Using history for popularization of mathematics

1
Using history for popularization of mathematics
Franka Miriam Brückler Department of
Mathematics University of Zagreb Croatia bruckler_at_
math.hr www.math.hr/bruckler/
2
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?

3
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

4
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

5

• making problems more interesting
• visually stimulating
• proofs without words
• giving some side-comments can enliven the class
  even when (or exactly because) its 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 -gt Greeks)
• showing there are things that cannot be done

6

• 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 dont 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)

7

• 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

8
Example 1 Completing a square / solving a
quadratic equation
al-Khwarizmi (ca. 780-850)
x2 10 x 39
x2 10 x 425/4 3925
(x5)2 64
x 5 8
x 3
9
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).

10

• 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.
11
Example 3 Homework problems (possible group
work)
? possible explorations of old books or specific
topics, e.g.
Fibonacci numbers and nature

• Fibonaccis biography
• rabbits, bees, sunflowers,pinecones,...
• reasons for seed-arrangement (mathematical!)
• connections to the Golden number, regular
  polyhedra, tilings, quasicrystals

12
Flatland

• 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)

13
Example 4 Proofs without words
? Pythagorean number theory

• 2(12...n)n(n1) 135...(2n-1)n2

14
Connections with other sciences Example
Chemistry
Polyhedra Plato and Aristotle - Molecules
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 F32. If we
count the vertices, well obtain the number V60.
And there are E90 edges. If we check the number
V-EF we obtain V-EF60-90322. This doesnt
seem interesting until connected to the Euler
polyhedron formula which states taht V-EF2 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!
15
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!)
16
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 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 didnt have my meal yet.
So he walked in the direction to the
restaurant...
17
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 wouldnt allow him to reach
this fame and thus he concluded his boat will
safely reach England!
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 couldnt come because he
just died. Van der Waerden replyed sending a
telegram expressing his deep sympathy to Siegel
about this stroke of the fate...
18

• 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 ax5bx4cx3dx2exf0.
• 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.

19
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).

20
Conclusion

• There 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

21
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.
22

• ? 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

23
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. Šikic 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

24

• 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 its 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

25

• Poucak (a journal for school math teachers)
  uses portraits of great mathematicians on
  their leading page and occasionally have texts
  about them
• Osjecka matematicka 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

26

• 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, ...)

27
Example Connecting mathematicians with their
biographies(university fair in Zagreb)
28
Marin Getaldic (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

29
Ruder Boškovic (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

30

• 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

31
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

32
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/Fract
  ions/egyptian.html Egyptian Fractions
• http//www.mcs.surrey.ac.uk/Personal/R.Knott/Fibon
  acci/ Fibonacci Numbers and the Golden Section

33
Links II

• http//www.maths.tcd.ie/pub/HistMath/Links/Culture
  s.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/unit
  1/INTRO.html Math in Art and Architecture
• http//www.georgehart.com/virtual-polyhedra/paper-
  models.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)

34
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. NortonComp. 1997

35

• 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 3 books above are by Dover Publications
• IN MATHE WAR ICH IMMER SCHLECHT
• Alberecht Beutelspacher,
• Vieweg 2000

36

• 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 CD-Player
• Ed. M. Aigner, E. Behrends
• Vieweg 2000
• THE MATHEMATICAL TOURIST Snapshots of modern
  mathematics
• Ivars Peterson,
• Freeman and Comp. 1988