No. 75 November 2010

This and earlier issues of the Newsletter can be downloaded from our website http://www.clab.edc.uoc.gr/hpm/

mathematician's scholarly work in the history Professor R. C. Gupta receives of mathematics. Kenneth O. May Prize Prof. Gupta's major contributions in the Professor Radha Charan Gupta received field include work on the history of the Kenneth O. May Prize for the History of development of trigonometry in India. He Mathematics at the International Congress of authored the chapter 'Historiography of Mathematicians (ICM) on 27 August 2010. Mathematics in India' in: Dauben and Scriba Prof. Gupta was chosen for the 2009 prize (eds.) Writing the History of Mathematics: Its jointly with Prof. Ivor Grattan-Guinness of Historical Development, Birkhauser, 2002. He UK by the International Commission for the was the President of the Association of History of Mathematics. Prof. Kim Plofker Mathematics Teachers of India from 1994 presented the award. until recently. He also founded the journal Ganita Bharati (meaning “Indian Mathematics”) and has written several articles in the journal. He is also a frequent contributor to the HPM Newsletter (including this issue).

New Books

De grands défis mathématiques: d’Euclide à Condorcet [On major challenges of mathematics: from Euclid to Condorcet] Evelyne Barbin (ed.) The Prize is named after the mathematician Paris: Vuibert; Adapt-Snes, 2010, 176pp, and historian Kenneth O. May, founder of the ISBN 9782356560100;9782311000191 International Commission for the History of Mathematics and its journal Historia The output of IREM colleagues is admirable Mathematica. The prize, instituted in 1989, and enviable. This latest volume, under the consists of a bronze medal and is given once editorship of Evelyne Barbin, provides nine in four years in appreciation of a incidents or problems that stimulated mathematical responses. The purpose of the

HPM Newsletter No. 75 November 2010 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ 1 descriptions here is to provide original ways of obtaining 5, something which students material that can be used in the classroom of can easily correct. Gérard Harmon treats early upper secondary schools, that is, in those years approaches to Bayes’ Theorem by Condorcet immediately preceding university entrance. (1805) and then Lacroix (1816) both The claim is slightly misleading in that the considering possible outcomes of taking black article by Evelyne Barbin describes work with or white balls from an urn. I also enjoyed third year university students preparing to reading the background to modern digital type teach and the historical time frame of the fonts and the use of Bézier curves which articles extends to before Euclid and after extended Loïc Le Corre’s lesson on Dürer’s Condorcet. The work is divided into four geometric representation of fonts. sections: measurement of magnitudes, The educational context – defending the representing magnitudes, probability, choice of material – makes us aware of French approximations to curves. concerns that would not trouble the British However entertaining and stimulating, I secondary mathematics teacher, more often feel that French studies linking history to accustomed to a pragmatic sloppiness of pedagogy would not appeal to the Anglo- approach to teaching mathematics, particularly Saxon reader and I even wonder how much with regard to notation. Vive la différence! they might be taken up in the French Chris Weeks, classroom, save for the enthusiasts who have Great Britain written these pieces. It is therefore encouraging to find, alongside suggested Tinhlelo exercises based on historical texts, or at least Interweaving Art and Mathematics: Colourful inspired by them, some illustrations of work Basket Trays from the South of Mozambique done by students themselves. Thus Dominique Paulus Gerdes Tournès from Réunion has examples of Lulu, 2010 students’ constructions of solution curves of differential equations, using Euler’s method, “Tinhlelo” exist in each family of the vast which they did following a guided reading of Mozambique. We need to know that there are an extract of Institutionum calculi integralis no factories for the production of the (1768). From Patrick Guyot of Bourgogne we “tinhlelo.” “Tinhlelo” knowledge is have an account of students working on the endogenous knowledge that repeats itself, problem of inscribing a square inside a recreates itself, and adorns itself in the triangle which includes images of the students’ multiplicity of its models, sizes, colours, and initial naïve attempts. Indeed, it is clear that in functions. No conventional school orients this almost all cases the writers are telling us of learning. The families, the remarkable, curious material they themselves have used. and master artisans, are the ones who are But this collection of articles is more than a creating and maintaining these products of description along the lines of a simple ‘these eloquent usefulness in our families. The have worked for us’. Each piece comes with families recreate the art of this mathematical an introduction of the mathematical, historical weaving! and educational context and some offer further (From the foreword) reflections and all have source bibliographies. There are some nice discoveries for me. Reports on new books are welcome. Leibniz does not appear among the founders of probability calculus but Renaud Chorlay has used a letter from him describing the expected outcomes of a simple game of dice which is easy to read and has the pedagogic advantage that his assumptions are wrong. He fails to count both (3,2) and (2,3) as distinct

HPM Newsletter No. 75 November 2010 2 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ Conference reports • History and epistemology as tools for an interdisciplinary approach in the teaching 6th European Summer University on the and learning of mathematics and sciences; History and Epistemology in Mathematics • Cultures and Mathematics; Education • Topics in the history of mathematics 19-23 July 2010 Vienna education. Each theme had a plenary lecture, several These Summer Universities started in workshops, oral presentations and posters, and Montpellier (France) in 1993 and from the some also had panel discussions. Since it was beginning provided an occasion for impossible to attend all the conferences and researchers and teachers, not only to hear workshops, we shall only comment here on about new experiences and ideas and share the ones we found most useful or interesting. their own, as in an ordinary international meeting, but above all to learn from each other, as in an open school inspired by cooperative learning. The main organizers were Evelyne Barbin, from University of Nantes (France); Manfred Kronfeller, from Vienna University of Technology (Austria); and Costantinos Tzanakis from University of Crete (Greece) and supported by the international Scientific Program Committee composed of 28 people from 18 different countries. There were 154 participants from 34 different countries, of whom about 75% made contributions (talks, workshops, posters). The conference was sponsored by the Austrian Federal Ministry of Science and Research, the Government of the City of Vienna, the Vienna

