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What is this about?• Why should pupils and studentslearn history of mathematics?• Why should teachers use history ofmathematics in schools?• How can it be done?• How can it improve the public imageof mathematics?
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Advantages of mathematicianslearning history of math• better communication with non-mathematicians• enables them to see themselves as part of thegeneral cultural and social processes and not tofeel “out of the world”• additional understanding of problems pupils andstudents have in comprehending some mathematicalnotions and facts• if mathematicians have fun with their disciplineit will be felt by others; history of math provideslots of fun examples and interesting facts
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History of math for school teachers• plenty of interesting and fun examples to enliventhe classroom math presentation• use of historic versions of problems can makethem more appealing and understandable• additional insights in already known topics• no-nonsense examples – historical are perfectbecause they are real!• serious themes presented from the historicalperspective are usually more appealing and ofteneasier to explain• connections to other scientific disciplines• better understanding of problems pupils have andthus better response to errors
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• making problems more interesting• visually stimulating• proofs without words• giving some side-comments can enliven the classeven when (or exactly because) it’s notrequested to learn... e.g. when a math symbolwas introduced• making pupils understand that mathematics isnot a closed subject and not a finished set ofknowledge, it is cummulative (everything thatwas once proven is still valid)• creativity – ideas for leading pupils to askquestions (e.g. we know how to double a sqare,but can we double a cube -> Greeks)• showing there are things that cannot be done
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• history of mathematics can improve theunderstanding of learning difficulties; e.g. the useof negative numbers and the rules for doingarithmetic with negative numbers were far fromeasy in their introducing (first appearance in India,but Arabs don’t use them; even A. De Morgan in the19thcentury considers them inconceavable; thoughbegginings of their use in Europe date fromrennaisance – Cardano – full use starts as late asthe 19thcentury)• math is not dry and mathematicians are humanbeeings with emotions anecdotes, quotes andbiographies• improving teaching following the natural processof creation (the basic idea, then the proof)
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•for smaller children: using the development ofnotions•for older pupils: approach by specific historicaltopics•in any case, teaching history helps learning howto develop ideas and improves the understandingof the subject•it is good for giving a broad outline or overviewof the topic, either when introducing it or whenreviewing it
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x2+ 10 x = 39x2+ 10 x + 4·25/4 = 39+25(x+5)2= 64x + 5 = 8x = 3al-Khwarizmi (ca. 780-850)Example 1: Completing a square /solving a quadratic equation
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Exampl e 2: The Bridges of KönigsbergThe problem as such is a problem in recreational math.Depending on the age of the pupils it can be presented just asa problem or given as an example of a class of problemsleading to simple concepts of graph theory (and evenintroduction to more complicated concepts for giftedstudents).
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The Bridges of Koenigsberg can also be a goodintroduction to applications of mathematics, in thiscase graph theory (and group theory) in chemistry:Pólya – enumeration of isomers (molecules which differ only in theway the atoms are connected); a benzene molecule consists of 12atoms: 6 C atoms arranged as vertices of a hexagon, whose edges arethe bonds between the C atoms; the remaining atoms are either H orCl atoms, each of which is connected to precisely one of the carbonatoms. If the vertices of the carbon ring are numbered 1,...,6, then abenzine molecule may be viewed as a function from the set {1,...,6} tothe set {H, Cl}.Clearly benzene isomers are invariant underrotations of the carbon ring, and reflections ofthe carbon ring through the axis connecting twooppposite vertices, or two opposite edges, i.e.,they are invariant under the group of symmetriesof the hexagon. This group is the dihedral groupDi(6). Therefore two functions from {1,..,6} to {H,Cl} correspond to the same isomer if and only ifthey are Di(6)-equivalent. Polya enumerationtheorem gives there are 13 benzene isomers.
