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Teaching a Course in the History of Mathematics Victor J. Katz University of the District of Columbia V. Frederick Rickey U. S. Military Academy. Start reading now !. Seminar Rules Apply. Ask any question at any time But, heed the schedule Email addresses are fred-rickey@usma.edu

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  1. Teaching a Course in the History of MathematicsVictor J. KatzUniversity of the District of ColumbiaV. Frederick RickeyU. S. Military Academy

  2. Start readingnow !

  3. Seminar Rules Apply • Ask any question at any time • But, heed the schedule • Email addresses are fred-rickey@usma.edu vkatz@udc.edu

  4. Outline • How to Organize a Course • Approaches to Teaching History • Resources for the Historian • Student Assignments • How to Prepare Yourself

  5. 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

  6. 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

  7. III. Resources for the Historian • Books, journals, and encyclopedias • Web resources • Caveat emptor

  8. 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

  9. 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

  10. Are there other topics you would like us to discuss? • Note: We are not teaching history here

  11. 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

  12. 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?

  13. 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?

  14. 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

  15. 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

  16. 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.

  17. 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.

  18. I.4    Types of History Courses • Survey • Theme • Topics • Sources • Readings • Seminar

  19. I.5 Survey Texts • Boyer • Burton • Calinger • Cooke • Eves • Grattan-Guinness • Katz • Hodgkin • Suzuki

  20. 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

  21. 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

  22. I.7. The design of your syllabus • Text • Aims • Outline • Readings • Assignments • Texts • Plagiarism

  23. I. 8    Is a Field Trip Feasible? • Visit a rare book room • Visit a museum • Visit a book store

  24. 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.

  25. 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

  26. 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”

  27. II.1. vs. External History • Cultural background • Mathematics is invented • History written by historians • Mathematics influenced by ambient culture (Story of Maclaurin) • Biographies

  28. 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?

  29. 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

  30. Examples of ideas that probably were transmitted • Basic ideas of equation solving • Trigonometry, both plane and spherical • Basic concepts of combinatorics

  31. 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”.)

  32. II.4. Ideas from non-Western sources

  33. 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

  34. 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.

  35. 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.

  36. 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.

  37. Teaching a Course in the History of MathematicsVictor J. KatzUniversity of the District of ColumbiaV. Frederick RickeyU. S. Military Academy

  38. 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.”

  39. 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.

  40. 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.”

  41. Islamic Proof

  42. 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?

  43. 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?

  44. 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

  45. Books on Ethnomathematics • Marcia Ascher, Ethnomathematics: A Multicultural View of Mathematical Ideas (1991) • Marcia Ascher, Mathematics Elsewhere: An Exploration of Ideas Across Cultures (2002)

  46. II.6. TEACHING 20TH AND 21ST CENTURY MATHEMATICS New concepts: • Set Theory and Its Paradoxes • Axiomatization • The Statistical Revolution • Computers and Computer Science

  47. Recently resolved problems: II.6. TEACHING 20TH AND 21ST CENTURY MATHEMATICS • Four Color Problem • Classification of Finite Simple Groups • Fermat’s Last Theorem • Poincaré Conjecture

  48. 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)

  49. III. Resources for the Historian • Books, journals, and encyclopedias • Web resources • Caveat emptor

  50. 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)

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