Submission 1 Title: Temesvár letter Appendix Author keywords: Temesvár letter Non-Euclidean Geometry The Temesvár letter (3 November 1823) is the first, written document of the discovery of the Non-euclidean Geometry by János Bolyai. The letter contains the famous sentence “semmiből egy ujj, más világot teremtettem” (from nothing I have created a new, Abstract: different word, translation made by the author), we consider the evidence of discovering the Non-Euclidean Geometry. Unfortunately, the first detailed version of his theory, presented to his superior, captain Johann Wolter Edler von Eckwehr (probably in 1825-26), has been lost. His notices however prove that his basic ideas were put on paper since 1820. Submitted: Jan 13, 22:23 GMT Last update: Jan 13, 22:23 GMT Authors first name last name email country organization Web page corresponding?

Péter Körtesi [email protected] Hungary university of Miskolc http://www.uni-miskolc.hu/~matkp ✔

Submission 2 Title: Riemann Sums Belong at the End of Integral Calculus, Not the Beginning. Teaching Integral Calculus Author keywords: History of Integration Differential Equations The typical integral calculus class starts with a lengthy definition of a definite integral utilizing Riemann sums. This occurs even though the integration of differentials, integral notation, and the fundamental theorem of calculus predate the birth of Riemann, who was more concerned with the representation of functions by trigonometric series. This talk proposes that it is pedagogically advantageous to follow history and that the topics of an integral calculus course can be rearranged to reflect this. Students will first Abstract: learn that integration (sum) is naturally the inverse operation of differentiation (difference) and will utilize this approach to solve various "inverse problems" (differential equations). This helps justify the learning of various integration techniques. Students will find Leibniz' proof of the fundamental theorem of calculus very natural, as it re-emphasizes that an integral is a sum of differences. This will lead toward a study of definite integrals and eventually Riemann sums. This is when the theory can be introduced. The author will make available materials to follow this approach. Submitted: Jan 23, 18:27 GMT Last update: Jan 23, 18:27 GMT Authors last first name email country organization Web page corresponding? name Robert Rogers [email protected] United States SUNY Fredonia ✔

Submission 4 Title: Figurate numbers. A Bridge between History and Learning of Mathematics Paper: (Jan 30, 19:46 GMT) Bruner’s representation theory Tobias Mayer’s Mathematischer Atlas Author keywords: figurate numbers space numbers mathematical problems for secondary school students It is necessary to rethink the main principles of the Hungarian mathematics teaching, to apply new methods and new contents, to renew the training of the teachers in the spirit of Tamás Varga. Nowadays the Hungarian mathematics teachers are uncertain in consequence of the bad PISA results. They want to teach better, but they need some help. In this presentation I want to share my teaching experiences and give a new approach to the practice of mathematics instruction, Abstract: connecting a problem of the history of mathematics with the modern learning of mathematics. We present some problems of Mayer’s Mathematischer Atlas (1745), and analyze his method in discussing figurate and space numbers. We deal with some other mathematical problems which were posed for secondary school students (AMC 10, Mason’s problem, KöMal problems, Viviani’s theorem). Submitted: Jan 30, 19:46 GMT Last update: Jan 30, 19:46 GMT Authors last Web first name email country organization corresponding? name page Institute of Mathematics, University of Debrecen, Tünde Kántor [email protected] Hungary ✔ Hungary

