8th Conference on History of Mathematics & Teaching of …hmtm/abstract.pdf · 2014. 5. 23. · Kapus Martin Catholic University in Ružomberok, SK [email protected] Klukovits

8th Conference on History of Mathematics & Teaching of Mathematics

8th HMTM

Mathematics – a common language for Europe for a thousand years

21–25 May, 2014

Cluj-Napoca / Kolozsvár / Klausenburg

Cooperating partners: Faculty of Mathematics and Computer Science of the Babeş-Bolyai University in Cluj, School of Mathematics of the University of St Andrews, Department of Analysis of the University of Miskolc, Department of Mathematics, Teacher Training College of the of the Eötvös Loránd

University in Budapest, Institute of Mathematics, and Institute of Informatics of the University of Debrecen, János Bolyai Mathematical Society, Hungary, Romanian Mathematical Society, Farkas Gyula Society for Mathematics and Computer Science, Cluj Organizing Committee: Chairman: Edmund F. Robertson (efr(at)st-and.ac.uk) Members: John O'Connor, University of St Andrews (joc(at)st-and.ac.uk) Péter Körtesi, Miskolc University (matkp(at)uni-miskolc.hu) Katalin Munkácsy, Eötvös University, Budapest (katalin.munkacsy(at)gmail.com) Tünde Kántor, University of Debrecen (tkantor(at)math.klte.hu) Lajos Klukovits, University of Szeged (klukovits(at)math.u-szeged.hu) Michael Lambrou, University of Crete (lambrou(at)math.uoc.gr) Katarzyna Winkowska Novak, University of Social Sciences and Humanities, Warsaw Local organizers: Judit Robu, Babeş-Bolyai University Cluj (robu(at)cs.ubbcluj.ro) Anna Soós, Babeş-Bolyai University Cluj (asoos(at)math.ubbcluj.ro) Invited Speakers Edmund ROBERTSON, John O'CONNOR, András PRÉKOPA, Dorel DUCA, Ferenc SZENKOVITS, Katalin MUNKÁCSY, Doru ŞTEFĂNESCU, Michael LAMBROU.

CONTENTS1

Preamble

Programme ............................................................................................................. 5 Participants ............................................................................................................. 6

Talks

Sudhakar Agarkar, Pedagogical Implications of Bhaskaracharya’s Lilavati................................... 7

Evgenios Avgerinos, Athanasios Karageorgiadis, On a new era in Mathematics Assessment with the assistance of Computers and New Technologies...................................................................................... 8

Jaroslav Baričák , Proposal of methodology of constructivist teaching Euclidean theorems using digital technologies ........................................................................................... 9

José Carlos Cortes Zavala, Elipsografos Using Learning Activities to Support the Process Demonstration in Analytical Geometry................................................................................... 10

José Carlos Cortes Zavala, Numeric and graphics for function approach, highlighting the importance of linearity and slope ........................................................................................... 11

Edith Debrenti, The Arithmetics from Kolozsvár .................................................................... 12

Dorel Duca, Adrian Petruşel, Mathematics Education in Romanian at Babeş-Bolyai University Cluj-Napoca .................................................................................. 13

Stefanie Eminger, Ferdinand Rudio (1856-1929): A Historian of Mathematics.......................... 14

Lajos Klukovits, Problems and Methods in Diophantus’ Arithmetica....................................... 15

Lilla Korenova, Modeling historical mechanisms for drawing curves with GeoGebra in high school .............................................................................................................. 16

Péter Körtesi Samu von Borbély, his life and activity 1907 Torda (Turda) – 1984 Budapest ............................................................. 17

Péter Körtesi János Bolyai’s ownershipsignature................................................................. 18

Michael Lambrou, Orthocentre: The Intriguing Story of a Theorem of Classical Geometry from Antiquity to Oblivion and Rediscovery .......................................................... 19

1 Abstracts are listed in alphabetic order according to the corresponding author.

8th Conf. on Hist. of Math. & Teaching of Math., May 21–25, 2014, Cluj-Napoca, Romania 4

Elizabeth Lewis, C.-V. Mourey’s single science of algebra and geometry................................ 20

Maria Mihaylova, Ion Mazilu, Active learning and activity theories in mathematics ..................................... 21

John O’Connor, An Elementary Approach to Hensel’s p-adic Arithmetic ............................... 22