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Fibonacci numbersand natureExampl e 3: Homework problems (possible: groupwork) possible explorations of old books or specific topics, e.g.Fibonacci’s biographyrabbits, bees, sunflowers,pinecones,...reasons for seed-arrangement(mathematical!)connections to the Golden number,regular polyhedra, tilings, quasicrystals
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FlatlandFlatland. A Romance of Many Dimensions. (1884) byEdwin A. Abbott (1838-1926).ideas for introducing higher dimensionsalso interesting social implications (connections tohistory and literature)
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2(1+2+...+n)=n(n+1) 1+3+5+...+(2n-1)=n2 Pythagorean number theoryExampl e 4: Proofs without words
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Connections with other sciences – Example: ChemistryWhat is a football? A polyhedron made up of regular pentagons andhexagons (made of leather, sewn together and then blouwn up tu aball shape). It is one of the Archimedean solids – the solids whosesides are all regular polygons. There are 18 Archimedean solids, 5 ofwhich are the Platonic or regular ones (all sides are equal polygons).There are 12 pentagons and 20 hexagons on thefootball so the number of faces is F=32. If we countthe vertices, we’ll obtain the number V=60. Andthere are E=90 edges. If we check the number V-E+F we obtainV-E+F=60-90+32=2.This doesn’t seem interesting until connected to theEuler polyhedron formula which states taht V-E+F=2for all convex polyhedrons. This implies that if weknow two of the data V,E,F the third can becalculated from the formula i.e. is uniquelydetermined!Polyhedra – Plato and Aristotle - Molecules
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In 1985. the football, or officially: truncated icosahedron, cameto a new fame – and application: the chemists H.W.Kroto andR.E.Smalley discovered a new way how pure carbon appeared. Itwas the molecule C60with 60 carbon atoms, each connected to 3others. It is the third known appearance of carbon (the first twobeeing graphite and diamond). This molecule belongs to the classof fullerenes which have molecules shaped like polyhedronsbounded by regular pentagons and hexagons. They are namedafter the architect Buckminster Fuller who is famous for hisdomes of thesame shape. The C60is the only possible fullerenewhich has no adjoining pentagons (this has even a chemicalimplication: it is the reason of the stability of the molecule!)
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Anecdotes enliven the class show that math is not a dry subject andmathematicians are normal human beeings withemotions, but also some specific ways of thinking can serve as a good introduction to a topicNorbert Wiener was walking through a Campus whenhe was stopped by a student who wanted to know ananswer to his mathematical question. Afterexplaining him the answer, Wiener asked: When youstopped me, did I come from this or from the otherdirection? The student told him and Wiener sadi:Oh, that means I didn’t have my meal yet. So hewalked in the direction to the restaurant...
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In 1964 B.L. van der Waerden was visiting professor in Göttingen. Whenthe semester ended he invited his colleagues to a party. One of them,Carl Ludwig Siegel, a number theorist, was not in the mood to come and,to avoid lenghty explanations, wrote a short note to van der Waerdenkurz, saying he couldn’t come because he just died. Van der Waerdenreplyed sending a telegram expressing his deep sympathy to Siegelabout this stroke of the fate...Georg Pólya told about his famous english colleague Hardy the follow-ingstory: Hardy believed in God, but also thought that God tries to makehis life as hard as possible. When he was once forced to travel fromNorway to England on a small shaky boat during a storm, he wrote apostcard to a Norwegian colleague saying: “I have proven the Riemannconjecture”. This was not true, of course, but Hardy reasoned this way:If the boat sinks, everyone will believe he proved it and that the proofsank with him. In this way he would become enourmosly famous. Butbecause he was positive that God wouldn’t allow him to reach this fameand thus he concluded his boat will safely reach England!
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It is reported that Hermann AmandusSchwarz would start an oral examinationas follows:Schwarz: “Tell me the general equationof the fifth degree.”Student: “ax5+bx4+cx3+dx2+ex+f=0”.Schwarz: “Wrong!”Student: “...where e is not the base ofnatural logarithms.”Schwarz: “Wrong!”Student: ““...where e is not necessarilythe base of natural logarithms.”
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Quotes from great mathematicians ideas for discussions or simply for enlivening the class•Albert Einstein (1879-1955)Imagination is more important than knowledge.•René Descartes (1596-1650)Each problem that I solved became a rule which servedafterwards to solve other problems.•Georg Cantor (1845-1918)In mathematics the art of proposing a question must be heldof higher value than solving it.•Augustus De Morgan (1806-1871)The imaginary expression (-a) and the negative expression-b, have this resemblance, that either of them occurring asthe solution of a problem indicates some inconsistency orabsurdity. As far as real meaning is concerned, both areimaginary, since 0 - a is as inconceivable as (-a).
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ConclusionThere is a huge ammount of topics from history which cancompletely or partially be adopted for classroom presentation.The main groups of adaptable materials areanecdotes quotesbiographies historical books and papersoverviews of development historical problemsThe main advantages are (depending on the topic andpresentation)imparting a sense of continuity of mathematicssupplying historical insights and connections of mathematicswith real life (“math is not something out of the world”)plain fun
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General popularizationThere is another aspect of popularization ofmathematics: the approach to the general public.Although this is a more heterogeneous object ofpopularization, there are possibilities for bringingmath nearer even to the established math-haters.Besides talking about applications of mathematics,there are two closely connected approaches: usage ofrecreational mathematics and history of mathematics.The topics which are at least partly connected to his-tory of mathematics are usually more easy to be ad-apted for public presentation. It is usually more easyto simplify the explanations using historical approachesand even when it is not, history provides the frame-work for pre-senting math topics as interestingstories.
