Mathematical Time Capsules: Historical Modules for the Mathematics Classroom

304 Pages·2011·5.891 MB·English

by  Dick Jardine, Amy Shell-Gellasch (eds.)

Mathematical Time Capsules
Historical Modules for the Mathematics Classroom
c 2011by
(cid:13)
TheMathematicalAssociationofAmerica(Incorporated)
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PrintISBN978-0-88385-187-6
ElectronicISBN978-0-88385-984-1
PrintedintheUnitedStatesofAmerica
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10987654321
Mathematical Time Capsules
Historical Modules for the Mathematics Classroom
Edited by
Dick Jardine
Keene State College
and
Amy Shell-Gellasch
Beloit College
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31. ResourcesforCalculusCollection,Volume5:ReadingsforCalculus,UnderwoodDudley,Editor.
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33. ResearchIssuesinUndergraduateMathematicsLearning:PreliminaryAnalysesandResults,JamesJ.KaputandEdDubinsky,
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35. YouretheProfessor,What Next?IdeasandResourcesfor PreparingCollegeTeachers,TheCommitteeonPreparationfor
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68. FromCalculustoComputers:Usingthelast200yearsofmathematicshistoryintheclassroom,AmyShell-GellaschandDick
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Preface
Mathematical Time Capsules offers teachers historical modules for immediate use in the mathematics classroom.
Relevanthistory-basedactivitiesforawiderangeofundergraduateandsecondarymathematicscoursesareincluded.
ThegenesisofthisvolumewasaContributedPapersSessiononUsingHistoryofMathematicsinYourMathematics
Courses, organizedbythe editorsat theJointMathematics Meetings,San Antonio,Texas, inJanuary of2006.That
session was very well attended, which prompted Andrew Sterrett from MAA publications to suggest that we put
togetheroursecondvolumefortheMAANotesseries.
Purpose
Forawidevarietyofreasons,instructorsarelookingforwaystoincludethehistoryofmathematicsintheircourses.
Itisnotuncommontosee requestsfor“howto”postedtotheHistoryofMathematics SpecialInterestGroupofthe
MAA(www.homsigmaa.org)emaillist,suchasthis2008posting:
...IamanewcomertoHOM.Whereandhowshouldanewcomerbegin?Rightnow,Iwouldliketoinclude
HOMinameaningfulwayinthecourses thatweteach.Weteach coursesfromcollegearithmetictolinear
algebra.
Inresponsetosuchinquiries,wehopetoservethebroadermathematicalcommunitybyofferingpracticalsuggestions
onhowtousethehistoryofmathematicsquicklyandeasilyinthemathematicsclassroom.
A time capsule can be defined as a container preserving articles and records from the past for scholars of the
future. Of course our volume does not fit that precise definition, but readers who open this book will find articles
and activitiesfrom mathematics historythatenhance the learning of topics typicallyassociated withundergraduate
orsecondary mathematics curricula. Each capsule presents onetopicor perhaps a few related topics, ora historical
threadthatcanbeusedthroughoutacourse.Thecapsuleswere writtenbyexperiencedpractitionerstoprovideother
teachers withthe historicalbackground,suggested classroom activities,and furtherreferences and resources onthe
chaptersubject.Aninstructorreadingacapsulewillhaveincreasedconfidenceinengagingstudentswithatleastone
activityrichinthehistoryofmathematicsthatwillenhancestudentlearningofthemathematicalcontentofthecourse.
Mostofthehistoricaltopicscontainedinacapsule can beimplementedinoneclass periodwithminimaladditional
preparationonthepartoftheteacher.
How to use Mathematical Time Capsules
Teaching styles have been categorized along a spectrum from lecture-oriented practices at one extreme to student-
centeredapproachesinwhichtheteacherguidesstudentworkintheclassroom.MathematicalTimeCapsulesrespects
the diversity of teaching styles which individual teachers adopt. Some of the capsules are, in some sense, ready-
madelecturestheinstructorcanadoptandadaptasappropriate.ExamplesofthoseincludeVictorKatz’s“Copernican
Trigonometry,”RogerCooke’s“NumericalSolutionofEquations,”orJimTattersall’s“FindingtheGreatestCommon
DivisorandMore... .”Othercapsulesclearlyengagethestudentsmoreactively,suchasVickyKlima’s“ADifferent
SortofCalculusDebate.” Butthecapsulesshouldnotbecategorized asappropriateforonepedagogicalapproachor
theother. Forexample, “FindingtheGreatest CommonDivisorand More...” couldbe adapted foruse as a student
projecttobepresentedbythestudent(s)inclassaftertheEuclideanalgorithmiscovered.
Thereadermaynotethattheauthorsofthecapsulesdemonstrateavarietyofapproachestointegratinghistory.The
differencesareconsistentwiththenatureofthisvolume,createdwithrespectforthediversityofourauthorsandour
vii
viii Preface
readers.Weacknowledgethatmanyteachersprefertodeveloptheirowncoursematerials,andweencouragereaders
tomodifytheofferingsoftheauthors.
MathematicalTime Capsules is organized in three sections. The first capsules have as their target mathematical
topicsthatare usuallyaddressed incourses taughtinsecondary school, at two-yearcolleges, or duringthe firsttwo
years of the undergraduate mathematics curriculum. These courses are notoften taken by mathematics majors, and
include,forexample,algebra,geometry,mathematicsforelementaryteachers,trigonometry,orprecalculus.
Thethirdsectionofcapsulesaddresstopicsincludedincoursestraditionallytakenbymathematicsmajors,suchas
calculus,differentialequations,numbertheory,abstractalgebra,differentialequations,andanalysis.