Nicolae Pop, On the Painlevé’s Paradox in Contact Problems for Rigid Body Dynamics . 23

András Prékopa, Hatvanöt év a felsőoktatásban ........................................................................ 24

Filip Prikler, Stefan Kriss, Martin Kapus, Using materials From Some Selected Mathematical Textbooks from Teacher Institute in spisska kapitula............................................................................. 25

Edmund Robertson, William Oughter Lonie – St Andrews Educational Reformer........................ 26

József Kolumbán, Ferenc Szenkovits, Remarkable Hungarian Mathematicians at the University of Cluj ................ 27

Doru Ştefănescu, A Historical Excursion on Polynomial Root Bounding.................................. 28

Valeriu Ungureanu, A Brief History of Pareto-Nash-Stackelberg Games ...................................... 23

List of Authors........................................................................................................... 31


PROGRAMME

May 21 (Wednesday):

• Arrival, registration: 18:00–21:00 in the hall of Hotel Universitas (Str. Pandurilor Nr. 7, Cluj Napoca, tel.: +40-264-429788; +40-264-584315).

• Get together party (19:00–21:00 in the hall of Hotel Universitas)

May 22 (Thursday):

• Arrival, registration 8:00–12:00 at the conference site (main building of the university, second floor, Str. M, Kogalniceanu 1).

• Opening ceremony (10:30–10:45) • Invited Lectures (11:00–13:00) • Coctail party (13:00 Piramida Restaurant) • Invited Lectures (15:00–16:20) • Section talks (16:40–18:00) • Cultural event: Ruggero Leoncavallo: Pagliacci (18:30–... Hungarian Opera)

May 23 (Friday):

• Section talks (9–10:20) • Invited Lectures (11:00–13:00) • TDK meeting in Hungarian Poster Session (13:00–14:00) • Invited Lectures (15:00–16:40) • Section talks (17:00–19:00) • Banquett (19:00–22:00 Piramida Restaurant)

May 24 (Saturday):

• Trip (8:00–20:00) Turda (Torda); Alba Iulia (Gyulafehérvár); Aiud (Nagyenyed)

May 25 (Sunday):

• Departure


PARTICIPANTS

Agarkar Sudhakar VPM's Academy of International Education and Research, India

[email protected]

Jaroslav Baričák Comenius University in Bratislava, SK [email protected]

Cortes Zavala José Carlos

Universidad Michoacana, México [email protected]

Debrenti Edith Partium Christian University Oradea, RO [email protected]

Dillingerová Monika Comenius University in Bratislava, SK [email protected]

Dorel Duca Babeş-Bolyai University, RO [email protected]

Eminger Stefanie University of St Andrews, UK [email protected]

Gutu Valeriu Moldova State University, MD [email protected]

Kapus Martin Catholic University in Ružomberok, SK [email protected]

Klukovits Lajos University of Szeged, HU [email protected]

Korenova Lilla Comenius University in Bratislava, SK [email protected]

Körtesi Péter University of Miskolc, HU [email protected]

Kriss Stefan Catholic University in Ružomberok, SK [email protected]

Lambrou Michael University of Crete, GR [email protected]

Lewis Elizabeth University of St Andrews [email protected]

Mihaylova Mariya University of Ruse, BG [email protected]

Munkacsy Katalin Eötvös Loránd University, HU [email protected]

O'Connor John University of St Andrews [email protected]

Opinca Carolina Moldova State University, MD [email protected]

Petkova Magdalena University of Ruse, BG [email protected]

Pop Nicolae North University Center at Baia Mare, Technical University of Cluj-Napoca and Institute of Solid Mechanics of the Romanian Academy, Bucharest, RO

[email protected]

Prékopa András Eötvös Loránd University, HU [email protected]

Prikler Filip Catholic University in Ružomberok, SK [email protected]

Robertson Edmund University of St Andrews [email protected]

Ştefănescu Doru SSMR, RO [email protected]

Szenkovits Ferenc Babeş-Bolyai University, RO [email protected]

Trifu Mircea SSMR, RO

Ungureanu Valeriu Moldova State University, MD [email protected]

Vasileva-Ivanova Ralitsa

University of Ruse, BG [email protected]


