The Number Line through Guided Inquiry

148 Pages·2021·6.304 MB·English

by  David M. Clark, Xiao Xiao

AMS / MAA TEXTBOOKS VOL 69
The Number Line
through Guided Inquiry
David M. Clark and Xiao Xiao
π
The Number Line through
Guided Inquiry

AMS/MAA TEXTBOOKS
VOL 69
The Number Line through
Guided Inquiry
David M. Clark
Xiao Xiao
MAATextbooksEditorialBoard
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LibraryofCongressCataloging-in-PublicationData
Names:Clark,DavidMichael,1947-author.∣Xiao,Xiao,1983-author.
Title:Thenumberlinethroughguidedinquiry/DavidM.Clark,XiaoXiao.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2022]∣Series: AMS/MAAtext-
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Contents
Preface vii
To the Student xvii
Acknowledgement xxi
1 Rational Numbers and Missing Numbers 1
1.1 Integers 2
1.2 RationalNumbers 3
1.3 MissingNumbers 4
1.4 NumberLines 7
1.5 ManyNumberLines 9
1.6 HistoricalNote: OtherMissingNumbers 10
2 Limit Points and Sequences 13
2.1 IntervalsandLimitPoints 15
2.2 SequencesandConvergence 16
2.3 HistoricalNote: Zeno’sParadox 19
3 Decimal Representations of Numbers 21
3.1 InfiniteDecimalRepresentations 22
3.2 RationalNumbersasInfiniteDecimals 24
3.3 ExistenceandUniqueness 27
3.4 HistoricalNote: TheHindu-ArabicNumerals 30
4 Complete Number Lines 35
4.1 TheCompletenessAxiom 36
4.2 TheSquareRootof17 37
4.3 HistoricalNote: TheArchimedeanDilemma 38
5 Continuity 41
5.1 DefinitionsandExamples 42
5.2 GeneratingContinuousFunctions 46
5.3 MissingNumbersasIntermediateValues 47
5.4 UniformContinuity 49
5.5 HistoricalNote: ContinuityviaInfinitesimals 51
v
vi Contents
6 Calculus 53
6.1 Integrals 54
6.2 Derivatives 58
6.3 TheFundamentalConnection 61
6.4 HistoricalNote: CalculusviaInfinitesimals 62
7 Log and Exponential Functions 65
7.1 RationalExponents 66
7.2 NaturalLogarithm 67
7.3 ExponentsRevealMoreMissingNumbers 69
7.4 HistoricalNote: The19th-CenturyTransition 72
8 The Real Number Line 75
8.1 TheNon-NegativeRealNumbersℝ 76
≥0
8.2 TheRealNumberLineℝ 85
8.3 AllNumberLines 89
8.4 HistoricalNote: TheHyperrealNumbers 90
9 The Price of Completeness 93
9.1 Cantor’sSetTheory 94
9.2 Lebesgue’sMeasureTheory 97
9.3 HistoricalNote: MoreInfinities! 98
A Historical Summary 101
B The Reals via Dedekind Cuts 105
B.1 RationalandGhostDownsets 106
B.2 ConstructionoftheNumberLine 107
B.3 RationalandIrrationalNumbers 112
C Guidelines for the Instructor 115
C.1 ChapterOptions 115
C.2 TeachingthroughGuidedInquiry 116
Bibliography 121
Index 123
Preface
This book is designed as a replacement for conventional real analysis for fu-
turemathematicsteachers. Therealnumbersystemandfunctionsonthereal
numbersarecentraltoolsfortheunderstandingofawiderangeofquantitative
subjects. Consequently these topics are taught to high school and beginning
college students, even though these students are not yet prepared to under-
stand them in any significant depth. This text will give future teachers both
that depth of understanding and the pedagogical and communication skills
thatcomewithguidedinquirylearning. Itisnotintendedtopreparestudents
forgraduatestudyinmathematicswherethecontentsofatraditionalrealanal-
ysiscourseareneeded.
Inspiteofitsphysicallyleanpresentation,thistextiswrittenforaone,two
orthreesemestercoursesequencedependingonhowfarstudentswanttogo
with it. Generally it will be the most advanced undergraduate mathematics
course they will have. In preparation for using this book, we advise that stu-
dentshaveayearofcalculusandanintroductorycoursethatfamiliarizesthem
withthebasicpropertiesoftheintegersandrationalnumbers,propertiesofsets
andmathematicalinduction. Agoodprevioustheoremprovingcourselikeax-
iomaticgeometrywouldverymuchenhancetheirabilitytogofurtherintothis
book.
Thisbookwasdesignedtofulfillthreegoals.
● ReasoningandProof. Asisneededineverydiscipline,itwillhelpstudents
develop a mastery of the particular methodology used in mathematics to
determinewhatistrue.
● Active Learning. It will guide students in proving theorems and solving
problems by having them prove theorems and solve problems, and then
present their results for feedback and discussion with the instructor as a
mentor.
● Content. It will provide students with a working knowledge of just that
part of real analysis that directly underlies the material on numbers and
functionsthattheywillteachinhighschool.
Ourthreegoalsforfutureteachers,originallyarticulatedin[6],lackedwide
spreadsupportuntilveryrecently. Theyarenowfullyconsistentwiththemost
currentrecommendationsoftheprofessionalmathematicscommunity.
vii
viii Preface
Ournationalstandardsforcollegemathematicsinstructionandforteacher
preparation in mathematics fall under the purview of the Conference Board
fortheMathematicalSciences. TheCBMS,formallyincorporatedin1960,de-
scribesitselfas“anumbrellaorganizationconsistingofseventeenprofessional
societies all of which have as one of their primary objectives the increase or
diffusionofknowledgeinoneormoreofthemathematicalsciences.”