Submission 5 Title: Serbian Mathematics Institutions 1841-1941 Mathematics Voivodina Author keywords: Norma Mathematical Seminar Belgrade University This lecture presents a historical review of the development of mathematics institutions in Voivodina and Serbia from the time around the middle of the 18th century until the first half of the 20th century, when the results of the work of several Serbian mathematicians were already well known and recognized in the mathematical world. The living and educational circumstances among the who lived north of the rivers Sava and Danube – within the Habsburg Monarchy – were significantly different from the circumstances on the south – within the Ottoman Empire. In 18th and 19th century Voivodina, a multinational and multicultural southern region of the Austro-Hungarian Monarchy and a part of the monarchy's defense zone towards the Ottoman Empire, was regarded as a province by all its citizens except the Serbs. The religious and national communities of Hungarians, Germans and Croats had their cultural and educational centers - Pest, Vienna and - where they could obtain higher education and therefore did not feel the need for such institutions in Voivodina. During the late 18th and early 19th century Novi Sad (today the principal city in Voivodina) gradually emerged as a prospective cultural and educational hub of the Serbs, not only from Voivodina, but also for the Serbs living under the Ottoman rule. In the Habsburg Monarchy, and in Voivodina as its province, the school curricula, textbooks, and teaching methods were determined by the monarch's order and the teaching of mathematics was in a modest but systematic fashion included in the programs of the Serbian elementary schools. The first mathematics book written in Voivodina was the New Serbian Arithmetic by Vasilije Damjanović printed in Venice in 1767. Its content does not show particularly high mathematical standards but it had educational and enlightening significance. Similar books by Avram Mrazovič, Atanasije Demetrovič Sekereš and Jovan Došenović, written in Voivodina during the 18th and 19th century have the same importance. It was only after grammar schools were established in Novi Sad (1810) and the nearby Sremski Karlovci (1791) that the firm foundation was laid for the institutional development of mathematics in Voivodina. From the very beginning arithmetic and Abstract: 'mathezis' (algebra and geometry) were regular subjects in the grammar school curricula. These were mainly taught using the translated textbooks written by Franc Močnik, a Slovenian mathematician who, as a school counsellor and inspector, played an important role in the development of mathematics education in primary and secondary schools in the Austro-Hungarian Monarchy. At that time the institutions of great importance were the first teacher training school (Norma) founded in Sombor in 1778 by Avram Mrazovič and its higher form named Preparandija, founded in 1812 in Sent Andreja (Hungary) and moved to Sombor in 1816.

On the other side, the complete illiteracy and educational backwardness were predominant among the Serbs under the Ottomans. After the first uprisings against the Turks in 1804 and 1815 and reestablishing Serbian state the organization of primary schools commenced immediately, although it was not until the Hatisherif of 1830 that the Serbs were allowed to form their own schools as well as other state institutions. During the next period, which can be considered very short in the overall history of any scientific discipline, Serbia, once an underdeveloped country in all aspects of education, succeeded in establishing a position in the world of mathematics which could be considered equal to the most developed European countries. Apart from the works and results on several prominent Serbian mathematicians (Mihailo Petrović (1868-1943), Jovan Karamata (1902-1967), Vojislav G. Avakumović (1910-1990)), the overall development of mathematics in Serbia has so far been little known to the general mathematical public. Institutional development of mathematics in Serbia rests on two national institutions: Lyceum, school of higher education founded in 1838 (after 1863 the Higher (Great) school and after 1905 the ) and the Society of Serbian Letters, founded in 1841 (after 1886 the Serbian Royal Academy of Sciences and today the Serbian Academy of Arts and Sciences). Dimitrije Nešić (1836-1904), professor of mathematics and rector of the Higher School, founded the first mathematics library in Serbia in 1871. In time, as a result of the collaboration between the Academy and the University the library had become the main place for mathematicians to gather and work in Belgrade and became known as the Mathematical Seminar of the University of Belgrade. The year 1896 is considered to be the year it was founded and when it began its activities as an institution. The period between the two world wars is the most significant period in the development and institutionalization of the activities of the Mathematics Seminar and Petrović's school of mathematics, which represent the root of the overall development of mathematics in Serbia. After World War II, the Mathematics Seminar developed into the most significant Serbian institution of mathematics under the name of Mathematical Institute. The Institute was founded in 1946 under the authority of Serbian Academy of Sciences. Submitted: Feb 04, 16:22 GMT Last update: Feb 04, 16:22 GMT Authors last first name email country organization Web page corresponding? name Faculty of Technical Sciences, University of Novi Aleksandar Nikolic [email protected] Serbia ✔ Sad

