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History of Mathematics for PS421

History of Mathematics for PS421. Maurice OReilly CASTeL, Dept of Mathematics, St Patrick’s College Drumcondra. The plan. HoM & me History in Maths Education HoM Timeline & Themes HoM Resources HoM & Ireland HoM Issues Conclusion. 1. HoM & me. Gerdes on ethnomathematics, c. 1992

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History of Mathematics for PS421

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  1. History of Mathematics for PS421 Maurice OReilly CASTeL, Dept of Mathematics, St Patrick’s College Drumcondra History of Mathematics for PS421 20/4/2012

  2. The plan • HoM & me • History in Maths Education • HoM Timeline & Themes • HoM Resources • HoM & Ireland • HoM Issues • Conclusion History of Mathematics for PS421 20/4/2012

  3. 1. HoM & me • Gerdes on ethnomathematics, c. 1992 • Baghdad & Monasterboice, 1997 • IMTA in Dundalk, 1996-7 • Computus (Ó Cróinín, McCarthy) • Stevick on bookarts • ICMI & ICME, Monterrey, Mexico, 2008 • van Maanen at MEI 3, Dublin, 2009 • HPM ESU 6, Vienna, 2010 • WG12, CERME7, Rzeszów, 2011 • PS421, 2012 History of Mathematics for PS421 20/4/2012

  4. Gerdes on ethnomathematics, c. 1992 History of Mathematics for PS421 20/4/2012

  5. Baghdad & Monasterboice, 1997 History of Mathematics for PS421 20/4/2012

  6. Where do the history of maths and the curriculum meet? NUMBERS ALGEBRA PROBABILITY GEOMETRY CALCULUS FUNCTIONS IMTA in Dundalk, 1996-7 History of Mathematics for PS421 20/4/2012

  7. Computus (Ó Cróinín, McCarthy) History of Mathematics for PS421 20/4/2012

  8. Stevick on Early Irish & English Bookarts History of Mathematics for PS421 20/4/2012

  9. ICMI & ICME, Monterrey, Mexico, 2008 History of Mathematics for PS421 20/4/2012

  10. x² + 10x = 39 van Maanen at MEI 3, Dublin, 2009 History of Mathematics for PS421 20/4/2012

  11. HPM ESU 6, Vienna, 2010 History of Mathematics for PS421 20/4/2012

  12. WG12, CERME7, Rzeszów, 2011 History of Mathematics for PS421 20/4/2012

  13. PS421, DCU 2012 History of Mathematics for PS421 20/4/2012

  14. The plan • HoM & me • History in Maths Education • HoM Timeline & Themes • HoM Resources • HoM & Ireland • HoM Issues • Conclusion History of Mathematics for PS421 20/4/2012

  15. 2. History in Maths Education • International setting • International Commission on Mathematical Instruction (ICMI) • History and Pedagogy of Mathematics (HPM) • ICMI Study: History in Mathematics Education • European Summer School (ESU) on History and Epistemology in Mathematics Education • Congress of the European Society for Research in Mathematics Education (CERME) • Why history? History of Mathematics for PS421 20/4/2012

  16. A view of the international setting History of Mathematics for PS421 20/4/2012

  17. International Commission on Mathematical Instruction (ICMI) at ICM, Rome 1908 by Felix Klein reconstituted/reborn 1952 first ICME, Lyon 1969 History of Mathematics for PS421 20/4/2012

  18. International Study Group on the Relations betweenthe History and Pedagogy of Mathematics (HPM) Not: but … History of Mathematics for PS421 20/4/2012

  19. formally affiliated to ICMI at ICME 3, Karlsruhe 1976 • conferences: • satellites at ICME (starting ICME 5, Adelaide 1984) • European Summer Schools (ESU, since Montpellier 1993) • working groups at ICME & CERME • 10th ICMI Study devoted to H in ME • HPM Newsletter – online History of Mathematics for PS421 20/4/2012

  20. 10th ICMI Study • Napier & Briggs meeting, Edinburgh 1616 • Does HoM have a role in ME? • Relation between how students achieve understanding and the historical construction of mathematical thinking • Integration of HoM into teaching – including use of original sources • Use of technology, e.g. GeoGebra • HoM on the Web, & its use for ME History of Mathematics for PS421 20/4/2012

  21. Examples: • Letter from Leibniz to Riccati, c. 1715 on 1-1+1-1+ … = ½ • Cayley’s 1878 paper on groups of order 6: a,b: a² = b³ = 1, ab = ba • Hamilton’s perception of multiplication of complex numbers: z1z2 History of Mathematics for PS421 20/4/2012

  22. ESU 6, Vienna 2010 • Uffe Janqviston history as tool or history as goal • Michael Fried on how can ME accommodate HoM • Michael Glaubitz& James Kiernan on using original sources • Olivier Bruneauon ICT & pedal curves, C 17th- 19th • Man KeungSiuon Sino-European dialogue, 1587 … History of Mathematics for PS421 20/4/2012

  23. WG12, CERME7, Rzeszów, 2011 History of Mathematics for PS421 20/4/2012

  24. Why history? “… mathematics, as persented to students, fails on two major grounds which you’ve pointed out and that it’s taught in a very dry formal way without any explanation of the origins and the motivations of where things are coming from. That is a terrible mistake and shouldn’t happen, but people are human.” Michael Atiyah @ RIA IMS Bulletin #58 (2006), p49

