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Using history for popularization of mathematics Franka Miriam Brückler Department of Mathematics University of Zagreb Croatia bruckler@math.hr www.math.hr/~bruckler/ What is this about? • Why should pupils and students learn history of mathematics? • Why should teachers use history of mathematics in schools? • How can it be done? • How can it improve the public image of mathematics? Advantages of mathematicians learning history of math • better communication with non-mathematicians • enables them to see themselves as part of the general cultural and social processes and not to feel “out of the world” • additional understanding of problems pupils and students have in comprehending some mathematical notions and facts • if mathematicians have fun with their discipline it will be felt by others; history of math provides lots of fun examples and interesting facts History of math for school teachers • plenty of interesting and fun examples to enliven the classroom math presentation • use of historic versions of problems can make them more appealing and understandable • additional insights in already known topics • no-nonsense examples – historical are perfect because they are real! • serious themes presented from the historical perspective are usually more appealing and often easier to explain • connections to other scientific disciplines • better understanding of problems pupils have and thus better response to errors making problems more interesting visually stimulating proofs without words giving some side-comments can enliven the class even when (or exactly because) it’s not requested to learn e.g when a math symbol was introduced • making pupils understand that mathematics is not a closed subject and not a finished set of knowledge, it is cummulative (everything that was once proven is still valid) • creativity – ideas for leading pupils to ask questions (e.g we know how to double a sqare, but can we double a cube -> Greeks) • showing there are things that cannot be done • • • • • history of mathematics can improve the understanding of learning difficulties; e.g the use of negative numbers and the rules for doing arithmetic with negative numbers were far from easy in their introducing (first appearance in India, but Arabs don’t use them; even A De Morgan in the 19th century considers them inconceavable; though begginings of their use in Europe date from rennaisance – Cardano – full use starts as late as the 19th century) • math is not dry and mathematicians are human beeings with emotions anecdotes, quotes and biographies • improving teaching following the natural process of creation (the basic idea, then the proof) •for smaller children: using the development of notions •for older pupils: approach by specific historical topics •in any case, teaching history helps learning how to develop ideas and improves the understanding of the subject •it is good for giving a broad outline or overview of the topic, either when introducing it or when reviewing it Example 1: Completing a square / solving a quadratic equation al-Khwarizmi (ca 780-850) x2 + 10 x = 39 x2 + 10 x + 4·25/4 = 39+25 (x+5)2 = 64 x + = x = Example 2: The Bridges of Königsberg The problem as such is a problem in recreational math Depending on the age of the pupils it can be presented just as a problem or given as an example of a class of problems leading to simple concepts of graph theory (and even introduction to more complicated concepts for gifted students) The Bridges of Koenigsberg can also be a good introduction to applications of mathematics, in this case graph theory (and group theory) in chemistry: Pólya – enumeration of isomers (molecules which differ only in the way the atoms are connected); a benzene molecule consists of 12 atoms: C atoms arranged as vertices of a hexagon, whose edges are the bonds between the C atoms; the remaining atoms are either H or Cl atoms, each of which is connected to precisely one of the carbon atoms If the vertices of the carbon ring are numbered 1, ,6, then a benzine molecule may be viewed as a function from the set {1, ,6} to the set {H, Cl} Clearly benzene isomers are invariant under rotations of the carbon ring, and reflections of the carbon ring through the axis connecting two oppposite vertices, or two opposite edges, i.e., they are invariant under the group of symmetries of the hexagon This group is the dihedral group Di(6) Therefore two functions from {1, ,6} to {H, Cl} correspond to the same isomer if and only if they are Di(6)-equivalent Polya enumeration theorem gives there are 13 benzene isomers important for all public presentation since the patience-level for reading math texts is generally very low history of mathematics gives also various ideas for interactive presentations, especially suitable for science fairs and museum exhibitions Actions in Croatia • University fairs – informational posters (e.g women mathematicians, Croatian mathematicians); game of connecting mathematicians with their biographies; the back side of our informational leaflet has quotes from famous mathematicians • Some books in popular mathematics published in Croatia: Z Šikić: “How the modern mathematics was made”, “Mathematics and music”, “A book about calendars” •The pupils in schools make posters about famous mathematicians or math problems as part of their homework/projects/group activities The Teaching Section of the Croatian Mathematical Society decided a few years back to initiate publishing a book on math history for schools; the book “History of Mathematics for Schools” has just come out of print •The authors of math textbooks for schools are requested (by the Teaching Section of the Croatian Mathematical Society) to incorporate short historical notes (biographies, anecdotes, historical problems ) in their texts; it’s not a rule though • “Matka” (a math journal for pupils