Submission 6 Title: Life and work of mathematician Jovan Karamata (1902-1967) Jovan Karamata Tauberian theorems Author keywords: slowly varying functions regularly varying functions This lecture is about the life and work of one of the most renowned Serbian mathematician, Jovan Karamata. He was born in Zagreb on February 1st 1902. He studied mathematics at the Faculty of Philosophy of Belgrade University, graduated in 1925 when he was appointed as a teaching assistant to Professor Mihailo Petrović. He obtained his doctoral degree in 1926 and promoted to the position of assistant professor in 1930, an associate professor in 1937 and full professor at Belgrade Unviersity in 1950. He left Belgrade in 1951 when he was appointed a professor of the University in , where he stayed until the end of his life. Karamata was a prolific writer. In his works, he paid great attention to form and style. His most significant results are in the field of classical mathematical analysis, more precisely in Tauberian theorems and in the theory of summability in general. These results, as well as those related to slowly varying functions, Mercer's theorems, inequalities, trigonometrical integrals, Froullani's integrals etc., have frequently been quoted in various papers and monographs. The originality of the approach to the various subject in mathematics, as well as the simplicity and elegance in proofs of theorems, confirm in the best manner not only Karamata's exceptional mathematical talent, but also his broad mathematical interest. He died on August 14th, 1967. There are some results which represent real progress in science, results which are further developed and which stay forever remembered as lasting scientific value. We pointed out here three such results of Jovan Karamata. All three are from the late twenties and the early thirties of previous century and almost equally known today as in the time when were created. The first one is his proof of Abel's inverse statement i.e. the new proof of Littlewood's theorem. In spite of the efforts of many mathematicians (e.g. Landau, Hardy, Abstract: R. Schmidt), the proof of Littlewood's theorem remained rather difficult and far from the apparent for twenty years. But in 1930 in the journal Mathematische Zeitschrift appeared Karamata's two-page paper entitled Über die Hardy-Littlewoodschen Umkehrungen des Abelschen Stätigkeitssatzes which brought new proof to Littlewood's theorem, and created a sensation in mathematical circles. Karamata became world-renowned immediately. The editorial board of the Mathematische Zeitschrift, on the occasion of its sixtieth anniversary, quoted this Karamata's paper in its selection of 50 of the most important papers, out of many thousands published. Karamata's proof was characterized as surprisingly simple by Knopp or extremely elegant by Titchmarsh. In this, to the math world invaluable proof, he devised a new method of approach enabling other applications and results to follow. This method entered later in the well-known monographs of Titchmarsh, Knopp, Doetch, Widder, Hardy, Favard. Discovery and introduction of slowly varying and regularly varying functions is the second Karamata's famous result. Starting from two simple definitions of these functions, Karamata developed the whole new theory of slowly and regularly varying functions which has included the majority of the most important properties of these functions. These ideas appeared in 1930 in the paper entitled Sur une mode de croissance régulière des fonctions in, at that time, a less known Romanian journal Mathematica (Cluj). The intent of this paper was to generalize the Tauberian conditions in some inverse theorems of the Tauberian type for the Laplace transform. But it was soon realized that those functions could be successfully applied to many branches of mathematical analysis, not only when the mere convergence but also when other additional information were needed. These are, besides Abelian and Tauberian theorems, Mercerian theorems, Fourier analysis, analytic numbers theory, complex analysis, differential equations etc. Those results, however, were not noticed and evaluated in a suitable manner until 1966 when they appeared in Feller's well-known monograph An Introduction to Probability Theory and its Applications whose second volume contained the elements of Karamata's theory, but not always with the precise presuppositions and clear conditions. This book has shown the great potential of regular variation for probability theory and stochastic processes in general. Hence Karamata's theory has grown, beyond all his expectations into a great mathematical edifice whose significance is still paramount. Theory of regularly varying functions, in recent years, has been expanded to the functions of several variables. Using both the concepts of regular varying functions and generalization of Abel's and Tauber's theorems in Laplace-Stiltjes transform, Karamata obtained the third result of lasting value. This is the Hardy-Littlewood-Karamata theorem, today known as Karamata's Tauberian theorem, which extends the Hardy- Littlewood's result for the Laplace transform. Australian mathematician of Ukrainian origin Eugene Seneta characterized this theorem as one of the most famous and very widely useful theorems in probabilistic (among other) context, and N.H. Bingham marked it as one of the major results in the analysis of the previous century. Karamatae took an active part in the activities at the Belgrade University and in the work of the Serbian Academy of Science and its Mathematical Institute, thus contributing a great deal to the world reputation that Belgrade mathematics had in those days. Although his original results in many areas of mathematics earned him recognition in the world of mathematicians, it must be remembered that for history of Serbian mathematics there are equally significant the great number of his students, later eminent mathematicians - Vojislav Avakumović, Slobodan Aljančić, Bogdan Bajšanski, Ranko Bojanić, Miodrag Tomić, Vojislav Marić, Dušan Adamović, Dragoljub Aranđelović - recognized as "Karamata's (Yugoslav) school of mathematics". Even today, more than 50 years after Karamata's death, his work and his successors give specific character to development of Serbian top-level mathematics. Submitted: Feb 04, 16:28 GMT Last update: Feb 04, 16:29 GMT Authors last first name email country organization Web page corresponding? name Faculty of Technical Sciences, University of Novi Aleksandar Nikolic [email protected] Serbia ✔ Sad

