This document discusses using history of mathematics to popularize mathematics in schools and for the general public. It provides several advantages, such as helping teachers enliven classroom lessons with interesting examples and insights. Using history can improve understanding of mathematical concepts and make problems more appealing. The document also provides many specific examples of how to incorporate history, such as discussing famous problems and mathematicians, their biographies and quotes. It concludes that history is an effective way to present mathematics as interesting stories and connect it to other disciplines and real life.
This document discusses how using history can help popularize mathematics. It provides advantages of mathematicians and teachers learning history, such as better communication skills and understanding student difficulties. Teachers can use historical examples, problems, and biographies to make lessons more interesting. The document also discusses how history can improve the public image of math by showing it is a human endeavor and not dry/boring. It provides examples of how Croatia promotes math history awareness through publications, student projects, and curriculum.
Add math project 2018 kuala lumpur (simultaneous and swing)Anandraka
This document is a student's additional mathematics project work. It includes an introduction on the history of equations and their applications. It discusses how ancient Babylonians, Arabs, and Europeans contributed to solving quadratic equations over time. It also provides examples of how quadratic equations are used in various areas like determining the length of the hypotenuse and establishing the proportions of paper sizes. The student expresses frustration with lack of help from their teacher and tuition teacher in completing the project, but was able to finish it through their own research and discussion with friends.
1) Augustin-Louis Cauchy was a renowned French mathematician born in 1789 in Paris. He attended the top schools in Paris and graduated from the École Polytechnique with honors.
2) After working as a junior engineer for three years, Cauchy devoted himself full-time to mathematics. He wrote influential works on analysis, functions, and geometry and was appointed to the Académie des Sciences in 1816.
3) Cauchy's career was disrupted after the 1830 revolution in France. He lost his posts but continued his mathematical work in Italy and Prague, tutoring the Duke of Bordeaux until 1838. He remained active in mathematics until his death in 1857.
(V.prasolov,i. sharygin) problemas de geometría plana en inglésperroloco2014
The document summarizes a book on solid geometry problems by authors Prasolov and Sharygin. It contains 560 problems with complete solutions in solid geometry, aimed at students in secondary school. The problems cover a wide range of topics in solid geometry and are classified by theme and difficulty level. Reviewers praised the book for its comprehensive coverage of topics and carefully presented solutions.
1) In the 19th century, mathematics underwent significant changes with a new emphasis on rigor, structure, and abstract concepts.
2) This included the development of non-Euclidean geometry which showed that Euclid's parallel postulate is independent of the other postulates of geometry.
3) Algebra evolved from a focus on symbols and arithmetic to studying mathematical structures in more abstract ways, such as in Boolean and quaternion algebras.
Talk given at Some snapshots of Women in
Mathematical History Panel, CMS winter meeting 2021, December 7
Title: Women mathematicians with PhD in the interwar Poland
Abstract: In this talk we will present profiles of a few women who earned PhD degrees in mathematics or its applications in the academic institutions of Warsaw, Lwow and Poznan between the two world wars.
This document discusses how using history can help popularize mathematics. It provides advantages of mathematicians and teachers learning history, such as better communication skills and understanding student difficulties. Teachers can use historical examples, problems, and biographies to make lessons more interesting. The document also discusses how history can improve the public image of math by showing it is a human endeavor and not dry/boring. It provides examples of how Croatia promotes math history awareness through publications, student projects, and curriculum.
Add math project 2018 kuala lumpur (simultaneous and swing)Anandraka
This document is a student's additional mathematics project work. It includes an introduction on the history of equations and their applications. It discusses how ancient Babylonians, Arabs, and Europeans contributed to solving quadratic equations over time. It also provides examples of how quadratic equations are used in various areas like determining the length of the hypotenuse and establishing the proportions of paper sizes. The student expresses frustration with lack of help from their teacher and tuition teacher in completing the project, but was able to finish it through their own research and discussion with friends.
1) Augustin-Louis Cauchy was a renowned French mathematician born in 1789 in Paris. He attended the top schools in Paris and graduated from the École Polytechnique with honors.
