Using history for popularization of

mathematics

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

x2 + 10 x = 39

x2 + 10 x + 4·25/4 = 39+25

(x+5)2 = 64

x + 5 = 8

x = 3

al-Khwarizmi (ca. 780-850)

Example 1: Completing a square / solving a quadratic equation

Example 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 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: 6 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.

Fibonacci numbers and nature

Example 3: Homework problems (possible: group work)

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

FlatlandFlatland. A Romance of Many Dimensions. (1884) by Edwin A. Abbott (1838-1926).

ideas for introducing higher dimensionsalso interesting social implications (connections to history and literature)

2(1+2+...+n)=n(n+1) 1+3+5+...+(2n-1)=n2

Pythagorean number theory

Example 4: Proofs without words

Connections with other sciences – Example: Chemistry

What is a football? A polyhedron made up of regular pentagons and hexagons (made of leather, sewn together and then blouwn up tu a ball shape). It is one of the Archimedean solids – the solids whose sides are all regular polygons. There are 18 Archimedean solids, 5 of which are the Platonic or regular ones (all sides are equal polygons).

There are 12 pentagons and 20 hexagons on the football so the number of faces is F=32. If we count the vertices, we’ll obtain the number V=60. And there 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 the Euler polyhedron formula which states taht V-E+F=2 for all convex polyhedrons. This implies that if we know two of the data V,E,F the third can be calculated from the formula i.e. is uniquely determined!

Polyhedra – Plato and Aristotle - Molecules

In 1985. the football, or officially: truncated icosahedron, came to a new fame – and application: the chemists H.W.Kroto and R.E.Smalley discovered a new way how pure carbon appeared. It was the molecule C60 with 60 carbon atoms, each connected to 3 others. It is the third known appearance of carbon (the first two beeing graphite and diamond). This molecule belongs to the class of fullerenes which have molecules shaped like polyhedrons bounded by regular pentagons and hexagons. They are named after the architect Buckminster Fuller who is famous for his domes of thesame shape. The C60 is the only possible fullerene which has no adjoining pentagons (this has even a chemical implication: it is the reason of the stability of the molecule!)

Anecdotes enliven the class show that math is not a dry subject and mathematicians are normal human beeings with emotions, but also some specific ways of thinking can serve as a good introduction to a topicNorbert Wiener was walking through a Campus when he was stopped by a student who wanted to know an answer to his mathematical question. After explaining him the answer, Wiener asked: When you stopped me, did I come from this or from the other direction? The student told him and Wiener sadi: Oh, that means I didn’t have my meal yet. So he walked in the direction to the restaurant...

In 1964 B.L. van der Waerden was visiting professor in Göttingen. When the 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 Waerden kurz, saying he couldn’t come because he just died. Van der Waerden replyed sending a telegram expressing his deep sympathy to Siegel about this stroke of the fate...

Georg Pólya told about his famous english colleague Hardy the follow-ing story: Hardy believed in God, but also thought that God tries to make his life as hard as possible. When he was once forced to travel from Norway to England on a small shaky boat during a storm, he wrote a postcard to a Norwegian colleague saying: “I have proven the Riemann conjecture”. This was not true, of course, but Hardy reasoned this way: If the boat sinks, everyone will believe he proved it and that the proof sank with him. In this way he would become enourmosly famous. But because he was positive that God wouldn’t allow him to reach this fame and thus he concluded his boat will safely reach England!

It is reported that Hermann Amandus Schwarz would start an oral examination as follows:Schwarz: “Tell me the general equation of the fifth degree.”Student: “ax5+bx4+cx3+dx2+ex+f=0”.Schwarz: “Wrong!”Student: “...where e is not the base of natural logarithms.”Schwarz: “Wrong!”Student: ““...where e is not necessarily the base of natural logarithms.”

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 served afterwards to solve other problems.•Georg Cantor (1845-1918)In mathematics the art of proposing a question must be held of 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 as the solution of a problem indicates some inconsistency or absurdity. As far as real meaning is concerned, both are imaginary, since 0 - a is as inconceivable as (-a).

ConclusionThere is a huge ammount of topics from history which can completely or partially be adopted for classroom presentation.

The main groups of adaptable materials are

anecdotes quotes

biographies historical books and papers

overviews of development historical problems

The main advantages are (depending on the topic and presentation)

imparting a sense of continuity of mathematics

supplying historical insights and connections of mathematics with real life (“math is not something out of the world”)

plain fun

General popularizationThere is another aspect of popularization of

mathematics: the approach to the general public. Although this is a more heterogeneous object of popularization, there are possibilities for bringing math nearer even to the established math-haters.

Besides talking about applications of mathematics, there are two closely connected approaches: usage of recreational 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 easy to simplify the explanations using historical approaches and even when it is not, history provides the frame-work for pre-senting math topics as interesting stories.

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

• 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

Actions in Croatia

• 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 familyin 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 algebraback to Dubrovnik continues contacts (by mail)Nonnullae propositiones de parabola mathematical analysis of the parabola applied to opticsDe resolutione et compositione mathematica application of Viete-s algebra to geometry: predecessor of Descartes and analytic geometry

Ruđer Bošković (1711-1787)

mathematician, physicist, astronomer, philosopher, interested in archaeology and poetry also from Dubrovnik, educated at jesuit schools in Italy, later professor in Rome, Pavia and Milanofrom 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 republicgreat 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 matterworks in combinatorial analysis, probability theory, geometry, applied mathematics mathematical textbook Elementa universae matheseos (1754) contains complete theory of conicscan 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 “math-haters” in a not officially mathematical context which is easier to achieve then trying to present pure mathematical themes

•VITA MATHEMATICA Historical Research and Integration with TeachingEd. Ronald CalingerMAA 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 NUMBERSJan GullbergW.W. Norton&Comp. 1997

Bibliography