Gert Schubring

History of mathematics for trainee teachers

The movement to integrate mathematics history into the training of future teachers, and into the in-service training of current teachers, has been a theme of international concern over much of the last century. Examples of current practice from many countries, for training teachers at all levels, enable us to begin to learn lessons and press ahead both with adopting good practices and also putting continued research efforts into assessing the effects.

Societal, Structural, and Conceptual Changes in Mathematics Teaching: Reform Processes in France and Germany over the Twentieth Century and the International Dynamics

This paper studies the evolution of mathematics teaching in France and Germany from 1900 to about 1980. These two countries were leading in the processes of international modernization. We investigate the similarities and differences during the various periods, which showed to constitute significant time units and this in a remarkably parallel manner for the two countries. We argue that the processes of reform concerning the teaching of this major school subject are not understandable from within mathematics education or even within the school system. Rather, the evolution of the processes of reform prove to be intimately tied to changing conceptions of modernity according to respective social and cultural values and to changing epistemological conceptions of mathematics. It is particularly novel that we show the key impact of the changing social status of primary schooling for these modernization processes.

Researching into the History of Teaching and Learning Mathematics: the State of the Art

First of all, we should address the question: Why study the history of mathematics instruction? Evidently, such research is not undertaken for pure curiosity. Since the present situation is the product of a historical process, the evolution informs the mathematics educator regarding political, social and cultural constraints to improving mathematics instruction. Practically all the research questions in mathematics education have a historical dimension that too often, however, remains implicit, or is treated too superficially. Research can be improved by explicit consciousness for the history of teaching and learning mathematics. And, what is probably even more important, the history of mathematics instruction should constitute one of the dimensions of the professional knowledge of mathematics teachers. In order to be able to handle the problems they encounter in their professional life, mathematics teachers should know how their profession emerged historically, how it developed and which types of problems were encountered during this development, and what obstacles had to be overcome for the effective establishment of mathematics teaching. The history of their own profession should, hence, constitute part of what has been called the ‘meta-knowledge’ of mathematics teachers and should therefore constitute an element in the education of mathematics teachers.2 Within the desirable and in some places realized historical component in this teacher training, there should also be

The Cooperation between Hermann and Robert Grassmann on the Foundations of Mathematics

The innovations introduced and the conceptual development established by Hermann Grassmann in his Ausdehnungslehre transcended, as is well known, the conceptual level of contemporaneous mathematical thinking in several respects and in a remarkably profound manner. This great departure from the understanding of mathematical concepts in Grassmann’s time is particularly striking in his introduction of an n-dimensional geometry, i.e. the break with the traditional notion of a three-dimensional space (see Lewis 1996). Even Gaus did not discuss or mention this innovation when responding to Grassmann’s sending of the book to him.1

‘Experimental pedagogy’ in Germany, elaborated for mathematics – a case study in searching the roots of PME

PME, the International Group for the Psychology of Mathematics Education, was founded in 1976, at the Third International Congress on Mathematical Education in Karlsruhe, organised by the International Commission on Mathematics Instruction (ICMI). While PME is thus beyond coming of age and is reflecting its further orientation – due to the present “social turn” – the origins of investigating psychological aspects of mathematics learning have not yet been systematically studied. I am undertaking here a first such approach, concentrating on Germany, where the first pertinent monographs were published in 1913 and 1916. Different endeavours, focussing in particular on the notion of error, merged into the characteristic approach of ‘experimental pedagogy’. Given the key function of ICMI for founding PME, an additional aspect is whether the forerunner of ICMI: the Internationale Mathematische Unterrichtskommission (IMUK), founded in 1908, had an impact upon promoting research into the psychology of mathematics education. The pertinent research was effected by psychology; doing research themselves was still outside the horizon of mathematics educators. Perspectives of future research, in particular comparative ones, are outlined.

Gegenständliche und soziale Momente des Wissens als Kategorien für Untersuchungen zur Geschichte der Mathematik-Didaktik

The article discusses systematical dimensions or categories which might encourage approaches for investigating the historical development of didactics of mathematics. It is proposed to take the relation of the objective (gegenständliche; and the social function of knowledge as foundational basis for such dimensions. Considering didactical principles as the kernel of didactical theories, three examples of didactical principles are discussed regarding this basic relation: the genetic principle, the socratic method, and the principle of ascending from the simple to the complex. The traditionel german Gymnasium having an eminent impact on the development of didactics, the article gives some new approaches to the social function of the gymnasial mathematical knowledge.