Submission 7 Title: Developing a new card game for teaching functions in higher education real valued functions game Author keywords: gamification cooperative learning differentiation As gamification became a prospering trend in higher education researchers have also been paying increasing attention to a variety of innovative play-based methods. The needs of competence-based education have made it necessary to develop a new toolkit. We have found that after some modification the popular and well-known family board game Brainbox can be applicable in teaching higher mathematics. The basic game broadens the players’ knowledge of different topics, it can be played individually or in groups and makes self-checking possible. Brainbox is a useful tool of cooperative learning, and it is perfectly suitable for developing generic Abstract: competences. Using the idea of the basic game we have elaborated a new mathematical game called ”Functions in the Box”. Our game improves observation skills, supports the correct engravement of the basic characteristics of the most important function classes and demonstrates well the transformations' impact on certain properties of functions. The 105-card deck can be considered as a complex tool for differentiated education of Generation Z providing a combination of theoretical and practical knowledge and supports cooperative, playful learning processes. Submitted: Feb 06, 10:21 GMT Last update: Feb 06, 10:21 GMT Authors last Web first name email country organization corresponding? name page University of Miskolc, Institute of Szilvia Szilágyi [email protected] Hungary ✔ Mathematics University of Miskolc, Institute of Szilvia Homolya [email protected] Hungary ✔ Mathematics University of Miskolc, Institute of Attila Körei [email protected] Hungary ✔ Mathematics

Submission 8 Title: The Copson and Curle Lectures, University of St Andrews Paper: (Feb 18, 11:30 GMT) memorial lectures University of St Andrews Author keywords: Newby Curle Edward Copson curle memorial lecture (142), applied mathematics (110), prime number (90), copson lecture (70), newby curle (60), curle lecture (60), regius professor (50), first copson lecture (47), sir jame lighthill (47), professor edward copson (47), professor newby curle EasyChair keyphrases: (47), united kingdom mathematics trust (40), lecture theatre (40), fusion power (40), mathematical biology (40), fluid mechanic (40), quadratic equation (40) We discuss two named lecture series given at the University of St Andrews over the last thirty years. They are the Curle Lectures named after Professor Newby Curle, formerly Gregory Professor of Applied Mathematics at the University of St Andrews, and the Abstract: Copson Lectures named after Professor Edward Copson, formerly Regius Professor of Mathematics at the University of St Andrews. We will discuss the range of topics of the lectures and give brief biographical information about the lecturers. We will also discuss the impact of these lecture series both on the student population and also on the wider public in St Andrews. Submitted: Feb 18, 11:30 GMT Last update: Feb 18, 11:30 GMT Authors first name last name email country organization Web page corresponding? Colin Campbell [email protected] United Kingdom University of St Andrews ✔ Edmund Robertson [email protected] United Kingdom University of St Andrews

Submission 9 Title: The International Congresses of Mathematicians - politics and mathematics Paper: (Feb 18, 12:19 GMT) International Congresses of Mathematicians Author keywords: International Mathematical Union organisation difficulties international mathematical union (221), american mathematical society (95), mathematical society (85), german mathematical EasyChair keyphrases: society (79), world war (70), plenary lecture (70), international research council (63), international mathematical (60), field medal (50), felix klein (50), mathematician attending (40), jesse dougla (40) The International Congresses of Mathematicians began in 1897 and, except for breaks during the two World Wars, has continued to Abstract: be held regularly ever since. In this lecture I want to look more at the politics behind the organising of the International Congresses of Mathematicians than at the mathematical lectures at these congresses. Submitted: Feb 18, 12:19 GMT Last update: Feb 18, 12:19 GMT Authors first name last name email country organization Web page corresponding? Edmund Robertson [email protected] United Kingdom University of St Andrews ✔

Submission 10 Title: The History of Mathematics in Basic Education courses in Portugal - a reflection for the future of teacher education training of elementary school teachers Author keywords: history of mathematics history of mathematics as a tool in educational context History of Mathematics (HM) is a tool that can be very useful in an educational context. However, HM is a very broad corpus of knowledge and it is necessary to reflect on what HM should be taught in the initial basic teacher education programs. In this presentation, we make this reflection by trying to justify which HM contents are truly essential for teachers and their future professional practices. In our opinion, HM contents should focus on the following topics: diverse ancient peoples numbering systems, Abstract: different algorithms for performing arithmetic operations and solving equations, geometry topics, as well as knowledge of various local/national HM aspects. We also give a brief description of Portuguese initial basic teacher education programs, giving special focus on the mathematics syllabus in these programs. And we present also several references to international studies that show the importance of using HM in the classroom context. Submitted: Feb 26, 13:48 GMT Last update: Feb 26, 13:48 GMT Authors first name last name email country organization Web page corresponding? Helder Pinto [email protected] Portugal I. Piaget and CIDMA-UA ✔ Cecília Costa [email protected] Portugal UTAD and CIDTFF ✔