PEDAGOGICAL IMPLICATIONS OF BHASKARACHARYA’S LILAVATI

Sudhakar Agarkar

VPM's Academy of International Education and Reserch

As a part of a year long celebration of 900th birth anniversary of Bhaskaracharya the Vidya Prasarak Mandal, Thane (India) has initiated workshops on Lilavati, a book written by him in 1150. The book contains problems in elementary Arithmetic, Algebra and Geometry written in poetic form with ample use of alliterations metaphors, pun and tenderness. Bhaskaracharya displays various qualities of a good teacher in Lilavati. For Example, he addresses the learner with personal words like sakhe (a friend), bale (a little girl), etc. This approach is found to establish close links between the teacher and the learner. Problems posed in the book are not solved. Nonetheless, the author challenges the reader to undertake the task seriously and provides appropriate hints to deal with them. Moreover, he has discussed the topic in such a way that pre-requisites for solving a given problem are adequately developed.

Bhaskaracharya poses problems from everyday life of school children like calculating the area of geometrical shapes, calculating expenses to buy essential things for home or determining the time required to fill the water container. One more special feature of Lilavati is that many of the problems are framed based on the movements of animals and birds like a serpent, a monkey, bees, elephants, deers, a peacock, swans, etc. Moreover, there are problems related to mythological stories from religious scriptures of Hinduism like Ramayana and Mahabharata. The students at young age are found to appreciate these examples and stories.

Beginning from January 2014 about a dozen workshops on Lilavati have been arranged both in rural as well as urban parts of India using the problems and techniques suggested in Lilavati. It has been observed that these techniques prove useful in creating interest towards mathematics. They not only remove the fear of mathematics but motivates the students to solve mathematical problems with zeal and enthusiasm. The paper would share the first hand experiences of conducting workshops on Lilavati and bring out pedagogic implications of the book to the teaching of mathematics in 21st century.


ON A NEW ERA IN MATHEMATICS ASSESSMENT WITH THE ASSISTANCE

OF COMPUTERS AND NEW TECHNOLOGIES

Evgenios Avgerinos, Athanasios Karageorgiadis

Mathematics Laboratory and Multimedia, Dept of Education, University of the Aegean, Rhodes, Greece

Although assessment and teaching mathematics have discreet roles, assessment is very important and necessary for the learning process. Assessment had a key role in Ancient Athens' educational system where rich families had an educator or tutor, who was an educated slave and his role was to help the student with their lessons and to complement the Teacher’s work.

Nowadays, assessment is anywhere important and countries like Japan, Singapore, Hong Kong and Switzerland, who have invested a big amount of money in their educational systems and have embraced new assessment techniques using computers and new technology, have scored on the top places in international student assessment programs like PISA.

Following the result of these countries, the United States decided in 2013 to reform their educational system. Key aspect of their educational system reform is the assessment in mathematics using techniques of new technologies, like artificial intelligence.

All the current computer based assessment systems involve only very basic functions and they are not able to implement the above mentioned techniques like artificial intelligence.

A new system, which has been built in the basis of being able to implement new technologies like artificial intelligence in assessment Mathematics is presented in this paper.


PROPOSAL OF METHODOLOGY OF CONSTRUCTIVIST TEACHING

EUCLIDEAN THEOREMS USING DIGITAL TECHNOLOGIES

Jaroslav Baričák

University of Comenius in Bratislava

The paper is in the theoretical part focused on teaching Euclidean theorems in secondary schools, constructivist teaching methods and using digital technologies in teaching mathematics. The aim of our work was to create a methodology and electronic teaching materials for constructivist oriented teaching of Euclidean theorems in the digital environment. This metodology and electronic materials form the practical part of the work. The teaching of Euclidean theorems is planned for three lessons. One lesson is focused on explaining the curriculum, one lesson is for practise and the last lesson is for repeating and knowledge testing. The instructional materials for teachers contain dynamic drawings created using GeoGebra, collection of tasks created in the Open-Sankoré software, e-test created using Hot Potatoes, solutions of all tasks and methodological notes for the teachers.


