Conference on the History of Mathematics and Teaching of Mathematics
List of abstracts, May, the 20th 2008
HISTORY OF POLYNOMIALS
Mircea BECHEANU
University of Bucharest, Romania
bechmir@fmi.unibuc.ro
The History of Mathematics was less focused on the History of Algebra. For many centuries,
Algebra was considered rather a collection of methods and algorithms created with the aim to
make possible computations.
Our aim is to present a short incursion into the History of Algebra through polynomials, one
of the basic notions in Mathematics. We want to clarify how this notion appeared in
Mathematics and especially why its born was so late. The clear idea of "what it is polynomial"
is very connected to modern Algebra. Nevertheless, for many centuries they were a necessary
tool in mathematical discovering.
We present the evolution of mathematical ideas connected to polynomials starting from
Ancient Mesopotamia, then we pass to Ancient Greek Mathematics, Eastern and Arabic
Mathematics. Next, coming to Renaissance and Modern Time we present the evolution of the
algebraic formalism in connection with the problem of solving algebraic equations. At the end
we will mention some important problems in modern theory of polynomial rings.
UP-TO-DATANESS OF THE GREEK WORLD-VIEW
Mária BOTH1, László CSORBA2
1
Apor Vilmos Catholic College, Vác
bothmaria@invitel.hu
2
Eötvös Loránd University (ELTE) Budapest
csorbafl@invitel.hu
The lecturers are going to show how the historical method can be applied int the teaching of
science at high school and university. Their examples are based on their own books (ScienceNature-History I-II.) and their experience.
The first example shows how the gnomonic world-view came from the ancient Greek
geometry and how it can be used in philosophy, astronomy and geography. The knowledge
from ancient astronomy is a part of the present-day natural geographical world view.
The other example shows the birth of atom theory and chemistry. The Demokritos’ idea of the
atom is based on logics proving with the help of an indirect method. This idea has been
loaded with empirical contents by modern chemistry. In this way the Greek theory has been
connected with the permanence of relations declared by Phüthagoras.
The knowledge about different viewpoints leads pupils to the methods of scientific
argumentation; therefore, the historical knowledge is up-to-date advanced and useful even
now.
THE POSSIBILITIES FOR HISTORY OF SCIENCE IN EDUCATION AND
TRAINING IN HUNGARY
Mária Both1; László Csorba2; Gábor Zemplén3
1
Apor Vilmos Catholic College, Vác
bothmaria@invitel.hu
2
Eötvös Loránd University, Budapest
csorbafl@invitel.hu
3
Budapest University of Technology and Economics
The paper explores – via some selected and detailed case studies – how history of science is
incorporated in science education and in the education of science teachers in Hungary.
After the significant curricular reforms in 2005 the development of specific competences have
been prioritized and expressly referred to in the national curricula. We show how history of
science can be instrumental in reaching these aims in the high school diploma exams in
Biology, both for test-questions and for the interpretation of primary-source citations. We
discuss the specific needs that science-teacher training faces in order to develop new teaching
skills for nature of science, critical/reflective thinking, and socio-scientific issues, and give
examples of how current textbooks and teaching-guides utilize history of science for reaching
these aims. Based on our experience in both secondary schools, universities (colleges), and
curriculum-development, we assess the successes of the current practice. The assessment is
partly based on the evaluation of the effectiveness of our courses for science-teachers, and
partly on the evaluation of the theoretical underpinnings to the new curricular developments.
MATHEMATICAL DUELS: CONFLICTS MAKING HISTORY
Franka Miriam BRÜECKLER
University of Zagreb, Croatia
bruckler@math.hr
Almost all educated people know that Newton and Leibniz quarreled about who first
discovered calculus, and to many one or the other conflict between great mathematicians is
known. Some of the conflicts may be just anecdotes and the reasons for some conflicts were
more of a personal than of a mathematical background, but still they give interesting insights
the development of various mathematical ideas. This talk shall cover some of the examples,
with the aim to show how mathematical conflicts can help to enhance the teaching of
mathematics.
'THE EDINBURGH MATHEMATICAL SOCIETY EDUCATION COMMITTEE'
Colin CAMPBELL
University of St Andrews
cmc@st-and.ac.uk
The Edinburgh Mathematical Society, founded in 1883, is the principal mathematical society
for the university community in Scotland. As well as a Research Support Fund which gives
financial support to a variety of mathematical activities, including research visits, conferences
and publications, the Society has an Education Committee which is concerned with the whole
spectrum of mathematical education. In particular, this Committee manages the Schools
Enrichment Fund to support a range of mathematical activities at school level. I will describe
some of the work of the Education Committee of which I am at present the convener.