Convention Bureau, Casio Europe and Texas Of the plenaries, the most appreciated was Instruments. that by Michael Glaubitz from the Albert This report is from a “teacher’s point of Einstein Gymnasium in Hameln (Germany): view” and summarises the reactions of about The Use of Original Sources in The classroom 30 classroom teachers who attended the - Empirical Research Findings (theme 3). He conference. I would particularly like to thank proposed the hermeneutic approach to the use Alexandra Lux from BRG Purkersdorf of historical material claiming that reading an (Austria) in helping to prepare the report. original source can be an especially rewarding The main themes of this 6th ESU were: enterprise which is capable of substantially • Theoretical and or conceptual frameworks deepening mathematical understanding, for integrating history in mathematics enriching classroom activities and developing education; learners’ beliefs in mathematics. He set out, • History and epistemology implemented in from an empirical point of view, feasible mathematics education: Classroom conceptual designs, necessary preparations experiments & teaching materials, and prospective effects when reading an considered from either the cognitive original source in class using the example of or/and affective point of view; survey of quadratic equations. This was judged as curricula and textbooks; “probably the most important lecture for actual • Original sources in the classroom, and teaching in class”, and suggested further their educational effects; questions to be probed. We were really sorry

HPM Newsletter No. 75 November 2010 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ 3 that the same author’s workshop was Livia Giacardi from Università di Torino cancelled. (Italy) were judged very interesting. The first one, titled Practical Geometries in Islamic Countries: the example of the division of plane figures (theme 4) presented different geometric problems linked to various other subjects as among others, the practices of craftsmen, architect or jurists. The interest for the class practice is the wide range of procedures in which the whole mathematical knowledge seems to be involved. The second one, named From Rome to Rome: Events, People, and Numbers during ICMI's First Century (theme 6), was enjoyed because of his Another successful plenary lecture was giving insights to some key ideas that led to Michael Fried’s: History of Mathematics in the birth and to the growth of ICMI and to the Mathematics Education: Problems and changes in methodological approaches to Prospects (theme 1). The speaker from the problems that offered, especially to young Ben Gurion University of the Negev (Israel), teachers, a global perspective and a theoretical suggested that teaching maths using history basis. has to involve a redefinition of our general goals for mathematics education more than the concrete ways in which we bring history of mathematics into the classroom. History as a goal means history as a new view point on the mathematical landscape in which a central Our favourite panel discussion was the one idea is the presence of original texts in a form chaired by Costantinos Tzanakis with or another. contributions from Anne Boyé from Centre François Viète, Université de Nantes (France), Adriano Dematté from Università di Genova (Italy), and Ewa Lakoma from Military University of Technology Warsaw (Poland). The title was The history of mathematics in school textbooks. The close link with teachers’ everyday practice together with the well constructed discussion appealed to us, even though some detected a lack of optimism in some panellists. For a future ESU it would be good to have a panel discussion on the use of history for students who find mathematics difficult. (Could history of maths help them to do better?) The most interesting workshops we thought were: Maps, Narratives and Orientations: The use of Concept Maps for Also the two plenary lectures given by exploring our Mathematical Heritage in the Marc Moyon, from the Centre d’Histoire des Classroom (theme 2), by Leo Rogers from Sciences et d’Epistémologie de Lille1 & Department of Education, University of IREM de Lille; and Fulvia Furinghetti from Oxford (UK); Defining derivatives, integrals Università di Genova (Italy) together with and continuity in secondary school: a phased

HPM Newsletter No. 75 November 2010 4 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ approach inspired by history (theme 2), by special interest to classroom teachers could Hilde Eggermont from Sint-Pieterscollege, help the absolute beginners to choose from Leuven (Belgium) together with Michel such a widespread menu. Maybe, the Roelens from Katholieke Hogeschool Limburg workshops could be of two different types: the (Belgium); Good Old arithmetic (theme 3), by usual ones on one hand and some others more Frédéric Metin from IREM of Dijon, similar to the pre-service teachers trainee Université de Bourgogne (France); activities on the other. This would help young Pedagogical and Mathematical Games people or other new entries to participate in an throughout the Times: from Rithmomachia to active way and to make the experience more Hex (theme 3) by Jorge Nuno Silva from useful and nice. University of Lisbon (Portugal); Digitising the To conclude, let me offer a suggestion past mathematics by the future mathematics about the arrangements. I think that the reason (theme 3), by Snezana Laurence from Bath why in recent ESUs there have not been as Spa University, Culverhay Campus (UK); many teachers participating is also a problem Mathematisation of nature and new of sponsorship. The economic crisis has had conceptions of curves in the years 1630 the effect of drastically reducing the money (theme 4), by Evelyne Barbin; The mediaeval available for these activities, especially for geometries: a way to use the history of teachers (this is the case in Italy and must be mathematics in the classroom of mathematics the same for other European countries). So, by Marc Moyon. The main reason why these even if personally I always was very happy were so popular was because they related when we stayed in a big city like Prague or, directly to classroom work and/or offered this year, Vienna (I really enjoyed the operas useful material. I’ve seen and listened to), it might be better for We have some critical comments and future meetings to take place in quieter and suggestions. There were rather too many short less expensive places. May be it also help if oral presentations. We noticed a really there were a cheaper registration fee for widespread range of quality, from rather poor, classroom teachers. to quite useful. Since the Summer Universities declared aims are to provide a forum for presenting research in mathematics education and innovative teaching methods and to give the opportunity to mathematics teachers, educators and researchers to share their teaching ideas and classroom experience, it would be a good idea to reduce the number of oral presentations so as to make room for, say, “panel discussions workshops”, coordinated by the theme of the discussion’s proposer, in which each participant has the right to say his/her opinion. This would be an opportunity, especially for new comers, who often feel that they are the only ones not already known to everyone else, to be introduced to the researchers and teachers more interested in the same questions as them. It would help first- time teachers to identify and meet other teachers at the very beginning of the conference. Even though there should not be Finally, we would all like to thank any real separation between researchers and Manfred Kronfeller and the local organizing teachers, a clearer indication of sessions of committee for the marvellous work they did

