Table Of ContentFoundations for Undergraduate Research in Mathematics
Eli E. Goldwyn
Sandy Ganzell
Aaron Wootton
Editors
Mathematics
Research
for the Beginning
Student, Volume 1
Accessible Projects for Students
Before Calculus
Foundations for Undergraduate Research
in Mathematics
SeriesEditor
Aaron Wootton, Department of Mathematics, University of Portland, Portland,
USA
Eli E. Goldwyn • Sandy Ganzell (cid:129)
Aaron Wootton
Editors
Mathematics Research for
the Beginning Student,
Volume 1
Accessible Projects for Students Before
Calculus
Editors
EliE.Goldwyn SandyGanzell
Mathematics Mathematics&ComputerScience
UniversityofPortland St.Mary’sCollegeofMaryland
Portland,OR,USA St.Mary’sCity,MD,USA
AaronWootton
Mathematics
UniversityofPortland
Portland,OR,USA
ISSN2520-1212 ISSN2520-1220 (electronic)
FoundationsforUndergraduateResearchinMathematics
ISBN978-3-031-08559-8 ISBN978-3-031-08560-4 (eBook)
https://doi.org/10.1007/978-3-031-08560-4
MathematicsSubjectClassification:00A08,00B10,00-02
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Preface
It is our strong belief that opportunities for mathematical research should be
available to all students with an interest in mathematics—not just to those who
already have expertise in the field. In our experience, engaging with research has
led students to discover a talent and passion for mathematics they didn’t even
know they had. In fact, studies show that students who participate in early-career
STEM research are more likely to remain in school, are more likely to remain in
a STEM major, perform better in upper division courses, and are more interested
in postgraduate STEM educational opportunities.1 Research experience is also a
desirabletraitinprivateindustry,asitillustratesanindividual’sabilitiestoproblem
solve.
Thoughtherehasbeensignificantgrowthinresearchexperiencesinmathematics
forundergraduatesat4-yearcollegesandresearchuniversities,theseopportunities
are often only available to upper-level students or those with significant back-
groundknowledge.Similarresearchexperiencesinmathematicsforthebeginning
student—that is, those in community college and early career college students—
are much rarer, even though this group stands to benefit significantly from such
opportunities.
Perhapsthebiggestbarrierfacedbythebeginningstudentandtheirfacultymen-
torsisnotknowingwheretobegin.Accessibletopicsfornewstudentresearchersare
oftenhardtofind,evenforexperiencedteachers,andfindingunansweredquestions
that are well suited to student projects is time consuming. Accordingly, the main
goal of these two volumes is to expand research opportunities in mathematics for
the beginning student by removing this significant hurdle. Specifically, we seek
to provide community college students, early college students, and perhaps even
advanced high school students everything they need to initiate research projects
withorwithoutafacultymentor,andtofosterindependenceinresearch.
Each chapter in these two volumes is a self-contained, accessible article that
provides ample background material, recommendations for further reading, and
perhaps most importantly, specific projects that can be pursued immediately upon
readingthechapter.WhatmakesthesevolumesdifferentfromotherFURMvolumes
1Hewlett,JamesA.Broadeningparticipationinundergraduateresearchexperiences(UREs):The
expandingroleofthecommunitycollege.CBE-LifeSciencesEducation17.3(2018):es9.
v
vi Preface
isthatthechapters have been writtenforthebeginning studentbyminimizingthe
numberofprerequisites.Indeed,manyofthechaptersrequirenoprerequisitesother
thanadesiretopursuemathematics,andeventhemostambitiouschaptersrequire
no more than linear algebra or introduction to proofs, both of which are typically
sophomore-levelclasses.
This volume is geared towards students with minimal experience beyond tradi-
tionalhighschoolmathematics.Forthereader’sease,weroughlyorderthechapters
by prerequisite. The first four chapters require only high school algebra, the fifth
chapterprecalculus,thesixthalittleprobabilityandprogramming,theseventhand
eighthsomeexposuretocalculus,andtheninthabriefintroductiontoproof.
InChap.1,“GamesonGraphs,”Carlson,Harris,Hollander,andInskointroduce
three games on graphs: cops and robbers, hungry spiders, and broadcast domina-
tion. Each of these games allows for a multitude of research opportunities and
further generalizations; the games can be studied and analyzed from a variety of
perspectives,providingstudentsanimmediateareaofresearchwithverylittleneed
formuchtechnicalorsophisticatedmathematicalbackground.Anyonewhoenjoys
playing board or video games should read this chapter and think about how their
favoritegamescanbemodeledasagameonagraph.
In Chap.2, “Mathematics for Sustainable Humanity—Population, Climate,
Energy, Economy, Policy, and Social Justice,” Wang takes the readers to confront
global challenges facing humanity and our planet. Through a sequence of
sustainability and mathematical concepts, exercises, and real-world projects, the
readerisbroughtalongonthisjourneytodevelopanunderstandingofthecomplex
environmental, economic, and sociocultural interlinkages; to create powerful data
visualizations;tointerpretanddrawinferencesfromfunctions,equations,models,
and graphs; and to use the knowledge to shape competency and reach justified
decisions.
InChap.3,“MosaicsandVirtualKnots,”Ganzellintroducesthemathematicsof
knots. Of course, knots are important to sailors and rock climbers, but interesting
properties of knots can also be discovered from simple diagrams called mosaics,
formedbyplacingsquaretilesontoagrid.Nomathematicalknowledgeotherthan
basic arithmetic is needed. After learning some of the notation for drawing and
describing knots, the reader will explore how to create these mosaics, trying to be
asefficient aspossible.Then Ganzellintroduces anabstractionknown asavirtual
knot. Using mosaics to describe virtual knots is a relatively new idea with lots of
openquestionstopursue.