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important for all public presentationsince the patience-level for reading mathtexts is generally very low.history of mathematics gives also variousideas for interactive presentations,especially suitable for science fairs andmuseum exhibitions
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• University fairs – informational posters (e.g. womenmathematicians, Croatian mathematicians); gameof connecting mathematicians with their biographies;the back side of our informational leaflet hasquotes from famous mathematicians• Some books in popular mathematics published inCroatia: Z. Šikić: “How the modern mathematics wasmade”, “Mathematics and music”, “A book aboutcalendars”•The pupils in schools make posters about famousmathematicians or math problems as part of theirhomework/projects/group activitiesActions in Croatia
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• The Teaching Section of the Croatian MathematicalSociety decided a few years back to initiatepublishing a book on math history for schools; thebook “History of Mathematics for Schools” has justcome out of print•The authors of math textbooks for schools arerequested (by the Teaching Section of the CroatianMathematical Society) to incorporate short historicalnotes (biographies, anecdotes, historical problems ...)in their texts; it’s not a rule though• “Matka” (a math journal for pupils of aboutgymnasium age) has regular articles “Notes fromhistory” and “Matkas calendar” starting from the firstedition; they write about famous mathematicians andgive historical problems
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• “Poučak” (a journal for school math teachers) usesportraits of great mathematicians on their leadingpage and occasionally have texts about them•“Osječka matematička škola” (a journal for pupils andteachers in the Slavonia region) has a regular sectiongiving biographies of famous mathematicians;occasionally also other articles on history ofmathematics• The new online math-journal math.e has regulararticles about math history; the first number also hasan article about mathematical stamps• All students of mathematics (specializing forbecoming teachers) have “History of mathematics” asan compulsory subject
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•4th year students of the Department ofMathematics in Osijek have to, as part of theexam for the subject “History of mathematics”,write and give a short lecture on a subject formhistory of math, usually on the borderline topopular math (e.g. Origami and math, MathematicalMagic Tricks, ...)
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Marin Getaldić (1568-1627)Dubrovnik aristocratic familyin the period 1595-1601 travelsthorough Europe (Italy, France,England, Belgium, Holland, Germany) contacts with the best scientists of the time (e.g.Galileo Galilei)enthusiastic about Viete-s algebraback to Dubrovnik continues contacts (by mail)Nonnullae propositiones de parabola mathematicalanalysis of the parabola applied to opticsDe resolutione et compositione mathematica application of Viete-s algebra to geometry: predecessorof Descartes and analytic geometry
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Ruđer Bošković (1711-1787)mathematician, physicist,astronomer, philosopher, interestedin archaeology and poetryalso from Dubrovnik, educated atjesuit schools in Italy, laterprofessor in Rome, Pavia and Milanofrom 1773 French citizneship,but last years of his life spent inItaly contacts with almost allcontemporary great scientists andmember of several academies ofscience
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founder of the astronmical opservatorium in Breri.for a while was an ambassador of the Dubrovnik republicgreat achievements in natural philosophy, teoreticalastronomy, mathematics, geophysics, hydrotechnics,constructions of scientific instruments,...first to describe how to claculate a planetary orbit fromthree observationsmain work: Philosophiae naturalis theoria (1758) containsthe theory of natural forces and explanation of thestructure of matterworks in combinatorial analysis, probability theory,geometry, applied mathematicsmathematical textbook Elementa universae matheseos(1754) contains complete theory of conicscan be partly considered a predecessor of Dedekindsaxiom of continuity of real numbers and Ponceletsinfinitely distant points
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Improving the public imageof math using history:•everything that makes pupils more enthusiasticabout math is good for the public image ofmathematics because most people form theiropinion (not only) about math during theirprimary and secondary schooling;•besides, history of mathematics can give ideasfor approaching the already formed “math-haters” in a not officially mathematical contextwhich is easier to achieve then trying to presentpure mathematical themes
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•VITA MATHEMATICAHistorical Research and Integration with TeachingEd. Ronald CalingerMAA Notes No.40, 1996•LEARN FROM THE MASTERSeditors: F.Swetz, J.Fauvel, O.Bekken, B.Johansson, V.Katz,The Mathematical Association of America, 1995•USING HISTORY TO TEACH MATHEMATICSAn international perspectiveeditor: V.Katz,The Mathematical Association of America, 2000•MATHEMATICS: FROM THE BIRTH OF NUMBERSJan GullbergW.W. Norton&Comp. 1997Bibl iography
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