HPM Newsletter No. 75 November 2010 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ 5 and for the kindness and the smiles they gave Catepillan, Ximena and Waclaw to all of us even if they sometimes were under Szymanski, "Maya Calendar Conversions," stress. Thanks a lot to Evelyne Barbin and Loci (July 2010). Costas Tzanachis too, for their giving the Clark, K. M. (2010). Connecting local teachers the opportunity to contribute to the history, ancient history, and mathematics: the improvement of ESU’s. Eustis Elementary School pilot project. BSHM Bulletin: Journal of the British Society for the Caterina Vicentini History of Mathematics, 25(3), 132-143. Istituto d’Arte “Max Fabiani”, Gorizia (Italy) de Wreede, L. C. (2010). A dialogue on the Scienza under 18 isontina, Fogliano use of arithmetic in geometry: Van Ceulen's Redipuglia (Italy) and Snellius's Fundamenta Arithmetica et Geometrica. Historia Mathematica, 37(3), (Editor’s note: Impressions on the ESU6 can 376-402. also be read at the blog Descotes, D. (2010). An unknown Teacher Educator Bjørn mathematical manuscript by Blaise Pascal. http://teachereducatorbjorn.blogspot.com/searc Historia Mathematica, 37(3), 503-534. h?q=ESU Galuzzi, M. (2010). Newton's attempt to from where the photos are taken.) construct a unitary view of mathematics. Historia Mathematica, 37(3), 535-562. Guan, Y. (2010). A new interpretation of The editors welcome reports from Shen Kuo’s Ying Biao Yi. Archive for History conferences. of Exact Sciences, 64(6), 707-719. Hogendijk, J. P. (2010). The scholar and the fencing master: The exchanges between Joseph Justus Scaliger and Ludolph van Ceulen on the circle quadrature (1594-1596). Work in progress Historia Mathematica, 37(3), 345-375. Kourkoulos, M. & Tzanakis, C. (2010). We encourage young researchers in fields History, and students' understanding of related to HPM to send us a brief description variance in statistics. BSHM Bulletin: Journal of their work in progress or a brief description of the British Society for the History of of their dissertation. Mathematics, 25(3), 168-178. Krüger, J. (2010). Lessons from the early seventeenth century for mathematics curriculum design. BSHM Bulletin: Journal of the British Society for the History of Mathematics, 25(3), 144-161. Mancosu, P., & Arana, A. (2010). Have you read these? Descartes and the cylindrical helix. Mathematica, 37(3), 403-427. Argün, Z., Arıkan, A., Bulut, S., & Maronne, S. (2010). Contexts, emergence Sriraman, B. (2010). A brief history of and issues of Cartesian geometry: In honour of mathematics education in Turkey: K-12 Henk Bos's 70th birthday. Historia mathematics curricula. ZDM, 42(5), 429-441. Mathematica, 37(3), 341-344. Bos, E.-J. (2010). Princess Elizabeth of Maronne, S. (2010). The ovals in the Bohemia and Descartes' letters (1650-1665). Excerpta Mathematica and the origins of Historia Mathematica, 37(3), 485-502. Descartes' method of normals. Historia Bressoud, David M. (2010). Historical Mathematica, 37(3), 460-484. Reflections on Teaching Trigonometry. Musto, G. (2010). Mathematical timelines. Mathematics Teacher, 104(2), 106. BSHM Bulletin: Journal of the British Society

HPM Newsletter No. 75 November 2010 6 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ for the History of Mathematics, 25(3), 162- http://www.maa.org/sigmaa/hom 167. Papadopoulos, I. (2010). “Reinventing” HPM Americas Techniques For The Estimation Of The Area http://www.hpm-americas.org/ Of Irregular Plane Figures: From The Eighteenth Century To The Modern Italian Society of History of Mathematics Classroom. International Journal of Science http://www.dm.unito.it/sism/indexeng.html and Mathematics Education, 8(5), 869-890. Pesic, P. (2010). Hearing the Irrational: Association pour la Recherche en Music and the Development of the Modern Didactique des Mathématiques: Concept of Number. Isis, 101(3), 501-530. http://www.ardm.asso.fr/ Rabouin, D. (2010). What Descartes knew of mathematics in 1628. Historia Commission Française pour l'Enseignement Mathematica, 37(3), 428-459. des Mathématiques: Schwartz, Randy K., "Combining Strands http://www.cfem.asso.fr/ of Many Colors: Episodes from Medieval Islam for the Mathematics Classroom," Loci Instituts de Recherche sur l'Enseignement (August 2010). des Mathématiques (IREM): Wagner, R. (2010). The natures of numbers http://www.univ-irem.fr/ in and around Bombelli’s L’algebra. Archive for History of Exact Sciences, 64(5), 485-523. Canadian Society for History and Philosophy of Mathematics http://www.cshpm.org

Brazilian Society for History of Mathematics Have you been here? http://www.sbhmat.com.br Nuncius Newsletter http://brunelleschi.imss.fi.it/nuncius/inln.asp?c In this section we bring links related to the =5302 scope of the HPM from around the world. Please send suggestions. International History, Philosophy and Science Teaching Group Societies and organisations www.ihpst.org

Commission on the History of Mathematics Centre for the History of the Mathematical in Africa (including newsletter) Sciences. http://www.math.buffalo.edu/mad/AMU/amuc The Open University, UK hma_online.html http://puremaths.open.ac.uk/pmd_research/CH

MS/index.html Association des Professeurs de

Mathematiques de l'Enseignement Public Oxford Museum of the History of Science [APMEP] History site: www.mhs.ox.ac.uk/exhibits/ http://www.apmep.asso.fr/BMhist.html http://www.mhs.ox.ac.uk/measurer/text/title.ht

m British Society for the History of http://www.mhs.ox.ac.uk/geometry/title.htm Mathematics [BSHM] http://www.mhs.ox.ac.uk/scienceislam/ http://www.bshm.org

