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Teaching a Course in the History of Mathematics

Victor J. KatzUniversity of the District of Columbia

V. Frederick RickeyU. S. Military Academy

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Seminar Rules Apply• Ask any question at any time

• But, heed the schedule

• Email addresses are

fred-rickey@usma.edu

vkatz@udc.edu

Outline

I. How to Organize a Course

II. Approaches to Teaching History

III. Resources for the Historian

IV. Student Assignments

V. How to Prepare Yourself

I. How to Organize a Course

1. Who is your audience?2. What are their needs? 3. What are the aims of your course? 4. Types of history courses5. Textbooks for survey courses with comments6. Textbooks for other types of courses7. The design of your syllabus8. Is a field trip feasible? 9. History of Math Courses on the Web

II. Approaches to Teaching History

1. Internal vs. External History

2. Whig History

3. The Role of Myths

4. Ideas from non-Western sources

5. Teaching ethnomathematics

6. Teaching 20th and 21st century mathematics

III. Resources for the Historian

1. Books, journals, and encyclopedias

2. Web resources

3. Caveat emptor

IV. Student Assignments

1. Learning to use the library

2. What to do about problem sets?

3. Student projects

4. Possible student paper topics

5. Projects for prospective teachers

6. Exams

V. How to Prepare Yourself

1. Start a reading program now!

2. Collect illustrations

3. Outline your course day by day

4. Get to know your library and librarians

5. Advertising your course

6. Counteract negative views

7. 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

1. Who is your audience?2. What are their needs? 3. What are the aims of your course? 4. Types of history courses5. Textbooks for survey courses with comments6. Textbooks for other types of courses7. The design of your syllabus8. Is a field trip feasible? 9. 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 Rickey’s 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

1. Internal vs. External History

2. Whig History

3. The Role of Myths

4. Ideas from non-Western sources

5. Teaching ethnomathematics

6. 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

Gauss’s 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 Sun’s 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 Mathematics

Victor 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-Wafā’al-Būzjānī (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:

• Set Theory and Its Paradoxes

• Axiomatization

• The Statistical Revolution

• Computers and Computer Science

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

Recently resolved problems:

• Four Color Problem

• Classification of Finite Simple Groups

• Fermat’s Last Theorem

• Poincaré Conjecture

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

Unresolved problems:• Hilbert’s 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)

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

III. Resources for the Historian

1. Books, journals, and encyclopedias

2. Web resources

3. 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 Euler’s 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.

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

Rome RebornEarliest Extant Euclid, 888

The Euler Archive

http://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 Joyce’s History of Mathematics Homepage

• http://aleph9.clarku.edu/~djoyce/

mathhist/

IV. Student Assignments

1. Learning to use the library

2. What to do about problem sets?

3. Student projects

4. Possible student paper topics

5. Projects for prospective teachers

6. 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 Fermat’s 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 Newton’s 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 Cardano’s 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 Newton’s and Leibniz’s 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

1. Start a reading program now!

2. Collect illustrations

3. Outline your course day by day

4. Get to know your library and librarians

5. Advertising your course

6. Counteract negative views

7. 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

Pacioli’s Summa

Reisch’s 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-

Guinness’s 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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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 lunches.

Contact Duane Bollenbacher at bollenbacher@bluffton.edu or 419-358-3296

Further information will appear on the webpage of the Ohio Section of the MAA.

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History of Undergraduate Mathematics in America (HUMA II)

Meeting is planned for the summer of 2010 at West Point.

Your research contributions are welcome.

Contact Fred Rickey.