In Chap.4, “Graph Labelings: A Prime Area to Explore,” Donovan and
Wiglesworth introduce the reader to prime labelings of graphs. Research in prime
labelings asks the question, “Can the vertices of a graph be labeled in such a way
thatthelabelsofadjacentverticesarerelativelyprime?”Severalvariationsofprime
labelingsarethenintroducedwhiletheauthorsguidethereaderthroughexamples
andexistingresearch.Avarietyofresearchquestionsforeachmodification,afew
ofwhichhaveneverbeenexplored,arepresented,thusallowingthereadertomake
progressonnewproblems.
Preface vii
In Chap.5, “Acrobatics in a Parametric Arena,” Shelton, Henderson, and Geb-
hardtintroduceparametricfunctionsinanengagingandaccessibleway,examining
componentgraphswithsinglevariablecalculus,andindicatingonlineresourcesto
complete some exercises. To illustrate, they analyze sinusoidal fits to original 2D
data from motion capture of actual acrobatic movements of juggling sticks. They
alsoanalyzepublicdataofcellphonesubscriptionstojugglesharesinacompetitive
financialmarket,exploringvariousfitstothe3Ddata.Anumberofresearchprojects
are proposed to inspire the reader to perform their own imaginative parametric
acrobatics.
In Chap.6, “But Who Should Have Won? Simulating Outcomes of Judging
Protocols and Ranking Systems,” Lewis and Clifton introduce the mathematical
andcomputationalskillsneededtoaddresstheissuesoffairnessandbiasinvarious
contests. A brief introduction to probability is provided, followed by instruction
for performing simulation experiments in R (other programming software can
easilybesubstituted).Detailedexamplesandcodesnippetsprovidethereaderwith
the skills needed to tackle any of the five suggested research projects, ranging
from discovering how likely it is that the two highest-ranked teams going into a
tournament end up in the championship playoff to investigating the stability and
fairness of three different voting schemes for an election. Each of the proposed
projectscanbeadaptedtosuittheinterestsofthereader.
In Chap.7, “Modeling of Biological Systems: From Algebra to Calculus and
ComputerSimulations,”Dimitrovetal.presentanexpansiveviewofmathematical
tools as used in modeling a variety of biological and biomedical systems. The
first two projects model a biological and a biomedical system and are accessi-
ble to anybody with knowledge of high school algebra. The final two projects
build on more advanced population and infectious disease models and require
some knowledge of calculus. The chapter introduces computational thinking and
computer-basedsimulationsasimportanttoolsforapproachingproblemsinmodern
applied mathematics. Mathematical modelers leverage mathematical techniques
withthosecomputationaltoolstotackleproblemsthataremuchmorecomplexthan
previously possible, allowing them to address more realistic and relevant societal
issueswiththeirmodelingefforts.
With the massive impact that COVID-19 has had on the lives of students,
mathematicalepidemiologyisaparticularlytimelysubject.InChap.8,“Population
Dynamics of Infectious Diseases,” Ledder and Homp present an individual-based
epidemicmodelandacontinuous-timeSEIRepidemicmodel,usingapresentation
thatisaccessibletostudentswhohaveastrongbackgroundinalgebraandfunctions,
with or without calculus. Projects focus primarily on extending the models to
novelsituations,includingmorecomplicateddiseasehistoriesandincorporationof
mitigatingstrategiessuchasisolationandvaccination.Theprojectsarecategorized
accordingtothelevelsofcalculusandprogrammingthatarerequired.
In Chap.9, “Playing with Knots,” readers will learn about how mathematicians
think about knots. Given two knots, how can you tell if they’re the same or
different?Howhardisittounknotcertainknots,andhowcanwemeasureaknot’s
complexity?Asifthatweren’tintriguingenough,Henrichdescribesseveralgames
viii Preface
that can be played with knots. She reveals the winning strategies researchers have
uncoveredtoensurethat,whenplayingknotgamesagainstunwittingcompetitors,
theycanalwayswin.Butresearchershaveonlyjustscratchedthesurfaceoflearning
aboutknotgames.Canyoudiscoverwinningstrategiesforgamesonknotsthathave
notyetbeenexplored?
Portland,OR,USA EliE.Goldwyn
St.Mary’sCity,MD,USA SandyGanzell
Portland,OR,USA AaronWootton
Contents
Games on Graphs: Cop and Robber, Hungry Spiders, and
BroadcastDomination........................................................... 1
JoshuaCarlson,PamelaE.Harris,PeterHollander,andErikInsko
Mathematics for Sustainable Humanity: Population, Climate,
Energy,Economy,Policy,andSocialJustice.................................. 21
JueWang
MosaicsandVirtualKnots...................................................... 57
SandyGanzell
GraphLabelings:APrimeAreatoExplore .................................. 81
ElizabethA.DonovanandLesleyW.Wiglesworth
AcrobaticsinaParametricArena.............................................. 113
ThereseShelton,BonnieHenderson,andMichaelGebhardt
ButWhoShould HaveWon? SimulatingOutcomesofJudging
ProtocolsandRankingSystems ................................................ 137
AnnW.CliftonandAllisonL.Lewis
ModelingofBiologicalSystems:FromAlgebratoCalculusand
ComputerSimulations........................................................... 173
AlexanderDimitrov,GiovannaGuidoboni,WilliamHall,RaulInvernizzi,
SergeyLapin,ThomasMcCutcheon,andJacobPennington
PopulationDynamicsofInfectiousDiseases .................................. 227
GlennLedderandMichelleHomp
PlayingwithKnots............................................................... 287
AllisonHenrich
ix
