Title: Teaching a Course in the History of Mathematics Victor J. Katz University of the District of Columbia V. Frederick Rickey U. S. Military Academy
1Teaching a Course in the History of
MathematicsVictor J. KatzUniversity of the
District of Columbia V. Frederick RickeyU. S.
Military Academy
2Start readingnow !
3Seminar Rules Apply
- Ask any question at any time
- But, heed the schedule
- Email addresses are
- fred-rickey_at_usma.edu
- vkatz_at_udc.edu
4Outline
- How to Organize a Course
- Approaches to Teaching History
- Resources for the Historian
- Student Assignments
- How to Prepare Yourself
5I. How to Organize a Course
- Who is your audience?
- What are their needs?
- What are the aims of your course?
- Types of history courses
- Textbooks for survey courses with comments
- Textbooks for other types of courses
- The design of your syllabus
- Is a field trip feasible?
- History of Math Courses on the Web
6II. Approaches to Teaching History
- Internal vs. External History
- Whig History
- The Role of Myths
- Ideas from non-Western sources
- Teaching ethnomathematics
- Teaching 20th and 21st century mathematics
7III. Resources for the Historian
- Books, journals, and encyclopedias
- Web resources
- Caveat emptor
8IV. Student Assignments
- Learning to use the library
- What to do about problem sets?
- Student projects
- Possible student paper topics
- Projects for prospective teachers
- Exams
9V. How to Prepare Yourself
- Start a reading program now!
- Collect illustrations
- Outline your course day by day
- Get to know your library and librarians
- Advertising your course
- Counteract negative views
- Record keeping
10Are there other topics you would like us to
discuss?
- Note We are not teaching history here
11I. How to Organize a Course
- Who is your audience?
- What are their needs?
- What are the aims of your course?
- Types of history courses
- Textbooks for survey courses with comments
- Textbooks for other types of courses
- The design of your syllabus
- Is a field trip feasible?
- History of Math Courses on the Web
12I.1. Who is your audience?
- What level are your students?
- How good are your students?
- What type of school are you at?
- How much mathematics or general history do they
know? Answer Not enough! - Is the course for liberal arts students?
- What will they do after graduation?
13I.2. What are their needs?
- If your students are prospective teachers, what
history will benefit them? - Why are the students taking the course?
- How much fact do the students need to know?
- Is this a capstone course for mathematics majors
that is intended to tie together what they have
learned in other course?
14PROSPECTIVE HIGH SCHOOL TEACHERS
- Teach more mathematics
- Make sure to deal with the history of topics in
the high school curriculum - Discuss the use of history in teaching secondary
mathematics courses - Stress the connections among various parts of the
curriculum
15OTHER MATHEMATICS MAJORS
- History as a capstone course helps to tie
together what they have learned - Graduate school and academia
- Need to understand the development of ideas and
how to use these in future teaching - How and why abstraction became so important in
the nineteenth century
161.3. AIMS OF THE COURSE
- To give life to your knowledge of mathematics.
- To provide an overview of mathematics
- To teach you how to use the library and internet.
- To indicate how you might use the history of
mathematics in your future teaching. - To improve your written communication skills in a
technical setting.
17MORE AIMS
- To show that mathematics has been developed in
virtually every literate civilization in history,
as well as in some non-literate societies. - To compare and contrast the approaches to
particular mathematical ideas among various
civilizations. - To demonstrate that mathematics is a living field
of study and that new mathematics is constantly
being created.
18I.4 Types of History Courses
- Survey
- Theme
- Topics
- Sources
- Readings
- Seminar
19I.5 Survey Texts
- Boyer
- Burton
- Calinger
- Cooke
- Eves
- Grattan-Guinness
- Katz
- Hodgkin
- Suzuki
201.6. Textbooks for other courses
- Dunham, Journey through genius the great
theorems of mathematics - Berlinghoff and Gouvêa, Math through the ages A
gentle history for teachers and others - Bunt, Jones, and Bedient, The historical roots of
elementary mathematics - Joseph,The crest of the peacock non-European
roots of mathematics - Struik, A concise history of mathematics. New
York
21Sourcebooks
- John Fauvel and Jeremy Gray, The History of
Mathematics A Reader - Ronald Calinger, Classics of Mathematics
- Jacqueline Stedall, Mathematics Emerging A
Sourcebook 1540-1900 - Victor J. Katz, ed., The Mathematics of Egypt,
Mesopotamia, China, India, and Islam A
Sourcebook
22I.7. The design of your syllabus
- Text
- Aims
- Outline
- Readings
- Assignments
- Texts
- Plagiarism
23I. 8 Is a Field Trip Feasible?