2) After working as a junior engineer for three years, Cauchy devoted himself full-time to mathematics. He wrote influential works on analysis, functions, and geometry and was appointed to the Académie des Sciences in 1816.
3) Cauchy's career was disrupted after the 1830 revolution in France. He lost his posts but continued his mathematical work in Italy and Prague, tutoring the Duke of Bordeaux until 1838. He remained active in mathematics until his death in 1857.
(V.prasolov,i. sharygin) problemas de geometría plana en inglésperroloco2014
The document summarizes a book on solid geometry problems by authors Prasolov and Sharygin. It contains 560 problems with complete solutions in solid geometry, aimed at students in secondary school. The problems cover a wide range of topics in solid geometry and are classified by theme and difficulty level. Reviewers praised the book for its comprehensive coverage of topics and carefully presented solutions.
1) In the 19th century, mathematics underwent significant changes with a new emphasis on rigor, structure, and abstract concepts.
2) This included the development of non-Euclidean geometry which showed that Euclid's parallel postulate is independent of the other postulates of geometry.
3) Algebra evolved from a focus on symbols and arithmetic to studying mathematical structures in more abstract ways, such as in Boolean and quaternion algebras.
Talk given at Some snapshots of Women in
Mathematical History Panel, CMS winter meeting 2021, December 7
Title: Women mathematicians with PhD in the interwar Poland
Abstract: In this talk we will present profiles of a few women who earned PhD degrees in mathematics or its applications in the academic institutions of Warsaw, Lwow and Poznan between the two world wars.
A square meets all the properties of a rectangle - it has four sides, four right angles, opposite sides that are parallel and equal in length. Additionally, all four sides of a square are equal in length. In mathematics, categories are defined inclusively so that a square is considered a special case of a rectangle. This makes theorems and proofs simpler by avoiding separate cases for different shapes.
The document discusses Hilbert's view that unsolved mathematical problems are essential for advancing the field, as they test new methods and ideas and can lead to new discoveries and applications. He cites examples throughout history where difficult problems inspired major developments, such as the problem of the shortest line influencing many areas of math and Klein's work on the icosahedron connecting geometry, group theory, and more. Hilbert argues that unsolved problems keep mathematics alive and ensure its continued independent development.
- The document discusses figurate numbers and their use in teaching mathematics.
- It presents Tobias Mayer's 18th century work which used visual representations to teach mathematical concepts like plane and space figurate numbers.
- The author held a mathematical circle where they had high school students solve problems involving figurate numbers. Students successfully solved problems moving from concrete to abstract representations, matching Mayer's approach and Bruner's learning theory.
Mathematics is the study of patterns and relationships between numbers and shapes. While empirical evidence may be gathered, mathematical knowledge requires rigorous deductive proof based on agreed upon axioms and theorems. However, Gödel's incompleteness theorem showed that the axiomatic foundations of mathematics cannot be proven with absolute certainty from within the system. There is an ongoing debate around whether mathematical truths are discovered or invented by humans. Overall, mathematics relies on both deductive and empirical reasoning but cannot claim absolute certainty due its axiomatic foundations.
Benno artmann (auth.) euclid—the creation of mathematics-springer-verlag new ...Vidi Al Imami
This document provides background information on Euclid's Elements, a seminal work in mathematics from ancient Greece. It discusses the contents and organization of the 13 books that make up the Elements. The books cover topics in plane and solid geometry, numbers, proportions, commensurability, constructions, and the Platonic solids. The document also provides context on the origins of mathematics in ancient Greece and key figures like Pythagoras who influenced Euclid's work. It aims to help readers understand and appreciate Euclid's work, which set standards for deductive reasoning in mathematics that are still followed today.
This document is the editorial for issue 1 of the Recreational Mathematics Magazine. It provides information about the magazine's publication details, topics covered, and goals of focusing on imaginative and profound mathematical ideas in a fun way. The magazine aims to bring attention to recreational mathematics, which can reveal important insights, and to support this important subject area through high-quality publications.
