on the

Report

Nijmegen, Thursday October 29 -- Sunday November 1

Danny Beckers, Katja Peters, Carsten Vollmers

The ninth ``Novembertagung zur Geschichte der Mathematik'' (November Conference on the History of Mathematics) took place from 29-10 until 1-11-1998 in Nijmegen (The Netherlands). Although the weather was not optimally, the conference took place in a friendly atmosphere. The talks were interesting and provoked discussions, which often were continued at the dinner table.

The programme of the conference was followed almost exactly. Only the stroll through Nijmegen ended up in a caffee quite soon due to the atmospheric conditions. Abstracts, as far as they are available, may be found at the bottom of this page.

Financial contributions were made by the Dutch Society for the History and Social Function of Mathematics (GMFW) and the Nijmegen Department of Mathematics. The Department of Mathematics in Nijmegen also served as the host of this years conference by granting us access to their conference facilities. A final contribution was made by the Nijmegen Mathematics Student-association (Desda), who kindly provided us with their coffee-machine, thus taking care of the coffee-supply on sunday.

On friday the conference took place in the deans room of the Nijmegen science department: a very luxerious room with very comfortable chairs, nice view from the fourth floor down upon the garden of the science department, lots of space and an awfull tapistry at the wall. On saturday, due to security regulations within the building, this room was not available and we used one of the ordinary rooms on the ground level, with chairs less comfy etc. Although it suited our status much, much better, several of us were already longing back to the ``good old days''. Wenn on sunday (in this same room), due to the holidays of the canteen personel, the coffee came no longer in real cups, but had to be consumed from plastic bekers, this provoked one of the participants (who probably wants to remain anonymous) to the remark ``that in some way a continuous steadily decreasing function could be spotted in the quality of the facilities of this years conference''. The way in which this remark was made (teasing in a friendly way) characterises the atmosphere of the 1998 conference in Nijmegen.

Discovering the discovered integral: William Henry Young und das Lebesgue-Integral

In 1902 Henri Lebesgue (1875-1941) published his thesis containing a new theory of integration which was based on Borel's theory of measure. Independently of this William Henry Young (1863-1942) together with his wife Grace Chrisholm Young (1868-1944) developed a similar theory of measure and integration. Only after submitting their papers on this subject to the London Mathematical Society did they learn about Lebesgue's results. Consequently the Youngs decided to publish a revised version in which the concept of Lebesgue was taken into consideration and discussed. This parallel discovery will be analysed both from a mathematical and a psychological point of view. The previously unpublished primary sources from the private correspondence of the Youngs will be used to illuminate the collaboration between the Youngs and their reaction to Lebesgue.

Theorems with exceptions and the use of proof

In the early 19th century the concepts of theorem and proof in analysis differed from our present day standards in two related fundamental ways: The proofs used other standards of rigour, and theorems admitting exceptions were not uncommon. The standards of proof depended on the situation in which they were needed and almost incommensurable standards were applied in textbooks and state-of-the-art research. In the prehistory of proof analysis and in the earliest counter-examples in analysis interesting fields of historical enquiry emerge which can shed light on both the mathematical practice of the time and the general formation and evolution of mathematical concepts. I will focus on a famous footnote by Abel concerning an exception to a theorem of Cauchy and analyse the status of theorems with exceptions at the time of Abel.

Developmental research on pre-algebra. A proposed learning trajectory from arithmetic to elementary algebra for 11 to 14 year old students