Submission 11 Title: From the Theorem of the Broken Chord to the Beginning of the Trigonometry The Theorem of the Broken Chord Author keywords: Chord table Origins of Trigonometry The subject of this paper is to present Archimedes’ Broken Chord Theorem (3rd century BC) and its relationship to the origins of Trigonometry. We learn about this theorem from the medieval Arab mathematician Al Biruni in his treatise entitled Book on the Derivation of Chords in a Circle, which the Swiss Heinrich Suter (1848-1922) translated into German (Zurich, 1910). In this work of Al Biruni, we can find 22 proofs of the Broken Chord Theorem, amongst which three are attributed to Archimedes. Al Biruni’s methods follow Ptolemy in the Almagest, where he has an elaborate method of calculating the table of chords. In turn Ptolemy uses ideas of Hipparchus Abstract: who also constructed a table of chords which, however, is lost today. An analysis of techniques from antiquity shows that Archimedes’ Broken Chord Theorem leads to the same steps. In fact we can recognize the Broken Chord Theorem and other results of Archimedes in a proof of a theorem in the Almagest. The Broken Chord Theorem served Archimedes in his studies of Astronomy as an analogous formula to our sin(x-y)=sinxcosy-cosxsiny and since this formula gives the ability to construct a chord table, we can assume that Archimedes was in possession of one of these. So, in Archimedes’ work we detect the first glimmers of Trigonometry and we have every reason to ask ourselves: Is Archimedes the founder of Trigonometry? Submitted: Mar 04, 19:35 GMT Last update: Mar 04, 19:35 GMT Authors first name last name email country organization Web page corresponding? Maria Drakaki [email protected] Greece University of Crete ✔

Submission 12 Title: Selected Passages from the History of the Szeged Mathematical School Paper: (Mar 07, 10:27 GMT) Szeged Mathematical School Riesz Frigyes Haár Alfréd Author keywords: Kerékjártó Béla Szökefalvy Nagy Béla Rédei László Kalmár László The talk covers three important chapters from the history of the Szeged mathematical School. The first one is about the very beginnings, the creation of a new university in Cluj (Transylvania) and the foundation of the higher mathematical education and research in this region. After World War I the Hungarian part of it (most of the Hungarian professors and students) is forced to move out of the newly formed Rumanian State. After a short transitional period (in Budapest), they settled in Szeged, where a new university was founded. Abstract: The second chapter covers the first two decades marked by three outstanding professors: Frigyes RIESZ, Alfréd HAÁR and Béla KERÉKJÁRTÓ. They formed the so called first triumvirate.After the death of Haár (1933), moreover Kerékjártó (1938) and Riesz (1946) moved to the Budapest University a new, the second triumvirate has been formed. The members of it was Béla SZŐKEFALVI NAGY, László RÉDEI and László KALMÁR. Their period lasted until about 1970, the work of them is the content of the third chapter. Submitted: Mar 07, 10:27 GMT Last update: Mar 07, 10:27 GMT Authors first name last name email country organization Web page corresponding? Bolyai Institute of the University of Lajos Klukovits [email protected] Hungary ✔ Szeged

Submission 13 Title: Figurate numbers A Bridge between History and Learning of Mathematics Paper: (Mar 07, 10:34 GMT) Bruner’s representation theory Tobias Mayer’s Mathematischer Atlas Author keywords: figurate numbers space numbers mathematical problems for secondary school students canon ball (100), mason problem (100), figurate number (70), iconic representation (60), tobia mayer (60), space EasyChair keyphrases: number (60), mathematical problem (50), symbolic representation (50), space figurate number (47), old historical problem (47), mayer mathematischer atlas (47), nagy karoly mathematical student meeting (46) It is necessary to rethink the main principles of the Hungarian mathematics teaching, to apply new methods and new contents, to renew the training of the teachers in the spirit of Tamás Varga. Nowadays the Hungarian mathematics teachers are uncertain in consequence of the bad PISA results. They want to teach better, but they need some help. In this presentation I want to share my teaching experiences and give a new approach to the practice of mathematics Abstract: instruction, connecting a problem of the history of mathematics with the modern learning of mathematics. We present some problems of Mayer’s Mathematischer Atlas (1745), and analyze his method in discussing figurate and space numbers. We deal with some other mathematical problems which were posed for secondary school students (AMC 10, Mason’s problem, KöMal problems, Viviani’s theorem). Classification: A30, B50, C70 Submitted: Mar 07, 10:34 GMT Last update: Mar 07, 10:34 GMT Authors Web first name last name email country organization corresponding? page Institute of Mathematics, University of Tünde Kántor [email protected] Hungary ✔ Debrecen

