Edited by

Reinhard Laubenbacher and David Pengelley

Mathematical Sciences, New Mexico State University

May, 1995

(Revised May, 2000 - 2008)

This portfolio contains the papers produced by school teachers and
graduate
students during a one semester workshop at New Mexico State University
during the Spring of 1995, and a resulting graduate course in
subsequent
years. The workshop occurred in the form of the course MATH 495/501, *Workshop
for Teachers: Using Historical Sources in Teaching Mathematics*, and
evolved into the regular graduate course MATH 561, *The Role of
History
in Teaching Mathematics*.

The papers in this volume have been prepared as teaching resources, mostly built around original historical source material in mathematics. Each is a self-contained supplement ready for use with students, most comprising an original source, mathematical and historical annotation for teacher and students, a discussion of the context of the mathematics, guidance for the teacher on how and where to use the supplement, exercises, and suggestions for further reading. The papers are intended for all levels ranging from middle school to graduate-level mathematics. Also included in the portfolio is a set of assessment guidelines prepared by the participants and ourselves, with guidance from Bonnie Votaw, for assessing the effectiveness of these teaching materials with students.

Our desire to work with present and future teachers grew out of our use of original historical sources in the undergraduate curriculum. We have found that exposing students directly to historical sources in mathematics contributes greatly to motivation and understanding, and brings mathematics alive as an ongoing process of discovery. Our philosophy and experiences are expressed in the article ``Recovering Motivation in Mathematics: Teaching with Original Sources'', included in this portfolio.

The papers in this volume are outstanding examples of what can be achieved through enthusiasm and commitment by dedicated educators. We hope they represent the beginning of an ongoing enterprise to produce a rich collection of teaching resources based on historical texts.

The original workshop was made possible by funding from the NMSU Department of Mathematical Sciences and Dean of the College of Arts and Sciences, the Las Cruces Public Schools, and the New Mexico Center for Teaching Excellence. We owe special thanks to Bonnie Votaw, Amalya Nattiv, and Doug Kurtz, for recognizing the promise of this endeavor.

Reinhard Laubenbacher and David Pengelley

Department of Mathematical Sciences

New Mexico State University

Las Cruces, NM 88003

Tel. 505--646--3901; FAX 505--646--1064

e-mail reinhard@nmsu.edu davidp@nmsu.edu

May 3, 1995 (with later additions)

