Teaching a Course in the History of

MathematicsVictor J. KatzUniversity of the

District of Columbia V. Frederick RickeyU. S.

Military Academy

Start readingnow !

Seminar Rules Apply

- Ask any question at any time
- But, heed the schedule
- Email addresses are
- fred-rickey_at_usma.edu
- vkatz_at_udc.edu

Outline

- How to Organize a Course
- Approaches to Teaching History
- Resources for the Historian
- Student Assignments
- How to Prepare Yourself

I. 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

II. 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

III. Resources for the Historian

- Books, journals, and encyclopedias
- Web resources
- Caveat emptor

IV. Student Assignments

- Learning to use the library
- What to do about problem sets?
- Student projects
- Possible student paper topics
- Projects for prospective teachers
- Exams

V. 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

Are there other topics you would like us to

discuss?

- Note We are not teaching history here

I. 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

I.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?

I.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?

PROSPECTIVE 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

OTHER 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

1.3. 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.

MORE 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.

I.4 Types of History Courses

- Survey
- Theme
- Topics
- Sources
- Readings
- Seminar

I.5 Survey Texts

- Boyer
- Burton
- Calinger
- Cooke
- Eves
- Grattan-Guinness
- Katz
- Hodgkin
- Suzuki

1.6. 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

Sourcebooks

- 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

I.7. The design of your syllabus

- Text
- Aims
- Outline
- Readings
- Assignments
- Texts
- Plagiarism

I. 8 Is a Field Trip Feasible?

- Visit a rare book room
- Visit a museum
- Visit a book store

I.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.

II. 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

II.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

II.1. vs. External History

- Cultural background
- Mathematics is invented
- History written by historians
- Mathematics influenced by ambient culture

(Story of Maclaurin) - Biographies

II.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?

Examples 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

Examples of ideas that probably were transmitted

- Basic ideas of equation solving
- Trigonometry, both plane and spherical
- Basic concepts of combinatorics

II.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.)

II.4. Ideas from non-Western sources

Why 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

Chinese 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.

Chinese 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.

Indian 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.

Teaching a Course in the History of

MathematicsVictor J. KatzUniversity of the

District of Columbia V. Frederick RickeyU. S.

Military Academy

Islamic 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.

Islamic 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.

Islamic 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.

Islamic Proof

Why 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?

II.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?

Examples 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

Books on Ethnomathematics

- Marcia Ascher, Ethnomathematics A Multicultural

View of Mathematical Ideas (1991) - Marcia Ascher, Mathematics Elsewhere An

Exploration of Ideas Across Cultures (2002)

New concepts

II.6. TEACHING 20TH AND 21ST CENTURY MATHEMATICS

- Set Theory and Its Paradoxes
- Axiomatization
- The Statistical Revolution
- Computers and Computer Science

Recently resolved problems

II.6. TEACHING 20TH AND 21ST CENTURY MATHEMATICS

- Four Color Problem
- Classification of Finite Simple Groups
- Fermats Last Theorem
- Poincaré Conjecture

Unresolved 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)

III. Resources for the Historian

- Books, journals, and encyclopedias
- Web resources
- Caveat emptor

Twenty 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)

Books

- 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)

Books

- 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)

Books

- 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)

Books

- 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)

Ten 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

Popular 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.

III.1.a. Collection

Table of Contents

- Archimedes
- Combinatorics
- Exponentials and Logarithms
- Functions
- Geometric Proof

- Lengths, Areas, and Volumes
- Linear Equations
- Negative Numbers
- Polynomials
- Statistics
- Trigonometry

III.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

III.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

III.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,

III.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.

(No Transcript)

Rome Reborn Earliest Extant Euclid, 888

www.ibiblio.org/expo/vatican.exhibit/Vatican.exhib

it.html

The Euler Archivehttp//www.math.dartmouth.edu/

euler/

E53 Solutio problematis ad geometriam situs

pertinentis

http//www.math.ubc.ca/people/faculty/cass/Euclid

/byrne.html

http//www.lib.cam.ac.uk/RareBooks/PascalTraite/

David Joyces History of Mathematics Homepage

- http//aleph9.clarku.edu/djoyce/
- mathhist/

IV. Student Assignments

- Learning to use the library
- What to do about problem sets?
- Student projects
- Possible student paper topics
- Projects for prospective teachers
- Exams

IV.1. Learning to use the library

- Tour of library
- Your mathematician
- Prize nomination
- Vita

IV.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

Discussion Problems

- Compare and contrast methods
- Develop a lesson for the classroom based on a

particular historical development - Discuss the pedagogy of an old textbook

IV.3. Projects

- Written projects and/or oral reports?
- Joint or individual projects?

IV.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

IV.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.

IV. 6. Exams

- Mathematical Problems
- Short Answer Questions
- Multiple Choice Questions
- Essay Questions

Mathematical 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

Mathematical 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.

Short 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.

Short 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

_____________________.

Essay 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?

Essay 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.

V. 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

Start readingnow !

V.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

How 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

V.2. Collect illustrations

- Pictures of mathematicians.
- Title pages of famous works.
- Significant pages from important works.
- Maps.
- Quotations from famous mathematicians.

Emilie du Chatelet

Paciolis Summa

Reischs Margarita Philosophica

V.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

Descartes

Outline 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.

Scientific 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.

Powerpoint 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.

- 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

V.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

V.5. Advertising your course.

- Talk to former students
- Send email to majors
- Post flyers
- Talk to colleagues

V.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.

V.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)

Start readingnow !

- PowerPoint and additional information available

at - http//www.dean.usma.edu/
- departments/math/people/
- rickey/hm/mini/default.html

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