What's in Convergence? - Contents of Volume 19 - 2022

Editors:  Amy Ackerberg-Hastings, Janet Heine Barnett

Associate Editors:  Paul Bialek (through 1/31/22), Eugene Boman, Ximena Catepillan, Sloan Despeaux (through 1/31/22), Joel Haack (through 7/21/22), Toke Knudsen, Stacy Langton, Betty Mayfield, Michael Molinsky, Adam Parker, Andrew Perry, Adrian Rice, Amy Shell-Gellasch, Erik R. Tou, Laura Turner

Founding Editors: Victor Katz, Frank Swetz

Articles

Need the Area of a Triangle? The Pope Can Help! by Betty Mayfield
Gerbert d’Aurillac on finding the area of an equilateral triangle, with exploration activities for students. (posted 11/28/2022)

El Gabinete de Maravillas Matemáticas de Pantas: Imágenes e Historia de las Matemáticas, por Frank J. Swetz; traducido por Ximena Catepillán con la ayuda de Samuel Navarro
El autor discute cómo motivar e involucrar a sus estudiantes mediante el uso de imágenes, especialmente las de objetos históricos, manuscritos y textos, en la enseñanza de las matemáticas. Traducido al español de un artículo de Convergence publicado en 2015, “Pantas’ Cabinet of Mathematical Wonders: Images and the History of Mathematics.” (posted 08/07/2022)

Do Teachers Need to Incorporate the History of Mathematics in Their Teaching? by Po-Hung Liu
The author discusses five reasons for using the history of mathematics in its teaching and provides additional references written since the original publication of the article. (posted 06/06/2022)

HOM SIGMAA 2022 Student Paper Contest Winners
Read the winning papers from the 19th annual edition of this contest: “The Assumptive Attitudes of Western Scholars Regarding the Contributions of Mathematics from India: Assessing yukti-s from the Yuktibhāṣā of Jyeṣṭhadeva” by Rye Ledford (first prize) and “Estimations of \(\pi\): The Kerala School of Astronomy and Mathematics, the Gregory-Leibniz Series, and the Eurocentrism of Math History” by Sarah Szafranski (second prize). (posted 06/06/2022)

An Ancient Egyptian Mathematical Photo Album – Hieroglyph Numerals and More, by Cynthia J. Huffman
Photographs of ancient Egyptian hieroglyphs in authentic contexts that instructors can use when teaching numeration systems. (posted 4/9/2022)

Kepler and the Rhombic Dodecahedron, by Roberto Cardil
Resources for sharing Kepler's fascinating studies of the rhombic dodecahedron with students. (posted 3/19/2022)

The High School Mathematics Curriculum—What Can We Learn from History? by Robert Reys and Barbara Reys
The authors review several of the major programs for reform in American mathematics education that appeared between 1894 and 2010 and conclude that, while calls for change have been constant, the full implementation of different approaches is much more difficult to achieve. (posted 3/05/2022)

Reflections on Chinese Numeration Systems, by Frank J. Swetz
Recommends ancient Chinese rod numerals to the instructors of preservice elementary teachers as an alternative place-value numeration system for helping students understand the structures and operations of arithmetic. Includes historical descriptions and classroom suggestions. (posted 2/20/2022)

Building a Book: HathiTrust, Ancestry.com, Serendipity, and Lifetime Interests, by David Lindsay Roberts
Reveals how personal knowledge, changes in historical research methods, and unexpected discoveries came together in the preparation of a book on the history of American mathematics, and suggests how the lessons learned could be incorporated into history of mathematics and other courses. (posted 1/22/2022)

Ongoing Series

Quotations in Context, by Michael Molinsky
A series of columns that explores the origins and meanings of various quotations about mathematics and mathematicians.

• Series Introduction
• Albert Einstein
• Bertrand Russell
• Plato

HoM Toolbox, or Historiography and Methodology for Mathematicians
A series that guides readers through the basic principles and theoretical approaches for researching and writing the history of mathematics.

• Series Purpose and Structure, by Amy Ackerberg-Hastings
• Introduction to Historiography and Methodology, by Amy Ackerberg-Hastings

Keys to Mathematical Treasure Chests
A series that offers examples of how online databases of mathematical objects can be mined to unlock the collections that they preserve for use in research and teaching.

• Series Introduction, by Peggy Aldrich Kidwell
• 19th-century String Models, by Peggy Aldrich Kidwell
• Andean Khipus, by Manuel Medrano

Teaching and Learning the Trigonometric Functions through Their Origins, by Daniel E. Otero
A series of curricular units based on primary source texts for use in teaching and learning trigonometry.

• Series Introduction
• Episode 1 – Babylonian Astronomy and Sexagesimal Numeration
• Episode 2 – Hipparchus’ Table of Chords
• Episode 3 – Ptolemy Finds High Noon in Chords of Circles
• Episode 4 – Varāhamihira and the Poetry of Sines
• Episode 5 –  al-Bīrūnī Does Trigonometry in the Shadows
• Episode 6 –  Regiomontanus and the Beginnings of Modern Trigonometry

A Series of Mini-projects from TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources
A collection of student-ready projects for use in teaching standard topics from across the undergraduate curriculum.