- Visit a rare book room
- Visit a museum
- Visit a book store
24I.9. HM Courses on the Web
- Many individuals have placed information about
their courses on the web. - See the url on p. 1 of the handout, which will
take you to Rickeys pages on this minicourse. - Note especially the sources course of Gary
Stoudt, whose url is on p. 6 of the handout.
25II. Approaches to Teaching History
- Internal vs. External History
- Whig History
- The Role of Myths
- Ideas from non-Western sources
- Teaching ethnomathematics
- Teaching 20th and 21st century mathematics
26II.1. Internal History
- Development of ideas
- Mathematics is discovered (Platonism)
- History written by mathematicians
- Mathematics is the same, whether created in
Babylon, Greece, or France i.e., mathematics is
universal
27II.1. vs. External History
- Cultural background
- Mathematics is invented
- History written by historians
- Mathematics influenced by ambient culture
(Story of Maclaurin) - Biographies
28II.2. Whig History
- It pictures mathematics as progressively and
inexorably unfolding, brilliantly impelled along
its course by a few major characters, becoming
the massive edifice of our present inheritance. - Does history then only include ideas that were
transmitted somehow to the present or had
influence later on? - Or do we try to understand mathematical ideas in
context?
29Examples of ideas that were probably not
transmitted
- Indian development of power series
- Babylonian solution of quadratic equations
- Islamic work on sums of integral powers
- Chinese solution of simultaneous congruences
- Gausss notebooks
30Examples of ideas that probably were transmitted
- Basic ideas of equation solving
- Trigonometry, both plane and spherical
- Basic concepts of combinatorics
31II.3. The Role of Myths
- What myths do we tell?
- What myths do we want future teachers to tell
their students? - Do we tell the truth and nothing but the truth?
(We cannot tell the whole truth.)
32II.4. Ideas from non-Western sources
33Why non-Western Mathematics?
- Not all mathematics developed in Europe
- Some mathematical ideas moved to Europe from
other civilizations - Relevance of Islam, China, India today
- Mathematics important in every literate culture
- Compare solutions of similar problems
- Diversity of your students and your students
prospective students
34Chinese Remainder Theorem
- Why is it called the Chinese Remainder Theorem?
- The first mention of Chinese mathematics in a
European language was in 1852 by Alexander Wylie
Jottings on the Science of the Chinese
Arithmetic - Among the topics discussed is the earliest
appearance of what is now called the Chinese
Remainder problem and how it was initially solved
in fourth century China, in Master Suns
Mathematical Manual.
35Chinese Remainder Theorem
- We have things of which we do not know the
number if we count them by threes, the remainder
is 2 if we count them by fives, the remainder is
3 if we count them by sevens, the remainder is
2. How many things are there? - If you count by threes and have the remainder 2,
put 140. If you count by fives and have the
remainder 3, put 63. If you count by sevens and
have the remainder 2, put 30. Add these numbers
and you get 233. From this subtract 210 and you
get 23. - For each unity as remainder when counting by
threes, put 70. For each unity as remainder when
counting by fives, put 21. For each unity as
remainder when counting by sevens, put 15. If
the sum is 106 or more, subtract 105 from this
and you get the result.
36Indian proof of sum of squares
- A sixth part of the triple product of the
term-count n plus one, that sum plus the
term-count, and the term-count, in order, is the
total of the series of squares. Being that this
is demonstrated if there is equality of the total
of the series of squares multiplied by six and
the product of the three quantities, their
equality is to be shown.
37Teaching a Course in the History of
MathematicsVictor J. KatzUniversity of the
District of Columbia V. Frederick RickeyU. S.
Military Academy
38Islamic Proof
- This example is taken from the Book on the
Geometrical Constructions Necessary to the
Artisan by Abu al-Wafaal-Buzjani (940-997). He
had noticed that artisans made use of geometric
constructions in their work. But, A number of
geometers and artisans have erred in the matter
of these squares and their assembling. The
geometers have erred because they have little
practice in constructing, and the artisans have
erred because they lack knowledge of proofs.