HOMSIGMAA - History of Mathematics Special Interest Group of the MAA

HPM Newsletter No. 75 November 2010 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ 7 Topics and Resources Digitization of the oldest extant manuscript of Euclid's Elements Maths is good for you! http://librarieswithoutwalls.org/bookviewer/ http://www.mathsisgoodforyou.com/ With the heading 'History of mathematics for History of Statistics young mathematicians' Snezana Lawrence's http://www.stat.ucla.edu/history/ very attractive website contains many suggestions for lesson starters and background Images of Lobachevsky’s context information. http://www.ksu.ru/eng/museum/page0.htm

A list of resources on history of mathematics related to school in the Images of Mathematicians on Postage Norwegian language Stamps http://eleviki.wikidot.com/ressurser-om- http://members.tripod.com/jeff560/index.html matematikkhistorie-pa-norsk Photos of Mathematicians MATHS for EUROPE: The history of some http://www.math.uni- aspects of mathematics like: history of hamburg.de/home/grothkopf/fotos/math-ges/ mathematical persons, symbols, algorithms... Numdam-Digitization of ancient http://mathsforeurope.digibel.be/index.html mathematics documents http://mathsforeurope.digibel.be/list.htm http://www.numdam.org/en/ressnum.php http://mathsforeurope.digibel.be/olvp.htm http://mathsforeurope.digibel.be/olvp2.htm The Montana Mathematics Enthusiast http://mathsforeurope.digibel.be/olvp3.htm (journal) http://www.montanamath.org/TMME/ Ethnomathematics on the Web http://www.rpi.edu/%7Eeglash/isgem.dir/links Loci: Convergence: an online magazine of .htm the MAA providing resources to teach mathematics through its history About Medieval Arabic Numbers http://mathdl.maa.org/mathDL/46/ http://www.geocities.com/rmlyra/Numbers.ht ml International Journal for Mathematics http://www.geocities.com/rmlyra/arabic.html Teaching and Learning, http://www.cimt.plymouth.ac.uk/journal/defau Annotated Bibliography on Proof in lt.htm Mathematics Education http://fcis.oise.utoronto.ca/~ghanna/educationa Homepage of International Journal for the bstracts.html History of Mathematics Education http://www.tc.edu/centers/ijhmt/index.asp?Id= BibM@th Journal+Home http://www.bibmath.net/dico/index.php3?actio n=rub&quoi=0 Documents for the History of the teaching of mathematics in Italy Centro Virtual de Divulgación de las http://www.dm.unito.it/mathesis/documents.ht Matemáticas, esta siendo desarrollada por ml la Comisión de Divulgación de la Real Sociedad Matemática Española (R.S.M.E.) Ethnomathematics Digital Library http://www.divulgamat.net/index.asp http://www.ethnomath.org/

HPM Newsletter No. 75 November 2010 8 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ Some Japanese Mathematical Landscapes: Archimedes Project [Some famous The results of wandering in a beautiful mathematical books of the Renaissance country, with a mathematical eye, aided by a period are available on line, i.e. Pacioli’s digital camera, by A. Arcavi Summa] http://math.criced.tsukuba.ac.jp/museum/arcav http://archimedes2.mpiwg- i/arcavi_english/index.html berlin.mpg.de/archimedes_templates

Wann-Sheng Horng’s webpage Simon Stevin’s De Meetdaet [The Practice with HPM related materials in Chinese. of Measuring] http://math.ntnu.edu.tw/~horng/ http://www.math.leidenuniv.nl/~wiskonst/mee tdaet/index.html Fred Rickey's History of Mathematics Page and The Principal Works of Simon Stevin http://www.dean.usma.edu/math/people/rickey http://www.historyofscience.nl/works_detail.c /hm/default.htm fm?RecordId=2702

CultureMATH. Ressources pour les Mathematicians Gallery enseignants de Mathématiques http://www.math.uconn.edu/MathLinks/mathe www.dma.ens.fr/culturemath/actu/livres.htm maticians_gallery.php?Rendition=printerfrien dly The French INRP (National Institute for Pedagogical Research) is developing a website History of Mathematics on questions related to mathematics teaching: http://www.otterbein.edu/resources/library/lib EducMath pages/subject/mathhis.htm http://educmath.inrp.fr The Garden of Archimedes. A museum for Geometrical books and instruments from Mathematics 15th to 18th century http://web.math.unifi.it/archimede/archimede_ http://www.geometricum.com/ NEW_inglese/

David Henderson’ s Home Page Mathematical instruments [Educational and Historical Topics on http://brunelleschi.imss.fi.it/museum/esim.asp Geometry] ?c=500164 http://www.math.cornell.edu/~dwh/ and http://web.mat.bham.ac.uk/C.J.Sangwin/Slider Homepage of Albrecht Heeffer ules/sliderules.html http://logica.ugent.be/albrecht/ and http://www.mhs.ox.ac.uk/epact/catalogue.php? Homepage of Jens Høyrup ENumber=52265 http://www.akira.ruc.dk/~jensh/ Homepage of Eleanor Robson L'Enseignement Mathématique, Archive http://www.hps.cam.ac.uk/dept/robson.html http://retro.seals.ch/digbib/vollist?UID=ensma t-001 Flickr group for HPM related photos http://www.flickr.com/groups/812621@N24/ Homepage of Prof. Leo Corry http://www.tau.ac.il/~corry/ Monuments on Mathematicians http://www.w-volk.de/museum/exposi.htm Opera Mathematica of Christoph Clavius http://mathematics.library.nd.edu/clavius/ Video on the history of mathematics