• Series Introduction, by Janet Barnett, Kathy Clark, Dominic Klyve, Jerry Lodder, Daniel E. Otero, Nick Scoville, and Diana White
• The Derivatives of the Sine and Cosine Functions: A Mini-Primary Source Project for Calculus 1, by Dominic Klyve
• Why be so Critical? Nineteenth Century Mathematics and the Origins of Analysis: A Mini-Primary Source Project for Introductory Analysis Students, by Janet Heine Barnett
• Connecting Connectedness: A Mini-Primary Source Project for Topology Students, by Nicholas A. Scoville
• Generating Pythagorean Triples: A Mini-Primary Source Project for Mathematics Majors, Elementary Teachers and Others, by Janet Heine Barnett
• Euler's Rediscovery of e: A Mini-Primary Source Project for Introductory Analysis Students, by Dave Ruch
• How to Calculate \(\pi\): Machin's Inverse Tangents, A Mini-Primary Source Project for Calculus 2 Students, by Dominic Klyve
• Henri Lebesgue and the Development of the Integral Concept: A Mini-Primary Source Project for Undergraduate Analysis Students, by Janet Heine Barnett
• Seeing and Understanding Data: A Mini-Primary Source Project for Students of Statistics, by Charlotte Bolch and Beverly Woods
• The Origin of the Prime Number Theorem: A Primary Source Project for Number Theory Students, by Dominic Klyve
• The Cantor Set Before Cantor: A Mini-Primary Source Project for Analysis and Topology Students, by Nicholas A. Scoville
• Euler’s Calculation of the Sum of the Reciprocals of the Squares: A Mini-Primary Source Project for Calculus 2 Students, by Kenneth M Monks
• Completing the Square: From the Roots of Algebra, A Mini-Primary Source Project for Students of Algebra and Their Teachers, by Daniel E. Otero
• Regression to the Mean: A Mini-Primary Source Project for Statistics Students, by Dominic Klyve
• Investigations Into d'Alembert's Definition of Limit: A Mini-Primary Source Project for Students of Real Analysis and Calculus 2, by David Ruch
• Braess’ Paradox in City Planning: A Mini-Primary Source Project for Multivariable Calculus Students, by Kenneth M Monks
• Topology from Analysis: A Mini-Primary Source Project for Topology Students, by Nick Scoville
• Babylonian Numeration: A Mini-Primary Source Project for Pre-service Teachers and Other Students, by Dominic Klyve
• Wronskians and Linear Independence: A Theorem Misunderstood by Many – A Mini-Primary Source Project for Students of Differential Equations, Linear Algebra and Others, by Adam E. Parker
• Bhāskara’s Approximation to and Mādhava’s Series for Sine: A Mini-Primary Source Project for Calculus 2 Students, by Kenneth M Monks
• The Logarithm of -1: A Mini-Primary Source Project for Complex Variables Students, by Dominic Klyve
• Gaussian Guesswork: Three Mini-Primary Source Projects for Calculus 2 Students, by Janet Heine Barnett
• Fourier’s Heat Equation and the Birth of Modern Climate Science: A Mini-Primary Source Project for Differential Equations and Multivariable Calculus Students, by Kenneth M Monks
• How to Calculate \(\pi\): Buffon's Needle – A Mini-Primary Source Project on Geometric Probability for Calculus 2 Students, Pre-service Teachers and Others, by Dominic Klyve
• Solving Linear Higher Order Differential Equations with Euler and Johann Bernoulli: A Mini-Primary Source Project for Differential Equations Students, by Adam E. Parker
• Fourier’s Infinite Series Proof of the Irrationality of e: A Mini-Primary Source Project for Second-Semester Calculus Students, by Kenneth M Monks

Mathematical Treasures

Mathematical Treasures, by Frank J. Swetz

Index to Mathematical Treasures Collection: Images of historical texts and objects from libraries, museums, and individuals around the world for use in your classroom!