Associate Professor Anita Wasilewska gave a lecture on "Descriptive Granularity" in the Distinguished Lecturer Series - Leon The Mathematician.
More Information available at:
http://dls.csd.auth.gr
The document discusses the relationship between mathematics and the natural sciences. It notes that mathematics has been remarkably successful at describing natural phenomena, which is unexpected given that mathematics is a product of human thought while nature exists independently. The document raises questions about how the "laws of nature" can be exact copies of the patterns humans discover in their abstract systems of thought. It calls this alignment of the two fields the "unreasonable effectiveness of mathematics."
Mathematics is connected to and used in many other subjects. In social studies, graphs and charts are used to compare and interpret quantitative information. In science, measurements are taken to track things like plant growth over time. Chemistry involves applications of graph theory and group theory to model molecular structures like benzene isomers. The discovery of fullerenes also linked mathematics to carbon molecule shapes modeled as polyhedra. Overall, mathematics underpins science, is present in everyday activities, and has deep historical connections to other domains of knowledge.
This document provides a history of the development of fractal geometry. It discusses how early 20th century mathematicians like Weierstrass, Cantor, Hausdorff, Julia, Fatou, and Lévy laid important foundations through their work on non-differentiable functions and self-similar sets, even if they did not use the term "fractal". It then describes how Benoit Mandelbrot in the 1970s unified these concepts and defined fractals as sets with non-integer Hausdorff dimensions. His work built directly on the earlier contributions around self-similarity, dimension, and iterative functions. The document traces the lineage of ideas that ultimately led to the definition and study of fractals as a field
This document contains a collection of mathematical problems that were historically used to discriminate against Jewish applicants during oral entrance exams for the mathematics department at Moscow State University in the Soviet Union. The problems were designed to have simple solutions but be very difficult to find. The document includes 21 such problems, along with hints and full solutions. It aims to preserve these problems and their solutions for historical and mathematical value.
This course introduces students to the art of mathematical proofs through examples from geometry, set theory, number theory, and other areas of mathematics. It begins with the basics of logic and proof techniques before exploring direct proofs, indirect proofs, proofs by induction, and other methods. The goal is for students to appreciate the beauty and creativity involved in rigorous mathematical arguments.
Diophantus was a Greek mathematician who lived in Alexandria during the 3rd century CE. He wrote the Arithmetica, which was divided into 13 books and introduced symbolic notation for unknowns and exponents. The Arithmetica contained problems involving determining integer and rational solutions to polynomial equations. Pappus of Alexandria lived in the 4th century CE and wrote the Synagoge or Collection, which contained summaries of earlier mathematical work across various topics, including constructions, number theory, and properties of curves and polyhedra. The Collection helped preserve important mathematical concepts and problems from antiquity.
This document discusses problems that could help identify students with strong mathematical abilities who may pursue careers involving serious mathematics. It presents 12 sample problems meant to capture elementary paradigms of mathematical thinking. The problems vary in difficulty and are not intended as tests, but to expose students to non-standard problems and assess how they approach such problems. Comments are provided on each problem to indicate skills and concepts they involve. The overall goal is to help teachers recognize mathematically gifted students and encourage more students to study mathematics.
Elementary geometry from an advanced standpoint(Geometría Elemental Desde Un ...Minister Education
Edwin Moise, este maravilloso libro, es lo mejor de este autor, espero que lo disfruten :) ... Y sepan valorar :)
Sigueme En twitter: https://twitter.com/bertoromer
Euclid of Alexandria was a Greek mathematician from approximately 325-265 BC who is considered the founder of geometry. In his seminal work Elements, he deduced the principles of what is now called Euclidean geometry from a small set of axioms and postulates. Elements was one of the most influential works in the history of mathematics and established principles of logic and mathematical proof still used today.