ELIPSOGRAFOS USING LEARNING ACTIVITIES TO SUPPORT THE

PROCESS DEMONSTRATION IN ANALYTICAL GEOMETRY

José Carlos Cortes Zavala

Universidad Michoacana México

In this paper are given to know results when applied to eight high school students two learning activities of analytical geometry in which elipsografos (specific artifacts) are used. Activities had the objective of supporting the process of demonstrating in analytical geometry, specifically with the theme of ellipse. Implemented a worksheet for each elipsografo, which guide students through instruction in the handling of the Elipsografo and questions relating to the appliance in such way that discovered the mathematical model immersed in each of the elipsografos and that the exploration to build a mathematical demonstration that allows them to build the formal concept of ellipse.

To carry out this work we use two mathematical artifacts that draw an ellipse, these are commonly called „Elipsografos”, the first is the Elipsografo of levers and connecting link of Inwards and the second Elipsografo Antiparalelogramo Articulated Van Schooten.

Alternative complementary theoretical and practical perspectives in mathematics education proposed by several researchers (Mariotti et al 1997; Bartolini et al 2003, 2004; Bartolini 2007, Boero et al, 1996, 1997,. Arzarello, Robutti 2004, Jill et al 2002), argue for the introduction into the classroom of „historical contexts of scientific recreation experience”, particularly those having to do with the practice of using mechanical geometry and articulated models or machines to draw or trace, as a means of generating ideas or complex mathematical notions. In addition, other researchers state that „Supervised sessions, students are able to deploy mathematical tools that are triggered by the understanding of concepts” ( Hoyos, and Genèves Capponi, 1998), made the above will work in a collaborative creativity and ingenuity in students is promoted. That is guided through the manipulation of the artifacts are expected to explore the student immersed in the engine mathematical model, this will allow the reversibility of knowledge according to the mentioned by Piaget (Piaget, 1950).


NUMERIC AND GRAPHICS FOR FUNCTION APPROACH, HIGHLIGHTING THE

IMPORTANCE OF LINEARITY AND SLOPE

José Carlos Cortes Zavala

Universidad Michoacana México

In this paper a numerical approach for performing graphics functions is proposed. Is initiated by introducing the ideas of differences, increments and ratio increases and as the foregoing construction performs graphed functions. It was designed and developed software to support the introduction of these ideas. To address the issue are presented theoretical ideas, a statement of what is proposed in the software and some results.

Several researchers point to the importance of introducing the concept of rate of change as a bridge to reach the intuitive concept of derivative. Based on this initial idea design and software development, which we called „Functions and Derivatives”. The proposed software activities that highlight differerence, increments, and rationale increments aspects first with a numerical approach and subsequently using graphical approaches highlight visual ideas involving numerical – graphical relationship is incorporated.

The numerical and graphic treatment is rarely used and when it is only serves as an introduction without a proper connection to the algebraic process. Proposals such as Duval (1988.1993 and 1995), Confrey (1993), Scher (1993), Mejia (1997), Hitt (2002) and Pluvinage (2005 ) mention the importance it has for the learner, the graph management and numeric. Numeric, graphic and algebraic aspects are representations of mathematical objects and each presents certain information object also allow certain types of cognitive activities in the subject. When only one type of representation is used there is a risk, as mentioned by Duval (1988), to confuse the object representation, so as methodology, this research proposes the use of multiple representations of an object.

Each representation leaves visible types of information, but also hides another and each of them allows us to perform certain types of operations. Both textbooks and in classroom work, these representations are given partially concealing information that allows us to make the right connections; Taking into account the above is that we propose the construction of the graph starting from the numerical, but emphasizing that there is hidden information in such numerical representation.


THE ARITHMETICS FROM KOLOZSVÁR

Edith Debrenti

Partium Christian University Oradea

The Arithmetics from Kolozsvár is part of the Scientific Collections of the Reformed College of Sárospatak. The work was published in Cluj in 1591. Thanks to modern technology, this treasure has been made available through digitalization by the Hungarian Institute for the History of Science.

Hungarian mathematics started to develop in the 16th – 18th centuries. The books published during this period are very important form the point of view of science because their curriculum content and didactic considerations formed the basis for elementary mathematics textbooks published later. Six important Hungarian works were born in this period, these are also mentioned.

The Arithmetics from Kolozsvár is a higher level book, it has functioned not only as a textbook. It was often used by merchants, who could thus study arithmetic operations in their native language. Historically, this book has a great scientific importance because in was in this time that the Hungarian mathematical terminology started to emerge.

The research method employed by us was document analysis. The following items were examined using Fischerné Dárdai’s (2008) criteria of evaluation: the syllabus (topics in the book), questions and exercises, the learnability of the pedagogical text with regards to terminology, as well as illustrations.

