bis 1967 aus persönlicher Sicht

This is a report on the history of pedagogy of mathematics in Germany from 1945 to 1967 containing a lot of autobiographical remarks.

Alchemia Adama Kochańskiego SJ

The subject of this paper is the issue of alchemy in the scientific activity of Adam A. Kochanski. Polish Jesuit, Adam Kochanski S J (1631-1700) was an unusual scholar in the Polish science of the 17th century. Kochanski was an excellent mathematician, one of his greatest achievement in mathematics is the solution of the famous squaring of the circle. He published his mathematical works in "Acta Eruditorium", one of the first scientific journals in 17th century Europe. Kochanski was a professor of mathematics in Germany, Italy, Prague, Olomouc, Wroclaw and Warszawa. An interesting part Kochanski's activity is his correspondence with many scholars, mathematicans and philosophers, for example the exchange of letters with German philosopher and mathematican Gottfried W. Leibniz.

Meadows on China: A Centennial Review

In november 1841 a young Englishman named Thomas Taylor Meadows, who had spent some three years studying chemistry and mathematics in Germany, happened to attend Professor Karl Friedrich Neumann's lectures on the Chinese language at the University of Munich. The young man “almost immediately gave up every other study” and prepared for the British service in China. Hardly more than a year later, at the beginning of 1843, Meadows arrived at the new crown colony of Hongkong. When George Tradescant Lay opened the first British consulate at Canton on July 23, Meadows was Senior Assistant and a year later became Interpreter. This made him, in modern parlance, the chief British intelligence officer at the leading treaty port. After a stretch of seven and one-half years at Canton, during which he reported the beginning in Kwangsi of the great domestic uprising later known as the Taiping Rebellion, Meadows became Interpreter at Shanghai (Jan. 1, 1852), and by the time the Taiping horde erupted down the Yangtze from Wuhan to Nanking in early 1853, he was well prepared to study them at first hand.

Modern Mathematics in Germany

Since I am speaking to you about modern German mathematics, I wish to call your attention to a most important subject, namely to the world which surrounds us, and to our scientific knowledge of its extent in space and time. There can be no doubt about the fact that this world we are living in is not the best of all possible worlds. Financial, industrial, and political dieases, you know them just as well on the other side of the great pond as we do on this side. Nevertheless in one respect we are living just now in a golden age. The world of science is in an excellent state and few eras have seen as important successes of mathematics and physics as ours.

The Function Concept in School Mathematics

There are certain topics connected with the teaching of mathematics which seem to present themselves periodically for review. The aims or objectives of mathematical teaching is one of them ; the foundations of school mathematics is another. I have selected for review to-day one of the fundamental concepts of mathematics, that of functionality.May I begin my discussion with a brief historical survey ? The idea that the function concept should have a prominent, even a dominant, place in the mathematical programme of the schools may be said to have originated with Klein. Others before him, in France, England and America, as well as in Germany, had advocated the inclusion of “ variables ” and “ functions ” in the school programme, but Klein was the first to press the view that “ functional thinking ” should be made the binding or unifying principle of school mathematics. Klein was equally distinguished as a mathematician and as a leader in the pedagogy of mathematics in Germany, but some years elapsed before his views became generally accepted, even in his own country.