HPM Newsletter No. 75 November 2010 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ 9 http://www.youtube.com/watch?v=wo- also of diameter d. Also let the cylinder be 6xLUVLTQ circumscribed by a cube of side d such that their bases and tops lie in the same planes. It International Journal for the History of was known that mathematics Education, Vol.of the cylinder /4 (5) http://www.comap.com/historyjournal/ Vol.of the cube Now let Vol.of the sphere We would like to provide a more (6) Vol.of the cylinder comprehensive list of websites containing resources useful to researchers and students From these two equations we see that (not necessarily in English). If there are any Vol. of sphere = /4 (7) you use, or you know are useful for students or researchers, please send your The mathematically exact value of k is easily recommendations to the editors. seen to be 2/3. The JZSS uses the simple approximation for calculating the area of a circle. 3 Therefore a suggested use of the very high value for getting (4) from (1) is Notices rejected as unlikely. In fact during the first century of CE and in the time of Zhang Heng (78-139 CE), the ratio in (5) was taken to be Volume of a sphere in Ancient China and India ¾ in China. At that time, it was thought that the ratio k also has the same value or ¾. Professor R. C. Gupta, Ph. D. (Hist. Of Math.) Ganita Bharati Academy Thus the early Chinese mathematicians were R-20, Ras Bahar Colony, JHANSI-284003, led, by (7), to the empirical rule India Vol. of a sphere, (8) The famous Greek mathematican Archimedes (died 212 BCE) had already found the correct Or, 9/16 (9) formula which is indeed the JZSS formula (4). This formula already greater than the true value of 4/3 (1) the volume but the calendrist Zhang Heng 0.5236 very nearly (2) thought otherwise and made it even worse by taking for the volume of a sphere of diameter d (=2r). On the other hand, the popular ancient Chinese 9/16 1/16 (10) work Jiu Zhang Suan Shu (JZSS) which 5/8 (11) reached its present form in the first century CE, contains two numerical problems in which However, this, compared with (8), leads to d is calculated by using the rule √10 which is better than 3 and better even than another of Zhang Heng’s values / (3) . A comment, attributed to Liu Hui (3rd cent. This clearly implies the formula CE), on the JZSS sphere problem shows that (4) (4) implies (6) with . According to Li Chunfeng (602-670), Zu Geng (=Zu Xuan), Suppose the sphere is inscribed in a right son of Zu Chongzhi (429-500), had stated that circular cylinder whose height is d and base is Liu Hui (like Zhang Heng) took (6) with

HPM Newsletter No. 75 November 2010 10 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ . There is some indication to show that In India by Śrīdhara (about 750 CE) in his Liu Hui considered this assumption to be Triśatikā (rule 47), by Pherū in his Gaņita- wrong although he was unable to derive the sāra (III, 46) and by others. However, if a rule correct rule. (to be modified) was presumed to involve the In India, Āryabhaţa (born 476 CE) in his square of π or of circular circumference, the work Āryabhaţīya (II,7) gives the rule for the adjustment was done by the factor 10/9. Thus volume of sphere as we find Mahāvīra giving his accurate rule for the volume of a sphere as (GSS, VIII, 28½) √ exactly (12) (17) where A is the area of a central section of the sphere (i.e. ). His commentator 5 (18) Bhāskara I (629 CE) quotes a rule (in Sanskrit) which happens to be the same as V1 in (11). which gives a formula equivalent to (4). The For Mahāvīra 3 and (4) were rough or rule was regarded empirical and so Āryabhaţa practical, and 10 was accurate. It may attempted to find a better one. √ be pointed out that (18) can also be obtained Mahāvīra (about 850 CE) in his Gaņita- sāra Sańgraha (=GSS, VIII, 28) has also from Āryabhaţa’s rule (12) by taking 3 , given the same rule and clearly stated it to be and using the usual Indian rule practical (i.e. not exact or accurate). The same √ , with a=1 and x=2. rule also appears in the Tiloyasāra (gāthā 96) of Nemicandra (about 975 CE), and in the In China, the exact Archimedean formula Gaņita-sāra (V, 25) of Ţhakkura Pherū (14th (1) was proved by Zu Geng (6th cent. CE). He cent.) who gave it the form used it for both 3 and for ·1 1 (13) convenience, although he knew even the far better value from his father. For the A detailed discussion of all these Indian simple value 3, we have simply rules show that (4) should be interpreted as (8) with 3. (19) The modification of empirical rules by adjusting them to suitable values of π is an If this is adjusted to a Jaina value , we ancient practice. For example, get

/2 (14) · (20) for the area of a circular segment (of chord c and height h) was supposed to be based on which is found in the Triśatikā (rule 56) of 3 for which it gives exact value in the Śrīdhara. The value is commonly case of a semicircle. It was modified or derived from the usual Jaina value √10 adjusted to in the form (used in their sacred works) by using the approximation · (15) √ , with a = 3 and x = 1. In the Mensurae attributed to Heron (1st cent. CE). More interestingly, (14) wa The formula (20) is also found in the adjusted to √10 (a popular Jaina value) in Siddhānta Śekhara (XIII, 46) of Śrīpati (11th the form cent.), and in the Mahāsiddhānta (XVI, 108) of Āryabhaţa II whose date, following recent th /2 · (16) research, has been changed from the 10 to the 15th century.

HPM Newsletter No. 75 November 2010 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ 11 The famous Indian mathematican Bhāskara 6. D. B. Wagner, “Liu Hui and Tsu Keng- th II (12 cent.) not only gave the correct chih on The Volume of a Sphere”, formulas for the surface and volume of a Chinese Science III (1978), 59-79. sphere but also supplied justifications. His Līlāvatī (rule 201) contains 4 (21) (22) Announcements of events The name of Archimedes, and his beautiful proof, is hardly ever met in ancient China or India. Was the ancient orient unaware of Greek achievements, or was it a case of older methods and practices being preferred even where new ones were known? The Jaina Pherū CERME 7 used the square form of the Jainisation February 9–13, 2011 adjustment factor 10/9 to improve the ancient Rzeszów, Poland formula (for surface of sphere) CERME is a Congress designed to foster a communicative spirit. It deliberately and (23) distinctively moves away from research which can be derived from a rule of presentations by individuals towards Mahāvīra (GSS, VII, 25); here C is the collaborative group work. Its main feature is a circumference of the great circle. Pherū also number of thematic groups whose members will work together in a common research area. used the same factor for V4 or which he Researchers wishing to present a paper at the perhaps picked up from (20). But in this case Congress should submit the paper to one of he was confused because here it is π which is 2 these groups. In 2011, Working Group 12 involved and not π . Had he used the (linear) (WG 12) will be on “History in Mathematics Jainisation factor 10/9, his result would Education”, see directly below. have been far better. Website: http://www.cerme7.univ.rzeszow.pl/