  25. Consider student activities: Women in Mathematics Day: Ireland UL 27th April 2010 National Committee for Commemorative Plaques in Science and Technology

  26. The plan HoM & me History in Maths Education HoM Timeline & Themes HoM Resources HoM & Ireland HoM Issues Conclusion History of Mathematics for PS421 20/4/2012

  27. 3. HoM Timeline & Themes • Beginnings • Greece, India, China • Arabic & Persian mathematics • Europe to early 16th century • Solving polynomials • Calculus & physics • Rigour & Foundations • Abstraction, specialisation & new synergies History of Mathematics for PS421 20/4/2012

  28. Beginnings • Plimpton 322 • c. 1800 BCE • Mesopotamia • Cuneiform • Sexagesimal • Pythagorean triples History of Mathematics for PS421 20/4/2012

  29. Rhind papyrus • c. 1650 BCE by the scribe Ahmes • Hieratic Egyptian script • Calculations with fractions; areas & volumes 710 = 23+130 History of Mathematics for PS421 20/4/2012

  30. “Pythagoras’ theorem” from Euclid Archimedes and  using inscribed & circumscribed polygons GREECE  from length, area or volume? History of Mathematics for PS421 20/4/2012

  31. Brahmagupta’s theorem (7th cent CE) Number system with place values and zero Negative numbers Trigonometry INDIA |AF|=|FD| History of Mathematics for PS421 20/4/2012

  32. Jiuzhang Suanshu or The Nine Chapters on the Mathematical Art Variety of practical problems with solutions c. 100 BCE “Pythagoras” 500-200 BCE CHINA History of Mathematics for PS421 20/4/2012

  33. Al-Khwarizmi’s al-jabr wa-l-muqabala x² + 10x = 39 Textual (rather than symbolic), but ‘justified’ by geometry. Specific examples, yet general approach: algorithm

  34. Omar Khayyam’s solution of x³+cx=d 1048-1131 CE The parabola x²=-√c y intersects the semi-circle (having radius dc) at B & D. x=|BE| is the solution of the cubic.

  35. Europe in the Middle Ages c. 500-1500 Trivium (logic, rhetoric & grammar) Quadrivium (arithmetic, geometry, music & astronomy) Boethius (480-524) Isidore of Seville (560-636) – Etymologiae Gathering universal knowledge Calendar studies important Translations: geometry, number systems Greek knowledge through Arab world, extended by them Leonardo di Pisa (Fibonacci, c. 1170-1240) Nicole Oresme (1320-1382) – kinematics, exponentials Scotii Charlemange Bede

  36. Solution of cubic & quartic 1540-1640 Tartaglia (1499-1557) & Cardano (1501-1576) Ars Magna, 1545 what about Fermat, Wallis, Pascal, … and logarithms – who thought of them? Algebrization of geometry La Géometrie, 1637

  37. Calculus … 1660-1780 Johann Bernoulli (1667-1748) & l’Hôpital (1661-1704) Leibniz (1646-1716) & Newton (1642-1727) The priority dispute!! what about other Bernoullis, Taylor, … doubting Berkeley … also d’Alembert & Agnesi? … becomes analysis Differential equations; books

  38. Leibniz’ characteristic triangle transmutation rule and Gregory-Madhava-Leibniz series:

  39. Political turmoil in France 1780-1820 • Lagrange (1736-1813) • Theory of equations • Monge (1746-1818) • Analytic & differential geometry) • Laplace (1749-1827) • Statistical inference • Legendre (1752-1833) • Number theory … but maths is healthy number theory, congruences, fundamental theorem of algebra, non-Euclidean and differential geometry, etc., etc. Is England still in the races?

  40. FTA Complex numbers? w=z²+z+1 replace 2 by 3, 4, … As r=|z| increases, the locus of w must pass through 0, and so …

  41. The plan HoM & me History in Maths Education HoM Timeline & Themes HoM Resources HoM & Ireland HoM Issues Conclusion History of Mathematics for PS421 20/4/2012

  42. 4. HoM Resources • Original sources • HoM books • The Web History of Mathematics for PS421 20/4/2012

  43. The plan HoM & me History in Maths Education HoM Timeline & Themes HoM Resources HoM & Ireland HoM Issues Conclusion History of Mathematics for PS421 20/4/2012

  44. 5. HoM in Ireland • Early medieval Ireland • Irish mathematicians • Foreign mathematicians in Ireland History of Mathematics for PS421 20/4/2012

  45. Mathematics in Early Medieval Ireland Two areas: the Computus & Geometry

  46. Imago LeonisEchternach Gospels f.75v Insular manuscripts St MatthewBook of Moling , f.12v DIT, February 2008 Fr Ingram lecture, IMTA, Carlow, 2000

  47. Carpet page (Luke)Lindisfarne f.138v Galway Computus conference, 2008

  48. Hamilton in Cabra plaque Seán Keating’s version tea @ Dunsink Enormous insights: Conical refraction General theory of dynamics Quaternions graffiti

  49. Casey in Reykjavík!? • New triangle geometry of: • Lemoine • Brocard • Casey 1820-1891

  50. Alastair Wood on Stokes George Gabriel Stokes 1819 - 1903 An Irish Mathematical Physicist, IMS Bulletin #35

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