Keywords: history of mathematics, arithmetics, operations.


MATHEMATICS EDUCATION IN ROMANIAN AT BABEŞ-BOLYAI

UNIVERSITY CLUJ-NAPOCA

Dorel Duca, Adrian Petruşel

Babeş-Bolyai University Cluj-Napoca

In this paper, we will present the most important moments of the evolution of the mathematical education and research activities of the Romanian line of study in Babeş-Bolyai University Cluj-Napoca. We will focus on the most important moments of the development of the mathematical studies in Romanian taking into account the following parameters: teaching, research, math journals, Professors.

We will also point out that, despite of the vicissitudes of the life, the university education and the research activities in the field of mathematics at Cluj were permanently on an ascendent slope. The Professors of the Cluj University, regardless of nationality, always worked, with abnegation and responsibility, on the development of the university and for of the perennial values promoted by it.


FERDINAND RUDIO (1856-1929): A HISTORIAN OF MATHEMATICS

Stefanie Eminger

University of St Andrews

Ferdinand Rudio, professor of mathematics at the Federal Polytechnic in Zurich, is best known as initiator and first general editor of Leonhardi Euleri Opera Omnia. However, his interest in the history of mathematics extended beyond Euler’s works; he published a number of biographies of mathematicians as well as treatises related to the history of the problem of squaring the circle. For this he produced his own translations of original texts, most notably those by the Greek mathematician and philosopher Simplicius. Rudio wrote with a view to spark interest in the history of mathematics not only among mathematicians, but also the general public, and to provide schoolteachers with teaching resources. The talk will focus on Rudio’s publications to this effect.


PROBLEMS AND METHODS IN DIOPHANTUS’ ARITHMETICA

Lajos Klukovits

Bolyai Institute, University of Szeged

Dioophantus of Alexandria was active around 250 AD. His most famous work Arithmetica consists of 13 books originally, but 10 of them are known yet. The main topic of these books is indeterminate equations and sets of such equations. In spite of a long lasting delusion, Diophantus have looked for rather one rational solution than all integed solutions. He never stated and proved general theorems, did not formulate general methods, but his calculations via definite numbers bear some general features.

We will present a couple of problems from the Arithmetic with their original solutions (that were verbal, but we will use formulates) and pointing out its general character.


MODELING HISTORICAL MECHANISMS FOR DRAWING CURVES WITH

GEOGEBRA IN HIGH SCHOOL

Lilla Korenova

Comenius University in Bratislava

Modeling historical mechanisms for drawing curves such as the parabola, ellipse, hyperbola are interesting for the students. Addressing these issues with GeoGebra combines the history of mathematics with the latest competencies in ICT and helps the students to develop a better understanding and build a positive attitude towards mathematics. In this article we will show several examples on how to use these tasks as a teaching aid in the high school mathematical education. Using modelling technology we are able create a virtual mechanism, for example a Pantoghraph which then helps in teaching concepts like similar figures, or a Franz Van Schooten's Conic Section Drawer which makes the teaching of conic sections much more interesting.


SAMU VON BORBÉLY, HIS LIFE AND ACTIVITY

1907 TORDA (TURDA) – 1984 BUDAPEST

Péter Körtesi

University of Miskolc

He finished his secondary school in his natal town, and in Cluj, where he already is mentioned for his student presentation about the relativy theory. He started his engineering studies in Budapest, and has finished them in Berlin-Charlottenburg, where he met for a short period Einstein aswell [1].

He did his Ph.D in Berlin Technical University [2], I nyear 1938, his scientific advisor was Rudolf Rothe. The research area he practiced was related to aerodynamics and applied mathematics. He held a position in Berlin.

During the Second World War he was lecturing at Cluj, but he was arrested by the Nazies in 1944, and transported in Germany to work as an applied mathematician. After the war ended, he was invited to Cluj, and his was to help an opening lecture at the newly formed Bolyai University in 1946 [3]. Soon after, in 1949 he left Cluj to move at Miskolc, where he founded the Mathematics Chair of the Faculty of Mechanical Engeering of the University of Miskolc.

In the year1955 he came to Budapest Technical University, where he followed Egerváry Jenő, as chair of Mathematics, and taught here till he retired, while between 1960-64 he was appointed director of the Institute of Mathematics at the Otto von Guericke University in Magdeburg.