of about gymnasium age) has regular articles “Notes from history” and “Matkas calendar” starting from the first edition; they write about famous mathematicians and give historical problems • • “Poučak” (a journal for school math teachers) uses portraits of great mathematicians on their leading page and occasionally have texts about them •“Osječka matematička škola” (a journal for pupils and teachers in the Slavonia region) has a regular section giving biographies of famous mathematicians; occasionally also other articles on history of mathematics • The new online math-journal math.e has regular articles about math history; the first number also has an article about mathematical stamps • All students of mathematics (specializing for becoming teachers) have “History of mathematics” as an compulsory subject •4th year students of the Department of Mathematics in Osijek have to, as part of the exam for the subject “History of mathematics”, write and give a short lecture on a subject form history of math, usually on the borderline to popular math (e.g Origami and math, Mathematical Magic Tricks, ) Example: Connecting mathematicians with their biographies (university fair in Zagreb) Marin Getaldić (1568-1627) Dubrovnik aristocratic family in the period 1595-1601 travels thorough Europe (Italy, France, England, Belgium, Holland, Germany) contacts with the best scientists of the time (e.g Galileo Galilei) enthusiastic about Viete-s algebra back to Dubrovnik continues contacts (by mail) Nonnullae propositiones de parabola mathematical analysis of the parabola applied to optics De resolutione et compositione mathematica application of Viete-s algebra to geometry: predecessor of Descartes and analytic geometry Ruđer Bošković (17111787) mathematician, physicist, astronomer, philosopher, interested in archaeology and poetry also from Dubrovnik, educated at jesuit schools in Italy, later professor in Rome, Pavia and Milano from 1773 French citizneship, but last years of his life spent in Italy contacts with almost all contemporary great scientists and member of several academies of science founder of the astronmical opservatorium in Breri for a while was an ambassador of the Dubrovnik republic great achievements in natural philosophy, teoretical astronomy, mathematics, geophysics, hydrotechnics, constructions of scientific instruments, first to describe how to claculate a planetary orbit from three observations main work: Philosophiae naturalis theoria (1758) contains the theory of natural forces and explanation of the structure of matter works in combinatorial analysis, probability theory, geometry, applied mathematics mathematical textbook Elementa universae matheseos (1754) contains complete theory of conics can be partly considered a predecessor of Dedekinds axiom of continuity of real numbers and Poncelets infinitely distant points Improving the public image of math using history: •everything that makes pupils more enthusiastic about math is good for the public image of mathematics because most people form their opinion (not only) about math during their primary and secondary schooling; •besides, history of mathematics can give ideas for approaching the already formed “mathhaters” in a not officially mathematical context which is easier to achieve then trying to present pure mathematical themes Links I •http://student.math.hr/~bruckler/ostalo.html •http://archives.math.utk.edu/topics/history.html Math Archives •http://www.mathforum.org/library/topics/history/ Math Forum •http://www-history.mcs.st-andrews.ac.uk/history/ MacTutor History of Mathematics Archive •http://www.maa.org/news/mathtrek.html Ivars Peterson's MathTrek •http://www.cut-the-knot.org/ctk/index.shtml Cut the Knot! An interactive column using Java applets by Alex Bogomolny •http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fractions/egyptian html Egyptian Fractions •http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/ Fibonacci Numbers and the Golden Section Links II • http://www.maths.tcd.ie/pub/HistMath/Links/Cultures.html History of Mathematics Links: Mathematics in Specific Cultures, Periods or Places • http://math.furman.edu/~mwoodard/mqs/mquot.shtml Mathematical Quotation Server • http://www.dartmouth.edu/~matc/math5.geometry/unit1/INTR O.html Math in Art and Architecture • http://www.georgehart.com/virtual-polyhedra/papermodels.html Making paper models of polyhedra • http://www.mathematik.uni-bielefeld.de/~sillke/ A big collection of links to math puzzles • http://mathmuse.sci.ibaraki.ac.jp/indexE.html Mathematics Museum Online (japan) • http://www.math.de/ Math Museum (Germany) Bibliography •VITA MATHEMATICA Historical Research and Integration with Teaching Ed Ronald Calinger MAA Notes No.40, 1996 •LEARN FROM THE MASTERS editors: F.Swetz, J.Fauvel, O.Bekken, B.Johansson, V.Katz, The Mathematical Association of America, 1995 •USING HISTORY TO TEACH MATHEMATICS An international perspective editor: V.Katz, The Mathematical Association of America, 2000 •MATHEMATICS: FROM THE BIRTH OF NUMBERS Jan Gullberg W.W Norton&Comp 1997 •THE STORY OF MATHEMATICS From counting to complexity Richard Mankiewicz, Orion Publishing Co 2000 •GUTEN TAG, HERR ARCHIMEDES A.G Konforowitsch, Harri Deutsch 1996 •ENTERTAINING SCIENCE EXPERIMENTS WITH EVERYDAY OBJECTS; MATHEMATICS, MAGIC AND MYSTERY; SCIENCE MAGIC TRICKS; ENTERTAINING MATHEMATICAL PUZZLES; and other books by Martin Gardner the books above are by Dover Publications •IN MATHE WAR ICH IMMER SCHLECHT Alberecht Beutelspacher, Vieweg 2000 •THE PENGUIN DICTIONARY OF CURIOUS AND INTERESTING NUMBERS David Wells, Penguin Books 1996 •WHAT SHAPE IS A SNOWFLAKE? Ian Stewart, Orion Publ 2001 •ALLES MATHEMATIK Von Pythagoras zum CDPlayer Ed M Aigner, E Behrends Vieweg 2000 •THE MATHEMATICAL TOURIST Snapshots of modern mathematics Ivars Peterson, Freeman and Comp 1988
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