References CERME7: Working Group 12 1. L. Berggren et. al, Pi: A Source Book. History in mathematics education Springer, New York, 2004. Leaders 2. R. C. Gupta, “Volume of a Sphere in Uffe Thomas Jankvist (Denmark) [email protected] Ancient Indian Mathematics”, Journal Snezana Lawrence (UK), Jan van Maanen of the Asiatic Society, XXX (1988), (The Netherlands), Constantinos Tzanakis 128-140. (Greece). 3. R. C. Gupta, “Techniques of Ancient Scope and Focus of WG12 Empirical Mathematics”, Indian The integration of history of mathematics in Journal of History of Science 45 mathematics education has received increasing (2010), 63-100. attention recently. This attention is reflected by the many publications both in journals and 4. J.-C. Martzloff, A History of Chinese books and the increasing activities at Mathematics. Springer, Berlin, 1997. international conferences and meetings. 5. D. B. Wagner, “Doubts Concerning the However, empirical research and coherent Attribution of Liu Hui’s Commentary theoretical/conceptual frameworks within this (on JZSS)”, Acta Orientalia, 39 area have been emerging relatively recently. (1978), 199-212. The purpose of this CERME WG is to provide

HPM Newsletter No. 75 November 2010 12 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ a forum to approach mathematics education in guidelines-for-authors. To submit it, you must historical context, dedicated primarily to email your paper as a WORD document to theory and research on all aspects of the role, Uffe Th. Jankvist at [email protected], AND effect and efficacy of history in mathematics at the same time to the conference secretariat education. at [email protected] Call for papers and poster proposals If possible please also send a pdf version in WG12 in particular welcomes empirical and addition to the WORD document. theoretical research papers, but to some degree Reviews and Decisions also methodological and developmental Each paper will be peer-reviewed by at least papers, (10 pages max) and poster proposals (2 two persons from among those who submit pages) related to one or more of the following papers to this Working Group and the leaders. issues – although any paper/poster of Please expect to be asked to review up to three relevance to the overall focus of the group will papers yourself between 15th September and be taken into consideration: 22nd October 2010. It may be necessary for 1. Theoretical, conceptual and/or you to revise your paper before final methodological frameworks for including acceptance. Please reserve some time to do history in mathematics education; this in the second half of November. The 2. Relationships between (frameworks for group leaders will decide about the acceptance and empirical studies on) history in of posters. mathematics education and theories and Important dates frameworks in other parts of mathematics 15th September 2010: Deadline for submission education; of papers. 3. The role of history of mathematics at 1st October 2010: Deadline for submission of primary, secondary, and tertiary level, both poster proposals from the cognitive and affective points of 22nd October 2010: Deadline for reviewers to view; submit their reviews. 4. The role of history of mathematics in pre- 1st December 2010: Deadline for revisions to and in-service teacher education, from papers cognitive, pedagogical, and/or affective points of view; 5. Possible parallelism between the historical Colloque « L’enseignement des development and the cognitive mathématiques, des mathématiques du development of mathematical ideas; quotidien à la théorie en l’honneur de 6. Ways of integrating original sources in Nicolas Rouche » classrooms, and their educational effects, March 16-18, 2011 preferably with conclusions based on Mons, Belgium (March 16th), Lille, France classroom experiments; (March 17th-18th) 7. Surveys on the existing uses of history in curricula, textbooks, and/or classrooms in This symposium in honor of Nicolas Rouche primary, secondary, and tertiary levels; will be in French, so the information is also 8. Design and/or assessment of given in French: teaching/learning materials on the history Quels qu’ils soient et quels que soient les of mathematics; systèmes éducatifs dont ils sont des maillons, 9. Relevance of the history of mathematical les enseignants de mathématiques sont obligés practices in the research of mathematics à un moment ou un autre de s’interroger sur le education. sens de leur enseignement en terme éducatif et Papers and poster proposals should use the politique, en terme de rapport à la réalité (celle CERME7 WORD template, and conform to des problèmes pratiques qui se posent à toute the guidelines at société). Cette réalité s’invite aussi par le biais www.cerme7.univ.rzeszow.pl/?id=cerme- des étonnements ou questions, venant de