In 2001 the CBMS published a manual, The Mathematical Education of
Teachers,withspecificrecommendationsfortheteachingrealanalysis[7,pp.
42–43]. These recommendations reflected the view of teaching mathematics
thathadbeendominantinovertheprevioustwodecades. Thisviewsawmath-
ematicsprimarilyasacollectionoftoolstosolveproblemsinscienceanden-
gineering. Teacherswereexpectedtolearn as manyof these toolsas possible
sothattheywouldbeabletopassthemontotheirownclasses.
Since2001anumberofdevelopmentshaveledthemathematicscommu-
nity to make some significant modifications of those recommendations. In
2012theCBMSissuedanupdatedTheMathematicalEducationofTeachersII
[8]. Readingthisrevisedmanual,wewerepleasedtoseethattheyhadincor-
porated these new developments and aligned their recommendations for un-
dergraduate mathematics instruction and teacher preparation with the three
abovegoalsthathadbeenusedfortheinitialdraftofthismanuscript.
We would like to outline the genesis of these new developments and cite
the parts of the 2012 CBMS recommendations that pertain to the teaching of
real analysis and to our three central goals for teacher preparation. We will
then identify a wider audience for the book and end with a summary of the
sequenceofchapters. AppendixCwillgiveavarietyofsuggestionsastohow
thisbookcanbeused.
1. ReasoningandProof
Eachdisciplinehasitsownparadigmforestablishingtruth,beitlaboratory
experiments,statisticalanalysisofdataormaybesubjectivearguments. What-
evertheparadigm,collegelevelmasteryofasubjectshouldincludeamastery
of its method of establishing truth. In mathematics we establish a statement
tobetruebygivingaproof. Mathematicsgraduateswithaproficiencyinprov-
ingtheoremswill,withsufficientbackgroundknowledge,beabletofigureout
whatisandisnotmathematicallytruebythemselves,andwillbeabletocrit-
ically judge the validity of what they hear. These skills extend well beyond
mathematicsperse.
In2012thenewCBMSrecommendationsincludedastrongendorsement
oftheReasoningandProof goalpursuedinthistext[8,p. 56].
Experiencewithreasoningandproof. Reasoningisessentialforall
mathematicalprofessions,especiallyforteaching. Makingsenseofmath-
ematicsmakesiteasiertounderstand,easiertoteach,andintellectually
satisfying for all students, including high school students who have no
Preface ix
intention of going into technical fields. And proof is essential to all of
mathematics. Accordingly,reasoningandproof,whileitisoftenthefo-
cusofaparticularcourse,shouldbepresent,notalwaysatthesamelevel
ofrigor,inmostundergraduatecourses.
2. ActiveLearning
How,then,dowebestteachreasoningandproof? Thetraditionalresponse
was to explain and demonstrate these concepts through clear and well orga-
nizedlectures. Ontheotherhand,itiswidelyacknowledgedthatlearningto
ride a bicycle or play the piano is best achieved by directly engaging in those
activities. In recent years there has been a growing awareness that the same
principleappliestoalllearning. Inordertoestablishtruthintheirdiscipline,
collegestudentsmustdotheexperiments,carryoutthestatisticalanalysesor
makethesubjectivearguments. Oursecondgoalistousethesamepedagogy,
broadlyknownasactivelearning.
Traditionalteachingfocusesattentiononinstructors,whosejobitistodis-
seminatetheirknowledgeandexpertisetothestudents. Activelearningshifts
the roles of instructors and students, putting the students at the center of the
learningprocess. Itistheinstructor’stasktoguidethestudentstowardagoal
that they have the tools to achieve but have not been told how to achieve. It
isthestudent’stasktocarryoutaninvestigationbyinquiringintodifferentas-
pectsofthegoalinordertobuildanunderstandingofitscontextuntilthatgoal
is achieved. The use of active learning in mathematics has gone under many
names,1 eachreflectingadifferentimplementationofthispedagogy. Wefind
thephraseguidedinquirylearning (GIL)tobeagoodgeneraltermforallim-
plementationsofactivelearninginmathematics,asitrefersdirectlytotherole
oftheinstructor(aguide)andtheactivityofthestudents(toinquire).
In1998HarryLucas’AustinbasedEducationalAdvancementFoundation
beganholding annual meetings tosupport theuse and dissemination of GIL.
WithoutthefundingandinspirationfromMr. Lucasthismovementwouldnot
bewhereitistoday. In2009theMathematicsAssociationofAmerica(MAA),
ourprimarystewardofundergraduatemathematicsteaching,publishedatext
ontheuseofGILbyCoppin,Mahavier,MayandParker[9]. In2011theMAA
became a joint sponsor of the annual EAF conferences, thereby officially en-
dorsingtheGILmovement.
Also in 2011 Sandra Laursen [16] reported on a large scale control study
donebyherteamattheUniversityofColoradoBoulderthatdocumentedthe
effectivenessofGIL(IBL)attheuniversitylevel. Itrevealedsignificantbene-
fitsrealizedbystudentsinGILclassesoverthoseinnon-GILclasses,withthe
greatest improvements shown for women and traditionally under achieving
groups;see[23],[15]and[16,p. xii,ES10.0].
1Texasmethod, Mooremethod, modifiedMooremethod, inquiry-basedlearning, guided
inquirylearning,learningbyinquiry,inquiryorientedinstruction,…