TEACHING DIFFERENTIAL GEOMETRY OF PLANE CURVES WITH MATHCAD
Nicolae DANET
Technical University of Civil Engineering of Bucharest
ndanet@cfdp.utcb.ro}
The aim of this paper is to present the benefits of using Mathcad for teaching differential
geometry of plane curves to civil engineering students. Being easy to combine math equations
and graphical representations with text regions in Mathcad, the teacher can deliver good
lectures without using the software like a black box.
TWO MAHEMATICAL GEMS BEHIND A TANGRAM
Fernando Castro GUTIÉRREZ
Universidad Pedagógica Experimental Libertador, Venezuela
fercasgu@hotmail.com
From the times of the Greeks the presence of the Golden ratio between segments has been a
seal of beauty and harmony. This is the emblematic case of the Parthenon in Athens. But what
about the presence of the Golden Number as a ratio between areas of plane regions ?. In
1984 the German mathematician Georg Brugner created an interesting 3-pieces Tangram.
Through the geometric Tangram construction we can find the Pythagoras theorem and a
Golden Section between areas. A search in the Kasimir Malevich paintings ,some decorative
Spanish bricks and book design reveals the presence of the Golden Number as a ratio
between the areas of plane regions.
EULER, SEGNER, TOBIAS MAYER AND DEBRECEN
Tünde KÁNTOR
University of Debrecen, Hungary
tkantor@math.klte.hu
„Read Euler, read Euler, he is the master of us all.”
Lagrange
This lecture presents a connection among Euler and his contemporaries Segner, Tobias Mayer
and some aspects concerning Debrecen. We discuss his influence on the growth of European
mathematics and physics. Euler was one of the most important mathematicians and physicians
of the Enlightenment. The year 2007 marks the 300th anniversary of Euler’s birth. In
Debrecen we organized lectures and an exhibition of remembrance. Why? The causes are
very obvious.
1. Segner was student of the ancient Calvinist College in Debrecen and later he was employed
as a doctor in Debrecen. Segner invented a simple reaction waterwheel. Euler presented
Segner’s turbine at the Academy of Berlin. This work influenced Euler to work on turbines.
Euler used Segner’s results when he created a crude turbine. We could find a very big
correspondence between Segner and Euler, Segner wrote 159 letters to Euler.
2. Euler’s lunar theory was used by Tobias Mayer in constructing his famous tables of the
moon. We could find Euler’s correspondence (10) with Tobias Mayer (21) from1751 to 1755.
3. Tobias Mayer and Segner were at the same time professors at the University of Göttingen.
Their common work was to build up and to equip the new observatory. Instead of Segner
Tobias Mayer got the director’s chair of the observatory. Euler solved the problem, he
advised the authorities that Segner should occupy Ch. Wolff’s chair in Halle.
4. In Debrecen, in the Great Library of the Calvinist College we found 12 original works of
Euler. Their owners were some peregrine students, who became later well-known outstanding
personalities. The most interesting is that we find a coloured exemplar of Tobias Mayer’s
Mathematischer Atlas (Pfeffel, Augsburg, 1745), and a lot of his maps as part of the Atlas
Scolasticus and the Atlas Geographicus Maior (1748-1751).
THE ROLE OF G. B. HALSTED IN THE RECOGNITION OF JÁNOS BOLYAI'S
ACHIEVEMENTS
Zoltán KÁSA
Sapientia University, Tg-Mures
kasa@cs.ubbcluj.ro
In the summer of the year 1896 George Bruce Halsted (1853-1922), professor at Texas
University in Austin, who had published in English the Lobachesky's treatise and the Bolyai's
Appendix, made a trip to Marosvásárhely and Kolozsvár and after this to Kazan. In The
University Of Texas Magazine was published a paper by J. A. Lomax, the editor in chief,
about this very interesting summer trip. In the same journal Halsted published two papers on
his visit in Transylvania.
In this paper we discuss how this visit was reacted in the local newspapers.
ON TWO LONG LASTING DELUSIONS IN THE HISTORY OF EQUATIONS
Lajos KLUKOVITS
University of Szeged, Hungary
klukovits@math.u-szeged.hu
The first is the misbelief that Al Khwarizmi was the inventor of two powerful method −
called by him as al-jabr and al-muquabla − in solving (quadratic) equations.
We will present several examples of two millennia earlier i.e. examples from the
period aof the Old Babylonian Empire.
The second is that either nothing remarkable happened in algebra between
Leonardo of Pisa and Luca Pacioli, or the 16th century Italian \\\"maestro\\\"-s and
mathematicians have first disproved the opinions of the mediaeval Islamic scholars
that the cubic and the biquadratic equations can be solved geometrically only.
We will show results due to Italian \\\"maestro\\\"-s (master Dardi of Pisa, master Benedetto
of Florence) and the famous renaissance painter Pierro della Francesca of the 14th and 15th
centuries.