REMARK: This submission was extended and re-submitted (see submission No 4.) Submission 14 Title: Lost books and forgotten theorems from ancient Greek Geometry Greek Geometry Lost Author keywords: Books forgotten theorems From ancient Greek books on Geometry today only a small fraction survives. They are authored by about 20 geometers, some of whom are outstanding mathematicians, such as Archimedes, Apollonius, Euclid, and Pappus. On the other hand we know the names of more than 300 ancient Greek geometers, some of whom are also first rate, but their work is lost. In the first part of the talk we will firstly discuss how the surviving Geometry books reached us and secondly which are our original sources that give information about the lost ones. For some of the lost texts we have only the title or a summary or a few fragments, but nothing beyond that. It is certain that some of these are a great loss, as time did not spare even exceptional geometers such as Pythagoras, Archytas, Eudoxus, Archimedes, Apollonius and Euclid. Abstract: In the second part of the talk we will discuss some of the surviving but forgotten geometric results from Greek antiquity. The reasons that they were forgotten is either because they were not noticed as much as they should or because in the course of techniques have changed, so they are obsolete in spite of their quality. For example we have a delightful theorem by Heron of Alexandria that is mentioned with a full proof, in the Arabic commentary by An-Nayrizi of Euclid’s Elements. We shall discuss this theorem and give its proof. Other forgotten theorems include Ptolemy’s geometric method described in full in his Almagest, of calculating sin1o which today is done using Taylor series. Another is the brilliant method of Archimedes to determine the surface area of a sphere, which today is done by integration via the Fundamental Theorem of Calculus. We will discuss several such forgotten results. Submitted: Mar 12, 18:53 GMT Last update: Mar 12, 18:53 GMT Authors last first name email country organization Web page corresponding? name Michael Lambrou [email protected] Greece University of Crete ✔

Submission 15 Title: George Polya’s influence on mathematics competitions in the USA modern heuristic Author keywords: problem solving competitions Soon after George Polya settled in Palo Alto, he published his most widely read book How to Solve It: A New Aspect of Mathematical Method, and started to collaborate on running a high school mathematics competition at Stanford University. Polya also served in constructing the first post-war contest for the William Lowell Putnam Mathematical Competition, and was a member of the Putnam Abstract: Prize Committee from 1948 until 1950. Polya’s international perspective to learning and doing mathematics made a palpable impact on the justifications for conducting mathematics competitions, the types of problems posed, and the ways to effectively prepare for the competitions. I plan to highlight samples of this impact in the USA during my presentation. Submitted: Mar 14, 05:11 GMT Last update: Mar 14, 05:11 GMT Authors last first name email country organization Web page corresponding? name Agnes Tuska [email protected] United States California State University, Fresno ✔

Submission 16 Title: Theoretical research frameworks on history of mathematics in mathematics education: current needs and emergent perspectives history of mathematics mathematics education Author keywords: Theoretical frameworks epistemology methodology In this communication, we will emphasize on current needs and issues lived in the community of researchers concerned with history of mathematics in mathematics education. Considering recent works focusing on these current needs and issues, we will focus on a convergent and recurrent point: the need for proper theoretical and/or conceptual frameworks in the field of research. Firstly, we will discuss more precisely about the needs for theoretical frameworks. In order to do so, we will try to highlight some suggesting elements around what a proper theoretical or conceptual framework in the field should be made of and to make clearer the possible uses that could be made of such framework in terms of research perspectives. Secondly, we will introduce and discuss about five propositions that can be found in literature and that are carrying element for a possible theoretical framework in the field: - The humanist perspective, - The hermeneutic perspective, Abstract: - The discursive and pragmatic perspective, - The sociocultural perspective, - The dialogical and ethical perspective. For each of these perspectives, we will try to highlight some underpinned epistemological and pedagogical fundaments or assumptions. Then, we will discuss the possible implications of these fundaments and assumptions on how the role and the potential of history of mathematics for teaching and learning mathematics are understood and on how history could/should be effectively convoked in mathematics classrooms. Finally, we will raise some possible methodological issues and concerns for each perspective. When it’s possible, references to studies based on these perspectives will be described in order to illustrate our point. We hope that our communication will help to understand more precisely the current needs and issues lived in the community of research concerned with history of mathematics in mathematics education and will help to support researchers that are trying to build rigorous, coherent and instructive studies in the field. Submitted: Mar 14, 23:21 GMT Last update: Mar 14, 23:21 GMT Authors first name last name email country organization Web page corresponding? Université du David Guillemette [email protected] Canada Québec à ✔ Montréal