Many secondary school students experience great difficulty in learning how to construct and solve equations. Arithmetic and elementary algebra appear to be a world apart: the former deals with straightforward calculations with known numbers, the latter requires reasoning about unknown or variable quantities and recognizing the difference between specific and general situations. In recent years, research has been done on the difficulties students have with translating arithmetic word problems into algebraic equations. In the transition from arithmetic to algebra there is claimed to be a discrepancy called epistemological break point (Chevallard 1991) or didactical cut (Filloy & Rojano 1989), hampering manipulations of algebraic expressions. A good starting point for an investigation into this issue could be a return to the roots. By looking into the past we can gain insight into the differences and similarities between arithmetic and algebra and learn from the experiences of others. Recent research on the advantages and possibilities of using and implementing history of mathematics in the classroom has led to a growing interest in the role of history of mathematics in the learning and teaching of mathematics. Streefland emphasized the value of ``reciprocal shifting'' changing one's point of view, looking back at the origins in order to anticipate. Such a change of perspective can propel the learning process of the researcher, the teacher and the student alike. Inspired by Streefland's work as well as the HIMED (History in Mathematics Education) movement, a developmental researchproject called ``Reinvention of Algebra as started at the Freudenthal Institute in 1995 to investigate which didactical means will enablestudents to make a smooth transition from arithmetic to elementaryalgebra. Specifically, the ``invention'' of algebra from a historical perspective will be compared with possibilities of ``re-invention'' by the students.

``Reinvention of Geometry'', a study into the use and value of history in teaching geometry.

Several arguments are being used in the discussion on the use and value of history of mathematics in mathematical classrooms in general and in teaching geometry specifically. These arguments correspond with different ways in which the history of mathematics can be used in classrooms. I intend to give first a survey of the recent literature pertaining to this discussion. (In August 1998 I just started my PhD-project, so I would appreciate it if you supplement this survey with other sources). In my view the arguments in favour of the didactical use of the history of mathematics split into three categories: moral, didactical and mathematical arguments. A second point I want to address is the position of the teachers who want to use the history of mathematics in classroom. How do they find useful material? Do teachers know enough about the history of mathematics and where to find and how to use primary sources to teach their pupils? Can historical articles and books of historians of mathematics be usefully applied by teachers in their lessons? It seems to me that there is a gap between the two groups, i.e. historians and teachers and that they have a difficulty in reaching each other. In general, historians think in terms of mathematical and historical correctness and rigour; teachers look for practical ideas that they can use. In the talk I shall discuss this phenomenon by studying examples of material used in teaching geometry. Afterwards I would like to discuss with you the responsability of historians in writing historical texts, useful for an educational setting.

Minimum Spanning Tree: A Neglected Paper

In this talk, I would like to show the relationship between different papers dealing with the MST problem. The Borùvka's algorithm will be described at greater length. (The reason is the language of origin.) I shall also mention other algorithms and try to describe shortly their method (Kruskal, Prim, Dijkstra). The connection of the MST problem with cluster analysis will also be considered.

Die Mathematisierung von Brot und Bier im Alten Ägypten

Innerhalb der altägyptischen mathematischen Aufgabentexte nehmen die sogenannten Brot- und Bieraufgaben aufgrund ihrer Häufigkeit einen herausragenden Platz ein. Die dort verwendeten Anweisungen und die zugehörigen Rechnungen sind für einen heutigen Bearbeiter nicht ohne weiteres zu verstehen, für eine der vorkommenden Formulierungen konnte bisher keine befriedigende Deutung gefunden werden. Eine Berücksichtigung nichtmathematischer Texte ermöglicht eine neue Interpretation.

A History of the Minimax Theorem: a journey through different mathematical contexts.

In his book

``Die Mathematisierung der Getriebenkinematik im 19ten Jahrhundert''

Die Mathematisierung der Getriebenkinematik im 19ten Jahrhundert ist ein Prozess der auf der einen Seite eng zusammenhängt mit der Industriellen Revolution und auf der anderen Seite mit Entwicklungen in der Geometrie. Ich werde das zeigen an Hand der Problematik der Geradfuehrungen.

Norbert Wiener und die Mathematisierung der Brownschen Bewegung

Mit dem Artikel `Differential Space' gelang Wiener 1923 der Durchbruch in seiner Mathematisierung der Brownschen Bewegung. Der Vortrag beschäftigt sich mit der Frage, inwieweit sich Einflüsse aus der Physik, insbesondere die Arbeiten Einsteins und Perrins, in den Beweisideen und -strukturen des Artikels `Differential Space' wiedererkennen lassen.