He became a member of the Hungarian Academy of Science, where his opening lecture was held in 1948, with the title Közönséges differenciálegyenletek parametrális feladatainak grafikus megoldásairól [4].

He is buried in his families grave in the city of the two Bolyai, Targu-Mures (Marosvásárhely, Neumarkt).

Some of his papers published at Cluj [1]:

A grafikus analízis két alapműveletéről (1945, Über einem Integraphenebener Vektoren (Acta Bolyai 1. 1946) and Bevezető a felsőbb matematikából 1945.

1. Romániai magyar irodalmi lexikon, [on-line. http://lexikon.kriterion.ro/szavak/453/] 2. Dissertation in Berlin: ÜbereinenspeziellenGrenzfall der instationären raumlichen

Tragflügelströmung[on-line: http://genealogy.math.ndsu.nodak.edu/id.php?id=19606] 3. Maurer Gyula: Először a nyilvánosság előtt Borbély Samu beszéde, Erdélyi Napló, XVI

évf. 5., [on line: http://www.hhrf.org/erdelyinaplo/cikk_nyomtatas.php?id_cikk=4426] 4. MTA [on-line: http://mta.hu/iii_osztaly_cikkei/borbely-samu-127990]


JÁNOS BOLYAI’S OWNERSHIPSIGNATURE

Péter Körtesi

University of Miskolc

Recently, in November 2013 it was initiated in Cluj-Napoca a nice event, the Bolyai-Dinner, to offer a „white table” session to those researchers and teachers who contributed to the activities around the two Bolyai, have studied their life or their papers, manuscripts, have organized different scientific or popularizing events – contributing to the development of heritage of two Bolyai.

The meeting was initiated by the Pannon University Veszprém, and hosted by the Babes-Bolyai University. The participants had a very nice dinner and festivity together, the presence of personalities like András Prékopa or Samu Benkő and many other well-known people contributed to this unforgettable night.

Some hours before the dinner, I did enter a small antiquariat beside the university building, where I discovered an interesting book, the shopkeeper first recommended me as an ex-libris of János Bolyai, but which in reality is a book with his ownership-signature, data and a short quotation from Latin poet Horace. Unfortunately, I had no opportunity to buy the book on spot (I had not enough money, and I had some doubts about the conditions the book overcame to the antiquariat), consequently I offered to a small group of people at the Bolyai dinner to „investigate” the case, and to make steps to buy it for a public library.

The case was overtaken by the journalist Csaba Szabó from Cluj, and actually, after about one month joint „struggle” [1] of journalists, mathematicians, librarians and even politicians, the book has been acquired and offered to the Teleki-Bolyai Library in Tg- Mures, where it really belongs [2].

In my presentation I would like to tell the story of that month of „struggle”.

1. [on-line. http://mkdsz.hu/content/view/32930/210/]

2. [on-line:http://www. e-nepujsag.ro/op/article/egy-könyvritkaság-helyére-került]


ORTHOCENTRE: THE INTRIGUING STORY OF A THEOREM OF CLASSICAL

GEOMETRY FROM ANTIQUITY TO OBLIVION AND REDISCOVERY

Michael Lambrou

University of Crete

The history of the theorem about the concurrency of the altitudes of a triangle is discussed, from the great Greek geometers of antiquity to the Renaissance and early modern times. The theorem does not appear explicitly in the Elements of Euclid but Proclus has a direct quote that it was known to the author. It was certainly known to Archimedes who uses it without proof in his Book of Lemmas, to Apollonius and to Pappus. Moreover Menelaus proves the spherical triangle version of it.

It turns out that in spite of the fact that the theorem is a simple and important result in euclidean geometry which is included in all modern Geometry textbooks, and was recorded in printed books in the West as early as 1533 and was known to Regiomontanus and van Caulen, there where long subsequent periods of time that it was forgotten or ignored. It reappears in the mid 18th century. In the paper we also give possible restorations, based on the texts, of the original and of other lost proofs such as the ones no doubt known to Apollonius, Menelaus, Pappus and Regiomontanus. There are at least a dozen elegant proofs or parts of proofs of the theorem in the classical texts.