Submission 17

Title: Methodological Landmarks in the History of Heuristic from Didactic Perspectives

Paper: (May 20, 17:26 GMT)(previous versions)

Heuristic

mathematical discovery

historical reconstruction

George Pólya Author keywords: Imre Lakatos

historiography of mathematics

methodology of teaching mathematics

social role of mathematics

problem solving (185), mathematical method (110), problem solving method (79), rational reconstruction (70), historical nature (60), EasyChair keyphrases: mathematical heuristic (50), subject matter (50), teaching and learning (47), general method (40), mathematical discovery (40), problem situation (40), time dependent (40)

“Heuriskein” is a word known from very ancient times, although not so the theory of Heuristic which gained an ahistorical meaning in spite of the fact that the time-dependent characteristics of the term are related to “historeîn” since at least Herodotus and Thucydides. Its modern "methodological" sense that is used in educational contexts was reintroduced by George Pólya and is often mixed with Herbert Simon’s and Allen Newell’s intellectual legacy and their efforts of modelling discovery. Meant as the study of problem-solving methods or as a set of principles including reflection on practices that guide our ways of doing mathematics and scientific research, the term is often used in a Abstract: timeless sense which leads to reading the present into the past. I discuss some points which show that “heuristic” is embedded not just in the history of mathematics but also in historical methodology and educational practice. Making some essential distinctions in the meaning of the term, I point out, going beyond Pólya with Imre Lakatos, that the methodology of the reconstruction of thoughts overlap with the field of discovery meanwhile both are time-dependent. I spell out my case in relation to Pólya’s conception itself, with implications of interest in changes of educational practice that promoted its worldwide spread and its historical extensions. Underlining Pólya’s appreciation of primary sources, I call attention to differences in awareness of time contexts of mathematics before and after WW2 in Hungary, Europe and in the USA and relate significant changes in the social role of mathematics to the transformation of its educational conceptions in the last two centuries. The main point is that changes in the use of the history of mathematics in the teaching of mathematics and in its historiography can be considered as methodological landmarks both from didactic perspectives and for the practice of presenting mathematics itself.

Submitted: Mar 15, 15:03 GMT

Last update: May 20, 18:33 GMT

Authors

first last email country organization Web page corresponding? name name

Andras Research Centre for the Humanities,

Benedek [email protected] Hungary http://www.fi.btk.mta.hu/index.php/en/ ✔ G Institute of Philosophy

Submission 18 Title: How can we use the results of mathematics history in the teaching of calculus? We will send it later. Author keywords: second third Abstract: We will send it later. Submitted: Apr 19, 11:19 GMT Last update: Apr 19, 11:19 GMT Authors first name last name email country organization Web page corresponding? Katalin Munkácsy [email protected] Hungary Eotvos University, ELTE ✔

Eleonóra Stettner [email protected] Hungary Kaposvári Egyetem

Submission 19 Connection of old and new mathematics in works of Islamic mathematician with a look to application of history of mathematics in education of Title: mathematics Paper: (Apr 20, 00:27 GMT) history of mathematics Author mathematics education keywords: innovation We know that all over the world mathematics is a difficult subject. In the last fifty years different tools were used to understanding

Mathematics .for example mathematical software like Mathematica, Mathcad and so on, however one of the helpful tools is the history of Abstract: mathematics, which during last little decade become very important. With history of mathematics teacher can make change of style in teaching, when is necessary. With history of mathematics student can understand that science is work of the whole civilization. In this article we show how history of mathematics can help teacher and student in maths subject. Submitted: Apr 20, 00:27 GMT Last update: Apr 20, 00:27 GMT Authors first name last name email country organization Web page corresponding? Saeed Seyed Banihashemi [email protected] Japan Iran embassy ✔