As an interludebetween the first and third, we offer some ideas that can be applied to a wide variety of courses
throughouttheundergraduate(includingtwo-yearcollege)orsecondarycurriculum.Theseinterludecapsules(15,16,
and17)areofageneralpedagogicalnature,notmathematical,andcouldbeadaptedforuseinanycourse.
Wecouldhave arrangedthecapsules differently,andweaskthatthereadernotlimitinvestigatingtheofferingsin
thisbookthinkingthatlower-levelmaterialisinthefrontandupper-levelmaterialattheback.Forexample,ateacher
interested in historical ideas for a numerical methods course should consider Roger Cooke’s “Numerical Solution
ofEquations,”RandySchwartz’s“RuleofDoubleFalsePosition,”ClemencyMontelle’s“Roots,Rocks,andNewton-
RaphsonAlgorithmsforApproximatingp23000YearsApart,”andDickJardine’s“Euler’sMethodinEuler’sWords.”
Some teachers are interestedin havingstudents read originalsources in the historyof mathematics, and Montelle’s
“Amo,Amas,Amat!What’stheSumofThat?”andJardine’s“Euler’sMethodinEuler’sWords”provideopportunities
todojustthatwithbriefexcerptsinthewordsoftheoriginators.
Wherepossible,wegroupedthecapsuleswithinthesectionsaccordingtomathematicalsubjectarea.Asanexample,
there are three capsules that address Pythagorean triples, but each of those capsules approaches the topicin a very
differentwayandfromtheperspectiveofadifferenteraofthehistoryofmathematics.Thatlatternotionissignificant,
as it is importantfor students to see the evolutionof a mathematical idea and how it is viewed througha different
lens depending on how mathematics was done at various times in history. One of the goals of MathematicalTime
Capsules is to provide a vehicle for teachers to help their students learn the historical context for a mathematical
development while they are learning the specific mathematical concept. Learning the history of an idea promotes
deeperunderstandingoftheidea.
Additional purposes—assessment and teacher certification
Beyond a teacher’s personal interest in usingthe historyofmathematics inteaching, accreditingagencies and state
certifyingagenciesnowrequirepre-serviceteacherstobewell-versedinthehistoryofmathematicsinspecificcontent
areas.Therequirementthatschoolteachersdemonstrateunderstandingoftheconnectionbetweenamathematicstopic
andthehistoryofthatspecifictopicisexplicitlydocumentedinstateandnationalstandards.Itisbecominganimper-
ativethatmanycollegeteachersintroducethehistoryofmathematicintheirteachingofmathematicsinordertopass
muster forstateand nationalcertificationof teacher educationprograms. The NationalCouncilforAccreditationof
TeacherEducation(NCATE)andtheNationalCouncilofTeachersofMathematics(NCTM)havepublishedstandards
whichlistwithinthecontentareasarequirementthatcandidatesforteachercertificationdemonstratetheirknowledge
of the historical development of that content area. For example, the current content Standard 10 is on the subject
of algebra. For certification, in addition to demonstrating expertise in the usual algebra concepts, candidates must
“Demonstrateknowledgeofthehistoricaldevelopmentofalgebraincludingcontributionsfromdiversecultures”[2].
In this era of outcomes-based assessment, to satisfy state and nationalevaluators itis not sufficient to show that
historywasincludedinthesyllabusortoclaimthathistorywasincludedinalecture.Actualstudentwork(interestingly
for us interested in history, the student work is called an artifact in current assessment jargon) must be presented
to the accrediting agency or state department of education evaluators. The material presented herein will provide
teachers with actual activities that students can do. The products of these activities become the artifacts necessary
tovalidatethatstudentsare engagedinlearningthehistoricaldevelopmentofthemathematical topic.Mathematical
Time Capsulesprovides materials enhancingstudentunderstandingandinterestinmathematics, always keepingthe
learningofmathematicsclearlyatthecenterofeachcapsule’sfocus.
Preface ix
Acknowledgements
Wecouldnothavetakenonthisprojectwithouttheinterestandenthusiasmofthemanycontributorsfromcampuses
around the world. We are indebted to the successful leadership efforts of V. Frederick Rickey and Victor Katz in
bringingtogetheracommunityofeducatorsandencouragingourpassionforimprovingstudentlearningthroughthe
useofthehistoryofmathematics. Theeditorsofthisvolumemet as participantsintheNSFfundedInstituteforthe
HistoryofMathematicsanditsuseinTeaching,organizedbyFredandVictor,whichprovidedasoundfoundationfor
ourworkandanetworkoflike-mindedcolleaguesandfriends.WealsoofferspecialthankstoChrisArney,wholedthe
wayasDepartmentHeadoftheDepartmentofMathematicalSciencesattheUnitedStatesMilitaryAcademyatWest
Pointinnotonlydocumentingthatthehistoryofmathematicsisasignificantlearningoutcomeattheprogramlevelin
undergraduatemathematicsdepartmentsbutalsoprovidedanexemplarofeffectiveimplementation.Webothworked
forChrisandhemadearealdifferencebyencouragingandsupportingourearlyuseofthehistoryofmathematicsto
deepenstudentlearningofourdiscipline.Finally,andperhapsmostimportantly,wealsoacknowledgetheinterestand
enthusiasmofourstudents,wholetusknoweach semester thatweteach thatlearningthehistoryofmathematics is
importanttothem.
References
1. AmyShell-GellaschandD.Jardine,ed.,FromCalculustoComputers:UsingtheLast200YearsofMathematics
HistoryintheClassroom,MathematicalAssociationofAmerica,Washington,2005.
2. NCATE/NCTMProgramStandards(2003),ProgramsforInitialPreparationofMathematicsTeachers,Standards
forSecondaryMathematicsTeachers
www.nctm.org/uploadedFiles/Math_Standards/,
accessed May12,2009.