Complex numbers, neeton's method and fractalsannalf
This document presents work done as part of a "Maths and Reality" project studying geometric transformations and fractals. It summarizes the aims of the project, which are to study geometric transformations, represent reality through mathematical models using transformations, integrate traditional teaching with new technologies, build known fractals, and make graphic representations of fractals. It then provides details on specific fractals and methods analyzed as part of this work, including definitions of fractals, origins of fractal geometry, complex numbers, Newton's equations, and specific fractal examples generated through Maple code.
This document discusses the history of set theory and problems in the foundations of mathematics. It begins by covering the birth of set theory with George Cantor in the late 1800s. It then discusses paradoxes that arose in set theory, such as Russell's paradox, and early attempts to address these issues through axiomatization by mathematicians like Zermelo. The document also covers the foundational crisis in mathematics around this time and different viewpoints on the issue, such as formalism and intuitionism. It discusses Hilbert's program and Godel's incompleteness theorems. Finally, the document briefly touches on non-Euclidean geometry and number theory.
Johann Carl Friedrich Gauss was a German mathematician born in 1777 who made significant contributions to many fields including number theory, algebra, and geometry. He showed early potential in mathematics and attended several universities. Notable achievements include discovering magnetic monopoles do not exist and developing Gaussian elimination to solve systems of linear equations. Gauss also pioneered non-Euclidean geometry and its role in Einstein's theory of general relativity. He received several honors for his scientific work but was a perfectionist who did not publish all his discoveries.
This document provides an introduction to the textbook "A First Course in Topology: Continuity and Dimension" by John McCleary. The book uses the historical problem of the invariance of dimension to motivate the key concepts in topology. It covers topics like metric and topological spaces, connectedness, compactness, the fundamental group, and homology. The goal is to prove the invariance of dimension using these topological tools.
A square meets all the properties of a rectangle - it has four sides, four right angles, opposite sides that are parallel and equal in length. Additionally, all four sides of a square are equal in length. In mathematics, categories are defined inclusively so that a square is considered a special case of a rectangle. This makes theorems and proofs simpler by avoiding separate cases for different shapes.
The document discusses Hilbert's view that unsolved mathematical problems are essential for advancing the field, as they test new methods and ideas and can lead to new discoveries and applications. He cites examples throughout history where difficult problems inspired major developments, such as the problem of the shortest line influencing many areas of math and Klein's work on the icosahedron connecting geometry, group theory, and more. Hilbert argues that unsolved problems keep mathematics alive and ensure its continued independent development.
- The document discusses figurate numbers and their use in teaching mathematics.
- It presents Tobias Mayer's 18th century work which used visual representations to teach mathematical concepts like plane and space figurate numbers.
- The author held a mathematical circle where they had high school students solve problems involving figurate numbers. Students successfully solved problems moving from concrete to abstract representations, matching Mayer's approach and Bruner's learning theory.
Mathematics is the study of patterns and relationships between numbers and shapes. While empirical evidence may be gathered, mathematical knowledge requires rigorous deductive proof based on agreed upon axioms and theorems. However, Gödel's incompleteness theorem showed that the axiomatic foundations of mathematics cannot be proven with absolute certainty from within the system. There is an ongoing debate around whether mathematical truths are discovered or invented by humans. Overall, mathematics relies on both deductive and empirical reasoning but cannot claim absolute certainty due its axiomatic foundations.
Benno artmann (auth.) euclid—the creation of mathematics-springer-verlag new ...Vidi Al Imami
This document provides background information on Euclid's Elements, a seminal work in mathematics from ancient Greece. It discusses the contents and organization of the 13 books that make up the Elements. The books cover topics in plane and solid geometry, numbers, proportions, commensurability, constructions, and the Platonic solids. The document also provides context on the origins of mathematics in ancient Greece and key figures like Pythagoras who influenced Euclid's work. It aims to help readers understand and appreciate Euclid's work, which set standards for deductive reasoning in mathematics that are still followed today.
This document is the editorial for issue 1 of the Recreational Mathematics Magazine. It provides information about the magazine's publication details, topics covered, and goals of focusing on imaginative and profound mathematical ideas in a fun way. The magazine aims to bring attention to recreational mathematics, which can reveal important insights, and to support this important subject area through high-quality publications.