39Islamic Proof
- I was present at some meetings in which a
group of geometers and artisans participated.
They were asked about the construction of a
square from three squares. A geometer easily
constructed a line such that the square of it is
equal to the three squares, but none of the
artisans was satisfied with what he had done.
40Islamic Proof
- Abu al-Wafa then presented one of the
methods of the artisans, in order that the
correct ones may he distinguished from the false
ones and someone who looks into this subject will
not make a mistake by accepting a false method,
God willing. But this figure which he constructed
is fanciful, and someone who has no experience in
the art or in geometry may consider it correct,
but if he is informed about it he knows that it
is false.
41Islamic Proof
42Why Was Modern Mathematics Developed in the West?
- Compare mathematics in China, India, the Islamic
world, and Europe around 1300 - Europe was certainly behind the other three
- Ideas of calculus were evident in both India and
Islam - But in next 200 years, development of mathematics
virtually ceased in China, India, and Islam, but
exploded in Europe - Why?
43II.5. Teaching Ethnomathematics
- Mathematical Ideas of traditional peoples
- What is a mathematical idea?
- Idea having to do with number, logic, and
spatial configuration and especially in the
combination or organization of those into systems
and structures. - Can these mathematical ideas of traditional
peoples be related to Western mathematical ideas?
44Examples of Ethnomathematics
- Mayan arithmetic and calendrical calculations
- Inca quipus
- Tracing graphs among the Bushoong and Tshokwe
peoples of Angola and Zaire - Symmetries of strip decorations
- Logic of divination in Madagascar
- Models and maps in the Marshall Islands
45Books on Ethnomathematics
- Marcia Ascher, Ethnomathematics A Multicultural
View of Mathematical Ideas (1991) - Marcia Ascher, Mathematics Elsewhere An
Exploration of Ideas Across Cultures (2002)
46New concepts
II.6. TEACHING 20TH AND 21ST CENTURY MATHEMATICS
- Set Theory and Its Paradoxes
- Axiomatization
- The Statistical Revolution
- Computers and Computer Science
47Recently resolved problems
II.6. TEACHING 20TH AND 21ST CENTURY MATHEMATICS
- Four Color Problem
- Classification of Finite Simple Groups
- Fermats Last Theorem
- Poincaré Conjecture
48Unresolved problems
II.6. TEACHING 20TH AND 21ST CENTURY MATHEMATICS
- Hilberts 1900 list of Problems
- Which Problems Are Still Unresolved?
- See Ben Yandell, The Honors Class (2002)
- Clay Millennium Prize Problems
- Riemann Hypothesis
- Birch and Swinnerton-Dyer Conjecture
- See K. Devlin, The Millennium Problems
(2002)
49III. Resources for the Historian
- Books, journals, and encyclopedias
- Web resources
- Caveat emptor
50Twenty Scholarly Books
- Jens Høyrup, Lengths, Widths, Surfaces A
Portrait of Old Babylonian Algebra and Its Kin
(2002) - Eleanor Robson Mathematics in Ancient Iraq A
Social History (2008) - Kim Plofker Mathematics in India 500 BCE
1800 CE (2009) - Jean-Claude Martzloff, A History of Chinese
Mathematics, translated by Stephen S. Wilson
(1997)
51Books
- Victor J. Katz, ed., The Mathematics of Egypt,
Mesopotamia, India, China, and Islam A
Sourcebook (2007) - D. H. Fowler, The Mathematics of Plato's Academy
A New Reconstruction (1987, 1999) - S. Cuomo, Ancient Mathematics (2001)
- Reviel Netz, The Transformation of Mathematics in
the Early Mediterranean World From Problems to
Equations (2004)
52Books
- J. Lennart Berggren, Episodes in the Mathematics
of Medieval Islam (1986) - Glen Van Brummelen, The Mathematics of the
Heavens and the Earth The Early History of
Trigonometry (2009) - Jeremy Gray, Worlds Out of Nothing A Course in
the History of Geometry in the 19th Century
(2007) - Hans Wussing, The Genesis of the Abstract Group
Concept (1984) -
53Books
- Ivor Grattan-Guinness, ed. From the Calculus to
Set Theory, 1630-1910 An Introductory History
(1980) - C. H. Edwards, The Historical development of the
calculus (1979) - Judith V. Grabiner, The Origins of Cauchy's
Rigorous Calculus (1981) - Stephen M. Stigler, The History of Statistics
(1986)
54Books
- Gerd Gigerenzer et al, The Empire of Chance How
Probability Changed Science and Everyday Life
(1989) - Ed Sandifer, The Early Mathematics of Leonhard
Euler (2007) - Robert Bradley and C. Edward Sandifer, eds.,
Leonhard Euler Life, Work and Legacy (2007) - Ivor Grattan-Guinness, ed., Landmark Writings in
Western Mathematics, 1640-1940 (2005)
55Ten Popular Books
- Derbyshire, John. Prime Obsession Bernhard
Riemann and the Greatest Unsolved Problem in
Mathematics, 2003. - Dunham, William. The Calculus Gallery
Masterpieces from Newton to Lebesgue, 2005 - Havil, Julian. Gamma Exploring Eulers
Constant, 2003. - Maor, Eli Trigonometric Delights, 1998.