HPM Newsletter No. 75 November 2010 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ 13 jeunes enfants comme d’étudiants en thèse ; The Mathematics Education into the 21st elle s’hybride à la symbolisation, elle se Century Project has just completed its tenth transforme dans des réseaux de techniques et successful international conference in de théorisations. S’agit-il d’aménager la Dresden, Germany, following conferences in construction, avec les meilleurs procédés, Egypt, Jordan, Poland, Australia, Sicily, d’une science déductive déjà faite ou s’agit-il Czech Republic, Malaysia and the USA. Our d’enseigner à penser mathématiquement ? A project was founded in 1986 and is dedicated chaque niveau de rigueur et d’exigence, to the planning, writing and disseminating of comment la pensée mathématique s’ancre-t- innovative ideas and materials in Mathematics, elle dans les perceptions, les actions, les Statistics, Science and Computer Education. mouvements ? Quel part peut y prendre le jeu, The next conference is planned for September l’expérimentation ? Comment se construisent 10–16, 2011 in Grahamstown, South Africa. définitions et concepts dans leurs rapports à The chairman of the Local Organising l’intuition, aux problèmes et aux Committee is Professor Marc Schafer of démonstrations ? L’histoire des Rhodes University. The conference will open mathématiques et l’histoire de l’enseignement with an evening welcome reception on informent sur toutes ces questions, quelles Sunday, Sep 10th and will close with lunch on ressources offrent-elles aux enseignants ? Saturday, Sep 16th. De l’école élémentaire à l’université, The title of the conference is "Turning chercheur-e-s et/ou enseignant-e-s apporteront Dreams into Reality: Transformations and des éléments pour travailler ces questions dans Paradigm Shifts in Mathematics Education". tous les champs de la discipline (algèbre, Paper proposals are now invited on all géométrie, analyse…) par des conférences innovative aspects of mathematics, statistics, plénières, des exposés ou des ateliers de 1h30. science and computer education. Our Conférences invitées : Erich Wiitmann, conferences are renowned for their friendly Rudolf Bkouche, Jean Mawhin, Christine and productive working atmosphere. They are Decoq, Christiane Hauchart, Evelyne Barbin, attended by innovative teachers and Thérèse Gilbert, Luc Sinègre. mathematics educators from all over the Ce colloque est organisé à l’initiative de la world, 44 countries were represented at our Régionale Pays-Bas, (Groupe d’Enseignement last conference for example. Mathématique de Louvain la Neuve, Groupe Plenary speakers: Prof. Dr. Ludwig Paditz de Leuven, Freudenthal Institute et IREM de from Germany and Professor Ubiratan Lille), du CREM (Centre de Recherche sur D’Ambrosio from Brazil. l’Enseignement des Mathématiques, Nivelles), There will be an additional full social et des deux Commissions Inter-IREM programme for accompanying persons. "Histoire et Epistémologie" et "Géométrie". For ALL further conference details please Informations et appels à contributions à email Alan Rogerson, Chairman of the partir du 1er décembre 2010 : International Programme Committee, at http://irem.univ-lille1.fr/ [email protected]

11th International Conference of The Mathematics Education into the 21st Century Project: Turning Dreams into Reality: Transformations and Paradigm Shifts in Mathematics Education September 10–16, 2011 Rhodes University, Grahamstown, South Africa

HPM Newsletter No. 75 November 2010 14 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ A note from the Editors

The Newsletter of HPM is primarily a tool for passing on information about forthcoming events, recent activities and publications, and current work and research in the broad field of history and pedagogy of mathematics. The Newsletter also publishes brief articles which they think may be of interest. Contributions from readers are welcome on the understanding that they may be shortened and ICME 12 edited to suit the compass of this publication. July 8–15, 2012 Seoul, South Korea http://www.icme12.org/ First Announcement now available from http://www.icme12.org/eng/announ_first_welc ome.html

HPM 2012 July 16–20, 2012 Daejeon, South Korea

Photo from a meeting of some of the people responsible for the HPM 2012 (from left to right): Sunwook Hwang (chair of the Local Organising Committee (LOC), president of KSME), Jinho Kim (secretary of LOC), Evelyne Barbin (HPM AdB), Pamela Chae (Daejeon Convention Center), Sung Sook Kim (vice-president of KSME and member of LOC), Masami Isoda (HPM AdB), Chang Kyoon Park (president of KSHM and member of LOC), Sangki Choi (vice-chair of LOC).

ESU7 2014 To be announced...

HPM Newsletter No. 75 November 2010 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ 15 Distributors: If you wish to be a distributor in a new or unstaffed area please contact the editor.

Area Name and address Email address Argentina Juan E. Nápoles Valdés, Lamadrid 549, (3400) Corrientes, [email protected] ARGENTINA Australia G. FitzSimons, Faculty of Education, P.O.Box 6, Monash [email protected] University, 3800 Victoria, AUSTRALIA onash.edu.au Austria Manfred Kronfellner, Institute of Discrete Mathematics and [email protected] Geometry, Vienna University of Technology, Wiedner Haupstr. 8- 10, A-1040 Wien, AUSTRIA Belgium and Sylvia Eerhart, Freudenthal Instituut, Aïdadreef 12, 3561 GE [email protected] The Utrecht, THE NETHERLANDS Netherlands Canada Thomas Archibald, Mathematics Department, Acadia University, [email protected] Wolfville, NS B0P1X0, CANADA China Ma Li, Linkoping University, ITN, SE - 601 74 Norrkoping, [email protected] SWEDEN Eastern Europe France Evelyne Barbin, Centre François Viète, Faculté des sciences et des [email protected] techniques, 2 Chemin de la Houssinière, BP 92208, 44322 Nantes cedex, FRANCE Germany Gert Schubring, Inst. f. Didaktik der Math., Universitaet Bielefeld, gert.schubring@uni- Postfach 100 131, D-33501, Bielefeld, GERMANY bielefeld.de Iran Mohammad Bagheri, P.O.Box 13145-1785, Tehran, IRAN [email protected] Israel Ted Eisenberg, Mathematics Department, Ben Gurion University of [email protected] the Negev, Beer-Sheva 84105, ISRAEL [email protected] Italy Giorgio T. Bagni †, Department of Mathematics and Computer Science, University of Udine, Polo Rizzi, via delle Scienze 206, I- 33100 Udine, ITALY and Marta Menghini, Dipartimento di Matematica (Universita` La [email protected] Sapienza), Piazzale A. Moro 5, 00185 Roma ITALY Japan Osamu Kota, 3-8-3 Kajiwara, Kamakura [email protected] Kanagawa-ken, 247-0063 JAPAN Malaysia Mohamed Mohini, Department of Science and Mathematical [email protected] Education, Universiti Teknologi Malaysia, 81310 Johor, MALAYSIA Mexico Alejandro R. Garciadiego, Caravaggio 24, Col. Nonoalco Mixcoac [email protected] Del. Benito Juárez 03700 México, D. F. México Morocco Abdellah El Idrissi, E.N.S. B.P: 2400 Marrakech, C.P: 40 000, [email protected] MOROCCO New Zealand Bill Barton, Mathematics Education Unit, [email protected] Dept of Mathematics and Statistics University of Auckland, Private Bag 92-019, Auckland, NEW ZEALAND Other East Gloria Benigno, Department of Education, Culture and Sports, [email protected] Asia Region X, Division of Misamis Occidental, Oroquieta City, PHILLIPINES Russia Vasilii Mikhailovich Busev [email protected] Scandinavia Sten Kaijser, Department of Mathematics, P.O. Box 480, SE- 751 [email protected] 06 Uppsala, SWEDEN South America Marcos Vieira Teixeira , Departamento de Matemática , IGCE - [email protected] UNESP, Postal 178 13 500 - 230 Rio Claro, SP BRAZIL South Asia Prof. R. C. Gupta, Ganita Bharati Academy, R-20, Ras Bahar Colony, Jhansi-284003, U.P. INDIA