C.-V. MOUREY’S SINGLE SCIENCE OF ALGEBRA AND GEOMETRY

Elizabeth Lewis


I propose to discuss the innovative and far-reaching mathematics of the elusive Frenchman, C.-V. Mourey. In 1828 Mourey shared his results on the difficulties presented by the theory of algebra, in a book published in Paris under the title, La Vraie Théorie des quantités négatives et des quantités prétendues imaginaries. Seeking algebraic reform, Mourey had set out to discover a new set of definitions and fundamental principles as a basis for algebra. To this end he developed a theory of directed lines, which constituted a single science of algebra and geometry; and, as an application of the theory, he gave a proof of the fundamental theorem of algebra. I intend to reconsider Mourey’s motivations and to re-evaluate his mathematics. I will examine Mourey’s 1828 work from a modern-mathematics perspective and provide evidence of his intent on algebraic reform, thereby establishing him as more than a re-inventor of the doctrine of Argand. Of real interest, aside from the mathematics, is the issue of the total absence of biographical information on Mourey: for the past 186 years Mourey has remained an unknown to historians of mathematics. A portion of the talk will focus on this issue.


ACTIVE LEARNING AND ACTIVITY THEORIES IN MATHEMATICS

Maria Mihaylova1, Ion Mazilu2

1University of Ruse, 2Universitatea Tehnică de Construcţii Bucureşti

We will present the benefits of the problems, which are examined by Teo Poh Yew and application active learning and activity theories in mathematics. They have been adapted for elementary school education by The Students’ Academic and Scientific Research Laboratory in Mathematics, Informatics and Linguistics with the Faculty of Natural Sciences and Education in University of Ruse “Angel Kanchev”. Materials and patterns are designed by bachelor degree students, course Pre-school and Primary School Education, the University of Ruse “Angel Kanchev” and leading teachers Assist. Prof. Valentina Radeva, Department of Bulgarian Language, Literature and Arts and Master Maria Mihaylova – PhD student at the Department of Mathematics.


AN ELEMENTARY APPROACH TO HENSEL’S P-ADIC ARITHMETIC

John O’Connor


Hensel introduced the p-adic numbers in 1897 in an attempt to bring methods of power series into number theory.

The p-adics are a “non-standard” extension of the rational numbers and are a completion of the rationals which are different to the usual real numbers.

As such they seem like a very exotic construct, but I will attempt to show that they arise in (almost) as natural a way as working with ordinary decimals.


ON THE PAINLEVÉ’S PARADOX IN CONTACT PROBLEMS FOR RIGID

BODY DYNAMICS

Nicolae Pop

North University Center at Baia Mare, Technical University of Cluj-Napoca and Institute of Solid Mechanics of the Romanian Academy, Bucharest, Romania

The Painlevé’s classical paradox was introduced in 1895. Paul Painlevé published a simple example of a uniform rod sliding with friction on a horizontal plane, demonstrating the absence or the multiplicity of solutions in case of dry Coulomb friction depending on how the initial conditions are chosen. In this paper, it shows that, from mathematical point of view, David Stewart has resolved the paradox by ensuring that impulsive reactions are allowed by the mathematical formulation, and from mechanical point of view, F. Genot and B. Brogliato have studied the rod dynamics in the neighborhood of a singular point, when the rod is sliding.


HATVANÖT ÉV A FELSŐOKTATÁSBAN

András Prékopa

Eötvös Loránd University, Budapest

1949-ben harmadéves egyetemi hallgatóként kezdtem meg egyetemi oktatói pályafutásomat. A hirtelen felduzzasztott létszámú hallgatósághoz nem volt elég oktató és néhányunkat, akik demonstrátorok voltak, bevontak az oktatásba, mint előadókat. Az előadásban beszámolok ezekről az első évekről, majd arról, hogy miként vettem részt Budapesten az Eötvös Loránd Tudományegyetemen és a Műegyetemen új irányzatok bevezetésében. Végül személyes tapasztalatok alapján összehasonlítom a magyar és az amerikai felsőoktatási rendszert.


USING MATERIALS FROM SOME SELECTED MATHEMATICAL

TEXTBOOKS FROM TEACHER INSTITUTE IN SPISSKA KAPITULA

Filip Prikler, Stefan Kriss, Martin Kapus

Student

We present in our contribution some motivational examples from some selected historical mathematical textbooks, mostly from Franc Mocnik (1814-1892) and other authors from Teacher Institute in Spisska Kapitula. This year we celebrate 200th anniversary of born of this great mathematician and author of textbooks. Franc Mocnik was dominated author of textbooks in the second half of 19th century.