- Netz, Reviel and William Noel. The Archimedes
Codex, 2007
56Popular Books
- Nahin, Paul J. An Imaginary Tale The Story of
?1, 1998. - __________ When Least Is Best How
Mathematicians Discovered Many Clever Ways to
Make Things as Small (or as Large) as Possible,
2003. - Salsburg, David. The Lady Tasting Tea How
Statistics Revolutionized Science in the
Twentieth Century, 2001. - Sobel, Dava. Longitude The True Story of a
Lone Genius Who Solved the Greatest Scientific
Problem of His Time, 1995. - Wilson, Robin. Four Colours Suffice How the
Map Problem Was Solved, 2002.
57III.1.a. Collection
58Table of Contents
- Archimedes
- Combinatorics
- Exponentials and Logarithms
- Functions
-
- Geometric Proof
- Lengths, Areas, and Volumes
- Linear Equations
- Negative Numbers
- Polynomials
- Statistics
- Trigonometry
59III.1.b. History Journals
- Historia Mathematica
- Isis
- Archive for History of Exact Sciences
- The British Journal for the History of Science
- Annals of Science
- History of Science
60III.1.b. Popular Journals
- The MAA journals
- Mathematical Intelligencer
- Physics Teacher
- Scientific American
- Mathematical Gazette
- Bulletin of the British Society of the History of
Mathematics
61III.1.c. Encyclopedias
- Companion encyclopedia of the history and
philosophy of the mathematical sciences, edited
by I. Grattan-Guinness - Dictionary of Scientific Biography
- The Encyclopaedia Britannica, 11th ed
- The Encyclopedia of Philosophy
- Dictionary of American Biography,
62III.2. WEB RESOURCES
- These are so abundant and the search engines are
so good, that it seems futile to attempt anything
comprehensive. - Here are a few especially useful ones.
63(No Transcript)
64Rome Reborn Earliest Extant Euclid, 888
www.ibiblio.org/expo/vatican.exhibit/Vatican.exhib
it.html
65The Euler Archivehttp//www.math.dartmouth.edu/
euler/
E53 Solutio problematis ad geometriam situs
pertinentis
66http//www.math.ubc.ca/people/faculty/cass/Euclid
/byrne.html
67http//www.lib.cam.ac.uk/RareBooks/PascalTraite/
68David Joyces History of Mathematics Homepage
- http//aleph9.clarku.edu/djoyce/
- mathhist/
69IV. Student Assignments
- Learning to use the library
- What to do about problem sets?
- Student projects
- Possible student paper topics
- Projects for prospective teachers
- Exams
70IV.1. Learning to use the library
- Tour of library
- Your mathematician
- Prize nomination
- Vita
71IV.2. Problem Sets
- Solve a problem as it was solved in a particular
time period - Complete the development of a particular idea or
procedure - Solve an old problem using modern tools and
compare methods - Generalize an old problem solving procedure
72Discussion Problems
- Compare and contrast methods
- Develop a lesson for the classroom based on a
particular historical development - Discuss the pedagogy of an old textbook
-
73IV.3. Projects
- Written projects and/or oral reports?
