Mathematics Research for the Beginning Student, Volume 1: Accessible Projects for Students Before Calculus PDF
Preview Mathematics Research for the Beginning Student, Volume 1: Accessible Projects for Students Before Calculus
Foundations 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 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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
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