Using history for popularization 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 processe

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

19 th century considers them inconceavable; though

begginings of their use in Europe date from

rennaisance – Cardano – full use starts as late as

the 19 th 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 2 : The Bridges of K önigsberg

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: 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.

rabbits, bees, sunflowers,pinecones,

reasons for seed-arrangement

(mathematical!)

connections to the Golden number,

regular polyhedra, tilings, quasicrystals

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

ideas for introducing higher dimensions

also 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 obtain

V-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!)

 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 topic

Norbert 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 ).

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 popularization

There 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.

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 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

(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 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

“math-haters” in a not officially mathematical context which is easier to achieve then trying to present pure mathematical themes