Jardine D., Shell-Gellasch A. Mathematical Time Capsules: Historical Modules for the Mathematics Classroom

Washington: MAA, 2010. — 120 p. — (Mathematical Association of America Notes) – ISBN: 9780883851876, 0883851873Mathematical Time Capsules offers teachers historical modules for immediate use in the mathematics classroom. Relevant history-based activities for a wide range of undergraduate and secondary mathematics courses are included. Readers will find articles and activities from mathematics history that enhance the learning of topics typically associated with undergraduate or secondary mathematics curricula. Each capsule presents one topic, or perhaps a few related topics, or a historical thread that can be used throughout a course. The capsules were written by experienced practitioners to provide other teachers with the historical background, suggested classroom activities and further references and resources on the subject addressed. After reading a capsule, a teacher will have increased confidence in engaging students with at least one activity rich in the history of mathematics that will enhance student learning of the mathematical content of the course.The Sources of Algebra, Roger CookeEgyptian problems
Mesopotamian problems
“Algebra” in Euclid’s geometry
Chinese problems
An Arabic problem
A Japanese problem
Teaching note
Problems and Questions
Further reading
How to Measure the Earth, Lawrence D’AntonioHistorical Introduction
In the Classroom
Taking it FurtherNumerical solution of equations, Roger CookeThe ancient Chinese method of solving a polynomial equation
Non-integer solutions
The cubic equation
Problems and questions
Further reading
Completing the Square through the Millennia, Dick JardineHistorical preliminaries
Student activities
Summary and conclusionAppendix: Student activities
Adapting the Medieval “Rule of Double False Position” to the Modern Classroom, Randy K. SchwartzHistorical Background
In the Classroom
Taking It FurtherComplex Numbers, Cubic Equations, and Sixteenth-Century Italy, Daniel J. CurtinHistorical Background
In the ClassroomShearing with Euclid, Davida Fischman and Shawnee McMurranHistorical Background
In the ClassroomAppendixThe Mathematics of Measuring Time, Kim PlofkerHistorical Background
In the Classroom
Taking It FurtherClear Sailing with Trigonometry, Glen Van BrummelenHistorical Background
Navigating with Trigonometry
In the ClassroomAppendix: Michael of Rhodes: Did He Know the Law of Sines?
Copernican Trigonometry, Victor J. KatzHistorical Background
In the ClassroomCusps: Horns and Beaks, Robert E. BradleyHistorical Background
In the ClassroomNotes on Classroom UseThe Latitude of Forms, Area, and Velocity, Daniel J. CurtinHistorical Background
In the Classroom
Taking It FurtherCommentsDescartes’ Approach to Tangents, Daniel J. CurtinHistorical Background
In the ClassroomIntegration `a la Fermat, Amy Shell-GellaschHistorical Background
In the Classroom
Taking it FurtherSharing the Fun: Student Presentations, Amy Shell-Gellasch and Dick Jardine 117Getting Started
Presentations
AssessmentDigging up History on the Internet: Discovery Worksheets, Betty MayfieldIn the Classroom
Learning about History — and about the WebAppendixes
Who Was Gauss?
History of Matrices and Determinants
Stephen Smale, a contemporary mathematician
Nancy Kopell, a female mathematician
Newton vs. Leibniz in One Hour!, Betty MayfieldHistorical Background
In the Classroom
Taking it FurtherAppendix: Instructions to the class
Connections between Newton, Leibniz, and Calculus I, Andrew B. PerryHistorical Background.
Newton’s Work
Leibniz’s Work
Berkeley’s Critique
Later Developments
In The ClassroomA Different Sort of Calculus Debate, Vicky Williams KlimaHistorical Background
In the Classroom
ConclusionsAppendixes
Fermat’s Method Worksheet
Barrow’s Theorem Worksheet
The Debates: Roles, Structure, Hints
A ‘Symbolic’ History of the Derivative, Clemency MontelleThe Derivative
Isaac Newton (1643–1727):
Gottfried Wilhelm von Leibniz (1646–1716):
Joseph-Louis Lagrange (1736–1813)
Louis François Antoine Arbogast (1759-1803)Reflective QuestionsLeibniz’s Calculus (Real Retro Calc.), Robert RogersDifferential Calculus (Rules of Differences)AppendixAn “Impossible” Problem, Courtesy of Leonhard Euler, Homer S. WhiteHistorical Setting
In the ClassroomAppendix: Remarks on Selected ExercisesMultiple Representations of Functions in the History of Mathematics, Robert Rogers
Introduction (A Funny Thing Happened on the Way to Calculus)
Area of a Circle
Ptolemy’s Table
From Geometry to Analysis to Set TheoryThe Unity of all Science: Karl Pearson, the Mean and the Standard Deviation, Joe AlbreeKarl Pearson: Historical preliminaries
A data set
The Mean and the First Moment
The Second Moment and the Standard Deviation
In the Classroom
Conclusions
Activities and QuestionsFinding the Greatest Common Divisor, J.J. TattersallHistorical Background
More Historical Background
In The classroomTwo-Way Numbers and an Alternate Technique for Multiplying Two Numbers, J.J. TattersallHistorical Background
In The Classroom
Taking It FurtherThe Origins of Integrating Factors, Dick JardineHistorical preliminaries
Mathematical preliminaries: Integrating factors
Bernoulli’s and Euler’s use of integrating factors
Student activities
Summary and conclusionAppendix: Student activities
Euler’s Method in Euler’s Words, Dick JardineHistorical preliminaries
Euler’s description of the method
Student Activities
Summary and conclusionAppendixes
Student Assignments
Original source translation
Newton’s Differential Equation y ̇/x ̇= 1 - 3x + y + xx + xy, Hüseyin KoçakNewton’s differential equation
Newton’s solution
Phaser simulations
Remarks: Newton, Leibniz, and Euler
Suggested ExplorationsRoots, Rocks, and Newton-Raphson Algorithms for Approximating p2 3000 Years Apart, Clemency MontelleThe Problem
Solving √2 — Second Millennium C.E. Style
Time Warp: Solving √2 — Second Millennium B.C.E. Style
Taking it Further: Final ReflectionsPlimpton 322: The Pythagorean Theorem, More than a Thousand Years before Pythagoras, Daniel E.OteroHistorical Background
Reading the Tablet
Sexagesimal numeration
So What Does It All Mean?
Why Tabulate These Numbers?
Plimpton 322 in the ClassroomThomas Harriot’s Pythagorean Triples: Could He List Them All?, Janet L. BeeryMathematical Background
Historical Background
In the ClassroomAppendix: Who Was Thomas Harriot?
Amo, Amas, Amat! What’s the sum of that?, Clemency MontelleThe Harmonic Series
Bernoulli and the Harmonic Series
The Mathematical ExplanationThe Harmonic Series: A Primer, Adrian RiceHistorical preliminaries
Introducing the harmonic series
A “prime” piece of mathematicsLearning to Move with Dedekind, Fernando Q. Gouvêa
Historical Background: What Dedekind Did
In the Classroom 1: Transition to Proofs
In the Classroom 2: Abstract Algebra
Moving with DedekindAbout the Editors