They have reached several interesting but non-general results,
CHANGES IN THE ROLE OF ZERO
Éva KOPASZ
Eötvös József College, Hungary
kopasz.eva@ejf.hu
In this talk what Í intend to focus on is the changes in the role of zero in history. The problem
of zero dates back to history. At the very beginning it was not even denoted later on it was
used to substitute space. Even some fifty years ago axiom 1 by Peano was taught in a way that
1 is a natural number. Then we move on to the role of zero in mathematics and in other
branches of science and its relevance in everyday life. Further on the results of a survey
conducted at our college revealed how the students appreciate zero. Since students presented
their ideas related not only to mathematics but other branches of science as well. In this
presentation excerpts from their ideas will be shown.
REVISITING PYTHAGORAS
David LINGARD
Sheffield
davideuclid@yahoo.co.uk
Jacob Bronowski described the theorem attributed to Pythagoras as “… the most important
single theorem in the whole of mathematics.” That may seem an extravagant claim, but what
Pythagoras (and others before him) established is “… a fundamental characterisation of the
space in which we move and the first time that it is translated into numbers.” (1)
What is beyond doubt is the hugely rich field of mathematics that springs from this simple
theorem, the variety of proofs of which alone are sufficient for weeks of enjoyment.
This lecture will touch upon some of these proofs, but will also examine other pieces of
mathematics that can be developed from the theorem, a few problems that rely upon it for a
solution and some aspects of the history behind much of all of this.
Hopefully, this session will be of interest to the academic mathematician, the historian of
mathematics and those who teach the subject at any level.
(1) Bronowski, Jacob, The Ascent of Man, BBC Publications, London, 1973.
SPATIAL ABILITY IN FACULTY OF TECHNICAL ENGINEERING
ÁRVAINÉ MOLNÁR Adrien1, NAGYNÉ KONDOR Rita2
University of Debrecen, Hungary
1
mazg@freemail.hu, 2kondorri@freemail.hu
Our article reports about a survey about the topic of spatial ability and fundamental
knowledge on descriptive geometry. We surveyed the knowledge of first year architect
students in the University of Debrecen, Faculty of Technical Engineering, because it is very
important for the students to solving different spatial problems. We analysed results of the
test, with emphasis on gender differences.
THE ROLE OF THE HISTORY OF MATHEMATICS IN THE TEACHING AND
LEARNING OF MATHEMATICS - AN ICMI STUDY 1997–2000
Kati MUNKÁCSY
Eötvös Lóránd University, Budapest
katalin.munkacsy@gmail.com
I would like to show some documents of ICMI study, which was working from 1996
(Barcelona) to 2000 (Tokyo). This study was chaired by John Fauvel and Jan van Maanen.
The work is going on, now there are working groups, web sites, journals, conferences in this
field.
A TRAINING MODEL FOR STUDENTS OF TEACHER STUDIES IN THE AREA
OF DETECTION AND EDUCATION
Margita PAVLEKOVIC1, Zeljko GREGOROVIC2
University of J.J.Strossmayer in Osijek, Croatia
1
pavlekovic@ufos.hr, 2zgregorovic@ufos.hr
The poster presents a training model for students of teacher studies in the area of detection
and education of mathematically gifted children.
The immediate instruction by students of teacher studies and university instructors with fourth
grade pupils of Osijek elementary schools with special interest in mathematics is performed
systematically and continuously during the entire academic year within the Little School of
Mathematics at the methodology practicum of the Faculty of Teacher Education in Osijek
(Internet access, didactic packages, SMART Board, children’s mathematical magazines, as
well as suitable domestic and foreign materials).
Based on very extensive field research, collaboration of mathematics methodologists,
psychologists and scientists in the field of computer science, an expert system was created in
order to detect children’s gift in mathematics.
Additionally, the expert system enables students to monitor the growth of their competencies
and establish a sense of self-efficacy in the area of detection and education of mathematically
gifted children.
NUMERICAL SOLUTIONS OF SYLVESTER AND ALGEBRAIC RICCATI
EQUATION
Mojca Premus,
University of Ljubljana, Slovenia
mpremus@fgg.uni-lj.si
The paper is devoted to study of (numerical) SOR-like methods of solving Sylvester
Equation AX−XB = C; where A is a real m×m matrix; B is a real n×n; C; X a real m×n, and
the continuous-time algebraic Riccati equation (CARE):
ATX + XA − XSX + Q = 0;
where A; Q; S; X are real n×n matrices, proposed by Z. Woznicki. In comparison to other
numerical methods methods the described in this paper use simple algorithms. They also give
an low-computational ways of estimating errors.
CORRESPONDENCE OF MATHEMATICIANS I.
LÁSZLÓ KALMÁR’S CORRESPONDENCE WITH HUNGARIAN
MATHEMATICIANS
Péter Gábor SZABÓ
University of Szeged, Hungary
pszabo@inf.u-szeged.hu
László Kalmár (1905-1976) was the leader of Hungarian mathematical logic at the University
of Szeged. He was an excellent mathematician and one of the best-known pioneers of
Computer Science in Hungary.