Associate Professor Anita Wasilewska gave a lecture on "Descriptive Granularity" in the Distinguished Lecturer Series - Leon The Mathematician.
More Information available at:
http://dls.csd.auth.gr
The document discusses the relationship between mathematics and the natural sciences. It notes that mathematics has been remarkably successful at describing natural phenomena, which is unexpected given that mathematics is a product of human thought while nature exists independently. The document raises questions about how the "laws of nature" can be exact copies of the patterns humans discover in their abstract systems of thought. It calls this alignment of the two fields the "unreasonable effectiveness of mathematics."
Mathematics is connected to and used in many other subjects. In social studies, graphs and charts are used to compare and interpret quantitative information. In science, measurements are taken to track things like plant growth over time. Chemistry involves applications of graph theory and group theory to model molecular structures like benzene isomers. The discovery of fullerenes also linked mathematics to carbon molecule shapes modeled as polyhedra. Overall, mathematics underpins science, is present in everyday activities, and has deep historical connections to other domains of knowledge.
This document provides a history of the development of fractal geometry. It discusses how early 20th century mathematicians like Weierstrass, Cantor, Hausdorff, Julia, Fatou, and Lévy laid important foundations through their work on non-differentiable functions and self-similar sets, even if they did not use the term "fractal". It then describes how Benoit Mandelbrot in the 1970s unified these concepts and defined fractals as sets with non-integer Hausdorff dimensions. His work built directly on the earlier contributions around self-similarity, dimension, and iterative functions. The document traces the lineage of ideas that ultimately led to the definition and study of fractals as a field
This document contains a collection of mathematical problems that were historically used to discriminate against Jewish applicants during oral entrance exams for the mathematics department at Moscow State University in the Soviet Union. The problems were designed to have simple solutions but be very difficult to find. The document includes 21 such problems, along with hints and full solutions. It aims to preserve these problems and their solutions for historical and mathematical value.
This course introduces students to the art of mathematical proofs through examples from geometry, set theory, number theory, and other areas of mathematics. It begins with the basics of logic and proof techniques before exploring direct proofs, indirect proofs, proofs by induction, and other methods. The goal is for students to appreciate the beauty and creativity involved in rigorous mathematical arguments.
Diophantus was a Greek mathematician who lived in Alexandria during the 3rd century CE. He wrote the Arithmetica, which was divided into 13 books and introduced symbolic notation for unknowns and exponents. The Arithmetica contained problems involving determining integer and rational solutions to polynomial equations. Pappus of Alexandria lived in the 4th century CE and wrote the Synagoge or Collection, which contained summaries of earlier mathematical work across various topics, including constructions, number theory, and properties of curves and polyhedra. The Collection helped preserve important mathematical concepts and problems from antiquity.
This document discusses problems that could help identify students with strong mathematical abilities who may pursue careers involving serious mathematics. It presents 12 sample problems meant to capture elementary paradigms of mathematical thinking. The problems vary in difficulty and are not intended as tests, but to expose students to non-standard problems and assess how they approach such problems. Comments are provided on each problem to indicate skills and concepts they involve. The overall goal is to help teachers recognize mathematically gifted students and encourage more students to study mathematics.
Elementary geometry from an advanced standpoint(Geometría Elemental Desde Un ...Minister Education
Edwin Moise, este maravilloso libro, es lo mejor de este autor, espero que lo disfruten :) ... Y sepan valorar :)
Sigueme En twitter: https://twitter.com/bertoromer
Euclid of Alexandria was a Greek mathematician from approximately 325-265 BC who is considered the founder of geometry. In his seminal work Elements, he deduced the principles of what is now called Euclidean geometry from a small set of axioms and postulates. Elements was one of the most influential works in the history of mathematics and established principles of logic and mathematical proof still used today.