Mathematical Treasures added during 2022:

• Keduka Bukit Stone (Indonesian zero, 683–686)
• Boethius’ De Institutione Arithmetica (845, ca 500 original)
• Chaturbhuj Temple Inscription, Gwalior (Sanskrit zero, 876)
• Bakhshālī Manuscript (885–993)
• Ibn Luyūn al-Tujībī’s Didactic poem on surveying and measurement of areas (14th century)
• Rechenbuch associated with Friedrich Gerhart’s Practica des Algorismus Ratisbonensis (ca 1460)
• Arithmetic block book attributed to Ulrich Wagner (1471–1482)
• Ulrich Wagner’s Das Bamberger Rechenbuch (1482)
• Jacques Lefèvre d’Étaples’s edition of Euclid’s Elements (1516)
• Albrecht Dürer’s Vnderweysung der Messung mit dem Zirckel vnd Richtscheyt (1525)
• Peter Apian’s Instrument Buch (1533)
• Sebastian Münster’s Fürmalung vnd künstlich Beschreibung der Horologien (1537)
• Peter Apian’s Astronomicum Caesareum (1540)
• Augustin Hirschvogel’s Geometria (1543)
• German portfolio of geometric watercolors (mid-late 16th century)
• Johann Vögelin’s Elementale geometricum (1550)
• Sebastian Münster’s Rvdimenta Mathematica (1551)
• Juan Martínez Población’s De Vsv Astrolabi Compendivm (1553, 1518 original)
• Simon Stevin's De Beghinselen Der Weeghconst (1586)
• Levinus Hulsius’s Theoria Et Praxis Quadrantis Geometrici (1594)
• Paul Pfinzing’s Methodus Geometrica (1598)
• Jacques Chauvet's Petit traitté de la fortification moderne (1600)
• Levinus Hulsius’s Tractat der Mechanischen Instrumenten (1604)
• Li Zhizao’s Hun gai tong xian tu shuo (1605–1607)
• Simon Stevin’s Disme, The Art of Tenths (1608, 1585 original)
• Hugolinus Kneiff’s Mathematisches Übungsbuch (1614)
• Lucas Brunn’s Praxis Perspectivae (1615)
• Johann Lörer’s Nova Instrumentum Geometricum Perfectum (1616)
• Johann Lörer’s Planemetrische Beschreibung (1616)
• Peter Ryff’s Questiones Geometricae (1621)
• Giulio Aleni’s revision of Matteo Ricci’s Kunyu wanguo quantu (1620s, 1602 original)
• Giulio Aleni’s and Yang Tingyun’s Zhifang Waiji (1623)
• Lucas Brunn’s Euclidis Elementa Practica, oder Ausszug aller Problematum und Handarbeiten auss den 15. Büchern Euclidis (1625)
• Gérard Thibault’s Académie de l'Espée (1628)
• Simon Stevin’s Castrametatio Avraico-Nassovica (1631)
• John Wallis’s Operum Mathematicorum Pars Prima (1657), contributed by Sid Kolpas
• Vincenzo Viviani’s De Maximis et Minimis (1659)
• Jean François Nicéron’s La Perspective Curieuse (1663, 1638 original)
• Johann Hemeling’s Selbstlehrende Rechenschul (ca 1663, 1655 original)
• Johann Hemeling’s Neugemehrtes Selbstlehrende Rechenschul (1678, 1655 original)
• Pierre de Fermat’s Varia opera mathematica (1679)
• Giulio Troili’s Paradossi per pratticare la Prespettiua (1683, 1672 original)
• Reeve Williams’s translation of Claude Dechales’s Euclid’s Elements (1685)
• Raimondo Montecuccoli’s Memorie della guerra (1697, 1704, and 18th century translations and printings from ca 1661 original)
• Gottfried Wilhelm Leibniz’s “Responsio Addn. [Nicolas Fatio’s] Imputaziones”(1700), contributed by Sid Kolpas
• Markus Christian Ries’s Praxis Geometriae (1700)
• Zeh Sefer Uklidus ha-Haham (manuscript Hebrew-language Euclid’s Elements, 1704)
• Nicolas Bion’s Traité de la construction et principaux usages des instruments de mathematique (1709)
• Isaac Newton’s Artis analyticae specimina : sive, Geometria analytica (manuscript copy, ca 1710, 1670–1671 original)
• Von der rechten Arithmetica geometrica, oder Decimal-Rechnung (18th century)
• Johann Friedrich Rosenzweig’s Rechenbuch (1721)
• Kātib Çelebi’s Cihānnümā (1732, ca 1648 original)
• Fra A. L.’s Alfabeto geometrico in due parte diviso (1735)
• William Hogarth’s The Importance of Knowing Perspective (1754)
• Johann Baptist Roppelt’s Geometria theoretica et practica (1772)
• Johann Baptist Roppelt’s Praktische Abhandlung Von Den Granz-Zeichen Sammt Einer Geometrischen Unterweisung (1775)
• Johann Baptist Roppelt’s Introdvctio in mathesin qvarvmcvnqve scientiarvm cvltoribvs accommodata ad eivs qvoqve stvdium incitatoria (1777)
• Johann Hemeling’s Kleines Rechenbuch (1817, mid-17th century original)
• János Bolyai’s Appendix to Tentamen juventutem studiosam in elementa matheseos purae (1832)
• Matteo Ricci’s and Li Zhizao’s Yuan rong jiao yi (1847, 1608 original)
• John Bart Geijsbeek’s Ancient Double-Entry Bookkeeping (1914)
• Mattel’s Teen Talk Barbie (1992), contributed by Sid Kolpas