The Kalmár’s scientific legacy is an important source of the History of Mathematics and the
History of Computer Science. He had more than 700 corresponding partners, many Hungarian
and foreign mathematicians, and other scientists.
In our project, we published two books based on the Kalmár’s legacy. We worked up his
correspondence with 24 Hungarian mathematicians: with János Aczél, Lajos Dávid, Pál
Erdős, Lipót Fejér, István Fenyő, Géza Grünwald, Béla Gyires, György Hajós, János
Neumann, Andor Kertész, Dénes Kőnig, Imre Lakatos, Dezső Lázár, Tibor Radó, László
Rédei, Alfréd Rényi, Frigyes Riesz, János Surányi, Tibor Szele, Barna Szénássy, Béla
Szőkefalvi-Nagy, Pál Turán, Tamás Varga, and István Vincze. The books contain more than
500 letters with 1000 comments and many other documents, photos, and biographical data
[1,2].
References
[1] KALMÁRIUM. Kalmár László levelezése magyar matematikusokkal (Dávid Lajos, Erdős Pál, Fejér Lipót,
Grünwald Géza, Kertész Andor, Kőnig Dénes, Rédei László, Rényi Alfréd, Riesz Frigyes, Szele Tibor, Turán
Pál, Varga Tamás). Összeáll.: Szabó P. G. Szeged, 2005. Polygon. 476 p.
[2] KALMÁRIUM II. Kalmár László levelezése magyar matematikusokkal (Aczél János, Fenyő István, Gyires
Béla, Hajós György, Lakatos Imre, Lázár Dezső, Neumann János, Radó Tibor, Surányi János, Szénássy Barna,
Szőkefalvi-Nagy Béla, Vincze István). Összeáll.: Szabó P.G. Szeged, 2008. Polygon. 424 p.
CORRESPONDENCE OF MATHEMATICIANS II.
THE RIESZ BROTHERS’S CORRESPONDENCE
Péter Gábor SZABÓ
University of Szeged, Hungary
pszabo@inf.u-szeged.hu
The Riesz brothers, Frigyes Riesz (1880-1956) and Marcel Riesz (1886-1969) were great
mathematicians of the 20th century. Frigyes Riesz lived in Hungary (in Kolozsvár, Szeged,
and Budapest), his brother, Marcel Riesz in Sweden (in Stockholm and Lund). Their scientific
works have a great significance in many parts of mathematics, and their collected papers
published in two monographies [1,2].
In our project we started the collecting and reviewing the Riesz brothers’s correspondence in
Lund, and carried on the preliminary studies [3] by the late László Filep (1941-2004), who
worked both in Lund and in the Institute for the History of Hungarian Sciences.
We concluded that these materials definitely contain items of mathematical and historical
worth, which help us understand the history of mathematical problem solving in the twentieth
century [4]. This project was supported by the Grant OTKA K 67652.
References
[1] Riesz Frigyes összegyűjtött munkái I-II. (A Magyar Tudományos Akadémia megbízásából sajtó alá rendezte
Császár Ákos), Akadémiai Kiadó, Budapest, 1960.
[2] Marcel Riesz, Collected Papers, (Edited by Lars Gårding and Lars Hörmander), Springer-Verlag,
Berlin, Heidelberg, 1988.
[3] Filep László, Szemelvények Riesz Frigyesnek Riesz Marcellhez írott leveleiből, Műszaki Szemle 27. szám,
Historia Scientiarum – 1, 2004, 26-38.
[4] Riesz Frigyes és Riesz Marcell levelezése (Submitted for publication).
THE CONCEPT OF SYMMETRY AT THE AGE OF 10-11
SZILÁGYINÉ SZINGER Ibolya
Eötvös József College, Hungary
szilagyine.szinger.ibolya@ejf.hu
In the lower primary the basics of geometrical concepts are laid down. In this presentation the
development of the concept of symmetry is examined. In May-June 2006 the evolvement of
several geometrical concepts - among which the concept of symmetry as well - were
examined in an educational development experiment conducted with fourth class students.
The research question is how lower primary geometry teaching, particularly the concept of
symmetry is related to the levels formulated by van Hiele. Moreover to what extent are the
concrete activities effectively carried out at these levels in evolving the concept of symmetry.
Our hypothesis is that in the lower primary geometry teaching (classes 1-4) the first two
stages of the van Hiele levels can be put into practice. By the completion of lower primary
classes level 3 cannot be reached. Children do dot see the logical relationship between the
properties of a given shape. They cannot come to a conclusion from one property of shapes to
another.
In our presentation we present the developing teaching experiment and its observations which
we support with measurement results