His textbooks were translated to 13 languages.

Second motivation is that in 1819 was established the first Teacher Institute in Spisska Kapitula in territory of Ugria. We present also some examples from historical textbooks from the period 1918–1938.

These motivational examples are useful for future teachers in mathematics.


WILLIAM OUGHTER LONIE – ST ANDREWS EDUCATIONAL REFORMER

Edmund Robertson


William Oughter Lonie (1822–1894) was appointed Mathematics Headmaster at Madras College, St Andrews, Scotland in 1846. In this talk we will look at the system of education at Madras College and, in particular, Lonie’s style of teaching. His ideas influenced teaching in Scotland. We will look at the things taught by Lonie in mathematics classes at Madras at this time. We will also look at the improvements he wished to see in the education of children.


REMARKABLE HUNGARIAN MATHEMATICIANS

AT THE UNIVERSITY OF CLUJ

József Kolumbán, Ferenc Szenkovits

Babeş-Bolyai University Cluj-Napoca

We intend to provide a brief overview of the activities of the most remarkable Hungarian mathematicians who worked at the University of Cluj, from the beginning until our days.


A HISTORICAL EXCURSION ON POLYNOMIAL ROOT BOUNDING

Doru Ştefănescu

University of Bucharest

We give a historical presentation of results concerning the computation of bounds for real roots and for the absolute values of complex roots of polynomials with complex coefficients. We briefly discuss ideas of Newton, Descartes, Lagrange, Cauchy, Bret, Eneström and Kakeya. We emphasize the contributions to the study of polynomial roots made by mathematicians in Romania and Hungary during the 20th century.


A BRIEF HISTORY OF PARETO-NASH-STACKELBERG GAMES

Valeriu Ungureanu

State University of Moldova

Interactive decisions situations, which involve both sequential and simultaneous decisions made by independent and interdependent players with one or more objectives, can be modeled by:

− strategic games – Emile Borel (1920th, 1953), John von Neumann (1928, 1937, 1944, 1947), Oskar Morgenstern (1944, 1947), J. Nash (1951),

− extensive games and information, dynamic games – J. von Neumann and O. Morgenstern (1944, 1947), Harold William Kuhn (1953), Tamer Basar and Geert Jan Olsder (1999),

− Stackelberg games – Heinrich Freiherr von Stackelberg (1934), J. von Neumann and O. Morgenstern (1944, 1947), George Leitmann (1978), Wayne Bialas and Mark Chew (1982, 2002),

− Nash games – J. Nash (1951),

− Pareto games – Vilfredo Pareto (1906), D. Blackwell (1956), Born P., Tijs S.and van den Aarssen J. (1988),

− PN game – V. Pareto (1906), J. Nash (1951), L.S. Shapley (1959), Born P., Tijs S. and van den Aarssen J. (1988),

− NS, PS and PNS games (2000+).

A brief overview of PNS games and its history is provided.



LIST OF AUTHORS

Agarkar, Sudhakar, 7 Avgerinos, Evgenios, 8 Baričák, Jaroslav, 9 Cortes Zavala, José Carlos, 10, 11 Debrenti, Edith, 12 Duca, Dorel, 13 Eminger, Stefanie, 14 Kapus, Martin, 25 Karageorgiadis, Athanasios, 8 Klukovits, Lajos, 15 Kolumbán, József, 27 Korenova, Lilla, 16 Körtesi, Péter, 18 Körtesi, Péter, 17

Kriss, Stefan, 25 Lambrou, Michael, 19 Lewis, Elizabeth, 20 Mazilu, Ion, 21 Mihaylova, Maria, 21 O’Connor, John, 22 Petruşel, Adrian, 13 Pop, Nicolae, 23 Prékopa, András, 24 Prikler, Filip, 25 Robertson, Edmund, 26 Ştefănescu, Doru, 28 Szenkovits, Ferenc, 27 Ungureanu, Valeriu, 29

Documents

8th Conference on History of Mathematics & Teaching of …hmtm/abstract.pdf · 2014. 5. 23. · Kapus Martin Catholic University in Ružomberok, SK [email protected] Klukovits