- Joint or individual projects?
74IV.4. Possible Student Paper Topics
- Bourbaki
- Julia Robinson and Hilbert's tenth problem
- Alan Turing
- Dürer's Polyhedra
- The Four Color Problem
- Holbein's Ambassadors
- Daniel Bernoulli the spread of smallpox
75IV.5. Projects for Prospective Teachers
- Compare the Babylonian, Mayan, and Hindu-Arabic
place-value systems in their historical
development and their ease of use. Devise a
lesson on this. - Analyze the history of the limit concept from
Eudoxus to the mid-eighteenth century, including
Berkeley's criticisms and Maclaurin's response.
Create a lesson. - Discuss how the history of the solution of cubic
equations from the Islamic period through the
work of Lagrange can be used in algebra classes. - Compare the teaching of algebra (or geometry) in
the eighteenth century and the twentieth by
studying textbooks.
76IV. 6. Exams
- Mathematical Problems
- Short Answer Questions
- Multiple Choice Questions
- Essay Questions
77Mathematical Problems
- 1. Translate a Babylonian problem and solution
into modern terms - 2. Solve a quadratic problem of Abu Kamil by
first converting it into one of the six types of
quadratic equations and then using the method for
that type. For example, suppose 10 is divided
into two parts, each one of which is divided by
the other, and the sum of the quotients is 4 ¼.
Find the two parts. - 3. Use Fermats method to find the maximum of
- bx x3
78Mathematical Problems
- 4. Find the relationship of the fluxions of x
and y on the curve x2 xy y3 7 using one of
Newtons procedures. - 5. Derive the quotient rule of calculus by an
argument using differentials. - 6. Give a geometric argument using
differentials and similar triangles to show that - d(sin x) cos x dx.
- 7. Show that one can solve the cubic equation
- x3 d bx2 by intersecting the hyperbola
- xy d and the parabola y2 dx db 0.
79Short Answer Questions
- 1. Order the following mathematical discoveries
by their approximate date, beginning with the
earliest - a. Solution of the system of equations which we
express as xy a, x y b. - b. Earliest explicit expression of the
multiplicative rule for combinations. - c. Development of the base 60 place value system.
- d. Development of the base 10 place value system.
- e. First statement of the parallel postulate.
- f. First extant rigorous proof of the rule for
combinations expressed in b.
80Short Answer Questions
- 2. State one mathematical contribution of each of
the following Cardano, Bombelli, Viete,
Harriot. - 3. What is Cardanos Ars Magna and why is it
important? - 4. Trigonometry was originally developed to
_____________________.
81Essay Questions
- 1. Outline the major contributions to
trigonometry of the civilizations of Greece,
India, and Islam. - 2. Today, mathematics is often thought of as the
intellectual exercise of proving theorems using
logical reasoning and beginning with explicitly
stated definitions and axioms. Were the
Babylonians and the Egyptians, then, doing
mathematics? Explain. - 3. Describe the proof method called today the
method of mathematical induction. Was the proof
by Levi ben Gerson giving the formula for the
number of permutations of a set of n objects a
proof by mathematical induction? Why?
82Essay Questions
- 4. Is mathematics invented or discovered?
Discuss with reference to at least four
mathematical concepts discussed this semester. - 5. Compare and contrast Newtons and Leibnizs
versions of the calculus. In your answer,
include ideas in differentiation, integration,
solving differential equations, and applications
to physical problems. - 6. Compare and contrast the use of axioms by
Euclid with the use of axioms around the turn of
the twentieth century. For the latter period,
you may pick one or two axiom sets to provide a
focus for your discussion. - 7. Why is rigor in analysis so important?
After all, Newton and Leibniz worked out the
basics of the calculus without it. Give examples
to support your argument.
83V. How to Prepare Yourself
- Start a reading program now!
- Collect illustrations
- Outline your course day by day
- Get to know your library and librarians
- Advertising your course
- Counteract negative views
- Record keeping
84Start readingnow !
85V.1. Start a reading program now!