Submission 20 Title: Imre Vajkovics Elements of Arithmetics Educational reform Author keywords: Nagyszombati Egyetem - University of Trnava Kassa - Cassoviae - Kosice Imre Vajkovics (Oradea, July 22, 1715 - Kalocsa, November 28, 1798) Doctor of Theology and Philosophy, Priest of the Society of Jesus, later Grand Priest of Kalocsa has published and has presented his arithmetics in Kosice, 1753. on the 1st of June. Élementa arithmetiae numericae et literalis practicae et theoreticae in usum discentium proposita. Cassoviae, 1753. Abstract: Promotore Emer. Vajkovics, Promotore reverendo patre, Anno MDCCLIII, Junio, Die Prima. His presentation and manual edited in Kassa (Cassoviae, Kosice), may be consider as part of the reform of the Nagyszombati Egyetem, University of Trnava, initiated that year, in order to introduce natural science subjects in teaching, beside initial Theology and Philosophy education. Submitted: Apr 23, 12:17 GMT Last update: Apr 23, 12:17 GMT Authors last first name email country organization Web page corresponding? name Péter Körtesi [email protected] Hungary University of Miskolc ✔

Submission 21 Title: How to divide a cake fairly: The legacy of Dénes König Dénes König fair division Author keywords: Kuhn's algorithm Frobenius-König theorem Suppose that N players are planning to divide among themselves a continuous but highly heterogeneous object--say a fancy cake (but it could also be something more serious like a piece of land or even an entire territory). The division is said to be fair if each player receives a share that in his or her opinion has a value that is at least 1/N th of the total value of the "cake". For N = 2 players there is a well known procedure for dividing a cake fairly: one player cuts the cake in two, the other player chooses one of the pieces. A generalization of this idea for an arbitrary number of players was developed by the American mathematician Harold Kuhn in the Abstract: 1950's. Kuhn's algorithm is based on a theorem by the Hungarian mathematician Dénes König in the 1930's (the theorem is now mostly known as the Frobenius-König Theorem).

In this talk I will (a) give a brief description of Kuhn's algorithm, (b) describe the Frobenius-König theorem (no proof), and (c) talk about the life and legacy of Dénes König. Submitted: Apr 30, 17:59 GMT Last update: Apr 30, 17:59 GMT Authors first name last name email country organization Web page corresponding? Peter Tannenbaum [email protected] United States California State University, Fresno ✔

Submission 22

Title: Old and new Kepler-Poinsot solids

Platonic solids

Author keywords: Kepler-Poinsot solids

great dodecahedron

The five regular convex polyhedra are called the ‘Platonic solids’, because Plato wrote about them in a his Timaeus (c. 360 B.C.). If he was the first to have discovered them, or if it was his contemporary Theaetetus, is a matter of discussion, as is the issue of the even older carved stone balls found in Scotland and vaguely representing those polyhedra. In any case, our topic here are the Kepler-Poinsot solids, the four regular star polyhedra, called small and great stellated dodecahedron, great dodecahedron and icosahedron. The first two are said to have been discovered by Kepler in 1619, the last by Poinsot in 1809. However, the small stellated dodecahedron was represented by Paolo Uccello in the St. Mark's Basilica (Venice) and dates from the 15th century, Abstract: while Wenzel Jamnitzer depicted the great stellated dodecahedron in a book from 1568. Less well-known is that Lorenz Stör represented this solid in 1567. Jamnitzer also drew the great dodecahedron but some pretend he didn’t realize it, as he made many ‘artistic variations’ of the Platonic solids. However, in a Spanish book from 1688, entitled “The excellent arts of perspective”, an (unknown) author clearly describes how to draw it, over a century before Poinsot (this rare book was brought to my attention by A. R. Buitrago, Albacete, Spain). The topic seems of interest since even today, as new Kepler- Poinsot polyhedra were discovered, be it of infinite type. The generalization of Arthur Cayley’s formulas (19th century) for the regular Kepler-Poinsot polyhedra to these infinite cases still needs to be confirmed.

Submitted: May 18, 08:38 GMT

Last update: May 18, 08:38 GMT

Authors

first name last name email country organization Web page corresponding?

Dirk Huylebrouck [email protected] Belgium KU Leuven http://www.dirkhuylebrouck.be/ ✔

Abstract submitted via e-mail only

Dénes Nagy Anti-golden-sectionism

The GS is very important in math education, but we should use the "real examples", not the nonsense cases (from the Egyptian pyramids to Greek temples, from Leonardo to the cubists - because, with great probability the GS played no importance in these cases).