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Unformatted text preview: Using history for
popularization of mathem
atics 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 nonmathematicians
• 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
• nononsense 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 sidecomments 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
alKhwarizmi (ca. 780850)
x2 + 10 x = 39
x2 + 10 x + 4·25/4 = 39+25
(x+5)2 = 64
x+5=8
x=3 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: 6 C atomsarranged 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. Example 3: Homework problems (possible: group
work) possible explorations of old books or specific topics, e.g. Fibonacci numbers and nature
Fibonacci’s biography
rabbits, bees, sunflowers,pinecones,...
reasons for seedarrangement
(mathematical!)
connections to the Golden number,
regular polyhedra, tilings, quasicrystals Flatland Flatland. A Romance of Many Dimensions. (1884) by Edwin A. Abbott (18381926). ideas for introducing higher dimensions
also interesting social implications (connections to history and literature) Example 4: Proofs without words Pythagorean number theory 2(1+2+...+n)=n(n+1) 1+3+5+...+(2n1)=n2 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 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 VE+F we obtain
VE+F=6090+32=2.
This doesn’t seem interesting until connected to the Euler polyhedron formula which states taht VE+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! 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 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... Georg Pólya told about his famous english colleague Hardy the following 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! 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... 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 (18791955)
Imagination is more important than knowledge.
•René Descartes (15961650)
Each problem that I solved became a rule which served
afterwards to solve other problems.
•Georg Cantor (18451918)
In mathematics the art of proposing a question must be held of
higher value than solving it.
•Augustus De Morgan (18061871)
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). 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 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 mathhaters.
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 history of mathematics are usually more easy to be adapted for public presentation. It is usually more easy to
simplify the explanations using historical approaches
and even when it is not, history provides the framework
for presenting math topics as interesting stories. important for all public presentation since
the patiencelevel 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 mathjournal 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 (15681627)
Dubrovnik aristocratic family
in the period 15951601 travels thorough Europe (Italy, France, England, Belgium, Holland, Germany) contacts with the best scientists of the time (e.g. Galileo Galilei)
enthusiastic about Vietes 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 Vietes 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://wwwhistory.mcs.standrews.ac.uk/history/ MacTutor History of Mathematics Archive
•http://www.maa.org/news/mathtrek.html Ivars Peterson's MathTrek
•http://www.cuttheknot.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/INTRO.html Math in Art and Architecture
• http://www.georgehart.com/virtualpolyhedra/papermodels.html Making paper models of polyhedra
• http://www.mathematik.unibielefeld.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 3 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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This note was uploaded on 11/11/2011 for the course MATH 220 taught by Professor Kearn during the Fall '11 term at BYU.
 Fall '11
 Kearn
 Math, The Republic

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