Patterns for Studying History of Mathematics: A Case Study of Germany

The history of mathematics teaching and learning presents an already well-established field of interdisciplinary research. What is still necessary, however, is to deepen and improve the methodological approaches to this research. A major issue is constituted by the polarity between the essentially national or regional restriction of the research agendas and the interpretation of the findings in a broader comparative understanding. The polarity and complementarity between the global and the local is used here to unravel general dimensions for the apparently specific cases of mathematics teaching in the German states. Firstly, however, the case of Italy from 1860 is presented to show that a local case is not necessarily a case of a small region and that specific cultural and epistemological visions in a country can effect long-lasting impacts on teaching and learning. Then, a case from pre-modern times in Germany will be analysed, showing the processes driving modernisation. Eventually, the cultural and social divergences between German states in modern times will be used to reveal the role of mathematics education for preparing cultural and technological changes.

CÁLCULO EM MATEMÁTICA: UM ASSUNTO PARA O ENSINO EM GERAL OU ESPECÍFICO PARA O ENSINO TÉCNICO

Nuestro objetivo es analizar la trayectoria del Calculo Diferencial e Integral en la Educacion General, Educacion Secundaria, y en la Educacion Profesional, Educacion Secundaria Modalidad Tecnico Profesional, en Brasil en un estudio de caso. Para realizar esa tarea, desarrollaremos un panorama historico del siglo 20, teniendo como punto de partida la Conferencia Internacional sobre la Ensenanza de Matematicas en Paris, en 1914, la legislacion brasilena y los contextos politicos y economicos que, en algun momento, influenciaron en la inclusion o la extincion del calculo en los programas nacionales; y tambien la trayectoria del calculo en la disciplina de matematicas de la Escuela Tecnica Federal de Sao Paulo hasta su transformacion para el Centro Federal de Educacion Tecnologica, donde fue implantado la educacion secundaria.

120 Jahre Deutsche Mathematiker-Vereinigung: Neue Ergebnisse zu ihrer Geschichte

Das Jubiläum zum hundertjährigen Bestehen der Deutschen Mathematiker-Vereinigung im Jahre 1990 ist von der DMV in intensiver Weise begangen worden. Für ihre Jahrestagung 1990 ist sie zu ihren Ursprüngen nach Bremen zurückgegangen und hat dort die Tagung unter das Rahmenthema des Jubiläums gestellt. Insbesondere ist dort die Festschrift Ein Jahrhundert Mathematik 1890– 1990 vorgelegt worden, in der in ausführlicher Weise einerseits die Entwicklung der Teildisziplinen der Mathematik in diesem Zeitraum beschrieben worden ist und andererseits die Entwicklung der DMV. Die ausgezeichnete historische Darstellung Fachverband – Institut – Staat von Norbert Schappacher unter Mitwirkung von Martin Kneser bedeutete eine für die DMV bahnbrechende Leistung. Die Vorgeschichte der Gründung und die Gründung selbst wurden relativ kurz behandelt, gestützt auf die bekannten Quellen wie Gutzmer [4] und auf die neueren Arbeiten wie Tobies [16] und Purkert & Ilgauds [8]. Den Schwerpunkt bildete dagegen die Analyse der Entwicklungen während der NS-Zeit, insbesondere in Auswertung der Pionier-Forschungen von Herbert Mehrtens. Damit wurde erstmals die bislang von der DMV als „heißes Eisen“ nur ungern thematisierte kritische Phase ihrer Geschichte offen und aufrichtig beschrieben.

E. H. Dirksens Beiträge zu den Grundlagen der Analysis

Enno Heeren Dirksen (1788–1850) was one of several mathematicians originating from Eastern Frisia at the end of the eighteenth century to acquire prominence in neohumanist reformed Prussia. While striving—like so many of his contemporaries—for more rigour in the foundations of calculus, his work is significant due to his unusual emphasis on the semiotic aspects of the basic concepts. As a clear-sighted propagator of Cauchys conceptual innovations in Germany, his achievements in improving sign expressions of such concepts enabled him to clarify hitherto not differentiated multiple limit processes—although less successfully in constructive concept formation itself.The contribution is based on the analysis of Dirksens published texts and on the numerous texts which remained drafts for his projected voluminousOrganon der gesamten Analysis.