Complex numbers, neeton's method and fractalsannalf
This document presents work done as part of a "Maths and Reality" project studying geometric transformations and fractals. It summarizes the aims of the project, which are to study geometric transformations, represent reality through mathematical models using transformations, integrate traditional teaching with new technologies, build known fractals, and make graphic representations of fractals. It then provides details on specific fractals and methods analyzed as part of this work, including definitions of fractals, origins of fractal geometry, complex numbers, Newton's equations, and specific fractal examples generated through Maple code.
This document discusses the history of set theory and problems in the foundations of mathematics. It begins by covering the birth of set theory with George Cantor in the late 1800s. It then discusses paradoxes that arose in set theory, such as Russell's paradox, and early attempts to address these issues through axiomatization by mathematicians like Zermelo. The document also covers the foundational crisis in mathematics around this time and different viewpoints on the issue, such as formalism and intuitionism. It discusses Hilbert's program and Godel's incompleteness theorems. Finally, the document briefly touches on non-Euclidean geometry and number theory.
Johann Carl Friedrich Gauss was a German mathematician born in 1777 who made significant contributions to many fields including number theory, algebra, and geometry. He showed early potential in mathematics and attended several universities. Notable achievements include discovering magnetic monopoles do not exist and developing Gaussian elimination to solve systems of linear equations. Gauss also pioneered non-Euclidean geometry and its role in Einstein's theory of general relativity. He received several honors for his scientific work but was a perfectionist who did not publish all his discoveries.
This document provides an introduction to the textbook "A First Course in Topology: Continuity and Dimension" by John McCleary. The book uses the historical problem of the invariance of dimension to motivate the key concepts in topology. It covers topics like metric and topological spaces, connectedness, compactness, the fundamental group, and homology. The goal is to prove the invariance of dimension using these topological tools.
Methanex is the world's largest producer and supplier of methanol. We create value through our leadership in the global production, marketing and delivery of methanol to customers. View our latest Investor Presentation for more details.
UnityNet World Environment Day Abraham Project 2024 Press ReleaseLHelferty
June 12, 2024 UnityNet International (#UNI) World Environment Day Abraham Project 2024 Press Release from Markham / Mississauga, Ontario in the, Greater Tkaronto Bioregion, Canada in the North American Great Lakes Watersheds of North America (Turtle Island).
World economy charts case study presented by a Big 4
World economy charts case study presented by a Big 4
World economy charts case
World economy charts case study presented by a Big 4
World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4
World economy charts case study presented by a Big 4
World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4World economy charts case study presented by a Big 4study presented by a Big 4
Cleades Robinson, a respected leader in Philadelphia's police force, is known for his diplomatic and tactful approach, fostering a strong community rapport.
ZKsync airdrop of 3.6 billion ZK tokens is scheduled by ZKsync for next week.pdfSOFTTECHHUB
The world of blockchain and decentralized technologies is about to witness a groundbreaking event. ZKsync, the pioneering Ethereum Layer 2 network, has announced the highly anticipated airdrop of its native token, ZK. This move marks a significant milestone in the protocol's journey, empowering the community to take the reins and shape the future of this revolutionary ecosystem.
The E-Way Bill revolutionizes logistics by digitizing the documentation of goods transport, ensuring transparency, tax compliance, and streamlined processes. This mandatory, electronic system reduces delays, enhances accountability, and combats tax evasion, benefiting businesses and authorities alike. Embrace the E-Way Bill for efficient, reliable transportation operations.
1. Using hist ory for
popul arizat ion of
mat hemat ics
Franka Miriam Brückler
Department of Mathematics
University of Zagreb
Croatia
bruckler@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) 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
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 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)
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. 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
9. Exampl e 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. Fibonacci numbers
and nature
Exampl e 3: Homework problems (possible: group
work)
possible explorations of old books or specific topics, e.g.
Fibonacci’s 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)
14. 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
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 didn’t have my meal yet. So he
walked in the direction to the restaurant...
17. 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!
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: “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.”
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. • 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
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; 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
25. • “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
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, ...)
28. 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
29. 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
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. •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/Fractions/egyptian
.html Egyptian Fractions
•http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
Fibonacci Numbers and the Golden Section
Links I
33. 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/INTR
O.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. •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
Bibl iography
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