16 HPM Newsletter No. 75 November 2010 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ South East Nikos Kastanis, Department of Mathematics, Aristotle University [email protected] Europe of Thessaloniki, Thessaloniki 54006, GREECE Southern Paulus Gerdes, Mozambican Ethnomaths Research Centre, C.P. [email protected] Africa 915, Maputo, MOZAMBIQUE Spain and Carlos Correia de Sá, Departamento de Matemática Pura; [email protected] Portugal Faculdade de Ciências da Universidade do Porto; Rua do Campo Alegre, 687 P - 4169 - 007 Porto, PORTUGAL Taiwan Wann-sheng Horng, Math dept NTNU, 88 Sec.4, Tingchou Rd., [email protected] Taipei, TAIWAN Turkey Funda Gonulates, Bagazici Universitesi, [email protected] Egitim Fakultesi, Bebek- Istanbul, TURKEY United Snezana Lawrence, Bath Spa University, Bath, UK snezana@mathsisgoodforyou. Kingdom com United States David L. Roberts, Prince George ’s Community College, Largo, [email protected] of America Maryland, USA

HPM Newsletter No. 75 November 2010 HPM webpage: http://www.clab.edc.uoc.gr/hpm/ 17 Items for the Newsletter should be sent to the editors, Table of contents preferably by email (see addresses below). The Newsletter appears three times a year with the Professor R. C. Gupta receives… p. 1 following deadlines for next year. New Books p. 1 Conference reports p. 3 ESU6 p. 3 No. Deadline for material Sent to distributors Have you read these? p. 6 76 12 February 2011 1 March 2011 Have you been there? p. 7 77 12 June 2011 1 July 2011 Notices p. 10 78 12 October 2011 1 November 2011 Volume of a sphere in… p. 10 Announcements of events p. 12

Distributors p. 16 The Newsletter is the communication of the

International Study Group on the Relations between the

History and Pedagogy of Mathematics, an affiliate of The views expressed in this Newsletter may the International Commission on Mathematical not necessarily be those of the HPM Advisory Board. Instruction.

Please pass on news of the existence of this The Newsletter is free of charge, available upon request

newsletter to any interested parties. from the distributor for your area, and may be This and previous newsletters can be downloaded from our website: reproduced with acknowledgement.

http://www.clab.edc.uoc.gr/hpm/ Editors: Chris Weeks, [email protected] (Downeycroft, Virginstow, Beaworthy, GB - EX21 5EA, Great Britain) Bjørn Smestad, [email protected] (Faculty of Education and International Studies, Oslo University College, Postbox 4 St. Olavs plass, N-0130 Oslo, Norway)

HPM Advisory Board: Arcavi Abraham Weizmann Institute of Science, Rehovot, 76100, Israel Barbin Evelyne Universite de Nantes, IREM-Centre Francois Viete, France Booker George Griffith University, Brisbane, Australia Chorlay Renaud IREM, Université Paris 7, 175-179 rue Chevalleret, 75013 Paris, France D'Ambrosio Ubiratan Pontificia Universidade, Catolica de Sao Paulo,Brazil El Idrissi Abdellah Ecole Normale Supιrieure, BP 2400, ENS, Marrakech, CP. 40000 Maroc Fasanelli Florence American Association for the Advancement of Science, USA FitzSimons Gail Faculty of Education, PO Box 6, Monash University, Victoria 3800, Australia Furinghetti Fulvia Dipartimento di Matematica dell'Universita di Genova, Genova, Italy Horng Wann-Sheng Department of Mathematics, National Taiwan Normal University, Taiwan Hwang Sunwook Department of Mathematics, Soongsil University, Seoul, Korea Isoda Masami Graduate School of Comprehensive Human Science, University of Tsukuba, Japan Jahnke Niels Fachbereich Mathematik, Universität Duisburg-Essen, Germany Jankvist Uffe Department of Science, Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark Kaisjer Sten Department of Mathematics, University of Uppsala, Uppsala Sweden Katz Victor University of the District of Columbia, Washington, DC, USA Kronfellner Manfred Vienna University of Technology, Vienna, Austria Lawrence Snezana Bath Spa University, Bath, UK Pengelley David Department of Mathematical Sciences, New Mexico State University, Las Cruces, USA Radford Luis École des sciences de l'éducation, Université Laurentienne, Sudbury, Ontario, Canada Roque Tatiana Universidade Federal do Rio de Janeiro, Brasil Schubring Gert IDM, Universitat Bielefeld, Postfach 100 131, D-33501 Bielefeld, Germany Siu Man-Keung Department of Mathematics, University of Hong Kong, Hong Kong SAR, CHINA Smestad Bjørn Faculty of Education, Oslo University College, Norway Stein Robert California State University, San Bernardino, USA Tzanakis Constantinos Department of Education, University of Crete, Rethymnon 74100, Greece van Maanen Jan Freudenthal Institute, Utrecht University, The Netherlands Weeks Chris Downeycroft, Virginstow Beaworthy, UK Winicki Landman Greicy Department of Mathematics and Statistics, California State Polytechnic University, USA

18 HPM Newsletter No. 75 November 2010 HPM webpage: http://www.clab.edc.uoc.gr/hpm/