1. An Ancient Construction Yields Solutions to Cubic Equations (Ali
Ahmad, NMSU)

Algebra, Euclid and Omar Khayyam

2. Graphs and More Graphs (Daniel Fish, NMSU)

Statistics, Florence Nightingale

3. Stem and Leaf Diagrams (Daniel Fish, NMSU)

Statistics, John Tukey

4. Box Plots (Daniel Fish, NMSU)

Statistics, John Tukey

5. Restoring the Square - The Methods of Al-jabr (Mike Goar,
Mayfield
High School)

Algebra and Geometry, al-Khowarizmi

6. Getting to the Root of the Problem: A Hint from René
Descartes
(Mike Goar, Mayfield High School)

Algebra, René Descartes

7. Linear Programming and the Simplex Algorithm (Jeff Holland, NMSU)

Linear Algebra, George Dantzig

8. Lobachevskian Geometry (Jeff Holland, NMSU)

Geometry, Lobachevsky

9. The Bernoulli Numbers (Tuesday Johnson, NMSU)

Number theory, Jacob Bernoulli

10. Every Integer is a Sum of Four Squares (Tuesday Johnson, NMSU)

Number Theory, Leonhard Euler

11. Ptolemy's ``Table of Straight Lines in a Circle'' (Mary Joy,
Mayfield
High School)

Trigonometry, Claudius Ptolemy

12. Developing the Quadratic Formula (Glendel DeVon Love, Las Cruces
High School)

Algebra, al-Khowarizmi

13. The Euclidean Algorithm (Glendel DeVon Love, Las Cruces High
School)

Number Theory and Geometry, Euclid

14. Egyptian Mathematics (Cynthia Oszman, White Sands Middle School)

Arithmetic, The Rhind Papyrus

15. Mathematics Using Sumerian Economic Texts (Cynthia Oszman, White
Sands Middle School)

Arithmetic, Texts from the Drehem archive

16. Fibonacci's Rabbit Problem, and Consecutive Odd Numbers (Carmen
Pareo, Gadsden High School)

Number Theory, Leonardo Fibonacci

17. Operation Tangent (Michael Quintana, NMSU)

Tangent Lines to Curves, Rene Descartes and Pierre de Fermat

18. Diophantine Equations (Gina Rivera, Vista Middle School)

Algebra and Number Theory, Diophantus

19. Goldbach's Conjecture (Chatt Williamson, NMSU)

Number theory, Christian Goldbach

20. Euler's Formula (Chatt Williamson, NMSU)

Analysis of the Infinite, Leonhard Euler

21. The Cayley-Hamilton Theorem (Eric York, NMSU)

Linear Algebra, Arthur Cayley and William Rowan Hamilton

**Additions from May, 1997 (NMSU students)**

22. Cramer's Rule and the Determinant (Rebecca Brown)

Linear Algebra, Gabriel Cramer

23. Olivier and Abel on Series Convergence: An Episode from Early
19th
Century Analysis (Michael Goar)

Calculus and Real Analysis, Niels Abel

24. Historical Sources in Elementary and Intermediate Algebra (Karen
Trujillo)

Algebra, Ancient Chinese, Mayan, Greek sources, and Descartes

**Additions from May, 1998 ****(NMSU students)**

25. Arthur Cayley and the Origins of Group Theory: An Introduction
to
Abstract Algebra (Daniele Richardson)

Abstract Algebra, Arthur Cayley

26. Appendix to ``Restoring the Square: The Methods of Al-Jabr''
(Daniele
Richardson)

Algebra and Geometry, al-Khowarizmi

27. Using Diophantus of Alexandria to Teach Algebra (Shelly Hangen)

Algebra, Diophantus

28. Using Stigler's ``Diet Problem'' to Teach Linear Programming
(Rumiya
Masagutova)

Linear Programming, George Stigler

**Additions from May, 2000**** (NMSU students)**

29. Logic Through the Looking Glass: Learning the Basics of Symbolic
Logic Through the Works of Lewis Carroll (Gloria Johnson)

Symbolic Logic, Lewis Carroll

30. Navigation and Map Making (Karen Ondo)

Geometry, Trigonometry, and Calculus, Gerardus Mercator and Edward
Wright

31. Mayan Arithmetic (Elaine Cohen)

Arithmetic, Mayan numeration

32. Cantor's Set Theory (Murat Basaran)

Set Theory, Georg Cantor

33. Napierian Logarithms: Who, When and How (Mary Williams)

Logarithms, John Napier

34. Introducing Characteristics of Functions to Elementary School
Students
(Elaine Cohen)

Functions Concepts

35. The Theory of Galois: A Historical Approach (Rebecca Pablo)

Galois Theory, Solutions of Equations, Evariste Galois

Additions from Spring, 2002**
(NMSU students)**

Calculus, Leibniz and Newton

Additions from Spring, 2002 or 2004

37. The Beginnings of Graph Theory (Ivette Chuca)

Graph Theory, Leonhard Euler

38. Squaring the Hyperbola (Lisa Hanna)

Quadrature, Pierre de Fermat

39. Alexander Polynomial in Knot Theory (Hyunju Oh)

Topology, James Alexander

40. Kepler's Elliptical Paths for the Planets (Katherine Kanim)

Geometry, Johannes Kepler

41. Newton's Rules for Finding Areas (Andres Gonzales)

Calculus, Issac Newton

42. A Lesson Concerning Induction Guided by Pascal's Treatise on the Arithmetical Triangle, With a Short Exploration of Combinations (Joel Lucero-Bryan)

Discrete Mathematics and Mathematical Induction, Blaise Pascal

43. Francois Viete: An Algebra Teaching Module (Kerry McKee)

Algebra, Francois Viete

44. Babylonians and the Quadratic Formula (Diana Orrantia-Kotowski)

Algebra, Babylonian

Additions from Spring, 2004**
(NMSU students)**

Algebra, Egyptian

46. Regiomontanus and Trigonometry (Charles Mundy-Castle)

Trigonometry, Regiomontanus

47. Numeration: A Historical Perspective (Keith Rountree)

Numeration, Babylonian and Mayan

48. Early Methods for Calculating Limits: An Historical Approach (Eric Moreno)

Limits, Leonhard Euler

49. Origin of Calculus (Candelario Castaneda)

Calculus, Isaac Newton

Additions from Spring, 2006

50. The Algebra Behind the Geometry of Euclid (Gloria Valdez)

Geometric Algebra, Euclid

51. Logarithms (Alicia Blueyes)

Logarithms, John Napier

52. A Lesson on Continued Fractions (Karim Lagnaoui)

Continued fractions, Leonhard Euler

Additions from Spring, 2008

Group Theory, Otto Hölder and Arthur Cayley

54. A History of de Rham Cohomology (Guy Roger Biyogmam)

Differential Forms, Henri Poincaré, Elie Cartan, and Georges de Rham

55. Cubic Equations from Khayyam and Cardano (Arturo Dominguez)

Cubic Equations, Omar Khayyam and Girolamo Cardano

56. The Beauty of Symmetry in Science, Mathematics, and Art (Karen Lowe)

Basic algebra, Symmetry, M.C. Escher

57. Head of the House of Geometry Extends a Warm Invitation to the Wedding of their Famous Daughter, the Line, to the Distinguished Son of the House of Arithmetic, the Real Numbers, as Presented in Richard Dedekind's "Continuity and the Irrational Numbers" (Joel Lucero-Bryan)

Real numbers, Richard Dedekind

58. Algebra in Words and Pictures: From the Ancient Civilizations to the Medieval Era (German Moreno)

Basic algebra, Egyptian, Greek, and al-Khwarizmi

59. Lessons from Newton, d'Alembert, Cauchy: Limits (David Perez)

Limit, Isaac Newton, Jean d'Alembert, and Augustin Cauchy

60. Finding Cubic and Quartic Roots (Sanford Rogers)

Cubic and quartic polynomial equations, Joseph Louis Lagrange

**Appendices**

Recovering Motivation in Mathematics: Teaching with Original Sources

Assessment Guidelines

To main page on

Teaching with Original Historical Sources in Mathematics