- Read a survey text
- Read journal articles
- Read deeply in the history of a mathematical
field you know well
86How to Learn More History
- Go to talks at meetings
- Join the Canadian Society for the History and
Philosophy of Mathematics and the British Society
for the History of Mathematics - Join/start a seminar in the history of mathematics
87V.2. Collect illustrations
- Pictures of mathematicians.
- Title pages of famous works.
- Significant pages from important works.
- Maps.
- Quotations from famous mathematicians.
88Emilie du Chatelet
89Paciolis Summa
90Reischs Margarita Philosophica
91V.3. Outline Your Course Day by Day
- Decide on nature of course (Chronological,
Themed, Combination) - Pick out key general concepts and order them
- For each key concept, pick out specific topics to
cover - Choose a topic or topics for each available day
- Pick materials related to each chosen topic
- Give yourself flexibility, for undoubtedly you
will have planned too much
92Descartes
93Outline for day xx Descartes on Analytic
Geometry
- Biographical information worth mentioning
- Attended a good school. Recent work of Galileo
and his telescope was discussed. - A sickly lad. Lay in bed.
- Fly and analytic geometry. True?
- Importance of contact with the Dutch. Latin.
- Queen Christina of Sweden. Death.
94Scientific work of Descartes
- Philosophia mathematica. Newton used this title.
- Method. Cogito ergo sum. Tell Ari Katz joke.
- Geometry is an appendix. You can read it.
Translations. - Optics Snell's law, rainbow.
- Started analytic geometry. Oblique axes. xyz for
variables. Exponent notation - Curves geometric vs. mechanical. Examples.
- He went from geometry to algebra, not v.v.
- Had a method to solve any problem (and Newton
believed him!). - Folium of Descartes. Fermat has a better method
for tangents. - Says you can't do arc length. Set up for van
Heuraet and Newton.
95Powerpoint to Prepare
- Portraits
- Seated, Schooten, stamps, Vic Norton's cartoon on
aliasing. - Title pages
- Geometrie 1637, 1649, 1659. Newton read the
second Latin edition. - Translations Smith-Latham, Olscamp.
- Quotations
- Rules of problem solving. For prospective
teachers especially. - Newton on reading Descartes.
- This method will solve all problems.
- Selected Pages
- Solution of quadratic equations.
- Conchoid is geometric.
- Finding tangents by the two circle method.
- Heuraet on arc length from second Latin edition
(1659). - Folium of Descartes. 1638 definition. Graphs of
Newton and L'Hospital.
96- Things to take to class to pass around
- Smith-Latham translation of Geometry.
- Olscamp translation of the whole Method.
- Polya's How to Solve It.
- Things to read before class
- Sections of the text the students are to read.
- The DSB article on Descartes.
- Look at J. F. Scott's, The scientific work of
René Descartes, 1952 - Read section on Descartes in Grattan-Guinnesss
Landmarks
97V.4. Get to know your library and librarians
- Look at every book
- Find the specialized librarians
- Tell them your interests
- Ask them to help your students
- Be determined to find answers
- Visit a rare book room
98V.5. Advertising your course.
- Talk to former students
- Send email to majors
- Post flyers
- Talk to colleagues
99V.6. Counteract Negative Views
- Among some mathematicians the history of
mathematics is not regarded as a serious pursuit.
- It is worth your while to spend some time talking
to your colleagues about your course. Point out
to them that you are doing significant amounts of
mathematics in your course (give some
illustrations). Point out that it is not a course
in anecdotes. - Students must master a great deal of material and
they are required to write about mathematics in a
way that shows that they have mastered the
details.
100V.7 Record keeping
- Immediately record full reference for items you
photocopy - Record what references you use for each class
- Record how you could improve the class (and what
not to do again)
101Start readingnow !
102- PowerPoint and additional information available
at - http//www.dean.usma.edu/
- departments/math/people/
- rickey/hm/mini/default.html
103Advertisement
- Using Historical Topics of Mathematics
Effectively in the High School Curriculum - David Kullman (Miami University of Ohio) will
host a workshop on June 23-25, 2009 at Bluffton
University. The cost is 125 which includes
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bollenbacher_at_bluffton.edu or 419-358-3296 - Further information will appear on the
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104Advertisement
- History of Undergraduate Mathematics in America
(HUMA II) - Meeting is planned for the summer of 2010 at West
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- Contact Fred Rickey.